A Simplified ML Detection for Spatial Modulation with Multiple Active Transmit

Antennas

Lloyd Blackbeard 1, Hongjun Xu1 and Fengfan Yang2

School of Engineering1 University of KwaZulu-Natal, Durban, 4041, Republic of South Africa

2 Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

email: xuh@ukzn.ac.za

Abstract: Spatial modulation (SM) with multiple active transmit antennas (MASM) is a scheme

capable of higher spectral efficiency than conventional spatial modulation. In this paper, the

authors simulate MASM with optimal maximum likelihood (ML) detection, a decorrelator based

detector, an ordered block minimum mean square error detector (OB-MMSE) and with a

proposed simplified maximum likelihood detector. In simulations, the proposed detector

performs the same as ML detection down to bit error rates of 10−6 for three considered MASM

configurations, whilst simulations for the sub-optimal detectors are shown to perform worse

than the proposed detector simulations. The complexity of the four detectors is considered,

showing that the proposed simplified ML detector is less complex than ML detection and less

complex than the average complexity of OB-MMSE, for all but the lowest spectral efficiency

configuration. The proposed detector has a fixed complexity, contrary to OB-MMSE, which has

a variable complexity.

Index Terms— Bit Error Rate (BER), Spatial Modulation (SM), Multiple-Input-Multiple-Output

(MIMO), M -ary Quadrature Amplitude Modulation (M -QAM), Maximum Ratio Combining

(MRC), Maximum Likelihood (ML)

1

I. Introduction

Multiple-input multiple-output (MIMO) schemes can provide greater bandwidth efficiency than

traditional single-input single-output (SISO) schemes. A benchmark for MIMO schemes is

vertical Bell Laboratories layered space-time V-BLAST [1], also known as spatial multiplexing

(SMX). In [1], a number of transmit antennas simultaneously transmit M-ary quadrature

amplitude modulation (MQAM) symbols and thus V-BLAST suffers from inter-channel

interference (ICI), inter-antenna coupling (IAC) and also requires antenna synchronization.

Unfortunately, optimal maximum likelihood (ML) detection of V-BLAST is of a high complexity

and sub-optimal methods require the number of receive antennas to equal or outnumber the

number of transmit antennas.

Spatial modulation (SM), another MIMO scheme proposed in [2], intrinsically avoids ICI and IAC

and also does not require antenna synchronization. In [2], a transmit antenna is selected from

an array to transmit an MQAM symbol, with selections of both transmit antenna and MQAM

symbol conveying data. Space-shift keying (SSK) [3] is a simplified version of SM that uses on-

off-keying (OOK) in place of MQAM.

SM has been developed further since its inception: [4] and [5] convey one set of data by

antennas which convey real symbols and another set by antennas which convey imaginary

symbols, [6] improves performance by allowing the number of active transmit antennas to

change, [7] incorporates a property of MQAM constellations in a sub-optimal detector, [8]

features a detector using compressed sensing, [9] offers a low-complexity near-optimal

detector by feeding a sub-optimal detector into an optimal one, [10] combines trellis coding

with SM, [11] allows spatial constellations whose sizes are not powers of two and [12]

combines space-time block codes with SM.

Wang et al propose SM with multiple active transmit antennas (MASM) [14], in which there are

multiple active transmit antennas over the single active transmit antenna in conventional SM

2

[2]. In MASM, a group of active antennas transmit MQAM symbols and thus data is carried both

via the MQAM symbol and the group selection. Let N t, N p and N γ be the numbers of transmit

antennas, active transmit antennas and antenna groups respectively and M be the size of the

MQAM constellation. MASM has a spectral efficiency of log2 N γ +N p log2 M , where

log2 N γ ≤ ⌊ log2(N t

N p)⌋ , which is usually larger than SM at log2 M N t .

[14] presents a suboptimal detector for MASM based on decorrelation but did not explore

maximum likelihood (ML) detection, nor analyze ML performance. This motivates the authors

to, in this work, simulate MASM with ML detection and analyze the ML BER performance of

MASM systems.

