A Survey on MIMO Transmission with Discrete

Input Signals: Technical Challenges, Advances, and

Future Trends

Yongpeng Wu, Senior Member, IEEE, Chengshan Xiao, Fellow, IEEE,

Zhi Ding, Fellow, IEEE, Xiqi Gao, Fellow, IEEE, and Shi Jin, Member, IEEE

Abstract

Multiple antennas have been exploited for spatial multiplexing and diversity transmission in a wide range

of communication applications. However, most of the advances in the design of high speed wireless multiple-

input multiple output (MIMO) systems are based on information-theoretic principles that demonstrate how to

efficiently transmit signals conforming to Gaussian distribution. Although the Gaussian signal is capacity-achieving,

signals conforming to discrete constellations are transmitted in practical communication systems. The capacity-

achieving design based on the Gaussian input signal can be quite suboptimal for practical MIMO systems with

discrete constellation input signals. As a result, this paper is motivated to provide a comprehensive overview

on MIMO transmission design with discrete input signals. We first summarize the existing fundamental results

for MIMO systems with discrete input signals. Then, focusing on the basic point-to-point MIMO systems, we

examine transmission schemes based on three most important criteria for communication systems: the mutual

information driven designs, the mean square error driven designs, and the diversity driven designs. Particularly,

Y. Wu is with Institute for Communications Engineering, Technical University of Munich, Theresienstrasse 90, D-80333 Munich, Germany

(Email:yongpeng.wu2016@gmail.com).

C. Xiao is with Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409,

USA. (Email: xiaoc@mst.edu).

Z. Ding is with Department of Electrical and Computer Engineering, University of California, Davis, California 95616, USA. (Email:

zding@ucdavis.edu).

X. Gao and S. Jin are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, P. R. China.

(Emails: xqgao@seu.edu.cn; jinshi@seu.edu.cn).

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a unified framework which designs low complexity transmission schemes applicable to massive MIMO systems

in upcoming 5G wireless networks is provided in the first time. Moreover, adaptive transmission designs which

switch among these criteria based on the channel conditions to formulate the best transmission strategy are discussed.

Then, we provide a survey of the transmission designs with discrete input signals for multiuser MIMO scenarios,

including MIMO uplink transmission, MIMO downlink transmission, MIMO interference channel, and MIMO

wiretap channel. Additionally, we discuss the transmission designs with discrete input signals for other systems using

MIMO technology. Finally, technical challenges which remain unresolved at the time of writing are summarized

and the future trends of transmission designs with discrete input signals are addressed.

GLOSSARY

3GPP 3rd Generation Partnership Project

5G fifth Generation

ADC analog-to-digital converter

AMC adaptive modulation and coding

AF amplify-and-forward

BER bit error rate

BICM bit-interleaved coded modulation

BOSTBC block-orthogonal property of space time block coding

BC broadcast channel

CQI Channel Quality Indicator

CSI channel state information

CSIT channel state information at the transmitter

CDMA code-division multiple access systems

DFE decision-feedback equalization

DF decode-and-forward

EPI entropy power inequality

FER frame error rate

2

GSVD generalized singular value decomposition

H-ARQ hybrid automatic repeat request

IPCSIR imperfect channel state information at the receiver

IPCSIT imperfect channel state information at the transmitter

KKT Karush-Kuhn-Tucker

LCFE linear complex-field encoder

LOS line-of-sight

LTE long term evolution

LDPC low density parity check codes

ML maximum likelihood

MRC maximum ratio combining

MRT maximum ratio transmission

MSE mean squared error

MMSE minimum mean squared error

MB multichannel-beamforming

MAC multiple access channel

MIDO multiple-input double-output

MIMO multiple-input multiple-output

MIMOME multiple input multiple output multiple-antenna eavesdropper

MISO multiple input single output

MISOSE multiple input single output single-antenna eavesdropper

NVD non-vanishing determinants

OFDM orthogonal frequency division multiplexing

OFDMA orthogonal frequency division multiplexing access

OSTBC orthogonal space-time block code

PER packet error rate

3

PEP pairwise error probability

PGP Per-Group Precoding

PCSIT perfect channel state information at the transmitter

PCSIR perfect channel state information at the receiver

PSK phase-shift keying

pdf probability distribution function

PAM pulse amplitude modulation

QAM quadrature amplitude modulation

QOSTBC quasi-orthogonal space-time block code

SNR signal-to-noise ratio

SISO single-input single-output

SVD singular value decomposition

STBC space-time block code

ST-LCFE space time linear complex-field encoder

ST-LCP space time linear constellation precoding

STTC space time trellis codes

SDMA spatial division multiple access

SCSI statistical channel state information

SCSIT statistical channel state information at the transmitter

SCSIR statistical channel state information at the receiver

SER symbol error rate

TAST threaded algebraic space time

THP Tomlinson-Harashima precoding

V-BLAST vertical bell laboratories layer space-time

WSR weighted sum-rate

ZF zero-forcing

4

I. INTRODUCTION

The emergence of numerous smart mobile devices such as smart phones and wireless modems has led

to an exponentially increasing demand for wireless data services. Accordingly, a large amount of effort has

been invested on improving the spectral efficiency and data throughput of wireless communication systems.

In particular, multiple-input multiple-output (MIMO) technology has been shown to be a promising means

to provide multiplexing gains and/or diversity gains, leading to improved performance. By increasing the

number of antennas at the base stations and mobile terminals, communication systems can achieve better

performance in terms of both system capacity and/or link reliability [1]–[3]. So far, MIMO technology has

been adopted by third generation partnership project (3GPP) long-term evolution (LTE), IEEE 802.16e,

and other wireless communication standards.

A. Future Wireless Communication Networks

The unprecedented growth in the number of mobile data and connected machines ever-fast approaches

limits of 4G technologies to address this enormous data demand. For example, the mobile data traffic is

expected to grow to 24.3 Exabytes per month by 2019 [4], while the number of connected Internet of

Things (IoT) is estimated to reach 50 Billion by 2020 [5]. Also, emerging new services such as Ultra-

High-Definition multimedia streaming and cloud computing, storage, and retrieval require the higher cell

capacity/end-user data rate and extremely low latency, respectively. It will become evident soon that current

4G technologies will be stretched to near breaking points and still can not meet the quality of experience

necessary for these emerging demands. Therefore, the development of the fifth generation (5G) wireless

communication technologies is a priority issue currently for deploying future wireless communication

networks [6], [7].

As illustrated in Figure 1, the fundamental requirements for building 5G wireless networks are clear.

From system performance point of view, 5G wireless networks must support massive capacity (tens of

Gbps peak data rate) and massive connectivity (1 million /Km2 connection density and tens of Tbps/Km2

traffic volume density), extremely diverging requirements for different user services (user experienced data

rate from 0.1 to 1 Gbps), high mobility (user mobility up to 500 Km/h) and low latency communication

5

(1 ms level end to end latency). From transmission efficiency point of view, 5G wireless networks must

simultaneously realize spectrum efficiency, energy efficiency, and cost efficiency. The evolution towards

5G wireless communication will be a cornerstone for realizing the future human-centric and connected

machine-centric wireless networks, which achieve near-instantaneous, zero distance connectivity for people

and connected machines.

To achieve these requirements, the 5G air interface needs to incorporate three basic technologies:

massive MIMO, millimeter wave, and small cell [8]. For millimeter wave transmission, the millimeter

wave beamforming formulated by large antenna arrays, including both analog beamforming and digital

beamforming, is required to combat the higher propagation losses [9]. For small cell, cell shrinking also

needs the deployment of large and conformal antenna arrays whose cost are inversely with the infrastructure

density [10]. Therefore, MIMO technology will continue to play an important role in upcoming 5G and

future wireless networks.

10 COPYRIGHT © 2014 IMT-2020(5G) PROMOTION GROUP. ALL RIGHTS RESERVED.

5G Key Capabilities: The 5G Flower

Performance Requirements

Efficiency Requirements

Fig. 1: Fundamental requirements of 5G: The 5G flower in [11].

B. MIMO Transmission Design

While the benefits of MIMO can be realized when only the receiver has the communication channel

knowledge, the performance gains can be further improved if the transmitter also obtains the channel

6

knowledge and performs certain processing operations on the signal before transmission. For example,

MIMO transmission design can achieve double capacity at -5dB signal-to-noise ratio (SNR) and 1.5 b/s/Hz

additional capacity at 5 dB SNR for a four-input two-output system over independent identically distributed

(i.i.d.) Rayleigh flat-fading channels [12]. These are normal SNR ranges for practical applications such as

LTE, WiFi, and WiMax. The gains of MIMO transmission design in correlated fading channels are even

more obvious [13], [14]. For the massive MIMO technology in future wireless communication networks, it

is shown in [15] that with perfect channel knowledge, linear precoding at the transmitters and simple signal

detection at the receivers can approach the optimum performance when the number of the antennas at the

base station tends to infinite. A two step joint spatial division and multiplexing transmission scheme is

further proposed to achieve massive MIMO gains with reduced channel state information at the transmitter

(CSIT) feedback [16]. Therefore, based on some forms of CSIT, MIMO transmission design has a great

practical interest in wireless communication systems.

The typical MIMO transmission designs can be mainly classified into three categories [17]: (i) designs

based on mutual information; (ii) designs based on mean squared error (MSE); (iii) diversity driven

designs. The first category adopts mutual information criterions from information theory such as ergodic

or outage capacity for the transmission optimization. The second category performs the transmission

design by optimizing the MSE related objective functions subject to various system constraints; The third

category maximizes the diversity of the communication system based on pairwise error probability (PEP)

analysis.

C. MIMO Transmission Design with Discrete Input Signals

From information theory point of view, the Gaussian distributed transmit signal achieves the fundamental

limit of MIMO communication [1]. However, the Gaussian transmit signal is rarely used in practical

communication systems. This is mainly because of two reasons: (i) The amplitude of Gaussian transmit

signal is unbounded, which may result in a substantial high power transmit signal at the transmitter; (ii)

The probability distribution function (pdf) of Gaussian transmit signal is a continuous function, which

will significantly increase the detection complexity at the receiver. Therefore, the practical transmit signals

7

are non-Gaussian input signals taken from finite discrete constellations, such as phase-shift keying (PSK),

pulse amplitude modulation (PAM), and/or quadrature amplitude modulation (QAM). Figure 2 compares

the pdf of the standard Gaussian distributed signal and the probability mass function (pmf) of the QPSK

signal. From Figure 2, we observe that the QPSK signal is significantly different from the Gaussian signal.

Therefore, the communication based on discrete input signals will normally be quite different from that

of the Gaussian signal and needs specific transmission designs [17]–[21].

The main difficulties of MIMO transmission design with discrete input signals are threefold: (i) The

mutual information and the minimum mean squared error (MMSE) of MIMO channels with finite discrete

constellation signals usually lack explicit expressions. As a result, the elegant water-filling type solutions

for MIMO channels with Gaussian input signal does not exist anymore. Sophisticated numerical algorithms

should be designed to search for both the power allocation matrix and the eigenvector matrix. (ii)

The complexity of MIMO transmission design with finite discrete constellation signals normally grows

exponentially with the number of antenna dimension and the number of the users, which often makes

the implementation of the design prohibitive. (iii) Depending on different channel conditions, adaptive

transmission is needed in practical systems which determine the best transmission policy among various

criterions such as mutual information, MSE, diversity, etc. This process is challenging since it involves a

selection of a large number of parameters.

D. The Contributions of This Paper

In this paper, a comprehensive review of MIMO transmission design with discrete input signals is

presented and summarized in the first time for various design criterions. We first introduce the fundamental

research results for discrete input signals, e.g., the mutual information and the MMSE relationships [22]–

[24], the properties of the MMSE function [25], [26], the definition of MMSE dimension [27], etc. Then,

focusing on the basic point-to-point MIMO systems, we provide in details the transmission designs with

discrete input signals based on the criterions of mutual information (e.g., [17], [28]–[31]), MSE (e.g., [32]–

[34]), and diversity (e.g., [35]–[40]), respectively. In particular, a unified framework for the low complexity

designs which can be implemented in large-scale MIMO antenna arrays for 5G wireless networks is

8

Fig. 2: The pdf of the Gaussian signal and the pmf of the QPSK signal.

provided in the first time. In light of this, the practical adaptive transmission schemes which switch among

these criterions corresponding to the channel variations are summarized for both uncoded systems (e.g.,

[41]–[44]) and coded systems (e.g., [45]–[48]), respectively. The specific adaptive transmission schemes

defined in 3GPP LTE, IEEE 802.16e, and IEEE 802.11n standards are further introduced. Moving towards

multiuser MIMO systems, we introduce the designs based on the above-mentioned criterions for multiuser

MIMO uplink transmission (e.g., [20], [49]–[51]), multiuser MIMO downlink transmission (e.g., [21],

[52]–[54]), MIMO interference channel (e.g., [55]–[58]), and MIMO wiretap channel (e.g., [59]–[61]).

Finally, we overview transmission designs with discrete input signals for other systems employing MIMO

technology such as MIMO cognitive radio systems (e.g, [62]–[64]), green MIMO communication systems

(e.g., [65], [66]), and MIMO relay systems (e.g, [67]–[70]). The overall outline of this paper is shown in

Figure 3.

9

Background Introduction(Sec. I)

The importance

The chanellege

Core Objective

First comprehensive review for MIMO transmission

with discrete input signals

Key Model

MIMO systems

Organ

ization

Point-to-point MIMO(Sec. III)

Design Criterion

Mutual information driven design

Diversity driven design

Adaptive transmission

MSE driven Design

Uplink transmission

MIMO interference channel

Downlink transmission

MIMO wiretap channel

MIMO cognitive radio

MIMO relay

Green MIMO communication

Mutual information

MSE

Diversity

Others

Multiuser MIMO(Sec. IV)

Other MIMO Systems(Sec. V)

Conclusion(Sec. VI)

Basic relationshipsFundamental Results(Sec. II)

Fig. 3: The overall outline of this paper

10

II. FUNDAMENTAL RESULTS FOR DISCRETE INPUT SIGNALS

In this section, we briefly review the existing fundamental results for discrete input signals. To design

transmission schemes for MIMO systems with discrete input signals has to rely on these fundamental

results.

A. Fundamental Link between Information Theory and Estimation Theory

The relationship between information theory and estimation theory is a classic topic, which dates back

to 50 years ago for a relationship between the differential entropy of the Fisher information and the mean

square score function in [71]. This relationship is called De Bruijn’s identity. The representation of mutual

information as a function of causal filtering error was given in [72] as Duncan’s theorem. Other early

connections between fundamental quantities in information theory and estimation theory can be found in

[73]–[77] and reference therein.

1) Gaussian channel model: Recently, D. Guo et al. establish a fundamental relationship between

the mutual information and the MMSE [22], which applies for discrete-time and continuous-time, scalar

and vector channels with additive Gaussian noise. In particular, for the real-value linear vector Gaussian

channel y =√snrHx+n, D. Guo et al. find the following relationship regardless of the input distribution

x

d

dsnrI(x;√snrHx+ n

)=

1

2mmse(snr) (1)

where the mutual information is in nats and

mmse(snr) = E[∥∥Hx− E

[Hx

∣∣√snrHx+ n]∥∥2] . (2)

The right-hand side of Eq. (2) denotes the expectation of squared Euclidean norm of the error correspond-

ing to the best estimation of Hx upon the observation y for a given SNR.

Eq. (1) is proved based on an incremental-channel method in [22]. The key idea of this method is

to model the change in the mutual information as the input-output mutual information of an equivalent

Gaussian channel with vanishing SNR. The mutual information of this equivalent Gaussian channel is

11

essentially linear with the estimation error. Therefore, it is possible to establish a relationship between

the incremental of the mutual information and the MMSE.

For the continuous-time real-value scalar Gaussian channel Rt =√snrXt + Nt, t ∈ [0, T ], where Xt

is the input process and Nt is a white Gaussian noise with a flat double-sided power spectrum density of

unit height, D. Guo et al. establish another fundamental relationship between the causal MMSE and the

noncausal MMSE with any input process

cmmse (snr) =1

snr

∫ snr

0

mmse (γ) dγ (3)

where

cmmse (snr) =1

T

∫ T

0

cmmse (t, snr) dt (4)

cmmse (t, snr) = E[(Xt − E

[Xt

∣∣Y t0 ; snr

])2] (5)

mmse (snr) =1

T

∫ T

0

mmse (t, snr) dt (6)

mmse (t, snr) = E[(Xt − E

[Xt

∣∣Y T0 ; snr

])2]. (7)

Eq. (3) is proved in [22] by comparing Eq. (1) to Duncan’s Theorem. The fundamental relationships in

Eq. (1) and (3) and their generalizations in [22], are referred to the I-MMSE relationships. These I-MMSE

relationships can be applied to provide a simple proof of Shannon’s entropy power inequality (EPI) [78].

The main idea is to use [22, Eq. (182)] to transform the EPI into the equivalent MMSE inequality, which

can be proved much more easily. Similar approaches have been applied to prove the Costa’s EPI and the

generalized EPI in [79] and a new inequality for independent random variables in [80].

I-MMSE relationships in [22] can be generalized. For the complex-value linear vector Gaussian channel

with arbitrary signaling, D. P. Palomar et al. derive the gradient of the mutual information with respect to

the channel matrix and other arbitrary parameters of the system through the chain rule [23]. It is shown

in [23] that the partial derivative of the mutual information with respect to the channel matrix is equal

to the product of the channel matrix and the error covariance matrix of the optimal estimation of the

input given the output. Moreover, the Hessian of the mutual information and the entropy with respect to

various parameters of the system are derived in [24]. In a special case where the left singular vectors of

12

the precoder matrix coincide with the eigenvectors of the channel correlation matrix, it is proved in [24]

that the mutual information is a concave function of the squared singular values of the precoder matrix.

Another generalization direction is to extend the I-MMSE relationships to the abstract Wiener space for

the scalar Gaussian channel in [81] and the vector Gaussian channel in [82].

D. Guo et al. investigate the properties of the MMSE as a function of the SNR and the input distribution

for a real-value scalar Gaussian channel in [25]. It is shown in [25] that the MMSE is concave in the

input distribution for a given SNR and infinitely differentiable for all SNRs for a given input distribution.

An important conclusion in [25] is that the MMSE curve of a Gaussian input and the MMSE curve of a

non-Gaussian input intersect at most once throughout the entire SNR regime. The MMSE properties are

extended to a real-value diagonal MIMO Gaussian channel in [83], where each eigenvalue of the MMSE

matrix with a Gaussian input is proved to have at most a single crossing point with that of the MMSE

matrix with a non-Gaussian input for all SNR values.

More results for the mutual information and the MMSE properties for the scalar Gaussian channel are

obtained. Based on the Lipschitz continuity of the MMSE and I-MMSE relationship, Y. Wu et al. prove

that the mutual information with input cardinality constraints converges to the usual Gaussian channel

capacity as the constellation cardinality tends to infinity [84]. Y. Wu et al. further provide a family of

input constellation generated from the roots of the Hermite polynomials, which achieves this convergence

exponentially [85]. N. Merhav et al. introduce techniques in statistical physics to evaluate the mutual

information and the MMSE in [86]. N. Merhav et al. establish several important relationships between

the mutual infomration, the MMSE, the differential entropy, and the statistical-mechanical variables. Some

application examples in [86] shows how to use the statistical physics tools to provide useful analysis for

the MMSE. K. Venkat et al. investigate the statistic distribution of the difference between the input-output

information density and half the causal estimation error [87]. G. Han et al. propose a new proof of

the I-MMSE relationship by choosing a probability space independent of SNR to evaluate the mutual

information [88]. Based on this approach, G. Han et al. extend the I-MMSE relationship to both discrete

and continuous-time Gaussian channels with feedback in [88].

13

A random variable with distribution P is at the receiver. The receiver performs the MSE estimation of

this random variable under the assumption that its distribution is Q. This is referred to the mismatch

estimation. S. Verdu initializes a new connection between the relative entropy and the mismatched

estimation for a real-value scalar Gaussian channel [89]. The key relationship revealed in [89] is as

follows:

D (P ‖ Q) = 1

2

∫ ∞0

mseQ (γ)−mseP (γ) dγ (8)

where D (P ‖ Q) is the relative entropy. mseQ (γ) and mseP (γ) are MMSEs obtained by an estimator

which assumes the input signal is distributed according to Q and P for a real-value scalar Gaussian

channel, respectively. Eq. (8) is a generalization of Eq. (1) for the scalar Gaussian channel with a mismatch

estimation. Similarly, using the mutual information and the relative entropy as a bridge, T. Weissman

generalizes Eq. (3) into the case of mismatch estimation in [90]. The statistic properties of the difference

between the mismatched and matched estimation losses are studied in [87]. The relationship between

relative entropy and mismatched estimation is extended to the real-value vector Gaussian channel in [91].

For a more complicated case of mismatched estimation where the channel distribution is mismatched at

the estimator instead of the input signal distribution, W. Huleihel et al. derive the mismatched MSE for

the vector Gaussian channel using the statistical physics tools [92]. A brief summation of recent research

results of the I-MMSE relationships for Gaussian channel is given in Table II.

2) Other Channel Models: The I-MMSE relationships are investigated in other channel models. For

the scalar Poisson channel, D. Guo et al. prove that the derivative of the input-output mutual information

with respect to the intensity of the additive dark current equals to the expected error between the logarithm

of the actual input and the logarithmic of its conditional mean estimate (noncausal in case of continuous-

time) [93]. The similar relationship holds for the derivative of the mutual information with respect to

input-scaling, by replacing the logarithmic function with x log x [93]. Moreover, R. Atar et al. prove

that the I-MMSE relationships hold for Gaussian channel [22], [89], [90], including discrete-time and

continuous-time channel, matched and mismatched filters, also hold for scalar Poisson channel by replacing

the squared error loss with the corresponding loss function [94]. L. Wang et al. derive the gradient of

14

TABLE II: Recent research results of the I-MMSE relationships for Gaussian channelPaper Model Main Contribution

D. Guo et al. [22]Real-value MIMO Reveal a fundamental relationship between mutual information and MMSE

Discrete-time, Continuous-time Reveal a fundamental relationship between causal and noncausal MMSEs

D. P. Palomar et al. [23]Complex-value MIMO Derive the gradient of the mutual information

Discrete-time with arbitrary parameters of the systems

M. Payaro et al. [24]Complex-value MIMO Derive the Hessian of the mutual inforamtion and entropy

Discrete-time with arbitrary parameters of the systems

M. Zakai [81] Real-value scalar Extend I-MMSE relationship in [22] to the abstract Wiener space (scalar version)

E. M.-Wolf et al. [82] Real-value MIMO Extend I-MMSE relationship in [22] to the abstract Wiener space (vector version)

D. Guo et al. [25]Real-value MIMO Obtain various properties of MMSE

Discrete-time with respect to SNR and input distribution

R. Bustin et al. [83]Real-value diagonal MIMO

Obtain properties of the MMSE matrixDiscrete-time

Y. Wu et al. [84]Real-value scalar Prove the mutual information with input cardinality constraints converges to

Discrete-time the Gaussian channel capacity as the constellation cardinality tends to infinity

Y. Wu et al. [85]Real-value scalar

Prove the convergence speed in [84] is exponentialDiscrete-time

N. Merhav et al. [86]Real-value scalar Introduce techniques in statistical physics to

Discrete-time evaluate the mutual information and the MMSE

K. Venkat et al. [87]

Real-value scalar, mismatched caseDerive the statistical properties of difference between the

with/without feedbackinput-output information density and half the causal estimation error

Continuous-time, Discrete-time

G. Han et al. [88]Real-value scalar, feedback

Extend the I-MMSE relationship to channels with feedbackDiscrete-time, Continuous-time

S. Verdu et al. [89]Real-value scalar Initializes a new connection between the relative entropy

Discrete-time, mismatched case and the mismatched estimation

T. Weissman [90]Real-value scalar Establish a relationship between causal MMSE

Continuous-time, mismatched case and non-causal MMSE in mismatched case

M. Chen et al. [91]Real-value MIMO

Extend the relationship in [89] to MIMODiscrete-time, mismatched case

W. Huleihel et al. [92]Real-value MIMO Derive the MSE when the mismatched estimation

Discrete-time, mismatched case occurs for the channel distribution

15

mutual information with respect to the channel matrix for vector Poisson channel [95]. In addition, L.

Wang et al. define a Bregman matrix based on the Bergman divergence in [95] to formulate a unified

framework for the gradient of mutual information for both vector Gaussian and Poisson channels. The

Bergman divergence is also used to characterize the derivative of the relative entropy and the mutual

information for the scalar binomial and negative binomial channels in [96]. The I-MMSE relationship

for a scalar channel with exponential-distributed additive noise is established in [97]. For continuous-time

scalar channel with additive Gaussian/Possion noise in the presence of feedback, the relationships between

the directed information and the causal estimation error are established in [98].

Some work consider more general channel models. A scalar channel with arbitrary additive noise is

studied in [99]. It is found that the increase in the mutual information due to the improvement in the channel

quality equals to the correlation of two conditional mean estimates associated with the input and the noise

respectively. D. P. Palomar et al. represent the derivative of mutual information in terms of the conditional

input estimation given the output for general channels (not necessarily additive noise) [100]. The general

results in [100] embrace most of popular channel models including the binary symmetric channel, the

binary erasure channel, the discrete memoryless channel, the scalar/vector Gaussian channel, an arbitrary

additive-noise channel, and the Poisson channel. For a scalar channel with arbitrary additive noise, D. Guo

obtain the derivative of relative entropy with respect to the energy of perturbation, which can be expressed

as a mean squared difference of the score functions of the two distributions [101]. Furthermore, Y. Wu et

al. analyze the properties of the MMSE as a function of the input-output joint distribution [26]. It is proved

in [26] that the MMSE is a concave function of the input-output joint distribution. N. Merhav analyze

the MMSE for both cases of matched and mismatched estimations by using the statistical-mechanical

techniques [102]. The main advantage of the results in [102] is that they apply for very general vector

channels with arbitrary joint input-output probability distributions. By exploiting the statistical-mechanical

techniques, N. Merhav also establish relationships between rate-distortion function and the MMSE for a

general scalar channel [103]. Then, simple lower and upper bounds on the rate-distortion function can be

obtained based on the bounds of the MMSE. A brief summation of recent research results of the I-MMSE

16

TABLE III: Recent research results of the I-MMSE relationships for non-Gaussian channelsPaper Model Main Contribution

D. Guo et al. [93]Scalar Poisson Obtain the derivative of the mutual information with respect to the additive dark current

Discrete-time, Continuous-time Obtain the derivative of the mutual information with respect to the input-scaling

R. Atar et al. [94]Scalar Poisson, mismatched case

Obtain I-MMSE relationshipDiscrete-time, Continuous-time

L. Wang et al. [95]MIMO Poisson Obtain gradient of mutual information with

Discrete-time respect to the channel matrix

C. G. Taborda et al. [96]Scalar binomial and negative binomial Obtain derivative of the relative entropy and the mutual information

Discrete-time with respect to scaling factor

M. Raginsky et al. [97]Scalar, exponential noise

Obtain the I-MMSE relationshipDiscrete-time

T. Weissman et al. [98]Scalar Gaussian and Poisson Obtain the relationships between the directed information

Continuous-time with feedback and the causal estimation error channels

D. Guo et al. [99]Scalar, arbitrary additive noise

Obtain the I-MMSE relationshipDiscrete-time

D. P. Palomar et al. [100]General channel Obtain the derivative of mutual information in terms of

Discrete-time the conditional input estimation given the output

D. Guo et al. [101]Scalar, arbitrary additive noise Obtain derivative of relative entropy with respect

Discrete-time to the energy of perturbation the output

Y. Wu et al. [26]Scalar, arbitrary additive noise Analyze the properties of the MMSE as a function

Discrete-time of the input-output joint distribution

N. Merhavet al. [102]MIMO, general channel

Analyze the MMSEDiscrete-time, mismatched case

N. Merhavet al. [103]Scalar, general channel Establish relationships between rate-distortion

Discrete-time function and the MMSE

relationships for non-Gaussian channels is given in Table III.

B. Other Analytical Results with Discrete Input Signals

1) Asymptotic Results in Low and High SNR Regime: Shannon’s pioneering work [104] have shown

that the binary antipodal inputs are as good as the Gaussian input in low SNR regime. Recently, S.

Verdu investigate the spectral efficiency in low SNR regime for a general class of signal inputs and

channels [105]. In [105], S. Verdu provides two important criterions to characterize the spectral efficiency

in low SNR regime: the minimum energy per information bit required for reliable communication and the

wideband slope of the spectral efficiency. S. Verdu further define that any input signals which achieve the

17

same minimum energy per information bit achieved by the Gaussian input are first-order optimal. If the

perfect channel knowledge is available at the receiver, both BPSK and QPSK are first-order optimal for a

general MIMO channel. Also, S. Verdu define that any first-order optimal input signals which achieves the

wideband slope achieved by Gaussian input are second-order optimal. For a MIMO Gaussian channel with

perfect channel knowledge at the receiver, it is proved that equal-power QPSK is second-order optimal.

Moreover, V. V. Prelov et al. provide the second order expansion of the mutual information in low SNR

regime for very general classes of inputs and channel distributions [106].

For a parallel Gaussian channels with m-ary constellation inputs, A. Lozano et al. provide both lower

and upper bounds of the MMSE in high SNR regime [18]. For an arbitrary MIMO Gaussian channel,

F. Perez-Cruz et al. further derive high SNR lower and upper bounds of the MMSE and the mutual

information [19]. For a scalar channel with arbitrary inputs and additive noise, Y. Wu et al. define the

MMSE dimension when SNR tends to be infinity in [27]. The MMSE dimension represents the gain of

the non-linear estimation error over the linear estimation error in high SNR regime. For MIMO Gaussian

block Rayleigh fading channels with m-ary constellation inputs, it is proved that the constellation with

a higher minimum Euclidean distance achieves a higher mutual information in high SNR regime [107,

Appendix E]. For a scalar Gaussian channel with arbitrary input distributions independent of SNR, the

exact high SNR expansion of the mutual information is obtained in [108] in the first time. The obtained

asymptotic expression validates the optimality of the Gray code for bit-interleaved coded modulation

(BICM) in high SNR regime. By using Mellin transform, A. C. P. Ramos et al. derive low and high SNR

asymptotic expansions of the ergodic mutual information and the average MMSE for a scalar Gaussian

channels with arbitrary discrete inputs over Rician and Nakagami fading [109]. High SNR expansions are

derived for a MIMO Gaussian channel over Kronecker fading with the line-of-sight (LOS) [110]. A brief

summation of the recent analytic results for discrete input signals in asymptotic SNR regime is given in

Table IV.

2) Asymptotic Results in Large System Limits: Some work provide the analytic results for discrete input

signals in large system limit. A technique which has been widely used in statistical physics, referred to

18

TABLE IV: Recent analytic results for discrete input signals in asymptotic SNR regimePaper Model Main results in asymptotic SNR regime

S. Verdu [105]MIMO, low SNR Analyze the minimum energy per information bit

A general class of channels Analyze the wideband slope

V. V. Prelov et al. [106]MIMO, low SNR

Obtain the second order expansion of the mutual informationA general class of channels

A. Lozano et al. [18]Parallel MIMO Gaussian channels

Provide lower and upper bounds of the MMSEhigh SNR

F. P.-Cruz et al. [19]MIMO Gaussian channels Provide lower and upper bounds of

high SNR the MMSE and the mutual information

Y. Wu et al. [27]Scalar, arbitrary additive noise

Define the MMSE dimensionhigh SNR

D. Duyck et al. [107]MIMO Gaussian channels Prove the mutual information is a monotonously

block Rayleigh fading, high SNR increasing function of the minimum Euclidean distance

A. Alvarado et al. [108]Scalar Gaussian channels Obtain the exact high SNR expansion

high SNR of the mutual information

A. C. P. Ramos et al. [109]Scalar Gaussian channels, Rician fading Obtain asymptotic expansions of the

Nakagami fading, low SNR, high SNR ergodic mutual information and the average MMSE

M. R. D. Rodrigues [110]MIMO Gaussian channels, high SNR Obtain asymptotic expansions of

Kronecker fading with LOS ergodic mutual information and the average MMSE

replica method, is initially exploited to analyze the asymptotic performance of various detection schemes

of code-division multiple access systems (CDMA) in large dimension [111]–[114]. By using this replica

method, R. R. Muller derive an asymptotic mutual information expression for a spatially correlated MIMO

channel with BPSK inputs in the first time [115]. This expression is extended to arbitrary input at the

transmitter in [116]. C.-K. Wen et al. obtain the asymptotic sum rate expression for the Kronecker fading

channels for the centralized MIMO multiple access channel (MAC) and the distributed MIMO MAC

in [117] and [118], respectively. The asymptotic sum-rate expression including the LOS effect for the

centralized MIMO MAC is derived in [119]. C.-K. Wen et al. further obtain the asymptotic sum-rate

expression for the Kronecker fading channels for the centralized MIMO MAC with amplify-and-forward

(AF) relay [120]. Also, M. A. Girnyk et al. derive an asymptotic mutual information expression for i.i.d.

19

Rayleigh fading for K-hop AF relay MIMO channels [121]. For other performance criterions, R. R. Muller

et al. obtain an asymptotic expression for the average minimum transmit power for i.i.d. Rayleigh fading

MIMO channels with a non-linear vector precoding [122]. Similar problem is also considered for MIMO

broadcast channel (BC) in [123]. A more general analysis, referred to one step replica symmetry breaking

analysis, is used in [123] to reduce the asymptotic approximation errors in [122] for non-convex alphabet

sets. The effect of the transmit-side noise caused by hardware impairments is investigated in [124] for

point-to-point MIMO systems with i.i.d. discrete input signals and i.i.d. MIMO Rayleigh fading channels.

It is assumed that the receiver has perfect channel state information (CSI) but the transmitter does not

know the CSI. Also, the transmit-side noise is generally unknown at both transmitter and the receiver. As

a result, the receiver employs a mismatched decoding without taking the consideration of the transmit-side

noise. The general mutual information in [125] is adopted as the metric to evaluate the performance. An

asymptotic expression of the general mutual information is derived when both the numbers of transmit and

receive antennas go to infinity with a fixed ratio. Numerical results indicate that the derived asymptotic

expression provides an accurate approximation for the exact general mutual information and the effects of

transmit-side noise and mismatched decoding become significant only at high modulation orders. A brief

summation of recent research results of the analytic results (transmitter side) for discrete input signals in

large system limit is given in Table V.

3) Other Results: There are some other analytical research results for discrete input signals. For a

scalar Gaussian channel with a finite size input signal constraint, N. Varnica et al. propose a modified

Arimoto-Blahut algorithm to search for the optimal input distribution which maximizes the information rate

[126]. The corresponding maximum information rate is referred to constellation constrained capacity. J.

Bellorado et al. study the constellation constrained capacity of MIMO Rayleigh fading channel. It is shown

in [127] that for a given information rate, the SNR gap between the uniformly distributed signal and the

constellation constrained capacity when the constellation size tends to infinity is 1.53 dB. This is the same

with the QAM asymptotic shaping gap for the scalar Gaussian channel in [128]. Also, it is conjectured in

[127] that when the elements of input signal vector are chosen from a Cartesian product of one dimension

20

TABLE V: Analytic results (transmitter side) for discrete input signals in large system limitPaper Channel Criterion

R. R. Muller [115] Point-to-point MIMO, spatially correlated Rayleigh Asymptotic mutual information (BPSK)

C.-K. Wen et al. [116] Point-to-point MIMO, Kronecker fading Asymptotic mutual information

C.-K. Wen et al. [117] Distributed MIMO MAC, Kronecker fading Asymptotic sum-rate

C.-K. Wen et al. [118] Centralized MIMO MAC, Kronecker fading Asymptotic sum-rate

C.-K. Wen et al. [119] Centralized MIMO MAC, Kronecker fading with LOS Asymptotic sum-rate

C.-K. Wen et al. [120] Centralized MIMO MAC with AF relay, Kronecker fading Asymptotic sum-rate

M. A. Girnyk et al. [121] K-hop AF relay MIMO, i.i.d. Rayleigh fading Asymptotic mutual information

R. R. Muller et al. [122] Point-to-point MIMO, i.i.d. Rayleigh fading, vector precoding Asymptotic average minimum transmit power

B. M. Zaidel et al. [123] MIMO BC, i.i.d. Rayleigh fading, vector precoding (1RSB ansatz) Asymptotic average minimum transmit power

M. Vehkapera et al. [124]Point-to-point MIMO, i.i.d. Rayleigh fading, i.i.d. input signals

General mutual informationTransmit-side noise and mismatched decoding

PAM constellations, the constellation constrained capacity can be achieved by optimizing the distribution of

each element of the signal vector independently. This reduces the computational complexity significantly.

This conjecture is proved in [129]. M. P. Yankov et al. derive a low-computational complexity low bound

of the constellation constrained capacity for MIMO Gaussian channel based on QR decomposition [130].

For one dimension channel with memory and finite input state, D. M. Arnold et al. propose a practical

algorithm to compute the information rate [131]. For two dimension channel, tight low and upper bounds

of the information rate are derived in [132]. O. Shental et al. propose a more accurate computation method

for the information rate of two dimension channel with memory and finite input state based on generalized

belief propagation [133].

III. POINT-TO-POINT MIMO SYSTEM

In this section, we focus on the transmission designs for point-to-point MIMO systems with discrete

input signals. We classify these designs into four categories: i) design based on mutual information; ii)

design based on MSE; iii) diversity driven design; iv) adaptive transmission.

A. Design Based on Mutual Information

The mutual information maximization yields the maximal achievable rate of the communication link

with the implementation of reliable error control codes [134]. For point-to-point MIMO system, the

21

maximization of the mutual information has been obtained in [1]. It is proved in [1] that the mutual

information is maximized by a Gaussian distributed signal with a covariance structure satisfying two

conditions. First, the signal must be transmitted along the right eigenvectors of the channel, which

decomposes the original channel into a set of parallel subchannels. Second, the power allocated for

each subchannel is based on a water-filling strategy, which depends on the strength of each subchannel

and the SNR.

Although the Gaussian signal is capacity-achieving, it is too idealistic to be implemented in practical

communication systems. The discrete input signals are taken from a finite discrete constellation set, such

as PSK, PAM, or QAM. Therefore, some work begin to investigate MIMO transmission designs based

on the discrete constellation input mutual information.

1) Mutual Information Maximization: By exploiting the I-MMSE relationship in [22], A. Lozano et

al. propose a mercury water-filling power allocation policy which maximizes the input-output mutual

information for diagonal MIMO Gaussian channels with arbitrary inputs [18]. The essential difference

between the mercury water-filling policy and the conventional water-filling policy is that it will pour

mercury onto each of the subchannel to a certain height before filling the water (allocated power) to the

given threshold. Because of the poured mercury, the classical conclusion for the Gaussian input signal

that, the stronger subchannel receives more filling water (allocated power) does not hold any more. The

asymptotic expansions of the MMSE function and the mutual information in low and high SNR regime

are obtained in [18]. Based on these expansions, it is proved that allocating power only to the strongest

channel is asymptotically optimal in low SNR regime, which is the same with the conventional water-filling

allocation. For the optimal power allocation policy in high SNR regime, in contrast, the less power should

be allocated to the stronger subchannel for equal constellation inputs. This is significant different from the

conventional water-filling allocation. Also, it is revealed that the more power should be allocated to the

richer constellation inputs for equal channel strength at high SNR. For the non-diagonal MIMO Gaussian

channel with arbitrary inputs, an iterative numerical to search for the optimal linear precoder based on

a gradient decent update of the precoder matrix is proposed in [23]. Moreover, necessary conditions

22

for the optimal linear precoding matrix maximizing the mutual information of the non-diagonal MIMO

Gaussian channel with finite alphabet inputs are presented in [135], from which a numerical algorithm

is formulated to search for the optimal precoder. Simulations indicate that the obtained non-diagonal

and non-unitary precoder in [135] outperforms the mercury water-filling design in non-diagonal MIMO

Gaussian channel. The design in [135] has been extended to the case where only statistical CSI (SCSI)

is available at the transmitter [136]. In the mean time, F. Perez-Cruz et al. also establish necessary

conditions that the optimal precoder should satisfy for the non-diagonal MIMO Gaussian channel with

finite alphabet inputs through Karush-Kuhn-Tucker (KKT) analysis [19]. The MMSE function and the

mutual information of the non-diagonal MIMO Gaussian channel with finite alphabet inputs are expanded

in asymptotic low and high SNR regime. It is proved that in low SNR regime, the optimal precoder design

for finite alphabet inputs is exactly the same with the optimal design for Gaussian input. In high SNR

regime, low and upper bounds of the mutual information are derived. Base on the bounds, it is shown

that the precoder that maximizes the mutual information is equivalent to the precoder that maximizes

the minimum Euclidean distance of the receive signal points in high SNR regime. This result indicates

that the three important criterions in MIMO transmission design: maximizing the mutual information,

minimizing the symbol error rate (SER), and minimizing the MSE are equivalent in high SNR regime.

The optimal power allocation scheme for the general MIMO Gaussian channel is also obtained in [19].