The ML MASM detector requires an exhaustive search among all possible N γ MN p MASM

symbols to choose the most probable estimate. The detection complexity increases rapidly with

N p and M. For large M the complexity is extremely high. Although low-complexity, close to

optimal techniques such as sphere decoding exist for SM [15-16], such has not been applied to

MASM. In addition, to the best of the authors’ knowledge at the time of writing, only [17] has

been proposed to deal with the high complexity of MASM. However, in our simulations, [17] is

unable to achieve ML performance and has a significantly fluctuating complexity at different

signal to noise ratios (SNR). This motivates the development of a low-complexity, ML

performing detection scheme for MASM with a fixed complexity. In this paper, the authors

propose a simplified ML detection scheme for MASM based on the symbol cancellation method

in [1] and the multi-level subset searching method of [18]. In simulations, the proposed

technique is found to perform the same as the ML detection down to bit error rates in the

order of 10−6 for three considered configurations of MASM.

The paper is organized as follows: a transmission model of MASM is presented in section 2, with

models of the signal, channel, noise and received signal. ML based optimal detection,

suboptimal detection based on decorrelation [14], OB-MMSE detection [17] and the proposed

3

simplified ML detection schemes are described in section 3. A lower bound for the performance

of ML detection is derived in section 4. Section 5 presents a complexity analysis of the four

considered detectors. Simulation and complexity results are shown in section 6, and finally,

concluding remarks are made in section 7.

The following notation convention is used in this work unless otherwise specified:

(⋅ )−1, (⋅ )T , (⋅ )H , (⋅)† , E [ ⋅] ,∨⋅∨¿ and |⋅|F refer to the matrix inverse, transpose, Hermitian, Moore-

Penrose pseudoinverse, expectation, Euclidean norm and Frobenius norm operators

respectively; regular, bold face lower case, bold face upper case and capital script/cursive text

refer to scalars, vectors, matrices and sets respectively; subscripts (⋅ )ij denote the ith row, jth

column entry in the corresponding matrix and subscript (⋅ )i denotes the ith entry in the

corresponding vector or the ith column in the corresponding matrix.

II. System Model

[Fig. 1 Here]

The MASM transmission scheme is described in [14]. For convenience, it is described again

here. We consider a MIMO environment with N t transmit and N r receive antennas. In Fig. 1, N p

groups of log2 M bits are taken from the input bitstream and each is mapped to an MQAM

symbol for each of the N p ≤N t active transmit antennas. Another log2 N γ bits are used to select

one of N γ antenna groups that prescribe which of the N t transmit antennas are active. A

symbol to antenna mapper maps each MQAM symbol, respectively, to its designated transmit

antenna in the selected antenna group. This mapping creates a transmit MASM symbol vector x

which is transmitted across the wireless fading channel H to N r receive antenna, producing a

received signal vector y after additive white Gaussian noise (AWGN) is added. The Gray coded

MQAM alphabet S of size M has symbols si ,i ϵ [1: M ] that are normalized so that E [|si|2 ]=1/N p

. The set of antenna groups Γ has antenna group vectors γc1×N t c ϵ [1 : N γ ] with ones in positions

corresponding to active antennas and zeroes elsewhere. For example, if a black/white dot

represents 1/0, then the antenna group selected in Fig. 1 is [1,0,1,0,0], meaning that the first

and second antennas are active. We also define for a given γc, the numbers lk , k ϵ [1 : N p], 4

corresponding to the index of the k th active antenna – note that the antenna group which lk

belongs to is implicit. In Fig. 1, l1=1 and l2=3. Each γc has an associated rotation angle which is

applied to the symbols in the transmit vector1.

The set of MASM symbol vectors A has transmit symbol vectors aqN t × 1 , q ϵ [1 : MN p N γ ], each with

N p entries aqk , kϵ [1: N p], taken from S in positions corresponding to the active antennas and

zeros elsewhere. For example, Fig. 1 would have x=aq=[ aq1 ,0 , aq2 ,0,0 ] , aq1 , aq 2ϵ S. x is that aq

chosen for transmission with xk=aqk. The received signal is given by

y=Hx+n (1)

where H N r×N t is the complex MIMO channel matrix with entries hij corresponding to the

complex channel gain between the jth transmit antenna and the ith receive antenna, j∈[1: N t ],

i∈[1 :N r]. H has independent and identically distributed (i.i.d.) entries with complex Gaussian

distributions hij C N (0;1 ) ,∀ i , j. nNr ×1 is the noise vector with i.i.d. complex Gaussian

distributions ni C N (0 ;σ2 ) , ∀ i, where σ 2 is the variance of the noise describing the signal-to-

noise ratio (SNR).