For the real-valued non-diagonal MIMO Gaussian channels, M. Payaro et al. decompose the design of

the optimal precoder into three components: the left singular vectors, the singular values, and the right

singular vectors [137]. The optimal left singular vectors are proved to be the right singular vectors of

the channel matrix. Necessary and sufficient conditions of the optimal singular values are established,

from which the optimal singular values can be found based on the standard numerical algorithm such

as Newton method. The optimal solution of the right singular vectors is a difficult problem. Based on

standard numerical algorithm, a local optimum design can be found. Moreover, the optimal designs in

low and high SNR regime are analyzed in [137]. For the complex-valued non-diagonal MIMO Gaussian

channels, M. Lamarca proves that the optimal left singular vectors of the precoder is still the right singular

23

vectors of the channel matrix [138]. More importantly, when the entries of the channel matrix and the

input signals are all real-valued, M. Lamarca proves that the mutual information is a concave function of

a quadratic function of the precoder matrix in [138]. Then, an iterative algorithm is proposed to maximize

the mutual information by updating this quadratic function along the gradient decent direction. For the

complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs, C. Xiao et al. prove

that the mutual information is still a concave function of a quadratic function of the precoder matrix [17].

Then, a parameterized iterative algorithm is designed to find the optimal precoder which achieves the

global maximum of the mutual information. Moreover, the coded bit error rate (BER) performance of the

obtained precoder is simulated. The BER gains of the obtained precoder comparing to the conventional

designs coincides with that of the mutual information gains, which indicate that maximizing the mutual

information is also an effective criterion for optimizing the coded BER in practical systems. X. Yuan et

al. consider a joint linear precoding and linear MMSE detection scheme for finite alphabet input signals

with perfect CSIT (PCSIT) and perfect CSI at the receiver (PCSIR) [139]. The linear precoder is set to be

a product of three matrices: the right singular vector matrix of the channel, the diagonal power allocation

matrix, and the normalized discrete Fourier transform matrix. When the detector and the decoder satisfy

the curve-matching principle [139, Eq. (23)], an asymptotic achievable rate of the linear precoding and

linear MMSE detection scheme as the dimension of the transmit signal goes towards infinity is derived.

The derived achievable rate is a concave function of the power allocation matrix. As a result, the optimal

power allocation scheme for maximizing the achievable rate can be obtained based on the standard convex

optimization method. Very recently, K. Cao et al. propose a precoding design in high SNR regime by

transforming the maximization of the minimum Euclidean distance into a semi-definite programming

optimization problem [140]. Moreover, for a large number of antennas, the implementation complexity of

digital beamforming may be significantly high. To reduce the complexity, the MIMO beamforming can

be implemented in both the analog and digital domains, which is referred as hybrid beamforming [141].

R. Rajashekar et al. investigate the hybrid beamforming designs to maximize the mutual information for

millimeter wave MIMO systems with finite alphabet inputs [142].

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TABLE VI: MIMO transmissions with finite alphabet inputs based on mutual information maximizationPaper Model Main Contribution

A. Lozano et al. [18]Diagonal MIMO Obtain the optimal power allocation

PCSIT, PCSIR Obtain the optimal power allocation in low and high SNR regime

D. P. Palomar et al. [23]Non-diagonal MIMO

Propose a gradient descent algorithm to search for the optimal precoderPCSIT, PCSIR

C. Xiao et al. [135]Non-diagonal MIMO Establish necessary conditions for the optimal precoder

PCSIT, PCSIR Propose a numerical algorithm to search for the optimal precoder

C. Xiao et al. [136]Non-diagonal MIMO Establish necessary conditions for the optimal precoder

SCSIT, PCSIR Propose a numerical algorithm to search for the optimal precoder

F. Perez-Cruz et al. [19]Non-diagonal MIMO Establish necessary conditions for the optimal precoder

PCSIT, PCSIR Analyze the optimal precoder in low and high SNR regime

M. Payaro et al. [137]Real-valued non-diagonal MIMO Obtain the left singular vectors and the singular values of the optimal precoder

PCSIT, PCSIR Analyze the optimal precoder in low and high SNR regime

M. Lamarca [138]

Non-diagonal MIMOObtain the left singular vectors of the optimal precoder in complex-valued channels

Prove that the mutual information is a concave function of a quadratic function

PCSIT, PCSIRof the precoder matrix for real-valued non-diagonal MIMO channels

Propose an iterative algorithm to maximize the mutual information

C. Xiao [17]

Non-diagonal MIMOProve that the mutual information is still a concave function of a quadratic function

of the precoder matrix for complex-valued non-diagonal MIMO channels

PCSIT, PCSIRPropose a global optimal iterative algorithm to maximize the mutual information

Examine the coded BER performance of the obtained precoder

X. Yuan et al. [139]Non-diagonal MIMO Design the linear precoder by a product of three matrices

PCSIT, PCSIR Obtain an asymptotic achievable rate of this linear precoder and linear MMSE detection

C. Kao et al. [140]Non-diagonal MIMO Transform the maximization of the minimum Euclidean distance in high SNR regime

PCSIT, PCSIR into a semi-definite programming optimization problem

R. Rajashekar et al. [142]Non-diagonal MIMO Propose hybrid beamforming designs to maximize

PCSIT, PCSIR mutual information for millimeter wave MIMO systems

2) Low Complexity Design: Although the proposed algorithm in [17] finds the global optimal precoder,

capable of achieving the maximum of the mutual information for the complex-valued non-diagonal MIMO

Gaussian channel with finite alphabet inputs, its implementation complexity is usually difficult to afford in

practice systems. There are two main reasons for the high computational complexity. First, both the mutual

information and the MMSE function lack closed-form expression. Therefore, the Monte Carlo method

has to be used to evaluation both terms, which requires an average over enormous noise realizations in

order to maintain the accuracy. More importantly, the evaluation of the mutual information [135, Eq. (5)]

25

and the MMSE function [135, Eq. (8)] involves additions over the modulation signal space which goes

exponentially with the number of transmit antennas. This results in a prohibitive computational complexity

when the dimension of the transmit antennas is large.

By employing the Jensen’s inequality and the intergral over the exponential function, a lower bound

of the single-input single-output (SISO) mutual information with finite alphabet inputs is derived in [143,

Eq. (9)]. The evaluation of this lower bound does not require to calculate the expectation over the noise.

As a result, W. Zeng et al. extend this lower bound to the case of MIMO systems to solve the first factor

of the high computational complexity due to the Monte Carlo average [144]. An algorithm is proposed

by maximizing this lower bound. Numerical results indicate that the proposed design in [144] reduces the

computational complexity without performance loss. W. Zeng et al. further investigate the linear precoder

design with SCSI at the transmitter (SCSIT) over the Kronecker fading model [30]. A lower bound of the

ergodic mutual information which avoids the expectations over both the channel and the noise is derived.

Simulations over various fading channels show that with a constant shift, this lower bound provides an

accurate approximation of the ergodic mutual information. Moreover, it is proved that maximizing this

lower bound is asymptotically optimal in low and high SNR regime. An iterative algorithm is further

developed by maximizing this low bound. Numerical results indicate that the proposed design in [30]

achieves a near-optimal mutual information and BER performance with reduced complexity. The lower

bound in [30] is utilized to design the linear precoder for practical MIMO-orthogonal frequency division

multiplexing (OFDM) systems in [145].

When perfect instantaneous CSI is available at the transmitter, the MIMO channel can be decomposed

into a set of parallel subchannels. For the second factor of the high computational complexity, S. K.

Mohammed et al. propose X-Codes to pair two subchannels into a group [29]. X-Codes are fully

characterized by the paring and a 2 × 2 single angle parameterized rotation matrix for each pair. Based

on X-Codes structure, the mutual information can be expressed as a sum of the mutual information

of all the pairs. In this case, the calculation of the mutual information only involves additions over the

modulation signal space which goes polynomially with the number of transmit antennas. This significantly

26

reduces the computational complexity of evaluating the mutual information. Designs of the rotation angle

and power allocation within each pair and the optimization pairing and power allocation among pairs

are further proposed to increase the mutual information performance. T. Ketseoglou et al. extend this

idea to pair multiple subchannels based on a Per-Group Precoding (PGP) technique in [31]. The PGP

technique decomposes the design of the power allocation matrix and the right singular vector matrix

into the design of several decoupled matrices with a much small dimension. An iterative algorithm is

proposed to find the optimal solution of these decoupled matrices. Numerical results indicate that the

proposed algorithm achieves almost the same performance as the design in [17] but with a significant

reduction of the computational complexity. Moreover, T. Ketseoglou et al. propose a novel and efficient

method to evaluate the mutual information of the complex-valued non-diagonal MIMO Gaussian channel

with finite alphabet inputs based on the Gauss-Hermite quadrature rule [146]. This approximation method

can be combined with the PGP technique to further reduce the design complexity.

When only SCSI is available at the transmitter, a low-complexity iterative algorithm to find the optimal

design over the Kronecker fading model is proposed in [147]. The proposed algorithm relies on an

asymptotic approximation of the ergodic mutual information in large system limit, which avoids the

Monte Carlo average over the channel. Moreover, based on this approximation, the design of the power

allocation matrix and the right singular vector matrix of the precoder can also be decoupled into the design

of small dimension matrices within different groups. Simulations indicate that the proposed algorithm in

[147] radically reduces the computational complexity of the design in [30] by orders of magnitude with

only minimal losses in performance. Y. Wu et al. propose a unified low-complexity precoder design for the

complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs [148]. An asymptotic

expression of the ergodic mutual information over the jointly-correlated Rician fading model is derived in

large system limit. Combined with the approximation given in [30], the obtained asymptotic expression

can provide an accurate approximation of the ergodic mutual information without the Monte Carlo average

over the channel and the noise. The left singular matrix of the optimal precoder for this general model

is obtained. Structures of the power allocation matrix and the right singular matrix of the precoder are

27

further established. The proposed structures decouple the independent data streams over parallel equivalent

subchannels, which avoid a complete-search of the entire signal space for the precoder design. Based on

these, a novel low computational complexity iterative algorithm is proposed to search for the optimal

precoder with the general CSI availability at the transmitter, which includes the cases of PCSIT, imperfect

CSIT (IPCSIT), and SCSIT. Here we provide Example 1 and Example 2 to analytically and numerically

compare the complexity of the low-complexity algorithm in [148] and the complete-search design in [30],

respectively. As observed from Table VII-X, the computational complexity of the complete-search design in

[30] grows exponential and quickly becomes unwieldy. Simulations indicate that the proposed algorithm

in [148] drastically reduces the implementation complexity for finite alphabet precoder design, but in

such a way that the loss in performance—established based on the 3GPP spatial channel model [149]—is

minimal. Moreover, some novel insights for the precoder design with SCSIT are revealed in [148]. A

brief summation of the above-mentioned low-complexity MIMO transmissions with finite alphabet inputs

is given in Table XI.

Example 1: Assume Nt and Ns denote the number of transmit antenna and the number of streams in

each subgroup, respectively. Assume Ns = 2 and QPSK modulation. The numbers of additions required

for calculating the mutual information and the MSE matrix in the complete-search design in [30] and the

algorithm in [148] are listed in Table VII for different numbers of transmit antenna.

TABLE VII: Number of additions required for calculating the mutual information and the MSE matrix.

Nt 4 8 16 32

Complete-search design in [30] 65536 4.29 e+009 1.84 e+019 3.4 e+038

Algorithm in [148] 512 1024 2048 4096

Example 2: Let us evaluate the complexity of the algorithm in [148] for different values of Ns. Matlab is

used on an Intel Core i7-4510U 2.6GHz processor. Tables VIII–X provide the running time per iteration,

for various numbers of antennas and constellations, with × indicating that the time exceeds one hour.

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TABLE VIII: Running time (sec.) per iteration with BPSK.Nt Ns = 2 Ns = 4 Ns = Nt

4 0.0051 0.0190 0.0190

8 0.0112 0.0473 11.6209

16 0.0210 0.1939 ×

32 0.0570 0.4111 ×

TABLE IX: Running time (sec.) per iteration with QPSK.Nt Ns = 2 Ns = 4 Ns = Nt

4 0.1149 21.5350 21.5350

8 0.2029 23.3442 ×

16 0.3001 48.1725 ×

32 0.7094 98.7853 ×

TABLE X: Running time (sec.) per iteration with 16-QAM.Nt Ns = 2 Ns = Nt

4 28.0744 ×

8 58.3433 ×

16 106.6022 ×

32 233.2293 ×

B. Design Based on MSE

1) Linear Transmission Design: The MSE is an important criterion to measure the quality of the

communication link with discrete input signals. With PCSIT and PCSIR, an optimal linear transceiver

design to minimize the weighted sum MSE of all channel substreams under total average transmit power

constraint is proposed in [150], developing upon early work in [151]–[154]. The minimization of the sum

MSE of all channel substreams under maximum eigenvalue constraint is investigated in [155]. A unified

framework to optimize the MSE function of all channel substreams under total average transmit power

constraint for a multi-carrier MIMO system and linear processing at both ends of the link is established in

[32]. It is assumed in [32] that the signal constellation and coding rate for all the subchannels have been

determined before the transmission and the carrier-noncooperative scheme is adopted. The system model in

[32] is illustrated in Figure 4. It was proved in [32] that if the MSE function is Schur-concave, the optimal

29

TABLE XI: Low-complexity MIMO transmissions with finite alphabet inputsPaper Model Main Contribution

W. Zeng et al. [144]MIMO

Design based on a lower bound without Monte Carlo average over the noisePCSIT, PCSIR

W. Zeng et al. [30]MIMO Design based on a lower bound without Monte Carlo average

SCSIT, PCSIR over the channel and the noise

Y. R. Zheng et al. [145]MIMO-OFDM Evaluate MIMO transmissions with finite alphabet inputs

practical channel estimation in practical MIMO test-bed

S. K. Mohammed et al. [29]MIMO Decouple the mutual information into a sum of

PCSIT, PCSIR the mutual information of small dimension matrices

T. Ketseoglou et al. [31]MIMO Decompose the design of the power allocation matrix and the right singular vector matrix

PCSIT, PCSIR into the design of several decoupled matrices with a much small dimension

T. Ketseoglou et al. [146]MIMO Propose a novel and efficient method to evaluate the mutual information

PCSIT, PCSIR based on the Gauss-Hermite quadrature rule

Y. Wu et al. [147]MIMO Propose a design which reduces the computational complexity of the design in [30]

SCSIT, PCSIR by orders of magnitude with only minimal losses in performance

Y. Wu et al. [148]MIMO Propose a unified low-complexity precoder design which applies

general CSIT, PCSIR for the general CSI availability at the transmitter

solution is to diagonalize the channel in each subcarrier directly, and if the MSE function is Schur-convex,

the optimal solution is to pre-rotate the transmit symbol and then diagonalize the channel in each subcarrier.

The above two families of objective functions embrace most of popular criteria used in communication

systems, including the MSE, the SINR, and also the uncoded BER with identical constellations. The

design in [32] is extended to a general shaping precoder constraint in [156]. Moreover, D. P. Palomar

employs a primal decomposition approach to decompose the original optimization problem over multiple

(e.g.,multicarrier) MIMO channels in [157] into several subproblems, each of which can be independently

solved over a single MIMO channel. These subproblems are coordinated by a simple master problem.

As a result, the optimization problem in [32] can be solved in a simple and efficient way. In the mean

time, the transmission design which satisfies the specific MSE constraints with minimum transmit power is

investigated in [158]. The optimal solution is to pre-rotate the transmit signal by a particular unitary matrix

and then diagonalize the channel. Similar problem is also considered in [159] under the uncoded BER

constraints instead of the MSE constraints. It is assumed that the constellation used for each substream is

different. Then, the uncoded BER is neither Schur-convex nor Schur-concave function of the MSE. This

30

optimization problem is solved by the primal decomposition approach. Moreover, L. G. Ordonez et al.

propose an adaptive number of substreams transmission scheme to minimize the uncoded BER under fix

transmission rate constraint and total average transmit power constraint [160]. A widely linear MMSE

transceiver which processes the in-phase and quadrature components of the input signals separately is

designed in [161]. The widely linear processing is implemented by using a real-value representation of

the input signals. Recently, a linear precoder design to minimize the sum MSE of all channel substreams

under the joint total average transmit power constraint and maximum eigenvalue constraint is proposed

in [162].

Linear Precoder

MIMOChannel

noise

Linear Equalizer

Input k Output k

. . .. . .

Input 1

Input L

. . .

. . .

. . .. . .

Output L

Output 1

Fig. 4: Multi-carrier MIMO system with linear processing and carrier-noncooperative scheme.

However, perfect CSI is only reasonable for the slow fading scenario where the channel remains

deterministic for a long period. Otherwise, the CSI estimated at the transmitter and the receiver may

have mismatches with the actual channel. Therefore, S. Serbetli et al. begin to investigate the optimum

transceiver structure in sense of minimizing the sum MSE of transmit data streams via imperfect CSI at

both ends [33]. A MIMO system with correlated receive antennas is considered and the effect of channel

31

estimation is investigated in [33]. X. Zhang et al. investigate the statistically robust design of linear MIMO

transceivers based on minimizing the MSE function under total average transmit power constraint [163].

The statistically robust design exploits the channel mean and channel covariance matrix (equivalently, the

channel estimate and the estimation error) to design linear MIMO transceivers. Two cases are considered in

[163]: i) IPCSIT and imperfect channel state information at the receiver (IPCSIR) with no transmit antenna

correlation; ii) IPCSIT and PCSIR. For the former case, the optimal solution has the same structure as the

perfect CSI case by replacing the instantaneous channel realization matrix and the noise covariance matrix

with the channel mean matrix and equivalent covariance matrix (e.g., channel covariance matrix plus noise

covariance matrix), respectively. For the latter case, a lower bound of the average MSE of all substreams

is derived and the linear precoder minimizing this lower bound is obtained. The statistically robust design

for the IPCSIT and IPCSIR case with transmit antenna correlation is investigated in [164]. By finding the

optimal solution which satisfies the KKT conditions, the structures of the optimal precoder and equalizer

minimizing the sum MSE of transmit data streams under total average transmit power constraint are

obtained in [164]. In the mean time, J. Wang et al. investigate a deterministic robust design for MIMO

system under total average transmit power constraint [165]. The deterministic robust design assumes the

estimated imperfect CSI is within a neighborhood of actual channel. The case of IPCSIT and IPCSIR

is considered in [165]. Given a pre-fixed linear receiver structure, the optimal transmission direction and

the optimal power allocation for the linear precoder matrix minimizing the MSE of transmit data streams

are found in [165]. For the same scenario, the optimal linear precoder structure given a pre-fixed linear

receiver and the optimal linear equalizer structure given a pre-fixed linear precoder are obtained in [166].

Therefore, the precoder and the equalizer can be designed alternatively by an iterative algorithm. For

the scenario considered in [165], the optimal linear MIMO transceiver design is found in [167]. On the

other hand, to reduce the cost of analog-to-digital converter (ADC) chains at the receiver, MIMO analog

beamformers are designed in [168] to minimize both the output MSE of the digital beamformer and the

power consumption of the ADC by exploiting the statistical knowledge of the channel. A brief summation

of linear transmission designs based on MSE is given in Table XII, where the notations LP and LE denote

32

abbreviations for linear precoder and linear equalizer, respectively.

2) Non-linear Transmission Design: Above-mentioned work are based on the linear processing at both

ends of the link. However, research have shown that non-linear processing schemes can offer additional

advantages over linear processing schemes [34], [169]–[173]. Two typical technologies are Tomlinson-

Harashima precoding (THP) at the transmitter and decision-feedback equalization (DFE) at the receiver.

A unified transceiver structure of a MIMO system with THP and DFE is illustrated in Figure 5. Assuming

PCSIT and PCSIR, a THP precoding with linear equalization which minimizes the sum MSE of transmit

data streams is proposed in [34]. The work in [34] focuses on the design of the feedback precoder in

Figure 5. The permutation matrix and the precoder matrix in Figure 5 are set to be identical matrix. The

THP design with linear equalization is extended to MIMO-OFDM system in [169], where the input signal

is pre-ordered by a permutation matrix and an additional precoder matrix is designed to improve the

overall MSE performance. Two linear precoding designs with DFE at the receiver are proposed in [170]

and [171], which optimize the sum MSE and the MSE function of the transmit data streams, respectively.

A unified framework which is applicable for both THP with linear equalization and linear precoding with

DFE is established in [172]. The proposed designs in [172] optimize the MSE function of the transmit

data streams. It is proved that if the MSE function is Schur-convex, the optimal design forces the MSE

of each transmit data stream to be equal, and if the MSE function is Schur-concave, the optimal THP

leads to linear precoding and the optimal DFE leads to linear equalization. When only statistical CSI is

available at the transmitter, O. Simenone et al. propose a THP with linear equalization design based on

minimizing a lower bound of the sum MSE of the transmit data streams [173]. A brief summation of

non-linear transmission designs based on MSE is given in Table XIII, where the notations NLP and NLE

denote abbreviations for non-linear precoder and non-linear equalizer, respectively.

3) Performance Analysis: Since the publication of [32], some work begin to analyze the performance of

the uncoded MIMO systems within the linear processing framework (also refereed to MIMO multichannel-

beamforming (MB) systems [174]) in [32]. In [175], the first-order polynomial expansions for the marginal

pdfs of all of the individual ordered eigenvalues of uncorrelated central Wishart-distributed matrix are

33

TABLE XII: Linear transmission designs with discrete input signals based on the MSEPaper Systems Criterion Transceiver CSI Constraint

H. Sampath et al. [150] MIMOWeighted LP PCSIT

Sum powersum MSE LE PCSIR

A. Scaglione et al. [155]Multi-carrier

Sum MSELP PCSIT

Maximum eigenvalueMIMO LE PCSIR

D. P. Palomar et al. [32]Multi-carrier

MSE functionLP PCSIT

Sum powerMIMO LE PCSIR

D. P. Palomar [156] MIMO MSE functionLP PCSIT Covariance matrix

LE PCSIR shaping

D. P. Palomar [157]Multi-carrier MSE function LP PCSIT

Sum powerMIMO efficient solution LE PCSIR

D. P. Palomar et al. [158] MIMO Transmit powerLP PCSIT

MSELE PCSIR

D. P. Palomar et al. [159]Multi-carrier Uncoded BER LP PCSIT

Sum powerMIMO different constellations LE PCSIR

L. G. Ordonez et al. [160] MIMOUncoded BER LP PCSIT Sum power

adaptive streams LE PCSIR fixed rate

F. Sterle [161] MIMO Sum MSEWidely LP PCSIT

Sum powerWidely LE PCSIR

J. Dai et al. [162] MIMO Sum MSELP PCSIT Sum power

LE PCSIR maximum eigenvalue

S. Serbetli et al. [33] MIMOAverage sum MSE LP IPCSIT and IPCSIR

Sum powerchannel estimation effect LE no transmit correlation

X. Zhang et al. [163] MIMO Average MSE functionLP IPCSIT and IPCSIR

Sum powerLE no transmit correlation

X. Zhang et al. [163] MIMOAverage MSE function LP IPCSIT

Sum powerlower bound LE PCSIR

M. Ding et al. [164] MIMO Average sum MSELP IPCSIT and IPCSIR

Sum powerLE with transmit correlation

J. Wang et al. [165] MIMO Sum MSELP IPCSIT

Sum powerfixed LE IPCSIR

J. Wang et al. [166] MIMOSum MSE LP IPCSIT

Sum powernot optimal LE IPCSIR

H. Tang et al. [167] MIMOSum MSE LP IPCSIT

Sum poweroptimal LE IPCSIR

V. Venkateswaran et al. [168] MIMOMSE, ADC power LP SCSIT

Nooptimal LE SCSIR

34

PermutationMatrix Modulo b

Equalizer Matrix

s

nosie

a

FeedbackPrecoder

y

Precodermatrix c

Channel

zQuantizer

FeedbackEqualizer

Fig. 5: MIMO transceiver with THP and DFE.

TABLE XIII: Non-linear transmission designs with discrete input signals based on the MSE

Paper Systems Criterion Transceiver CSI Constraint

R. F. H. Fischer et al. [34] MIMO Sum MSENLP PCSIT

Sum powerLE PCSIR

A. A. D. Amico et al. [169]MIMO

Sum MSENLP, LP PCSIT

Sum powerOFDM LE PCSIR

F. Xu et al. [170] MIMO Sum MSELP PCSIT

Sum powerLE, NLE PCSIR

Y. Jiang et al. [171] MIMO MSE functionLP PCSIT

Sum powerLE, NLE PCSIR

M. B. Shenouda et al. [172] MIMO Sum MSENLP, LP, LE PCSIT

Sum powerLP, LE, NLE PCSIR

O. Simeone et al. [173] MIMOAverage sum MSE NLP, LP, LE SCSIT

Sum powerlower bound LP, LE, and NLE PCSIR

derived. Based on these results, the closed form expressions for the outage probability and the average

uncoded BER performance for each subchannel of MIMO MB systems with fixed power allocation over an

uncorrelated Rayleigh flat-fading channel in high SNR regime are obtained [175]. The obtained expressions

reveal both the diversity gain and the array gain in each subchannel. Also, tight lower and upper bounds

35

for the performance with nonfixed power allocation such as waterfilling power allocation are obtained. For

the correlated central Wishart-distributed matrix, the exact pdfs expressions for the ordered eigenvalues

are derived in [176]. These expressions allow us to analyze the MIMO MB systems over a semicorrelated

Rayleigh flat-fading channel under various performance criterions, including the outage probability and

the average uncoded BER. Simple closed-form first-order polynomial expansions for these marginal pdfs

are derived in [160], from which the diversity gain and the array gain of the MIMO MB systems are

indicated. In the mean time, the exact pdfs expressions for the uncorrelated noncentral Wishart-distributed

matrix are independently derived in [174] and [176], respectively. The first order polynomial expansions

for these marginal pdfs are independently obtained in [160] and [174], respectively. Performance of

MIMO MB systems over uncorrelated Rician fading channel can be analyzed similarly based on these

expressions. For MIMO MB over a more general Rayleigh-product fading channel, H. Zhang et al. derive

the first order polynomial expansions of the ordered eigenvalues of the channel Gram matrix [177]. The

diversity gain and the array gain for MIMO MB systems with fixed power allocation over the Rayleigh-

product fading channel are revealed in [177]. Later, the analysis for MIMO MB systems is extended to

scenarios in the presence of co-channel interference. With fixed power allocation, L. Sun et al. obtain

exact and asymptotic (low outage probability) expressions of outage probability of MIMO MB systems in

the presence of unequal power interferers [178], where both the desired user and the interferes experience

semicorrelated Rayleigh flat-fading. Also, S. Jin et al. obtain exact and asymptotic expressions of outage

probability of MIMO MB systems in the interference-limited (without noise) scenarios with equal power

interferers. It is assumed in [179] that the desired user experiences uncorrelated Rician fading and the

interferers experience uncorrelated Rayleigh fading. Y. Wu et al. obtain exact and asymptotic expressions

of outage probability of MIMO MB systems with arbitrary power interferers, where the desired user

experiences Rayleigh-product fading and the interferers experience uncorrelated Rayleigh fading [180]. A

brief summation of performance analysis of MIMO MB systems is given in Table XIV.

36

TABLE XIV: Performance analysis of MIMO MB systems

Paper user channel interferer channel interferer power outage probability

L. G. Ordonez et al. [175] Uncorrelated Rayleigh fading No No Asymptotic

L. G. Ordonez et al. [176]Semicorrelated Rayleigh fading

No No ExactUncorrelated Rician fading

L. G. Ordonez et al. [160]Semicorrelated Rayleigh fading

No No AsymptoticUncorrelated Rician fading

S. Jin et al. [174] Uncorrelated Rician fading No No Exact, asymptotic

H. Zhang et al. [177] Rayleigh-product fading No No Asymptotic

L. Sun et al. [178] Semicorrelated Rayleigh fading Semicorrelated Rayleigh fading Unequal Exact

S. Jin et al. [179] Uncorrelated Rician fading Uncorrelated Rayleigh fading EqualExact, asymptotic

Interference-limited

Y. Wu et al. [180] Rayleigh-product fading Uncorrelated Rayleigh fading Arbitrary Exact, asymptotic

C. Diversity Driven Designs

1) Early research for Open-loop Systems: For the open-loop MIMO system, it is usually assumed

that the CSI is available at the receiver but not at the transmitter. To acquire receive diversity, exploiting

diversity combining schemes at the receiver, such as equal gain combining, selection combining, and

maximum ratio combining (MRC), can be tracked back to the 60 years ago with references in the paper

[181]. Without CSIT, space-time techniques are designed to combat the deleterious effects of small scale

fading. These space-time techniques can acquire additional transmit diversity and thereby improve the

reliability of the link with discrete input signals. A simple and well known space-time technique is the

Alamouti orthogonal space-time block code (OSTBC) for a two transmit antennas and two receive antennas

system [35], as illustrated in Figure 6. For OSTBC in Figure 6, in time slot 1, the symbols s1 and s2 are

transmitted in antenna 1 and 2, respectively. In time slot 2, the symbol −s∗2 and −s∗1 are transmitted in

antenna 1 and 2, respectively. After passing the channel and the combiner in Figure 6, the signal s1,c and

s2,c sent to the maximum likelihood (ML) detector are given by

s1,c =(|h11|2 + |h12|2 + |h21|2 + |h22|2

)s0 + h∗11n11 + h12n

∗12 + h∗22n22 + h21n21 (9)

s2,c =(|h11|2 + |h12|2 + |h21|2 + |h22|2

)s1 − h11n∗11 + h∗12n12 − h22n∗22 + h∗21n21. (10)

37

(9) and (10) are equivalent to that of four-branch MRC. Therefore, it achieves a diversity of four for the

ML detector. Additional transmit diversity is obtained by utilizing OSTBC. For both real and complex

OSTBC, the construction of OSTBC for any number of transmit antennas and receive antennas is proposed

in [36]. The proposed design achieves maximum possible spatial diversity order with a low complexity

linear decoding. The space-time block code (STBC) is extended to single-carrier MIMO system in [37]

and MIMO-OFDM system in [182]. The proposed designs achieve the maximum diversity but with the

increase of the decoding complexity since the detector needs to jointly decode all subchannels. To solve

this problem, a subchannel grouping method with the maximum diversity performance and a reduced

decoding complexity is proposed in [182]. The transmission rate of above OSTBC design is less than 1

symbol per channel use (pcu) for more than two transmit antennas. Therefore, a quasi-orthogonal space-

time block code (QOSTBC), which provides 1 symbol pcu transmission rate and half of the maximum

possible diversity for four transmit antennas, is proposed in [183]. The QOSTBC divides the transmission

matrix columns into different groups, and makes the columns in different groups orthogonal to each

other. This quasi-orthogonal structure increases the transmission rate. Also, based on this structure, the

ML decoding at the receiver can be done by searching symbols in each group individually instead of

searching all transmit signals jointly. By rotating half of the transmit symbol constellations, QOSTBC

can achieve 1 symbol pcu rate and maximum diversity for four transmit antennas [184] and higher rate

than OSTBC and maximum diversity for MIMO systems [185]. Based on a different rotation method,

C. Yuen et al. construct a class of QOSTBCs for MIMO systems, whose ML decoding can be reduced

to a joint detection of two real symbols [186]. On the other hand, the space time trellis codes (STTC),

which achieves both maximum spatial diversity and large coding gains, is designed in [187]. However,

the decoding complexity of STTC grows exponentially with transmit rate and becomes significantly high

for high order modulation.

A space time linear constellation precoding (ST-LCP) is proposed in [188] as an alternative scheme

to obtain transmit diversity. The baseband equivalent framework of ST-LCP is illustrated in Figure 7.

The key idea of ST-LCP is to perform a pre-combination of the transmit signal by a linear precoder

38

s1

-s2*

s2

-s1*

h11

h22

h12h21

n21,n22

n11,n12

Channel Estimator

h11,h12

Channel Estimator

h21,h22

Combiner

h11,h12

s1,c,s2,c

h21,h22

Maxim

um liklihood

detector

s1,e,s2,e

Fig. 6: The OSTBC for a 2× 2 system.

matrix as in Figure 7. Then, the pre-processed data is mapped to the transmit antenna by a diagonal

matrix and a unitary matrix. Based on the average PEP expression, linear precoders which achieve the

maximum diversity and perform close to the upper bound of the coding gain are designed in closed-form.

Near optimal sphere decoding algorithms are employed to reduce the decoding complexity of ST-LCP.

Simulations indicate that ST-LCP achieve a better BER performance than classic OSTBC. On the other

hand, Liu et al. study ST-LCP for MIMO-OFDM system [189]. The authors in [189] combine ST-LCP

with a subcarrier grouping method to realize both the maximum diversity and the low decoding complexity.

Linear Precoder

ST-LCPMapper

MIMOChannel

noise

Detection input output

Fig. 7: The baseband equivalent framework of ST-LCP.

Although the above-mentioned work achieves good BER performance, their transmit rate can not exceed

1 symbol pcu. This limits the transmission efficiency of MIMO system. B. A. Sethuraman et al. construct

39

a high rate full diversity STBC using division algebras with high decoding complexity [190]. Also, many

work try to construct full rate (Nt symbol pcu) full diversity space time codes. One approach is to extend

the ST-LCP to the full rate full diversity design [191]. The key idea is to replace the linear precoder in

Figure 7 with a linear complex-field encoder (LCFE). The LCFE divides the N2t transmit symbols into Nt

layers, where Nt is the number of transmit antenna. In each layer, a closed form linear precoder matrix is

used to combine the Nt symbols. Then, the ST-LCP mapper in Figure 7 is replaced with space time LCFE

(ST-LCFE) mapper, where the pre-precessed N2t data is mapped into a Nt×Nt matrix. Finally, the mapped

data is transmitted by Nt antennas in Nt time slot. In this case, Nt symbol pcu transmission rate is achieved.

It is proved that the ST-LCFE design also obtains the maximum spatial diversity of MIMO system. The

price of the ST-LCFE is the exponential increase of the decoding complexity. Therefore, both diversity-

complexity tradeoff and modulation-complexity tradeoff schemes are proposed in [191]. Moreover, the

ST-LCFE design is extended to frequency-selective MIMO-OFDM and time-selective fading channels in

[191]. Another approach is to employ threaded algebraic space time (TAST) code [192]. When using

TAST code, the size of the information symbols alphabet is increased since the degree of the algebraic

number field increases. Therefore, full rate Nt symbol pcu and the maximum spatial diversity can be

achieved simultaneously. A golden STBC with non-vanishing determinants (NVD) property when the

signal constellation size approaches infinity is first constructed for 2×2, 3×3, and 6×6 antenna systems

in [193], [194], and extended to arbitrary number of antennas MIMO systems in [195]. The NVD property

guarantees that the STBC performs well irrespectively of the transmit signal alphabet size. The above-

mentioned work are some important initial space-time techniques with discrete input signals and PCSIR.

These space-time techniques focus on improving the transmit diversity, coding gain, and transmission rate

of the link. A brief comparison of these work is given in Table XV.

For the noncoherent scenario where the CSI is neither available at the transmitter nor at the receiver, V.

Tarokh et al. combine differential modulation with OSTBC to formulate transmission schemes, which can

still obtain transmit diversity with PSK signals [196], [197]. Based on unitary group codes, differential

STBCs to achieve the maximum transmit diversity of MIMO systems are proposed in [198], [199], where

40

TABLE XV: Some important initial space-time techniques with discrete input signals and perfect CSIRPaper System Coding gain Rate Decoding complexity

S. M. Alamouti [35] 2× 2 Not optimal 1 symbol pcu Low

V. Tarokh et al. [36] MIMO Not optimal < 1 symbol pcu Low

S. Zhou et al. [37] Single-carrier MIMO Not optimal < 1 symbol pcu High

Z. Liu et al. [182] MIMO-OFDM Large < 1 symbol pcu Lower than [37]

H. Jafarkhani [183] 4× 4 Not optimal (half diversity) 1 symbol pcu Reduced

N. Sharma et al. [184] 4× 4 Not optimal 1 symbol pcu Reduced

W. Su et al. [185] MIMO Not optimal ≤ 1 symbol pcu Reduced

C. Yuen et al. [186] MIMO Not optimal ≤ 1 symbol pcu Lower than [185]

V. Tarokh et al. [187] MIMO Large < 1 symbol pcu High

Y. Xin et al. [188] MIMO Near-optimal 1 symbol pcu High

Z. Liu et al. [189] MIMO-OFDM Not optimal 1 symbol pcu Lower than [37]

B. A. Sethuraman et al. [190] MIMO Near-optimal > 1 symbol pcu High

X. Ma et al. [191]MIMO, MIMO-OFDM

Not optimal Nt symbol pcu HighMIMO time selective channels

H. E. Gammal et al. [192] MIMO Not optimal Nt symbol pcu High

J.-C. Belfiore et al. [193] 2× 2 Large (NVD) 2 symbol pcu High

F. Oggier et al. [194] 2× 2, 3× 3, 6× 6 Large (NVD) Nt symbol pcu High

P. Elia et al. [195] MIMO Large (NVD) Nt symbol pcu High

the optimal modulation scheme for two transmit antennas is obtained in [199]. A double differential

STBC is designed for time selective fading channels in [200]. In addition, a differential STBC using

QAM signals is constructed in [201]. For the partially coherent scenario where the CSI is not available

at the transmitter and only imperfect CSI is available at the receiver, robust STBCs are designed in

[202]–[204]. The effects of mapping on the error performance of the coded STBC have been studied in

[205], [206]. A rate-diversity tradeoff for STBC under finite alphabet input constraints is obtained in [207]

and the corresponding optimal signal constructions for BPSK and QPSK constellation are provided. The

space-time techniques have also been extended to MIMO CDMA systems [208]–[210].

2) Recent Advance for Open-loop Systems: Recent space-time techniques with discrete input signals

focus on designing high rate full diversity STBCs with NVD and low decoding complexity. For a multiple

input single output (MISO) system equipped with a linear detector, a general criterion for designing full

41

diversity STBC is established in [39]. The STBC with Toeplitz structure satisfies this criterion. Also,

the rate of this Toeplitz STBC approaches 1 symbol pcu when the number of channel use is sufficiently

large. For MISO system equipped with ML detector, this Toeplitz STBC achieves the maximum coding

gain. A new 2 × 2 full rate full diversity STBC is proposed by reconstructing the generation matrix of

OSTBC [211]. The proposed STBC reduces the complexity of ML decoding to the order of the standard

sphere decoding. Another 2×2 full rate full diversity STBC is designed in [212]. During the ML decoding

exhaustive search process for this STBC, the Euclidean distance can be divided into groups and each group

is minimized independently. Henceforth, this STBC leads to an obvious decoding complexity reduction

comparing to the Golden code in [193] with a slight performance loss. A unified design framework for

full rate full diversity 2× 2 fast-decodable STBC is provided in [213]. The ML decoding complexity of

the designed code is reduced to the complexity of standard sphere decoding. For a 4 × 2 system, two

QSTBCs are combined to formulate a new 2 pcu symbol rate and half diversity STBC with simplified ML

decoding complexity [213]. K. P. Srinath et al. construct a 2× 2 STBC which has the same performance

as the Golden code in [193] and a lower ML decoding complexity for non-rectangular QAM inputs [214].

K. P. Srinath et al. further construct a 2 pcu symbol rate and full diversity 4 × 2 STBC which offers a

large coding gain with a reduced ML decoding complexity [214]. H. Wang et al. construct a QOSTBC

which experiences the same low decoding complexity as the design in [186], but has a better coding gain

for rectangular QAM inputs [215]. K. R. Kumar et al. design a STTC for MIMO systems based on lattice

coset coding [216]. The constructed code has a good coding gain and a reduced decoding complexity for

large block length with a DFE and lattice decoding.

In [217], a general framework of multigroup ML decodable STBC is introduced to reduce the decoding

complexity of MIMO STBC. Multigroup ML decodable STBC is a full diversity STBC. It allows ML

decoding to be implemented for the symbols within each individual group. As a result, multigroup ML

decodable STBC provides a tradeoff between rate and ML decoding complexity. A specific multigroup

ML decodable STBC based on extended Clifford algebras, named Clifford unitary weight, is constructed

to meet the optimal tradeoff between the rate and the ML decoding complexity in [218]. Moreover, the

42

structure of fast decodable code is integrated into the multigroup code to formulate a fast-group-decodable

STBC [219]. A new class of full rate full diversity STBC which exploits the block-orthogonal property

of STBC (BOSTBC) is proposed in [220]. Simulation shows that this BOSTBC achieves a good BER

performance with a reduced decoding complexity when QR decomposition decoder with M paths (QRDM

decoder) is employed at the receiver. Also, this block-orthogonal property can be utilized [221] to reduce

the decoding complexity of the Golden code in [193] with ML decoder. Rigorous theoretical analyses

for the block-orthogonal structure of various STBCs are given in [222]. Some fast-decodable asymmetric

STBCs are designed in [223]–[225]. In particular, R. Vehkalahti et al. establish a family of fast-decodable

STBCs from division algebras for multiple-input double-output (MIDO) systems [223]. These codes are

full diversity with rate no more than Nt/2 symbol pcu. By properly constructing the geometric structures

of these codes, the ML decoding complexity can by reduced by at least 37.5%. In addition, explicit

constructions for 4× 2, 6× 2, and 6× 3 codes are given in [223]. N. Markin et al. propose an iterative

STBC design method for Nt×Nt/2 MIMO systems [224]. The iterative design constructs two Nt/2×Nt/2

original codewords with algebraic codewords based on cyclic algebra to create a new Nt×Nt codewords.