III. Detection Algorithms

In this section, we describe, for MASM, the ML detector, the decorrelator-based detector [14],

the OB-MMSE detector [17] and then present the proposed simplified ML detector. It is

assumed that full channel knowledge is available at the receiver.

A. Maximum Likelihood (ML) Detection

The ML MASM detector fully exploits the advantages of MASM by performing a joint detection

of transmit antenna group and MQAM symbols. The ML estimation of x can be expressed as:~x=argmax

aqϵ AP ( y|aq ,H ) (2)

(2) is developed further into the following form [19]:

1 It was found when simulating the MASM system with ML detection that the rotation of the MQAM constellation by different angles for each antenna group as described in [14] made no difference to the performance. Since the simplified decoding method proposed in this paper is based on maximum likelihood, the rotation step is omitted for simplification.5

~x=argminaq ϵ A

‖y−H aq‖F2 (3)

For an MASM system with N p active transmit antennas, the ML detector requires an exhaustive

search among all possible N γ MN p symbol vectors to choose only one of them. Clearly, the

detection complexity increases rapidly with N p and M.

B. Decorrelator-Based Sub-Optimal Detection [14]

Since the ML detector for MASM has a very high detection complexity, [14] proposes a

decorrelator-based sub-optimal detector. The detector uses a zero-forcing (ZF) detector T [19]

to estimate both the transmit antenna group and the MQAM symbols. The detector T , the

pseudo-inverse of the channel matrix, and subsequent antenna detection are given by:

T=H †=(H H H )−1 H H (4)

{l1 ,…, lN p}=sort (argmax

k(Ty )k) (5)

where, in (5), the argmax function returns the largest N p entries, as opposed to the single

maximum entry, sort (⋅ ) arranges the input vector in ascending order and lk is the estimated

index of the k th transmit antenna. The active antennas can be estimated in such a way because

the inactive antennas transmit 0 and thus are expected to be the minimum entries in Ty [14].

In [14], no procedure is described to deal with the selection of an invalid antenna group, thus,

we create our own solution. We say that an invalid antenna group is selected when ∄ γc with

{l1 ,…,lN p }= {l1 ,…, lN p}. Let us define b as a vector with N t entries, containing the entries of Ty

sorted in descending order. If the first N p entries of b result in an invalid group, we select

another group from b in lexicographic order. If the second selection is also invalid, we continue

in lexicographic order until a valid group is found. Note that the procedure is repeated

(including the first selection) a maximum of (N t

N p)−N γ+1 times.

Assuming the estimate is correct, we estimate the transmitted MQAM symbols:

{ x1 ,…, x Np }=Q (Ty ) (6)

6

where Q (⋅) is the MQAM slicing function applied to each entry in the input vector [1]. For (6),

we choose the entries in Ty that correspond to the estimated active antennas. From {l1 ,…, lN p}

and { x1 ,…, xNp } , we create an estimate x of x.

C. Ordered Block Minimum Mean Square Error Detection (OB-MMSE) [17]

The OB-MMSE detector is another method to deal with the high complexity of ML detection, in

which the focus is to estimate the most likely transmitted antenna groups in descending order,

which is apt for the configurations of N t=16 ,32 considered in [17]. OB-MMSE begins by

creating a variable z i for each transmit antenna i∈1: N t by multiplying the pseudo-inverse of

each channel column hi by the received signal vector:

z i=(h i )† y (7)

A sorted list j of most likely antenna groups is then obtained using:

w c=∑k=1

N p

z lk l k∈ γ c∀ c (8)

[ j1 , j2 ,…, jN γ ]=arg sort (w ) (9)

A minimum mean square error (MMSE) detector is applied to each antenna group in the sorted

list, in turn, using iteration number q, beginning with the most likely, until a stopping criterion is

met. The MMSE detector and stopping criterion are written:

~sq=Q ((H γ jq

H H γ jq

+σ 2 I )H γ jq

H y ) (10)

‖y−H γ jq

~sq‖<V th (11)

Where V th=2 N r σ2 is the threshold distance and H γ jq

is a matrix containing the columns of H

corresponding to γ jq. If (11) is satisfied, the OB-MMSE detector constructs an estimate of the

transmitted MASM symbol using the current iteration of ~sq and γ jq. If all N γ groups are explored

and the stopping criterion is still not met, the detector falls back to ML detection as in (3).