The resulting code achieves Nt/2 symbol pcu rate and full diversity with fast decoding complexity. This

iterative design is used to construct new 4 × 2 and 6 × 3 STBCs in [224] based on some well known

STBCs such as Golden code in [193]. Moreover, a generalization framework of this iterative design is

used in [225] to construct 2 symbol pcu rate and full diversity STBC with large coding gain and fast

decoding complexity for MIDO systems. Based on this framework, explicit 4×2, 6×2, 8×2, and 12×2

STBCs are constructed in [225]. A brief comparison of above low decoding complexity STBC designs is

given in Table XVI.

Besides reducing the decoding complexity, there are some other space-time techniques recently, e.g.,

a full rate full diversity improved perfect STBC is designed in [226], which has larger coding gain than

the perfect STBC in [194]. A full rate STBC under the BICM structure is designed in [226] to optimize

the entire components of the transmitter jointly, including an error-correcting code, an interleaver, and a

symbol mapper. A new spatial modulation technology which only simultaneously activates a few number

43

TABLE XVI: Recent low decoding complexity space-time techniques with discrete input signalsPaper System Coding gain Rate Decoding complexity

J. Liu et al. [39] MISO NVD Approach 1 symbol pcu Linear

J. M. Paredes et al. [211] 2× 2 NVD 2 symbol pcu Reduced ML

S. Sezginer et al. [212] 2× 2 Not optimal 2 symbol pcu Reduced ML

E. Biglieri et al. [213]2× 2 Not optimal 2 symbol pcu Reduced ML

4× 2 Not optimal (half diversity) 2 symbol pcu Reduced ML

K. P. Srinath et al. [214]2× 2 NVD 2 symbol pcu Lower than Golden code [193]

4× 2 NVD 2 symbol pcu Reduced

H. Wang et al. [215] MIMO Larger than [186] 1 ≤ symbol pcu Same as [186]

K. R. Kumar et al. [216] MIMO Good High Reduced ML

S. Karmakar et al. [217] MIMO Not optimal Flexible Trade off with rate

G. S. Rajan et al. [218] MIMO Not optimal Flexible Trade off with rate

T. P. Ren et al. [219] MIMO Not optimal Flexible Lower than [213]

T. P. Ren et al. [220] MIMO Not optimal Nt symbol pcu Reduced QRDM

M. O. Sinnokrot et al. [221] 2× 2 Golden code [193] NVD 2 symbol pcu Reduced ML

G. R. Jithamithra et al. [222] MIMO Classic STBCs Flexible Reduced SD

R. Vehkalahti et al. [223] MIDO NVD Nt/2 symbol pcu Reduced ML

N. Markin et al. [224] Nt ×Nt/2 NVD Nt/2 symbol pcu Reduced ML

K. P. Srinath et al. [225]4× 2, 6× 2 NVD, large

2 symbol pcu Reduced ML8× 2, 12× 2 NVD, large

of antennas among the entire antenna arrays is described in [227]–[229]. Since the publication of [35],

space-time techniques with discrete input signals have been studied extensively around the world [38],

[230], [231].

3) Close-loop Systems: The space-time technologies achieve the transmit diversity without CSI at the

transmitter, which are applicable for open-loop systems. Alternatively, for close-loop MIMO systems in

Figure 8, the CSI is available at the transmitter either by a feedback link from the receiver (frequency

division duplex systems) or a channel estimation from the pilot signal of the receiver (time division duplex

systems). In this case, proper transmission designs can be formulated to exploit the close loop transmit

diversity of MIMO systems with discrete input signals. For example, a bit interleaved coded multiple

beamforming technology is introduced in [232] to achieve the full diversity and full spatial multiplexing

for MIMO and MIMO-OFDM systems over i.i.d. Rayleigh fading channels. The decoding complexity of

44

the bit interleaved coded multiple beamforming design is further reduced in [233]. An important criterion

to enhance the transmit diversity for the close loop MIMO systems is to maximize the minimum Euclidean

distance of the receive signal points. This criterion is directly related to the SER of the MIMO system

if the ML detector is used and is proved to achieve the optimal mutual information performance in high

SNR regime [19]. For a 2×2 system with BPSK and QPSK inputs, an optimal solution for a non-diagonal

precoder, which maximizes the minimum Euclidean distance of the receive signal points (max-dmin), is

proposed in [234]. It is shown in [234] that for BPSK inputs, the optimal transmission is to focus the

power on the strongest subchannel, which is the beamforming design. For QPSK inputs, the beamforming

design is optimal when the channel angle is less than a threshold. When the channel angle is larger

than the threshold, the optimal design can be found by numerically searching the angle of the power

allocation matrix. In [40], a suboptimal precoder design for MIMO systems with an even number of

transmit antenna and 4-QAM inputs is proposed. The authors transform the MIMO channels into parallel

subchannels by the singular value decomposition (SVD). Each two subchannels are paired into one group

to formulate a set of 2× 2 subsystems. Then, 2× 2 sub-precoders are designed similar as in [234]. The

design in [40] is denoted as the X-structure design. It is noted this X-structure design enjoys both reduced

precoding and decoding complexities. For M -QAM inputs, finding the optimal max-dmin MIMO precoder

is rather complicated and the optimal solution varies for different modulations. Therefore, Ngo et al.

propose a suboptimal design by selecting 2× 2 sub-precoders between two deterministic structures [235].

These sub-precoders are then combined together to formulate a MIMO precoder based on the X-structure.

The suboptimal design in [235] has good uncoded BER performance with a reduced implementation

complexity. An exact expression for the pdf of the minimum Euclidean distance under this suboptimal

precoder design is derived in [236]. Also, a max-dmin MIMO precoder design for a three parallel data-

stream scheme and M -QAM inputs is proposed in [237]. This work allows us to decompose arbitrary

MIMO channels into 2×2 and 3×3 subchannels and then design the max-dmin MIMO precoder efficiently.

On the other hand, a suboptimal real-value max-dmin MIMO precoder for M -QAM inputs is proposed in

[238]. The proposed design has a order of magnitude lower ML decoding complexity than the precoders

45

in [40], [235]. Moreover, it is proved in [238] that the max-dmin precoder achieves full diversity of MIMO

systems. In the mean time, two low decoding complexity precoders derived from a rotation matrix and

the X-structure, called X-precoder and Y-Precoder, are proposed in [29]. X-precoder has a closed-form

expression for 4-QAM inputs. Y-precoder has an explicit expression for arbitrary M -QAM inputs but with

a worse error performance than X-precoder. Overall, both X-precoder and Y-Precoder are suboptimal in

error performance and loses out in word error probability when comparing to max-dmin precoder [40],

[235]. Two suboptimal max-dmin MIMO precoder designs for M -QAM inputs by optimizing the diagonal

elements of the “SNR-like” matrix and the lower bound of the minimum Euclidean distance are proposed

in [239] and [240], respectively.

Linear Precoder

MIMOChannel

noise

Receiverinput output

Feedback CSIfrom Receiver

Fig. 8: The baseband equivalent framework of close-loop system.

An alternative approach of exploiting the close loop transmit diversity is to design the precoder based on

the lattice structure. A lattice-based linear MIMO precoder design to minimize the transmit power under

fixed block error rate and fixed total data rate is proposed in [241]. The precoder design is decomposed

into the design of four matrices. The optimal rotation matrix is given in a closed form structure, the bit

load matrix and the basis reduction matrix are optimized alternatingly via an iterative algorithm, and the

lattice base matrix is selected based on a lower bound of the transmit power. Linear precoders which

achieve full diversity of Nt × Nt MIMO systems and apply for integer-forcing receiver are designed in

[242] based on full-diversity lattice generator matrices. Some work also use lattice theory to design max-

dmin precoder. For a 2×2 system with a square QAM constellation inputs, the optimal real-value max-dmin

46

TABLE XVII: Techniques for the transmit diversity of close loop MIMO systems with discrete input

signalsPaper System Modulation minimum Euclidean distance Design complexity Decoding complexity

E. Akay et al. [232]MIMO

M -QAM Not optimal Low HighMIMO-OFDM

B. Li et al. [233] MIMO-OFDM M -QAM Not optimal Low Reduced

L. Collin et al. [234] 2× 2 BPSK, QPSK Optimal Low Low

B. Vrigneau et al. [40] MIMO QPSK Near-optimal Reduced Reduced

Q.-T. Ngo et al. [235] MIMO M -QAM Suboptimal Reduced Reduced

Q.-T. Ngo et al. [237] MIMO M -QAM Suboptimal Reduced Reduced

K. P. Srinath et al. [238] MIMO M -QAM Smaller than [40] Lower than [235] Lower than [235]

S. K. Mohammed et al. [29] MIMO M -QAM Not optimal Lower than [235] Lower than [235]

Q.-T. Ngo et al. [239] MIMO M -QAM Larger than [235] Reduced High

C.-T. Lin et al. [240] MIMO M -QAM Close to [29] Lower than [29] High

S. Bergman et al. [241] MIMO M -QAM Not optimal Reduced High

A. Sakzad et al. [242] Nt ×Nt M -QAM Not optimal Reduced Reduced

X. Xu et al. [243] 2× 2 Square M -QAM Optimal Low Low

X. Xu et al. [244] MIMO QPSK Larger than [235] Reduced High

D. Kapetanovic et al. [245] Nt × 2 Infinite Constellation Asymptotic optimal Low Low

D. Kapetanovic et al. [246] MIMO Infinite Constellation Asymptotic optimal Reduced High

precoder is found by expanding an ellipse within a 2-dimensional lattice to maximum extent [243]. For

MIMO systems with 4-QAM inputs, a real-value max-dmin precoder is directly constructed based on well-

known dense packing lattices [244]. The obtained precoder offers better minimum Euclidean distance and

BER performance than the precoder based on the X-structure [40]. For Nt × 2 systems with infinite

signal constellation, an explicit structure of the max-dmin precoder is obtained from the design of the

lattice generator matrix [245]. Numerical results indicate that the obtained precoder achieves good mutual

information performance with large QAM inputs. This precoder design is extended to MIMO systems

in [246]. A brief summation of transmission designs with discrete input signals to optimize the transmit

diversity for the close loop MIMO systems is given in Table XVII.

Some work combine the STBC with the linear processing step in close loop system to further improve

the performance of MIMO systems. The structure of a close loop STBC systems is illustrated in Figure

47

9. It is important to note that the main difference between the linear processing in Figure 7 and the

linear precoder in Figure 9 is the usage of CSI. The linear precoder in Figure 7 does not need any

knowledge of CSI and the linear processing in Figure 9 requires some forms of the CSI. With PCSIT,

H. J. Park et al. exploit the constellation precoding scheme in [188] to design a full-diversity multiple

beamforming scheme for uncoded systems [247]. For the cases of both PCSTT and SCSIT, D. A. Gore et

al. investigate antenna selection techniques to minimize the SER for MIMO systems employing OSTBC

[248]. Jongren et al. combine the beamforming design and the OSTBC with partial knowledge of the

channel over frequency-nonselective fading channel [249]. The linear precoder design for a STBC system

over a transmit correlated Rayleigh fading channel is proposed in [250]. The transmitter utilizes the

transmit correlation matrix to design a linear precoding matrix, which minimizes an upper bound of the

average PEP of the STBC system. This design is extended to transmit correlated Rician fading channel

and arbitrary correlated Rician fading channel in [251] and in [252], respectively. Linear precoding for

a limited feedback STBC systems is proposed in [253]. For the noncoherent scenario where the CSI is

not available at the receiver, linear precoder designs for differential STBC systems over Kronecker fading

channel and arbitrary correlated Rayleigh fading channels are provided in [254] and [255], respectively.

For the partially coherent scenario where the imperfect channel estimation is performed at the receiver and

the estimation error covariance matrix is perfectly known at the transmitter via feedback, STBC designs

are proposed in [256] and [257] for the i.i.d. Rayleigh fading channels based on criterions of Kullback-

Leibler distance and the cutoff rate, respectively. The design in [258] is extended to the correlated Rayleigh

fading channels where the transmit and receive correlation matrices are perfectly known at the transmitter.

Moreover, a novel and efficient bit mapping scheme is proposed in [258]. A linear precoder is further

designed in [259] to adapt to the degradation caused by the imperfect CSIR.

D. Adaptive MIMO Transmission

The above-mentioned multiplexing, hybrid, and diversity transmission schemes can be employed adap-

tively according to the channel condition. Adaptive MIMO transmission is an effective approach to increase

48

Linear Processing

STCode

MIMOChannel

noise

Receiverinput output

Feedback CSIfrom Receiver

Fig. 9: A close loop STBC system

the data rate and decrease the error rate for wireless communication systems [260]–[262]. Adaptive MIMO

transmission techniques are investigated for both uncoded and coded systems.

1) Uncoded Systems: The capacity-achieving coding usually has long (infinitely) codewords and brings

delay to the system. Therefore, for the delay-sensitive applications, the modulation part of physical layer

is usually designed separately before the outer error-correcting coding is applied, as shown in Figure

10. For the uncoded SISO system subject to the average power and the BER (instantaneous or average)

constraints, the maximization of the spectral efficiency is obtained by jointly optimizing the transmit

rate and power over Rayleigh fading channels in [263], [264] and Nakagmi fading channels in [265],

developing upon the previous adaptive transmission techniques [266]–[276]. Z. Zhou et al. investigate

adaptive transmission schemes for i.i.d. Rayleigh flat fading MIMO channels with perfect or imperfect

CSIT and CSIR [41]. With perfect CSIT and CSIR, the MIMO channels are decomposed into a set of

parallel SISO subchannels by SVD. Then, the transmit rate and power allocation of each subchannel

need to be optimally designed to maximize the average spectral efficiency under the average transmit

power and instantaneous subchannel BER constraints. It is proved that this optimization problem can be

transformed into several subproblems, each of which only involves one subchannel. Then, the adaptive

transmission designs for SISO channels in [263], [264] can be exploited to find the optimal solution.

Moreover, the effect of the discrete rate constraint and the imperfect CSIT on system performance are

evaluated in [41]. When only partial CSI, i.e., the channel mean feeback, is available at the transmitter,

49

S. Zhou et al. provide an adaptive two-dimensional beamformer to exploit the transmit diversity by

incorporating the Alamouti coded [42]. Based on this beamformer, the beamvectors, the power allocation

policy, and the modulation schemes are designed for MIMO and MIMO-OFDM systems in [42] and [277],

respectively, which maximize the average discrete spectral efficiency under the constant power and the

average BER constraints. To further study the impact of the CSI error on adaptive MIMO transmission,

S. Zhou et al. consider the case where the transmitter performs a pilot signal assisted channel prediction

and examine the effect of the prediction error on the BER performance for a one-dimensional adaptive

beamformer [278]. Numerical results show that when the normalized estimation error is below a critical

threshold, the channel prediction error has a minor impact on the BER performance. For correlated MISO

Rayleigh fading channels with a limited number of feedback bits at the transmitter, P. Xia et al. develop a

strategy to adaptively select the transmission mode to maximize the average spectral efficiency under the

average power and the average BER constraints [279]. For correlated MIMO fading channels where the

correlation matrices are available at the transmitter and equal power is allocated to each transmit antenna, a

strategy to determine the number of active transmit antenna and the corresponding transmit constellations

is developed by maximizing a low bound of the minimum SNR margin with a fixed data rate [280]. R.

W. Heath Jr et al. propose a simple MIMO adaptive transmission scheme by switching between spatial

multiplexing and transmit diversity [281]. With perfect CSIR, both the minimum Euclidean distances of

spatial multiplexing scheme and STBC scheme which achieve a fixed transmit rate are computed at the

receiver. The scheme with larger minimum Euclidean distances is chosen and the decision is feedback to

the transmitter via a low-rate feedback channel. For double-input multiple-output (DIMO) systems with

both PCSIT and SCSIT, an adaptive scheme switching between OSTBC and spatial multiplexing with

zero-forcing (ZF) receiver based on the average spectral efficiency is proposed in [282]. For an adaptive

MIMO-OFDM system employing maximum ratio transmission (MRT) and MRC transceiver structure,

the statistic properties of the number of bits transmitted per OFDM block and the number of outage

per OFDM block over i.i.d. Rayleigh fading channels are derived in [283]. A brief comparison of above

adaptive MIMO transmitter designs is given in Table XVIII.

50

TABLE XVIII: Adaptive MIMO transmitter designs for uncoded systemsPaper System Criterion CSI Constraint

Z. Zhou et al. [41]MIMO Maximize average PCSIT, IPCSIT Instantaneous BER

i.i.d. Rayleigh fading spectral efficiency PCSIR Average transmit power

S. Zhou et al. [42]MIMO Maximize average Channel mean feedback Average BER

i.i.d. Rician fading spectral efficiency PCSIR Constant transmit power

P. Xia et al. [277]MIMO-OFDM Maximize average Channel mean feedback Average BER

i.i.d. Rician fading spectral efficiency PCSIR Constant transmit power

S. Zhou et al. [278]MIMO Maximize average

Channel prediction errorInstantaneous BER

i.i.d. Rayleigh fading spectral efficiency Constant transmit power

P. Xia et al. [279]MISO Maximize average Limited feedback Average BER

i.i.d. Rayleigh fading spectral efficiency PCSIR Average transmit power

R. Narasimhan [280]MIMO Maximize minimal SCSIT Fixed data rate

correlated Rayleigh fading SNR margin PCSIR Equal power allocation

R. W. Heath Jr [281] MIMOSpatial Multiplexing Feedback decision

Fixed data rateOSTBC switch PCSIR

J. Huang [282]DIMO Spatial Multiplexing PCSIT, SCSIT

ZF receivercorrelated Rayleigh fading OSTBC switch PCSIR

K. P. Kongara [283]MIMO-OFDM

Performance AnalysisPCSIT MRT

i.i.d. Rayleigh fading PCSIR MRC

MIMOChannel

noise

Receiverinput outputModulation Precoder

Feedback from Receiver

Adaptive Design

Fig. 10: Adaptive MIMO transmission for uncoded systems

Some work study the adaptive MIMO transceiver designs. For perfect CSIT and CSIR, D. P. Palomar

et al. investigate the transmit constellation selection and the linear transceiver design to minimize the

51

TABLE XIX: Adaptive MIMO transceiver designs for uncoded systemsPaper System Criterion CSI Constraint

D. P. Palomar et al. [284] MIMOMinimize PCSIT Instantaneous BER

transmit power PCSIR Fixed transmit rate

A. Yasotharan et al. [285] MIMOMinimize PCSIT ZF transceiver, constant transmit power

average BER PCSIR Fixed transmit rate

S. Bergman et al. [286] MIMOMinimize PCSIT DFE receiver, constant transmit power

weighted MSE PCSIR Fixed transmit rate

C.-C. Li et al. [287] MIMOMaximize PCSIT Instantaneous BER

transmit rate PCSIR Constant transmit power

transmit power with fixed transmit rate, where each subchannel is under the same BER constraint [284].

Assuming the symbol error probability of each subchannel is the same, the gap approximation method

is used to select the transmit constellations. Then, necessary and sufficient conditions are established to

examine the optimality of the subchannel diagonalization design. Moreover, an upper bound of the transmit

power loss incurred by using the subchannel diagonalization design is presented. On the other hand, an

optimal bit loading scheme among subchannels is developed in [285]. The proposed scheme minimizes a

lower bound of the average BER of the system subject to the fixed transmit data, constant transmit power,

and ZF transceiver structure constraints. S. Bergman et al. further obtain the optimal transceiver design

and the bit loading scheme among subchannels with DFE receiver [286]. The p-norms of weighed MSEs

is minimized under the fixed transmit data and the constant transmit power constraints. The subchannel

diagonalization design is proved to be always optimal for the system model considered in [286]. C.-C.

Li et al. establish a duality relationship between the joint linear transceiver and bit loading design of

maximizing the transmit rate and that of minimizing the transmit power in [287]. Then, an algorithm

can be developed to find the optimal design for the transmit rate maximization by using the solution of

transmit power minimization problem. A brief comparison of above adaptive MIMO transceiver designs

is given in Table XX.

With perfect CSIR, Y. Ko et al. propose a rate adaptive modulation scheme combined with OSTBC for

i.i.d. Rayleigh fading MIMO channels [288]. Close-form expressions for the average spectral efficiency

52

and a tight upper bound of the average BER in presence of feedback delay are derived. Then, optimal SNR

switching thresholds for adaptive modulation, which maximize the average spectral efficiency under the

average BER, the outage BER, and the constant transmit power constraints, are designed at the receiver.