7

D. Simplified ML-Based Detection

Compared to ML detection, the decorrelator-based detector has a negligibly small complexity.

However, in simulations, the bit error rate (BER) performance of the decorrelator-based

detector is far worse than ML detection. In addition, again in simulations, the BER performance

of OB-MMSE detection is worse than ML and has a complexity which fluctuates widely with

SNR. Therefore, we propose a detector of simpler complexity than ML that can achieve ML

performance. We achieve lower complexity in the proposed scheme by reducing the set of

MASM symbol vectors that are evaluated by an ML detector - specifically, we create a subset

X '⊂A that contains only probable MASM symbols. The ML detector (3) is revised to:

~x=argminaqϵ X '

‖y−H aq‖F2 (12)

For the MASM configurations in this paper, and, more generally, for systems that do not have

N t ≫1, we have N γ ≪MN p. Thus, a reduction in antenna group candidates is not deemed a

priority. In light of this, X ' is created by sequentially considering each antenna group γc∈ Γ a

priori. In each consideration, we use the successive interference cancellation (SIC) detector of

[1] to find an estimate xc of x which contains entries xck for the k th active antenna. To improve

the reliability of the SIC detector, we use the multi-level subset searching method of [15] to find

those symbols in S which are adjacent to xck (neighbours) as alternative estimates. Finally, we

create X ' containing those aq which are allowed by the estimates and alternative estimates.

The SIC detector operates as follows: for each γc ϵΓ , we associate an N r ×N p transmission

matrix H γ c containing only those columns in H corresponding to γc. Execution is performed ∀ γ c

to output estimates xcϵ A of the transmit vector x. The pseudo-code of the proposed SIC

detection is given in (13):

Initialisation

c ←0

Outer Recursion

(13a)

i←0 H 1← H γ c

y1← y

(13b)

(13c)

8

Inner Recursion (13d)

i←i+1Gi← (H iH H i )

−1 H iH

t i←argmink

‖(Gi )k‖2z i← (Gi )ti y i

xci ←Q (zti )H i+1 ← ( H i )tiy i+1 ← y−ht i

xci

(13e)

(13f)

(13g)

(13h)

(13i)

(13j)

(13k)

Until i>N p

Until c>N γ

End

where the scalar xci refers to the ith symbol to be used in the construction of xc and i indicates

the ith recursion for a given γc. Note that trivial steps, not included in the pseudo-code, are

needed for proper ordering and distribution in the construction of xc. (H i )t i is the matrix H i with

column t iremoved, (Gi )t i is the t i

th row of Gi and Q (⋅) is an MQAM slicing function. After (13), we

are presented with N γ SIC estimates xc, one estimate per antenna group.

As discussed, the next step is to find the subset of MQAM symbols lying adjacent to xck ,∀ c , k.

We define the set of neighbours to an MQAM symbol si as (here, i refers to the ith MQAM

symbol in S, not the ith iteration of the SIC detector):

Ssi= {sm|‖sm−s i‖

2<d2∀ sm∈S },m∈[1: M ] (14)

where d is the radius within which the neighbours lie. In this paper, d is the distance between

two MQAM symbols lying diagonally adjacent in S. We expand xc to include neighbours and

the resultant set X c is:

9

X c={aq|aqk∈S x ck, aq∈ γ c , aq∈ A } (15)

where aq∈ γ c means that aq lies in the same antenna group as xc. Finally, the reduced set upon

which (12) is performed is given by:

X '={X1 ,…, XN γ} (16)

IV. Theoretical Performance Analysis of MASM with ML Detection

Considering that the final step of the proposed detector is ML detection and assuming that the

reduced set of MASM symbols to be searched by the final step includes the transmitted symbol,

the performance bound for ML detection is applicable to the proposed scheme. It is shown in

Fig. 2, the simulation results, that this assumption is correct. Thus, we derive an asymptotic

performance bound for MQAM MASM with ML detection in i.i.d. Rayleigh flat fading channel

conditions.

We note that ML detection performs a joint detection of MQAM symbols and transmit antenna

group. In order to derive a closed form BER expression we simplify the analysis by decoupling

transmit antenna group detection and symbol detection performance as in [9]. In the process of

doing this, we assume perfect MQAM symbol detection when deriving antenna group detection

and vice-versa. The overall bit error probability is bounded by

pe≥ pa+ pd−pa pd (17)

where pa is the bit error probability (BEP) for antenna group detection, pd is the BEP for MQAM

symbol detection and pe is the overall BEP for the whole system.

In the following two subsections, we derive pa and pe .

A. Antenna Group Detection

Noting (3) and the assumption of correct symbol detection, we can say that an antenna group

detection error will occur when:

‖y−∑k=1

N p

hlk xk‖F

2

>‖y−∑k=1

Np

hlkxk‖F

2

(18)

10

where l, l ϵΓ , hlk is the channel vector in the lkth column of H , corresponding to the k th active

transmit antenna, l ≠l and xk is the k th MQAM symbol in x. As such and similarly, lk corresponds

to the k th active transmit antenna in an incorrect antenna group.

For simplicity, we further assume that only one antenna is decoded incorrectly and, the BEP for

antenna group detection based on (18) is given by (intermediate steps can be found in the Equ.

(36) in [9])

pa=P (‖n‖F2>‖(h j−h j ) x+n‖F

2∨H ) (19)

where j , j∈[1 : N t], j ≠ j.

This problem is similar to the case in Equ.(14) in [9] which has solution Equ.(19) in [9] and is

written in (20):

pa ≤∑j=1

N t

∑j=1

N t N ( j , j )N t

⋅∑q=1

M μαNr ∑

w=0

Nr−1

(N r−1+ww )[1−μα ]w

M

(20)

where, μα=12 (1−√ σ α

2

1+σα2 ), σ α

2= p2 |x j|

2, N ( j , j ) is the number of bits in error between transmit

antenna index j and estimated transmit antenna index j and p is the SNR per active antenna. Note that the total power must be divided among all active antennas.

The solution Equ.(19) in [9] is for a single active transmit antenna and [9] has not dealt with the

case of multiple active transmit antennas. Since MASM has multiple active transmit antennas,

we adapt (20) to cater to MASM. We observe that the first factor in (20), the double

summation, can be written as E [ N ( j , j ) ] N t, which will be generalized to Na−bits N Γ−1. Here

N a−bitsis the average number of bits in error when a single active antenna is mistaken, and N Γ−1

is the average number of incorrect positions that can be occupied by the single active antenna

when the positions of the other N p−1 active antenna(s) are fixed. Similar to the discussion in

Appendix A of [14], we see that when finding the antenna error probability for MASM, we must

take into account the probability of any of the active antennas being mistaken. Thus, we

multiply our expression for pa by N p.

11

Let us define N c (lk ,l k' ) as a function which returns the number of bits in error when, for the c th

antenna group, the k th active antenna in position lk is mistaken for the k th inactive antenna in

position lk' . Using these definitions, we write Na−bits N Γ−1 as ∑

c=1

N γ

∑k=1

N p

∑k =1

N t −N p N c (lk , lk' )N γ N p

. Note that

N c ( lk ,l k' ) returns zero if an invalid γ is described. Also note that Na−bits N Γ−1 is dependent on the

selection of the antenna set.

Finally, we can write the BEP of antenna group detection as:

pa ≤∑c=1

Nγ

∑k=1

N p

∑k=1

N t−N p N c ( lk ,lk' )

N γ⋅∑

q=1

M μαN r ∑

w=0

N r−1

(N r−1+ww ) [1−μα ]w

M(21)

B. MQAM Symbol Error probability

Similar to (18) and under the assumption of perfect antenna group detection, we write that an

MQAM symbol error will occur when:

‖y−Hx‖F2 >‖y−H x '‖F

2 (22)

where x∈ A is the correct MASM symbol and x '∈ A is incorrect, but from the same antenna

group as x. For simplicity, we further assume that x differs from x ' by only one MQAM symbol,

giving the BEP for MQAM symbol detection, based on (22):

pd=P(‖n‖F2 >‖hk (xk−xk

' )+n‖F

2∨H ) (23)