The constellation chosen at the receiver is feeded back to the transmitter and the impact of feedback delay

on the average BER, throughout, and outage probability are analyzed. The design in [288] is extended

to spatially correlated Rayleigh fading channels in [289], where a low complexity method to determine

a set of near optimal SNR switching thresholds is provided. On the other hand, the design in [288] is

also extended to the case with the channel estimation error and the average transmit power constraint

in [290]. For MIMO-OSTBC systems, the channel estimation process of the rate adaptive modulation

scheme is investigated [291]. The transmit constellation and the allocation of the time and power between

the pilot and data symbols are optimally designed to maximize the average spectral efficiency under

the instantaneous BER constraint. The adaptive MIMO transmission scheme for the energy efficiency

maximization of MIMO-OSTBC systems with the channel estimation error subject to an instantaneous

BER constraint is proposed in [43]. By utilizing the imperfect CSIT, Q. Kuang et al. incorporate the

design of spatial power allocation between different transmit antennas into the adaptive modulation scheme

in [290] to further increase the average spectral efficiency [44]. A brief comparison of above adaptive

transmission designs for MIMO-OSTBC systems is given in Table XX.

Other adaptive MIMO transmission schemes with outdated CSIT and imperfect CSIT are investigated

in [292] and [293], [294], respectively. X. Zhang et al. and T. Delamotte et al. consider the adaptive

power allocation for each subchannel with the instantaneous BER constraint based on the modified water-

filling and mercury water-filling algorithms in [295] and [296], respectively. T. Haustein et al. develop

an adaptive power allocation strategy for each subchannel by minimizing the MSEs at the receiver under

the maximum average BER constraint [297]. Some adaptive MIMO transmission schemes by providing

the reconfigurable operation modes of the antenna arrays are discussed in [298], [299].

2) Coded Systems: The capacity expression in [1] has provided a fundamental transmission limit

for MIMO system. Since then, adaptive modulation and coding (AMC) schemes have been extensively

53

TABLE XX: Adaptive transmission designs for MIMO-OSTBC uncoded systemsPaper System Criterion CSI Constraint

Y. Ko et al. [288]MIMO Maximize average Feedback decision with delay Average BER, outage BER

i.i.d. Rayleigh fading spectral efficiency PCSIR Constant transmit power

J. Huang et al. [289]MIMO Maximize average Perfect channel estimation Instantaneous BER, average BER

correlated Rayleigh fading spectral efficiency Imperfect channel estimation Constant transmit power

X. Yu et al. [290]MIMO Maximize average

Imperfect channel estimationAverage BER

i.i.d. Rayleigh fading spectral efficiency Average transmit power

D. V. Duong et al. [291]MIMO Maximize average

Channel estimation optimizationInstantaneous BER

i.i.d. Rayleigh fading spectral efficiency Equal power allocation

L. Chen et al. [43]MIMO Maximize IPCSIT Instantaneous BER

i.i.d. Rayleigh fading energy efficiency PCSIR Equal power allocation

Q. Kuang et al. [44]MIMO Maximize average IPCSIT Average BER

i.i.d. Rayleigh fading spectral efficiency PCSIR Equal power allocation

investigated to approach this performance limit. M. R. McKay et al. consider the AMC for the correlated

Rayleigh fading MIMO channels [45]. The BICM is used for the coding. With SCSIT, two simple

transmission schemes are switched: statistical beamforming or spatial multiplexing with a zero-forcing

receiver. Then, the transmitter selects the best combination of the code rate, the modulation formate,

and the transmission scheme to maximize the data rate subject to a general BER union bound. Adaptive

DIMO switching between the spatial multiplexing and STBC based on the spectral efficiency maximization

for a non-selective channel with practical constraints, impairments, and receiver designs is investigated

in [46]. For spatial multiplexing or STBC, the spectral efficiency thresholds to select the best coding

rate and the modulation format is determined based on the packet error rate (PER) vs. instantaneous

capacity simulation curves (convolutional code). The selection is performed at the receiver with perfect

CSIR and then feeds back to the transmitter. Also, for a time- and frequency-selective channel, a new

physical layer abstraction and switching algorithm is proposed in [46], where a weighed sum of channel

qualities is minimized to reduce the variance of the channel qualities. P. H. Tan et al. investigate the

AMC for MIMO-OFDM systems in presence of channel estimation errors in slow fading channels [47].

The convolutional or low density parity check codes (LDPC) coded BER at the output of the detector

is evaluated via curving fitting method. The packet error rate (PER) for each modulation and coding

54

scheme is accurately approximated based on the obtained coded BER expression. Then, the code rate, the

modulation format, and the number of transmit streams are selected by maximizing the system throughout

under a fixed PER constraint. A supervised learning algorithm, called k-nearest neighbor, is used to select

the AMC parameters by maximizing the data rate under frame error rate (FER) constraints for MIMO-

OFDM systems with convolutional codes in frequency- and spatial-selective channels [300]. Assuming

perfect CSIT and CSIR, M. D. Dorrance et al. consider the AMC for a vertical bell laboratories layer

space-time (V-BLAST) MIMO system with rate-compatible LDPC codes and an incremental redundancy

hybrid automatic repeat request (H-ARQ) scheme [301]. A constrained water-filling algorithm for finite

alphabet input signals is provided to allocate power for each subchannel. Then, the modulation format used

for each subchannel is determined based on a look-up table. Z. Zhou et al. incorporate the BICM into the

adaptive transmission scheme in [41] to improve the BER robustness against the CSI feedback delay in

[48]. Moreover, BICM technology is modified by using a multilevel puncturing and interleaving technique

and combined with rate-compatible punctured code/turbo code to achieve a near full multiplexing gain.

An AMC strategy for BICM MIMO-OFDM systems with soft Viterbi decoding is proposed in [302].

A system performance metric called the expect goodput is defined in [302], which is a function of the

code rate, the modulation format, the subchannel power allocation, and the space-time-frequency precoder.

The transmitter determines these AMC parameters by maximizing the expect goodput via the feedback

of the perfect CSI and the effective SNR from the receiver. Spectral efficiencies of the adaptive MIMO

transmission schemes in real-time practical test-bed are shown in [303]. A brief comparison of above

AMC schemes for MIMO systems is given in Table XXI.

The current wireless standards such as 3GPP LTE, IEEE 802.16e, and IEEE 802.11n define concrete

implementation procedures for adaptive MIMO transmission in practice. We take the downlink trans-

mission in LTE standard as an example. The code rate and modulation format are selected based on

the received Channel Quality Indicator (CQI) user feedback. For example, the 4-bit CQI indices and

their interpretations in LTE [304] are given in Table XXII. Then, the modulated transmit symbols are

mapped into one or several layers. This antenna mapping depends on the Rank Indicator user feedback,

55

TABLE XXI: AMC schemes for MIMO systemsPaper System Criterion CSI Constraint

M. R. McKay et al. [45]MIMO Statistical beamforming SCSIT

BER union boundcorrelated Rayleigh fading spatial multiplexing switch PCSIR

Y.-S. Choi et al. [46]DIMO, non-selective Spatial multiplexing Feedback decision

Practical PERtime- and frequency-selective STBC switch PCSIR

P. H. Tan et al. [47]MIMO-OFDM Maximize

Channel estimation error Fixed PERslow fading system throughout

R. C. Daniels et al. [300]MIMO-OFDM Maximize average PCSIT

Fixed FERfrequency- and spatial-selective spectral efficiency PCSIR

M. D. Dorranceet al. [301]V-BLAST MIMO Maximize average PCSIT Rate-constrained

H-ARQ spectral efficiency PCSIR power, bit allocation

Z. Zhou et al. [48]MIMO Maximize average IPCSIT Instantaneous BER

i.i.d. Rayleigh fading spectral efficiency PCSIR Average transmit power

I. Stupia et al. [302] MIMO-OFDMMaximize PCSIT

Soft Viterbi decodingexpect goodput PCSIR

T. Haustein et al. [303]Practical MIMO Maximize average Channel estimation

Average BERtest bed spectral efficiency in practical systems

which includes a single antenna port, transmit diversity, and spital multiplexing [304, Table 7.2.3-0].

The precoding matrices for these transmission modes depend on the Precoding Matrix Indicator user

feedback, which are defined in details in [305, Sec. 6.3.4]. The adaptive MIMO transmission schemes in

IEEE 802.16e and IEEE 802.11n are similar. In IEEE 802.16e, 52 combinations of codes, code rate, and

modulation format are defined as in [306, Table 8.4]. The transmission mapping matrices includes [306,

Eq. (8.7)]: Matrix A, exploiting only diversity; Matrix B, combining diversity and spatial multiplexing;

Matrix C, employing spatial multiplexing. Then, a transmission mode is selected based on the link quality

evaluation. In IEEE 802.11n, 16 mandatory modulation and coding schemes are defined [307, Table 1].

A spatial multiplexing matrix is applied to convert the transmitted data streams into the transmit antennas

and an additional cyclic deay can be applied per transmitter to provide transmit cyclic delay diversity

[307, Figure 1]. Some practical AMC simulation results in these standards are shown in [262], [300],

[307]–[313].

56

TABLE XXII: 4 bit CQI table in LTE standardCQI index Modulation Code rate × 1024 Efficiency

0 out of range

1 QPSK 78 0.1523

2 QPSK 120 0.2344

3 QPSK 193 0.3770

4 QPSK 308 0.6016

5 QPSK 449 0.8770

6 QPSK 602 1.1758

7 16QAM 378 1.4766

8 16QAM 490 1.9141

9 16QAM 616 2.4063

10 64QAM 466 2.7305

11 64QAM 567 3.3223

12 64QAM 666 3.9023

13 64QAM 772 4.5234

14 64QAM 873 5.1152

15 64QAM 948 5.5547

Encoder MIMO

Channel

noise

Receiverinput outputModulation Precoder

Feedback from Receiver

Adaptive Design

Fig. 11: Adaptive MIMO transmission for coded systems

IV. MULTI-USER MIMO SYSTEMS

In this section, we introduce the transmission designs with discrete input signals for multiuser MIMO

systems. We discuss the following four scenarios: i) MIMO uplink transmission; ii) MIMO downlink

transmission; iii) MIMO interference channel; iv) MIMO wiretap channel.

57

A. MIMO Uplink Transmission

1) Mutual Information Design: For the multiple access channel (MAC), the maximum mutual infor-

mation region with uniformly distributed finite alphabet inputs is defined as the constellation-constrained

capacity region [314], [315]. For the MIMO MAC with PCSIT, M. Wang et al. derive the constellation-

constrained capacity region for an arbitrary number of users and arbitrary antennas configurations [20]. The

boundary of this region can be achieved by solving the weighted sum-rate (WSR) maximization problem

with finite alphabet and individual power constraints. Necessary conditions of the optimal precoders of all

users are established and an iterative algorithm is proposed to find these optimal precoders. Moreover, a

LDPC coded system with iterative detection and decoding is provided to examine the BER performance

of the obtained precoders. Numerical results indicate the obtained precoders achieve considerably better

performance than the non-precoding design and the Gaussian input design in terms of both mutual

information and coded BER.

For the MIMO MAC with SCSIT, asymptotic expressions for the mutual information of the MIMO

MAC with arbitrary inputs over Kronecker fading channels are derived in large system limits [118]. The

linear precoder design for the MIMO MAC is further developed based on the sum-rate maximization

[316]. The proposed design embraces the case where non-Gaussian interferers are present. For a more

general Weichelberger’s fading model in [317], W. Yu et al. investigate the linear precoder design based

on the WSR maximization [318]. The asymptotic (in large system limits) optimal left singular matrix of

each user’s optimal precoder is proved to be the eigenmatrix of the transmit correlation matrix of the

user. This results facilitates the derivation of an efficient iterative algorithm for computing the optimal

precoder for each user. A brief summation of above mutual information based precoder designs for the

MIMO uplink transmission with discrete input signals is given in Table XXIII.

2) MSE Design: For the MIMO MAC with PCSIT and PCSIR, E. A. Jorswieck et al. first design the

transmit covariance matrices of the users to minimize the sum MSE with the multiuser MMSE receiver

and individual user power constraints [50]. A power allocation scheme among users is developed under a

sum power constraint to further reduce the sum MSE. Based on the KKT conditions, the optimal transmit

58

TABLE XXIII: Mutual information based precoder designs for the MIMO uplink transmission with

discrete input signalsPaper Model Main Contribution

M. Wang et al. [20]MIMO MAC Propose an iterative algorithm to find

PCSIT, PCSIR the optimal precoders which maximize the WSR

C.-K. Wen et al. [118]MIMO MAC, Kronecker fading Derive asymptotic expressions for the mutual information

SCSIT, PCSIR of the MIMO MAC with arbitrary inputs

M. A. Girnyk et al. [316]

MIMO MAC, Kronecker fadingPropose an iterative algorithm to find

SCSIT, PCSIRthe optimal precoders which maximize the sum-rate

non-Gaussian interferers

Y. Wu et al. [318]MIMO MAC, Weichelberger’s fading Find the asymptotic optimal left singular matrix of each user’s optimal precoder

SCSIT, PCSIR Propose an efficient iterative algorithm to maximize WSR

covariance matrices and the power allocation scheme can be found by an iterative algorithm. In addition,

the achievable MSE region is studied in [50]. S. Serbetli et al. investigate how to determine the number

of transmit symbols for each user [319]. Z.-Q. Luo et al. extend the MMSE transceiver design to MIMO

MAC OFDM systems [320]. I.-T. Lu et al. further study the impact of the practical per-antenna power

and peak power constraints [321]. The MMSE transceiver design for MIMO MAC with channel mean

or covariance SCSIT feedback and PCSIR subject to individual user power constraints is investigated

in [322], [323]. For both sum and individual user power constraints, X. Zhang et al. study the average

sum MSE minimization problem for two user MIMO MAC with correlated estimation error IPCSIT and

IPCSIR [324]. P. Layec et al. further consider the K user MIMO MAC case with additional quantization

noise and individual user power constraints [325]. A near optimal closed-form transceiver structure, which

minimizes the average sum MSE for MIMO MAC with i.i.d estimation error IPCSIT and IPCSIR subject

to a sum power constraint, is provided in [326]. A brief summation of above MMSE precoder designs

for the MIMO uplink transmission with discrete input signals is given in Table XXIV.

3) Diversity Design: For open-loop systems, based on the framework for analyzing the dominant error

event regions in [327], M. E. Gartner et al. derive space-time/frequency code design criterions for two-

user fading MIMO MAC [328]. Moreover, an explicit two-user 2 × 2 coding scheme is constructed by

concatenating two Alamouti schemes with a column swapping for one user’s codeword to achieve a

59

TABLE XXIV: MMSE precoder designs for the MIMO uplink transmission with discrete input signalsPaper Model Main Contribution

E. A. Jorswieck et al. [50]

MIMO MAC

PCSIT, PCSIR Propose an iterative algorithm to minimize the sum MSE

Individual/sum power constraints

S. Serbetli et al. [319]

MIMO MAC

PCSIT, PCSIR Investigate how to determine the number of transmit symbols

Individual power constraints

Z.-Q. Luo et al. [320]

MIMO MAC OFDM

PCSIT, PCSIR Extend the MMSE transceiver design to OFDM systems

Individual power constraints

I.-T. Lu et al. [321]

MIMO MAC

PCSIT, PCSIR Propose iterative algorithms to minimize the sum MSE

Per-antenna/peak power constraints

[322], [323]

MIMO MAC

SCSIT, PCSIR Propose iterative algorithms to minimize the average sum MSE

Individual power constraints

X. Zhang et al. [324]

Two-user MIMO MAC

IPCSIT, IPCSIR Propose iterative algorithms to minimize the average sum MSE

Individual/sum power constraints

P. Layec et al. [325]

MIMO MAC

IPCSIT, IPCSIR, Quantization noise Propose iterative algorithms to minimize the average sum MSE

Individual power constraints

J. W. Huang et al. [326]

MIMO MAC

IPCSIT, IPCSIR Provide closed-form near optimal MMSE precoder designs

A sum power constraint

minimum rank of three. A algebraic construction of STBC based on diversity and multiplexing tradeoff

for MIMO MAC is proposed in [329]. Similar STBC designs are also provided for the asynchronous

single-input multiple-output (SIMO) MAC in [330], non-uniform user transmit power MIMO MAC in

[331], and the two-user MIMO multiple-access AF relay channel in [332]. A space-frequency code for

MIMO-OFDM MAC is designed in [49], which achieves full diversity for every user. However, the codes

in [49] suffer from high large peak-to-average power ratio since some elements in the codeword matrices

are zero. Another group of STBC for MIMO MAC is proposed in [333] by minimizing a truncated union-

bound approximation. Simulations show that the codes in [333] achieve better error probability than the

60

TABLE XXV: Space-time techniques for the MIMO uplink transmission with discrete input signalsPaper Model Main Contribution

M. E. Gartner et al. [328]MIMO MAC Derive space-time/frequency code design criterions

Two-user Construct an explicit coding scheme for two-user 2× 2

M. Badr et al. [329]–[332]MIMO MAC, asynchronous SIMO MAC, unbalanced MIMO MAC Construct STBCs based on

Two-user MIMO multiple-access AF relay diversity and multiplexing tradeoff

W. Zhang et al. [49] MIMO-OFDM MACDesign a space-frequency code

which achieves full diversity for every user

Y. Hong et al. [333] MIMO MACDesign STBCs based on

truncated union-bound approximation

[334], [335]MIMO MAC

Design sphere-decodable STBCsTwo-user

[336], [337]MIMO MAC

Design differential STBCsTwo-user

J. W. Huang et al. [338]MIMO MAC Investigate the linear precoding design

Outdated CSIT subject to individual user’s transmit power constraint

Y.-J. Kim et al. [339]DIMO MAC

Design a STBC with a linear detectionTwo-user, feedback

F. Li et al. [340], [341]Two-user MIMO MAC, K-user MIMO MAC

Propose full-diversity precoder designsPCSIT

codes in [49], [328]. H.-F. Lu et al. propose sphere-decodable STBCs for two-user MIMO MAC and

provide an explicit construction for 2 × 2 case [334]. STBCs for two-user MIMO MAC with reduced

average sphere decoding complexity are further constructed in [335]. Moreover, A differential STBC for

two-user MIMO MAC is proposed in [336]. Another differential STBC for two-user MIMO MAC with

low complexity non-coherent decoders is further designed in [337]. For close-loop systems, by exploiting

the outdated CSI, J. W. Huang et al. investigate the linear precoding design for MIMO MAC with OSTBC

to minimize PEP subject to individual transmit power constraint of each user [338]. Also, Y.-J. Kim et

al. propose a STBC for two-user DIMO MAC with a linear detection by utilizing the phase feedback

transmitted to one user [339]. By exploiting PCSIT, F. Li et al. propose full-diversity precoder designs for

two-user and K-user MIMO MAC in [340] and [341], respectively. A brief comparison of above work is

given in Table XXV.

4) Adaptive Transmission: For MIMO-OFDM MAC with PCSIT and PCSIR, Y. J. Zhang et al. propose

an adaptive resource allocation scheme when each user’s signal is transmitted along the maximum singular

61

TABLE XXVI: Adaptive transmissions for the MIMO uplink transmission with discrete input signalsPaper Model Main Contribution

Y. J. Zhang et al. [51]MIMO-OFDM MAC, perfect CSI Propose an adaptive resource allocation

Each user’s uncoded BER and data rate constraints to minimize the total transmit power

Y. J. Zhang et al. [342]MIMO-OFDM MAC, perfect CSI Propose a low complexity adaptive resource

Each user’s uncoded BER and data rate constraints allocation to minimize the total transmit power

R. Chemaly et al. [343]MIMO-OFDM MAC, SCSIT, PCSIR Propose an adaptive resource allocation

Each user’s uncoded BER and total power constraints to maximize the sum data rate

M. S. Maw et al. [344]MIMO-OFDM MAC, perfect CSI Propose an adaptive resource allocation considering the

Each user’s uncoded BER and total power constraints users’ fairness and maximizing the sum data rate

of its own channel [51]. The proposed scheme selects the subcarrier allocation, the power allocation, and

modulation modes to minimize the total transmit power subject to each user’s uncoded BER and data rate

constraints. It is assumed that these uplink transmission parameters are determined at the base station and

then sent to the users via the error-free channel. Simulation results indicate that within a reasonable region

of Doppler spread, the proposed scheme is also robust to the channel time variation. A low-complexity

design with a neighborhood search for the optimal resource allocation solution is further proposed in

[342]. For MIMO-OFDM MAC where each user employs the transmission design in [279] with its own

SCSI, R. Chemaly et al. propose an adaptive resource allocation scheme to maximize the sum data

rate subject to each user’s uncoded BER and the total power constraints [343]. By taking an additional

fairness consideration among different users, another adaptive resource allocation scheme is given in [344]

to maximize the sum data rate subject to each user’s uncoded BER and the total power constraints. A

brief comparison of above work is given in Table XXVI.