This probability is equivalent to the symbol error rate SERfor MQAM with N r receivers and ML

detection as given in [9], which is the following numerically integrated solution:

SER=ac {1

2 ( 2bp+2 )

N r

−a2 ( 1

bp+1 )N r

+(1−a )∑i=1

c−1

( S i

bp+S i )N r

+ ∑i=c

2c−1

( S i

bp+S i )N r} (24)

where a=1− 1√M

, b= 3M−1 , m=log2 M , Si=2sin2 θi, θi=

iπ4c , c is the number of summations

and p is the SNR per antenna. [9] shows that for c>10, there is a 0.0015dB, 0.0025dB and

0.0029dB error between the simulated and theoretical results for each of 4, 16 and 64 MQAM

constellations respectively.

12

Since gray mapping is used, we assume a symbol error results in a single bit error at high SNR.

Therefore, we write:

pd ≈ SER / log2 M (25)

V. Complexity Analysis

In this section, we analyze the computational complexity of the ML detector, the decorrelator-

based detector [14], the OB-MMSE detector [17] and the proposed detector. As in [9], the

computational complexity is in terms of complex multiplications and additions.

A. Computational Complexity of ML Detection

There are N γ MN p MASM symbols to be evaluated. Let us note the ML metric (3): we see that

for each evaluated symbol, we need N r N p complex multiplications in the matrix multiplication

H aq and N r complex multiplications in finding the Frobenius norm. The multiplication H aq

requires N r (N p−1 ) complex additions, the subtraction of this product from y requires N r

complex additions and finding the Frobenius norm requires N r N p+N r−1 complex additions.

We write these as:

δML−mult=N γ MN p N r (N p+1 ) (26)

δML−add=N γ MN p (N r N p+N r+1 ) (27)

Let us note that the number of complex multiplications and additions here reduce to the case in

Table 1 of [20] if we: i) choose a square channel matrix; ii) choose N t−N p=0.

B. Computational Complexity of Decorrelator-Based Detection [14]

Note that the matrix inverse operation in [14] uses Gaussian elimination, whilst [20], which will

be used for our analysis, uses LD LH decomposition. We begin with the computation of the

pseudo-inverse for (4). There are three sub-steps: matrix multiplication of the Hermitian of the

channel matrix with the channel matrix, the inverse operation and the multiplication of the

inverse matrix with the Hermitian of the channel matrix. These sub-steps are adapted from δ 1

in Section V-D. T is then multiplied by the received signal vector y for (5) and the complexity

13

for this step is adapted from δ 3 in Section V-D. A zero complexity slicing function Q (⋅) is

assumed [14].

The complexity for multiplication and addition are respectively given by:

δ original−mult=12 (3N r N t

2+N t3+3 N r N t+N t

2−2N t ) (28)

δ original−add=12 (3 N r N t

2+N t3+N r N t−2N t

2−3 N t ) (29)

C. Computational Complexity of OB-MMSE Detection [17]

The complexity of OB-MMSE can be broken into five steps: finding z i, finding w i, performing

MMSE, determining if the stopping criterion is met and performing ML decoding should no

suitable candidates be found. These steps have complexities represented here as

δ z , δw , δMMSE , δ stop and δML, with relevant suffixes in the subscripts to denote multiplication or

addition. We find in [17] that the ML and stopping criterion steps are not considered, thus for

fair comparison, the complexity of OB-MMSE is recalculated and found to be comprised of the

following:

δ z−mult=2N t (N r+1 ) (30)

δ z−add=N t ( Nr−1 ) (31)

δw−mult=0 (32)

δw−add=N γ (N p−1 ) (33)

δMMSE−mult=( 12

N p3 + 1

2N p

2 (3N r+1 )+ 32

N p N r) ρMMSE (34)

δMMSE−add=( 12

N p3 + 1

2N

p

2

(3 N r−2 )+12

N p (N r−1 )) ρMMSE (35)

δ stop−mult=N r (N p+1 ) ρMMSE (36)

δ stop−add=(N r N p+N r+1 ) ρMMSE (37)

δML−mult=N γ MN p N r (N p+1 ) ρML (38)

δML−add=N γ MN p (N r N p+N r+1 ) ρML (39)

14

Where ρMMSE , ρML are the average number of executions of the MMSE and ML step in each

frame, respectively, found via simulation.