B. MIMO Downlink Transmission

1) Mutual Information Design: For the 2-user MISO BC with PCSIT, PCSIR, and finite alphabet inputs,

the precoder designs to maximize the sum rate and the corresponding simplified receiver structure are

studied in [345]–[347]. Y. Wu et al. further investigate the linear precoder design to maximize the WSR

of the general MIMO BC [21]. Explicit expressions for the achievable rate region are derived, which is

applicable to an arbitrary number of users with generic antenna configurations. Then, it is revealed that the

62

TABLE XXVII: Mutual information based precoder designs for the MIMO downlink transmission with

discrete input signalsPaper Model Main Contribution

R. Ghaffar et al. [345]–[347]MISO BC, 2-user Propose the precoder designs and the corresponding

PCSIT, PCSIR simplified receiver structure to maximize the sum-rate

Y. Wu et al. [21]MIMO BC Propose an iterative algorithm to find

PCSIT, PCSIR the optimal precoders which maximize the WSR

W. Wu et al. [349]Cooperative multi-cell MIMO downlink Propose two low-complexity algorithms to find

PCSIT, PCSIR the optimal precoders which maximize the sum-rate

S. X. Wu et al. [350]MISO downlink multicasting Propose two transmit schemes

PCSIT, PCSIR to maximize the multicast rate

sum rate loss will occur in high SNR regime for the improper precoder design because of the non-uniquely

decodable transmit signals. An iterative algorithm is proposed to optimize precoding matrices for all users.

A downlink multiuser system with LDPC code and iterative detection and decoding is further developed.

Simulations illustrate that the proposed precoding design provides substantial WSR and coded BER gains

over the best rotation design for SISO BC [348], the non-precoding design, and the Gaussian input design.

W. Wu et al. investigate the linear precoder design for cooperative multi-cell MIMO downlink systems

with PCSIT and finite alphabet inputs [349]. To reduce the complexity of evaluating the non-Gaussian

interferers, W. Wu et al. use a Gaussian interferer to approximate the sum of non-Gaussian interferers

at each user’s receive side and propose two iterative algorithms to maximize the approximated sum-rate

under per base station power constraints. Comparing to the algorithm in [21], the proposed algorithms

in [349] reduce the implementation complexity but with negligible performance losses. S. X. Wu et al.

propose two transmit schemes to maximize the multicast rate of MISO downlink multicasting systems with

PCSIT and finite alphabet inputs [350]. A brief summation of above mutual information based precoder

designs for the MIMO downlink transmission with discrete input signals is given in Table XXVII.

2) MSE Design: For the linear processing, early research in [351], [352] consider to extend the MMSE

transceiver design in [32] to MIMO BC. For MIMO BC with PCSIT, PCSIR, and a sum power constraint,

S. Shi et al. establish a uplink-downlink duality framework between the downlink and uplink MSE feasible

63

regions [53]. Based on this framework, the downlink sum-MSE minimization can be transformed into an

equivalent uplink problem. Then, two globally optimum algorithms are proposed to find the optimal

MMSE transceiver design. S. Shi et al. further propose near-optimal low complexity iterative algorithms

to minimize the maximum ratio of MSE and given requirement under a sum power constraint and to

minimize the transmit power under various MSE requirements [353]. Moreover, R. Hunger et al. propose

a low complexity approach to evaluate the MSE duality [354], which is able to support the switched off

data streams and the passive users correctly. The MSE duality in [53] is extended to the imperfect CSI

case for MISO BC [355]. D. S.-Murga et al. provide a transceiver design framework for MIMO-OFDM

BC based on the channel Gram matrices feedback, where the CSI quantization error is also taken into

consideration [356]. As an example, a robust precoder design with fixed decoder which minimizes the

sum MSE subject to a sum power constraint is given in [356]. M. Ding et al. propose a robust transceiver

design for network MIMO systems with imperfect backhaul links [357], which minimizes the maximum

substream MSE subject to per base station power constraints.

For the robust design where the actual channel is considered to be within an uncertain region around the

channel estimate, a robust power allocation scheme among users is proposed for MISO BC to minimize

the transmit power subject to each user’s MSE requirement [358]. N. Vicic et al. extend this to robust

transceiver design for MISO BC and MIMO BC in [359] and [360], respectively. X. He et al. further

investigate this robust transceiver design subject to the probabilistic MSE requirement of each user [361].

A robust resource allocation design to optimize the function of worst case MSE subject to quadratic power

constraints for multicell MISO downlink transmission is proposed in [362].

Some work consider the MMSE precoder design with the fixed decoder. A. D. Dabbagh et al. obtain

an MMSE based precoding technique for MISO BC with IPCSIT and a sum power constraint [363]. The

limited feedback system model is also investigated in [363]. H. Sung et al. develop a generalized MMSE

precoder for MIMO BC with perfect and imperfect CSI [364]. The designs for the minimization of the

sum MSE subject to a sum power constraint and the minimization of the interference-plus-noise power at

the receiver subject to individual user’s power constraints are obtained. P. Xiao et al. propose an improved

64

MIMO BC MMSE precoder with perfect and imperfect CSI subject to a sum power constraint for improper

signal constellations such as amplitude shift-keying, offset quadrature phase shift keying, etc [365]. A brief

summation of above linear MMSE precoder designs for the MIMO downlink transmission with discrete

input signals is given in Table XXVIII. There are also some work studying the non-linear MMSE precoder

designs for the MIMO downlink transmission with discrete input signals [53], [366]–[371].

3) Diversity Design: In the downlink transmission, mitigating the multiuser interference is essential

for the reliable communication and this normally needs the availability of some forms of CSI. Therefore,

there are not much work for the open loop downlink systems. Q. Ma et al. propose a trace-orthogonal

space-time coding, which is suitable for MIMO BC [372]. The proposed space-time coding reduce the

complexity of ML with little performance loss. Some differential STBC designs for the data encoded

using layered source coding are proposed for MIMO BC [373], [374]. For the close loop systems,

the space-time techniques are usually combined with the linear precoding design to eliminate the co-

channel interference and improve the diversity performance. R. Chen et al. propose a unitary downlink

precoder design for multiuser MIMO STBC systems with PCSIT [375]. Using enough antenna dimension,

the proposed precoder can effectively cancel the co-channel interference at each user’s side. Then, the

additional antenna dimension can be used to select the precoder matrix for the diversity gain. R. Chen

et al. further provide a more general transmission design for MIMO BC [52], which combines the

unitary downlink precoder for co-channel interference mitigation with the space-time precoder design and

antenna selection technique for SER minimization. Simulations indicate that the proposed design achieves

significant diversity gains in terms of SER. D. Wang et al. combine nonlinear THP with two diversity

techniques: dominant eigenmode transmission and OSTBC to improve the BER performance of MIMO

BC [376]. B. Clerckx et al. investigate the transmission strategy for two-user MISO BC with outdated

CSIT and obtain the analytical expressions for the corresponding average PEP over correlated Rayleigh

fading channel [377]. By exploiting the further SCSIT, signal constellations are optimized to improve

the average PEP performance. D. Lee et al. analyze the outage probability of the average effective SNR

for MIMO BC employing OSTBC [378]. The analytical results indicate that the opportunistic scheduling

65

TABLE XXVIII: Linear MMSE precoder designs for the MIMO downlink transmission with discrete

input signalsPaper Model Main Contribution

S. Shi et al. [53]

MIMO BC

PCSIT, PCSIR Establish a uplink-downlink MSE duality framework

A sum power constraint

S. Shi et al. [353]MIMO BC Minimize the maximum ratio of MSE and given requirement

PCSIT, PCSIR Minimize the transmit power under various MSE requirements

R. Hunger et al. [354]

MIMO BCEstablish a MSE duality, which is able to support

PCSIT, PCSIRthe switched off data streams and the passive users correctly

A sum power constraint

M. Ding et al. [355]

MISO BC

IPCSIT, IPCSIR Extend the MSE duality in [53] to the imperfect CSI case

A sum power constraint

D. S.-Murga et al. [356]

MIMO-OFDM BCProvide a transceiver design framework

Channel Gram matrixbased on the channel Gram matrices feedback

A sum power constraint

M. Ding et al. [357]

Network MIMO systems

Imperfect backhaul links Minimize the maximum substream MSE

Per base station power constraints

M. Payaro et al. [358]

MISO BCPropose a robust power allocation scheme among users

IPCSIT, IPCSIRto minimize the transmit power

Each user’s MSE requirement

N. Vicic et al. [359]

MISO BCPropose a robust transceiver design

IPCSIT, IPCSIRto minimize the transmit power

Each user’s MSE requirement

N. Vicic et al. [360]MIMO BC Propose robust transceiver designs

IPCSIT, IPCSIR for several MSE-optimization problem

X. He et al. [361]

MIMO BCPropose a robust transceiver design

IPCSIT, IPCSIRto minimize the transmit power

Each user’s probabilistic MSE requirement

E. Bjornson et al. [362]

MISO BCPropose a robust resource allocation design

IPCSIT, IPCSIRto optimize the function of worst case MSE

Quadratic power constraints

A. D. Dabbagh et al. [363]

MISO BC, fixed decoder

IPCSIT, PCSIR Obtain an MMSE based precoding technique

A sum power constraint

H. Sung et al. [364]MIMO BC, fixed decoder Minimize the sum MSE subject to a sum power constraint

Perfect and imperfect CSI Minimize the interference-plus-noise power subject to individual user’s power constraints

P. Xiao et al. [365]

MIMO BC, fixed decoderPropose an improved MIMO BC MMSE precoder

Perfect and imperfect CSIfor improper signal constellations

A sum power constraint

66

TABLE XXIX: Space-time techniques for the MIMO downlink transmission with discrete input signalsPaper Model Main Contribution

Q. Ma et al. [372] MIMO BC Propose a trace-orthogonal space-time coding

[373], [374] MIMO BCPropose differential STBC designs

for the data encoded using layered source coding

R. Chen et al. [375]MIMO BC Combine a unitary downlink precoder

PCSIT, PCSIR with STBC

R. Chen et al. [52]MIMO BC Combine a unitary downlink precoder

PCSIT, PCSIR with STBC and antenna selection

D. Wang et al. [376]MIMO BC Combine THP with dominant eigenmode

PCSIT, PCSIR transmission and OSTBC

B. Clerckx et al. [377]MISO BC Optimize signal constellation

two-user, outdated CSIT, PCSIR to improve the average PEP

D. Lee et al. [378]MIMO BC Analyze opportunistic scheduling and diagonalization-precoding

PCSIT, PCSIR for MIMO BC with OSTBC

L. Li et al. [379]MIDO BC Combine a interference cancellation scheme

PCSIT, no CSIR with the Alamouti code

C.-T. Lin et al. [380]MIMO BC Extend X-Structure precoder

PCSIT, PCSIR to MIMO BC

scheme provide the effective SNR and the diversity gains compared to the block diagonalization-precoding

scheme. With PCSIT and no CSIR, L. Li et al. propose a interference cancellation scheme for MIDO BC,

where the transmitter sends the precoded Alamouti code to every user simultaneously [379]. Numerical

results indicate that the proposed design in [379] has better diversity gains than the block diagonalization

designs in [375], [378]. By using a regularized block diagonalization precoder to suppress the multiuser

interference in the first step, C.-T. Lin et al. extend the X-Structure precoder in [40] to MIMO BC in

[380]. A brief comparison of above work is given in Table XXIX.

4) Adaptive Transmission: For uncoded systems, most adaptive transmission literatures focus on the

adaptive resource allocation for multiuser MIMO-OFDM systems. Typically, with perfect CSIT and CSIR,

C.-F. Tsai et al. devise an adaptive resource allocation algorithm for multiuser downlink MISO orthogonal

frequency division multiplexing access (OFDMA)/spatial division multiple access (SDMA) systems with

multimedia traffic [54]. A dynamical priority scheduling scheme is developed to provide service to urgent

67

users based on time to expiration. Then, the resource for power, subchannel, and bit allocation is iteratively

allocated to maximize the spectrum efficiency subject to various quality of service constraints. C.-M. Yen

et al. extend this adaptive resource allocation design to multiuser downlink MIMO OFDMA systems

[381], where a utility function is defined to distinguish the real time service and the nonreal time service

more clearly. Other resource allocation schemes for practical multiuser MIMO-OFDM systems can be

found in [382]–[384].

For coded systems, Y. Hara et al. design an efficient downlink scheduling algorithm for multiuser

downlink MIMO systems by optimizing an equivalent uplink scheduling problem [385]. With perfect

knowledge of the channel, the proposed algorithm selects the user for each transmit beam, the transmit

weight for the user, and the corresponding modulation and coding scheme to maximize the overall

system throughout subject to PER constraints. Moreover, M. Fsslaoui et al. propose a two-step adaptive

transmission scheme for multiuser downlink MISO-OFDM systems [386]. In the first step, the users that

should be simultaneously transmitted are selected based on the degree of orthogonality criterion. In the

second step, the modulation and coding scheme for each user is determined adaptively by maximizing the

throughput while satisfying the effective SNR constraint for each user. The proposed design is simulated

for the IEEE 802.11ac. It is worth mentioning that as shown in [386, Fig. 4], even the adaptive transmission

technology is employed, there are still obvious gaps between the practical schemes and the theoretical

capacity achieved by Gaussian input. The design in [386] is extended to multiuser downlink MIMO-

OFDM systems and the fairness among users is taken into consideration additionally [387]. J. Wang

et al. propose another transmission scheme for multiuser downlink MIMO-OFDMA systems, where the

power allocation, the resource allocation policy, and the modulation and coding scheme for each user

are adaptively determined to maximize the system throughout [388]. The proposed design is simulated

for IEEE 802.16e and achieves good performance. The adaptive transmission for multiuser downlink

MIMO-OFDM systems with limited feedback is provided in [389], where only a single spatial stream is

transmitted for each user. A. R.-Alvarino et al. propose an adaptive transmission scheme for multiuser

downlink MIMO-OFDM systems with limited feedback, which allows multiple spatial streams for each

68

TABLE XXX: Adaptive transmissions for the MIMO downlink transmission with discrete input signalsPaper Model Main Contribution

C.-F. Tsai et al. [54]Uncoded MISO-OFDMA/SDMA, perfect CSI Devise an adaptive resource allocation algorithm

Various quality of service constraints to maximize the spectrum efficiency

C.-M. Yen et al. [381]Uncoded MIMO-OFDMA, perfect CSI Define a utility function to distinguish the

Various quality of service constraints real time service and the nonreal time service

Y. Hara et al. [385]Coded MIMO, perfect CSI Design an efficient downlink scheduling algorithm

PER constraints to maximize the overall system throughout

M. Fsslaoui et al. [386]Coded MISO-OFDM, perfect CSI Propose a two-step adaptive transmission scheme

Effective SNR constraints to maximize the spectral efficiency

M. Fsslaoui et al. [387]Coded MIMO-OFDM, perfect CSI Fairness among users is taken

Effective SNR constraints into consideration of system designs

J. Wang et al. [388]Coded MIMO-OFDMA, perfect CSI Propose an adaptive scheme for the power allocation,

Effective SNR constraints the resource allocation policy, and the modulation and coding scheme

Z. Chen et al. [389]Coded MIMO-OFDM, feedback CSI

Propose an adaptive scheme to maximize the spectrum efficiencyEffective SNR constraints

A. R.-Alvarino et al. [390]Coded MIMO-OFDM, feedback CSI Propose an adaptive scheme to maximize the overall system

FER constraints throughout, which allows multiple spatial streams for each user

user [390]. A greedy algorithm is used to select the users and the spatial modes. A machine learning

classifier is used to select the modulation and coding schemes for users by maximizing the overall system

throughout subject to FER constraints. A brief comparison of above work is given in Table XXX.

C. MIMO Interference Channel

1) Mutual Information Design: By directly extending the classical interference alignment scheme used

for Gaussian input signal, B. Hari Ram et al. and Y. Fadlallah et al. investigate the precoder designs for

K-user MIMO interference channel by maximizing the mutual information of the finite alphabet signal

sets in [391] and [392], respectively. Following the interference alignment scheme, the precoders in [391],

[392] only use half of the transmit antenna dimension to send the desired signals. H. Huang et al. consider

to relax this dimension constraint by allowing the interfering signals to overlap with the desired signal and

propose a partial interference alignment and interference detection scheme for K-user MIMO interference

channel with finite alphabet inputs [393]. For the interfering signals which can not be aligned at the each

receiver, the receiver detects and cancels the residual interference based on the constellation map. By

69

TABLE XXXI: Mutual information based precoder designs for the MIMO interference channel with

discrete input signalsPaper Model Main Contribution

[391], [392]MIMO interference channel Extend interference alignment scheme

Global PCSIT, PCSIR to precoder design with finite alphabet inputs

H. Huang et al. [393]MIMO interference channel Propose a partial interference alignment

Global PCSIT, PCSIR and interference detection scheme

[55], [394]MIMO interference channel Propose gradient decent based iterative algorithms

Global PCSIT, PCSIR to find optimal precoders of all transmitters

[395], [396]MIMO interference channel Propose a novel low-complexity

Global PCSIT, PCSIR fractional interference alignment scheme

treating the interference as noise, Y. Wu et al. and A. Ganesan et al. independently derive the mutual

information expressions for K-user MIMO interference channel with finite alphabet inputs in [55] and

[394], respectively. Then, gradient decent based iterative algorithms are proposed in [55], [394] to find

the optimal precoders of all transmitters. Instead of degree of freedom used for Gaussian input signal, B.

Hari Ram et al. define a new performance metric to analyze the transmission efficiency for K-user MIMO

interference channel with finite alphabet inputs: number of symbols transmitted per transmit antenna per

channel user (SpAC) [395]. It is revealed in [55], [394] that simple transmission schemes can achieve

the maximum 1 SpAC in high SNR regime. However, the joint detection of all the transmitter signals

including the desired signals and the interfering signals at each receiver is needed in [55], [394] to achieve

1 SpAC. B. H. Ram et al. further propose a fractional interference alignment scheme, which does not

decode any of the interfering signals [396]. The fractional interference alignment can achieve any value

of SpAC in range [0, 1] and a good error rate performance for both uncoded and coded BER. A brief

summation of above mutual information based precoder designs for the MIMO interference channel with

discrete input signals is given in Table XXXI.

2) MSE Design: For MIMO interference channel with global perfect CSI, H. Shen et al. investigate

linear transceiver designs under a given and feasible degree of freedom [56]. Two iterative algorithms

are proposed to minimize both the sum MSE and the maximum per-user MSE. The transceiver design

70

TABLE XXXII: MMSE transceiver designs for MIMO interference channel with discrete input signalsPaper Model Main Contribution

H. Shen et al. [56]MIMO interference channel

Minimize both the sum MSE and the maximum per-user MSEGlobal perfect/Imperfect CSI

C.-E. Chen et al. [397]MIMO interference channel

Minimize the maximum per-stream MSE of all usersGlobal perfect CSI

F. Sun et al. [398]MIMO interference channel

Minimize the MSE of the signal and interference leakageGlobal perfect CSI

F.-S. Tseng et al. [399]MIMO interference channel

Minimize both the sum MSE and the maximum per-user MSERandom vector quantization feedback

for MIMO interference channel with channel estimation error is also provided in [56]. With perfect CSI,

C.-E. Chen et al. extend the design in [397] to minimize the maximum per-stream MSE of all users. Also,

F. Sun et al. propose a low complexity transceiver design based on minimizing the MSE of the signal

and interference leakage at each transmitter [398]. The robust transceiver design for MISO interference

channel with random vector quantization feedback mechanism is proposed in [399], where the impact

of quantization error at the transmitter is considered. The optimal transmit beamforming vectors and the

receiver decoding scalars for both the sum MSE and the maximum per-user MSE minimization are found

by iterative algorithms. A brief summation of above work is given in Table XXXII.

3) Diversity Design: For the open loop system, L. Shi et al. design full diversity STBCs for two-user

MIMO interference channel with the group ZF receiver [57]. Y. Lu et al. combine the blind interference

alignment scheme with three STBCs for K-user MISO interference channel employing reconfigurable

antenna [400]. These combinations provide tradeoff between diversity, rate, and decoding complexity,

where threaded algebraic space-time codes have full diversity and high rate with high decoding complexity,

OSTBCs have full diversity and low rate with the linear decoding complexity, and Alamouti codes have

high rate and low linear decoding complexity but with full diversity loss. For the close loop system, when

the global CSI is available at all transmitters and receivers, A. Sezgin et al. combine the STBC with

interference alignment to achieve high diversity gains for MIMO interference channel [401]. However,

the feasibility condition investigation for MIMO interference channel reveals that the separation of space-

time coding and precoding design may not be optimal in general [402]. Moreover, the diversity gains of

71

TABLE XXXIII: Space-time techniques with discrete input signals for MIMO interference channelPaper Model Main Contribution

L. Shi et al. [57]Two-user MIMO interference channel

Design full diversity STBCsNo PCSIT, PCSIR

Y. Lu et al. [400]MISO interference, reconfigurable antenna Combine the blind interference alignment scheme

No PCSIT, PCSIR with three STBCs

A. Sezgin et al. [401]MIMO interference channel Combine STBC

Global PCSIT, PCSIR with the interference alignment design

H. Ning et al. [402]MIMO interference channel Reveals that the separation of space-time coding

Global PCSIT, PCSIR and precoding design may not be optimal

L. Li et al. [403]3-user 2× 2 interference channel Combine the antenna selection technique

Global PCSIT, PCSIR with the interference alignment design

various interference alignment schemes for MIMO interference channels are analyzed in [402]. L. Li et al.

combine the antenna selection technique with the interference alignment design to improve the diversity

gain of the worst case user for 3-user 2×2 interference channel [403]. A brief comparison of above work

is given in Table XXXIII.

4) Adaptive Transmission: B. Xie et al. investigate the adaptive transmission for uncoded MIMO

interference channels with imperfect CSIT [58]. Interference alignment based bit-loading algorithms are

proposed to minimize the average BER subject to a fixed sum-rate constraint. Based on this, an adaptive

transmission scheme switches among two interference alignment algorithms and the basic time-division

multiple access scheme to combat the CSI uncertainty is further provided. M. Taki et al. investigate the

adaptive transmission for coded MIMO interference channels with imperfect CSIT [404]. An interference

alignment based adaptive transmission scheme is proposed, which adaptively selects the coding, mod-

ulation, and power allocation across users to maximize the weighted sum rate subject to a sum power

constraint and BER constraints for every transmit stream.

D. MIMO Wiretap Channel

1) Mutual Information Design: For the wiretap channel, the finite-alphabet inputs may allow both the

receiver and the eavesdropper to decode a transmitted message accurately if the transmit power is high-

enough. Therefore, the secrecy rate may not be high at a high SNR. This phenomenon has been observed

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in SISO single-antenna eavesdropper wiretap channels via simulations [405]. Therefore, S. Bashar et al.

suggest to find an optimum power control policy at the transmitter for maximizing the secrecy rate [406].

For MISO single-antenna eavesdropper (MISOSE) wiretap channel, S. Bashar et al. investigate the power

control optimization issue at the transmitter based on the conventional beamforming transmission for the

Gaussian input case and develop a numerical algorithm to find the optimal transmission power [406]. This

idea is extended to MIMO multiple-antenna eavesdropper (MIMOME) wiretap channel by exploiting a

generalized SVD (GSVD) precoding structure to decompose the MIMOME channel into a bank of parallel

subchannels [407]. Then, numerical algorithms are proposed to obtain the adequate power allocated to

each subchannel. Although the precoding design in [407] achieves significant performance gains over the

conventional design that relies on the Gaussian input assumption, it is still suboptimal. The reasons are

twofold. First, part of the transmission symbols is lost by both the desired user and the eavesdropper

according to the GSVD structure. This condition may result in a constant performance loss. Second, the

power control policy compels the transmitter to utilize only a fraction of power to transmit in the high SNR

regime, which may impede the further improvement of the performance in some scenarios (see [407, Fig.

3] as an example). As a result, Y. Wu et al. further investigate the linear precoder design for MIMOME

wiretap channel when the instantaneous CSI of the eavesdropper is available at the transmitter [59]. An

iterative algorithm for secrecy rate maximization is developed through a gradient method. For systems

where the number of transmit antennas is less than or equal to the number of the eavesdropper antennas,

only partial transmission power is necessary for maximizing the secrecy rate at a high SNR. Accordingly,

the excess transmission power is used to construct an artificial jamming signal to further improve the

secrecy rate. By invoking the lower bound of the average mutual information in MIMO Kronecker fading

channels with the finite alphabet inputs [30], Y. Wu et al. also study the scenarios where the transmitter

only has the statistical CSI of the eavesdropper [59]. W. Zeng et al. study the precoder design for MIMO

secure cognitive ratio systems with finite alphabet inputs [408]. With the SCSIT of the eavesdropper,

an iterative algorithm is proposed to maximize the secure rate of the secondary user under the control

of power leakage to the primary users. The transmission design for MIMO decode-and-forward (DF)

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TABLE XXXIV: Mutual information based precoder designs for the MIMO wiretap channel with

discrete input signalsPaper Model Main Contribution

S. Bashar et al. [406]MISOME Find an optimum power control policy

SCSIT of the eavesdropper at the transmitter for maximizing the secrecy rate

S. Bashar et al. [407]MIMOME Propose a GSVD precoder structure to

PCSIT of the eavesdropper increase the secure rate performance

Y. Wu et al. [59]MIMOME Propose iterative algorithms for secrecy rate maximization

PCSIT and SCSIT of the eavesdropper Construct the jamming signal with the excess power

W. Zeng et al. [408]MIMO cognitive radio secure Propose an iterative algorithm to maximize

SCSIT of the eavesdropper the secure rate of the secondary user

S. Vishwakarma et al. [409]MIMO DF relay secure Design the optimal source power, signal beamforming

IPCSIT of all links and jamming signal matrix for worst case secrecy rate maximization

K. Cao et al. [410]MIMOME with a helper Design a secure transmission scheme

SCSIT of the eavesdropper with a help of source node

relay systems with the IPCST and finite alphabet inputs in presence of an user and J non-colluding

eavesdroppers is investigated in [409]. Very recently, s secure transmission scheme with a help of source

node is proposed in [410]. A brief summation of above mutual information based precoder designs for

the MIMO wiretap channel with discrete input signals is given in Table XXXIV.

2) MSE Design: For MIMOME where the perfect instantaneous CSI is available at all parties, H.

Reboredo et al. investigate the transceiver design minimizing the receiver’s MSE while guaranteeing the

eavesdropper’s MSE above a fixed threshold [61]. A linear zero-forcing precoder is used at the transmitter

and either a zero-forcing decoder or an optimal linear Wiener decoder is designed at the receiver. Moreover,

the design is generalized to the scenarios where only the statistical CSI is available. Simulations show

that maintaining the eavesdropper’s MSE above a certain level also limits eavesdropper’s error probability.

This design is further extended to MIMO interference channel with a multiple antenna eavesdropper [411].

M. Pei et al. investigate the masked beamforming design for MIMO BC in presence of a multiple antenna

eavesdropper [412]. It is assumed that the transmitter obtains the perfect/imperfect CSI of the desired

receivers but no CSI of the eavesdropper. The transmit power used to generate the artificial noise is

maximized subject to the MSE requirements of the desired receivers. Based on the same criterion, L.

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TABLE XXXV: MMSE transceiver designs for MIMO wiretap channel with discrete input signalsPaper Model Main Contribution

H. Reboredo et al. [61]MIMOME Minimizing the receiver’s MSE while guaranteeing

Instantaneous/statistical CSI the eavesdropper’s MSE above a fixed threshold

Z. Kong et al. [411]MIMO interference channel secure Minimizing the receivers’ sum MSE while guaranteeing

Perfect CSI the eavesdropper’s MSE above a fixed threshold

M. Pei et al. [412]MIMO BC secure, Perfect/imperfect CSI of receivers Maximize the transmit power to generate the artificial noise

No CSI of the eavesdropper subject to the MSE requirements of the desired receivers

L. Zhang et al. [413]MIMOME, nonlinear THP design Maximize the transmit power to generate the artificial noise

Perfect CSI of the receiver, no CSI of the eavesdropper subject to the MSE requirements of the receiver

Zhang et al. investigate the nonlinear THP design for MIMOME [413]. A brief summation of above work

is given in Table XXXV.

3) Diversity Design: For two transmit antennas, two receive antennas, multiple antennas eavesdropper

wiretap channel, S. A. A. Fakoorian et al. design a secure STBC by exploiting the perfect CSI of the

receiver but no CSI of the eavesdropper [414]. By combining QOSTBC in [183] with the artificial noise

generation, the proposed design in [414] effectively deteriorates the uncoded BER performance of the

eavesdropper while maintains the good uncoded BER performance of the receiver. For the same STBC

system, H. Wen et al. propose a cross-layer scheme to further ensure the secure communication [415].

T. Allen et al. investigate two/four transmit antennas, one receive antenna, and one antenna eavesdropper

wiretap channels employing STBCs, where no CSI of the receiver and the eavesdropper are available

at transmitter [416]. By randomly rotating the transmit symbols at each transmit antenna based on the

receive signal strength indicator, the proposed scheme provides full diversity of the receiver and enables

the eavesdropper’s diversity to be zero. For MISOSE wiretap channel with the same CSI assumption, S. L.

Perreau designs the transmit antennas weights based on STBC, which formulates beamforming along the

receiver’s direction while disables the eavesdropper’s ability to discriminate the transmit signal [417]. With

an assumption that the transmitter has the perfect CSI of the receiver based on the channel reciprocity

while both the receiver and the eavesdropper have no channel knowledge, X. Li et al. investigate the

secure communication for MISOME wiretap channel by using M-PSK modulation and OSTBC [60]. A

phase shifting precoder is designed to align the transmit signals at the receiver side so that the receiver

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TABLE XXXVI: Space-time techniques with discrete input signals for MIMO wiretap channelPaper Model Main Contribution

S. A. A. Fakoorian et al. [414]

Two transmit antennas and receive antennas Design a secure STBC

Multiple antennas eavesdropper with low uncoded BER of the receiver

Perfect CSI of the receiver, no CSI of the eavesdropper and high uncoded BER of the eavesdropper

H. Wen et al. [415] Same as [414]Propose a cross-layer scheme to

further ensure the secure communication

T. Allen et al. [416]

Two/four transmit antennas, one receive antennas Design a secure STBC

one antennas eavesdropper providing full diversity of the receiver

No CSI of the receiver and eavesdropper and zero diversity of the eavesdropper

S. L. Perreau [417]MISOSE Design the transmit antennas weights based on STBC

No CSI of the receiver and eavesdropper to make the eavesdropper unable to decode the signal

X. Li et al. [60]MISOME Design the secure communication scheme

No CSI of the receiver and eavesdropper by using M-PSK modulation and OSTBC

X. Li et al. [418]MISOME Propose transmit antennas weights design method

No CSI of the receiver and eavesdropper to deliberately randomize the eavesdropper’s signal

J.-C. Belfiore et al. [419]MIMOME Establish a lattice wiretap code design criterion minimizing

Perfect CSI of the receiver, no CSI of the eavesdropper the eavesdropper’s correctly decoding probability

T. V. Nguyen et al. [420]MISOME

Analyze the SER of confidential informationPerfect CSIT of the receiver, SCSIT of the eavesdropper

can perform a non-coherent detection for the transmit signals. At the same time, the information rate of

the eavesdropper can be reduced to zero by the proposed design. For the same channel model, another

transmit antennas weights design method to deliberately randomize the eavesdropper’s signal but not the

receiver’s signal is proposed in [418]. For MIMOME wiretap channel without the eavesdropper’s CSI,

J.-C. Belfiore et al. establish a lattice wiretap code design criterion, which minimizes the probability of

correctly decoding at the eavesdropper’s side [419]. T. V. Nguyen et al. analyze the performance of the

MISOME wiretap channel with perfect CSIT of the receiver and SCSIT of the eavesdropper [420]. The

SER of confidential information is derived and the corresponding secure diversity is obtained. A brief

comparison of above work is given in Table XXXVI.

V. OTHER SYSTEMS EMPLOYING MIMO TECHNOLOGY

In this section, we overview the transmission design with discrete input signals for other systems

employing MIMO technology. We consider the following three typical scenarios: i) MIMO Cognitive

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Radio Systems; ii) Green MIMO communications; iii) MIMO relay systems.

A. MIMO Cognitive Radio Systems

1) Mutual Information Design: W. Zeng et al. investigate the spectrum sharing problem in MIMO

cognitive radio systems with finite alphabet inputs [62]. With the global PCSIT, the linear precoder is

designed to maximize the spectral efficiency between the secondary-user transmitter and the secondary-

user receiver with the control of the interference power to primary-user receivers. This design is extended

to the secure communication of MIMO cognitive radio systems in [408]. However, these algorithms only

apply for the equip-probable finite alphabet inputs. By exploiting the I-MMSE relationship in [23], R. Zhu

et al. propose a gradient based iterative algorithm to maximize the secondary users’ spectrum efficiency

for arbitrary inputs [421].

2) MSE Design: B. Seo et al. investigate the transceiver design to minimize the MSE of the primary

user subject to the secondary user’s interference power constraint [422]. For MIMO cognitive networks,

X. Gong et al. jointly design the precoder and decoder to minimize the MSE of the secondary network

subject to both transmit and interference power constraints [423]. The transceiver designs for ad hoc

MIMO cognitive radio networks based on total MSE minimization are proposed in [64], [424]. For full-

duplex MIMO cognitive radio networks, A. C. Cirik et al. minimize both the total MSE and the maximum

per secondary users’ MSE [425].

3) Diversity Design: For cognitive radio networks, by considering multiple cognitive radios as a virtual

antenna array, W. Zhang et al. design space-time codes and space-frequency codes for cooperative spectrum

sensing over flat-fading and frequency-selective fading channels, respectively [63]. X. Liang et al. design

a flexible STBC scheme for cooperative spectrum sensing by dynamically selecting users to form the

clusters [426]. A robust STBC scheme based on dynamical clustering is further proposed in [427]. The

differential STBCs for cooperative spectrum sensing without CSI knowledge are also proposed in [428],

[429]. J.-B. Kim et al. analyze the outage probability and the diversity order of various spectrum sensing

methods for cognitive radio networks [430]. K. B. Letaief et al. design a space-time-frequency block

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code to exploit spatial diversity for cognitive relay networks with cooperative spectrum sharing [431]. A

distributed space-time-frequency block code is further proposed to reduce the detection complexity [432].

B. Green MIMO Communications

1) Mutual Information Design: M. Gregori et al. study a point-to-point MIMO systems with finite

alphabet inputs where the transmitter is equipped with energy harvesters [65]. A total number of N

channels are used and the non-causal knowledge of the channel state and harvested energy are available

at the transmitter. By obtaining the optimal left singular matrix and setting the right singular matrix to

be identity matrix, M. Gregori et al. propose an optimal offline power allocation solution to maximize

the spectral efficiency along N channel uses [65]. This design is extended to the SCSIT case in [433],

where an iterative algorithm to find the entire optimal precoder is proposed. The further exploitation of

partial instantaneous CSI along with the SCSI is studied in [434]. An resource allocation algorithm for

the downlink transmission of multiuser MIMO systems with finite alphabet inputs and energy harvest

process at the transmitter is proposed [435]. With the SCSIT and the statistical energy arrival knowledge,

the precoders, the MCR selection, the subchannel allocation, and the energy consumption are jointly

optimized by decomposing the original problem into an equivalent three-layer optimization problem and

solving each layer separately.

2) Diversity Design: Y. Tang et al. propose a STBC for geographically separated base stations, which

improves the energy efficiency at remote locations [436]. Y. Zhu et al. design a low energy consumption

turbo-based STBC [437]. A cooperative balanced STBC is proposed for dual-hop AF wireless sensor

networks to reduce the energy consumption [438]. M. Tomio et al. compare the energy efficiency of

transmit beamforming scheme and transmit antenna selection scheme [439]. It is revealed that transmit

antenna selection scheme is more efficient for most transmit distances since only a single radio-frequency

chain is used at the transmitter. Novel non-coherent energy-efficient collaborative Alamouti codes are

constructed by uniquely factorizing a pair of energy-efficient cross QAM constellations and carefully

designing an energy scale [66], [440].

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C. MIMO Relay Systems

1) Mutual Information Design: W. Zeng et al. investigate the linear precoder design for dual hop

amplify-and-forward (AF) MIMO relay systems with finite alphabet inputs [67]. By exploiting the optimal

precoder structure similar as the point-to-point MIMO case, a two-step algorithm is proposed to maximize

the spectral efficiency. The obtained precoder in [67] also achieves a good BER performance. Since the

publication of [67], the transmission design for the spectral efficiency maximization under finite alphabet

constraints has been investigated for various MIMO relays systems including: two-hop nonregenerative

three-node MIMO relay systems [441], non-orthogonal AF half-duplex single relay systems [442], MIMO

two-way relay systems [443], and MIMO DF relay secure systems [409].

2) MSE Design: Linear transceiver designs via MMSE criterion for MIMO AF relay systems are

proposed in [69], [444]–[450]. In addition, non-linear THP/DFE transceiver structures based on the

optimization of MSE function are designed for MIMO AF relay systems in [451]–[455]. The average

error probability of AF relay networks employing MMSE based precoding schemes is derived in [456].

At the same time, the MMSE based transceiver designs for non-regenerative MIMO relay networks are

provided in [457]–[461]. The MMSE transceiver designs are further investigated in multiuser MIMO relay

networks [462]–[466]. Two MMSE transceiver designs for filter-and-forward MIMO relay systems and

single-carrier frequency-domain equalization based MIMO relay systems are given in [467] and [468],

respectively.

3) Diversity Design: For AF relay systems with multiple antennas and finite alphabet inputs, various

transmission schemes are provided to optimize the diversity gain of the system in [68], [469]–[472].

Moreover, the diversity order and array gains for AF relay MIMO systems employing OSTBCs are derived

analytically in [473], [474]. For DF relay systems with multiple antennas and finite alphabet inputs, various

transmission schemes are proposed based on minimizing the error probability of the system in [470], [475],

[476]. Meanwhile, the error performance of DF relay MIMO systems are analyzed in [477], [478]. Further

transmission schemes to increase the reliability of MIMO relay networks with finite alphabet inputs are

designed and analyzed in [479]–[486]. The diversity designs for the non-coherent MIMO relay networks

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are investigated in [487]–[489]

4) Adaptive Transmission: I. Y. Abualhaol et al. propose an adaptive transmission scheme for MIMO-

OFDM systems with a cooperative DF relay [70]. The bit and power are adaptively allocated among

different subchannels to maximize the receive signal power under the constraint that each subchannel is

assigned either to the direct link or the relay link. D. Munoz et al. propose an adaptive transmission

scheme for MIMO-OFDMA systems with multiple relay nodes [490]. Based on the channel conditions,

the transmission mode adaptively switches between joint relay and antenna selection and space-frequency

block coding to minimize the sum BER of the direct link and the relay link subject to a total power

constraint.

D. Others

An optimal precoding technique for coded MIMO inter-symbol interference MAC with joint linear

MMSE detection and forward-error-correction codes is proposed in [491]. The geometric mean method,

which obtains an equal SINR for all transmit streams, is applied to single user and multiuser MIMO

systems [492]–[495]. There are some precoder designs for multiuser MIMO systems based on the receiver

side’s SINR [496]–[503]. There are some transmission designs for MIMO X channel based on the diversity

performance [504]–[509]. The diversity designs for cooperative systems, Ad Hoc networks, and distributed

antenna systems are given in [510], [511], and [512], respectively.

VI. CONCLUSIONS

This paper has provided a first comprehensive summary for the research on MIMO transmission design

with practical input signals. We introduced the existing fundamental understanding for discrete input

signals. Then, to facilitate a better clarification of practical MIMO transmission techniques, we provided

a detailed discussion of the transmission designs based on the criterions of mutual information, MSE,

diversity, and the adaptive transmission switching among these criterions. These transmission schemes were

designed for point-to-point MIMO systems, multiuser MIMO systems, MIMO cognitive radio systems,

80

green MIMO communication systems, MIMO relay systems, etc. Valuable insights and lessons were

extracted from these existing designs.

For the forthcoming massive MIMO systems in 5G wireless networks, the transmission designs with

discrete input signals will become more challenging. A unified framework for the mutual information

maximization design for point-to-point massive MIMO systems with discrete input signals was introduced

in this paper. However, there are a variety of open research issues currently. The tractable mutual

information maximization design for multiuser massive MIMO systems with discrete input signals is

still unknown. The robust MSE design for large scale antenna arrays with channel uncertainty due to pilot

contamination and other channel variation effects is less studied. The high rate low detection complexity

STBCs for large MIMO systems need to be designed. Most importantly, an integrity and efficient adaptive

transmission scheme for practical massive MIMO systems is still missing yet. This problem is extremely

challenging since it needs to determine the optimal resource allocation (time/space/frequency), the user

scheduling, the transmission mode selection, etc, in a significant large dimension. The characteristics of

massive MIMO channels should be exploited and heuristic methods from other research areas such as

machine learning may be helpful.

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