D. Computational Complexity of the Proposed Simplified ML detector

We tackle the complexity analysis of the proposed detector in four steps:

1) The first step is the calculation of the ZF receiver G (13f), which is further broken down

into three sub-steps: The multiplication of the Hermitian of the nulled channel matrix by

itself H iH H i, the calculation of the matrix inverse and the multiplication of said inverse

by the Hermitian of the nulled channel matrix.

In sub-step 1, the result contains N p2 entries, each requiring N r complex multiplications

and N r−1 complex additions. Noting symmetry, the complexity is reduced in light of

there being N p unique entries and N p2−N p

2 symmetrical entries.

Thus, we write δ a−b for this sub-step, where a and b indicate the step and sub-step

number and the suffix in the subscript denotes multiplication or addition:

δ 1−1−mult=N r( N p2 +N p

2 ) (40)

δ 1−1−add=(N r−1 )( N p2 +N p

2 ) (41)

In sub-step 2, we use LD LH decomposition as in [17] to give:

δ 1−2−mult=12

N p3+ 1

2N p

2−N p (42)

δ 1−2−add=12

N p3−1

2N p

2 (43)

For sub-step 3, the result contains N r N p entries, each requiring N p multiplications and

N p−1 additions:

δ 1−3−mult=N r N p2 (44)

δ 1−3−add=N r N p (N p−1 ) (45)

2) In the second step, we compute the symbol which has the largest post-detection SNR by

evaluating the ZF receiver G. Each of the N r N p entries in G is squared and each

15

subsequent row is summed before selecting the largest entry in the resulting column.

Thus, we write:

δ 2−mult=N r N p (46)

δ 2−add=(N r−1 ) N p (47)

3) Step three multiplies a row of G by y i. It is trivial that the complexity of this step is

written:

δ 3−mult=N r (48)

δ 3−add=(N r−1 ) (49)

4) Step four is the preparation for the next iterative step in which the contribution to y i by

the detected symbol in the current step is subtracted from y i. It is trivial to write:

δ 4−mult=N r (50)

δ 4−add=N r (51)

We now develop the complexity analysis which began with the four steps, noting:

1) We assume a zero complexity slicing function Q (⋅).

2) It is clear that the complexity for each recursive step reduces as the recursion number

increases, since the number of columns in H i decreases as i increases. We account for

this by replacing N p in iterated steps with N i ϵ [1 : N p ]. 3) Step four, being a preparation, need not be executed on the final iterative step.

Once the symbol has been estimated, we proceed with ML decoding the reduced set X '. The

complexity of this step is dependent on the average number of neighbours to a given symbol in

the MQAM constellation N s – it can be shown that this number is:

N s=4 N corners+4 N sides (√M−2 )+Nmiddles (√M−2 )2

M

(52)

where N corners, N sides and Nmiddles refer to the number of neighbours to a constellation point

(including itself) if the constellation point lies in the corner, on the side or in the middle of the

MQAM constellation respectively. These numbers, in turn, depend on d , the radius of a circle

encompassing the neighbour set of MQAM symbols with its centre at the estimated symbol.

Thus, we can write the total complexities for the simplified detection algorithm as:16

δ simplified−multiplications=N γ(N sN p N r (N p+1 )+ 1

2 ∑N i=1

N p

[N i3+N i

2 ( 3N r+1 )+N i (3N r−2 )+2N r ]+N r(N p−1))(53)

δ simplified−additions=N γ (N sN p (N r N p+N r+1 )+1

2 ∑N i=1

N p

[N i3+N i

2 (3 N r−2 )+N i (N r−3 )+2N r−2 ]+N r (N p−1 ))(54)

VI. Results and Discussion

In this section, simulation and complexity results are presented to compare the ML detector,

the decorrelator-based detector [14], the OB-MMSE detector [17] and the proposed detector.

In addition, the results of the theoretical bound are also presented. The three MASM

configurations considered for simulations are taken from [14] and are shown in Table 1.

The simulation environment regarding fading and noise is described in Section II and the

following parameters are assumed for simulation: a Gray coded M-QAM constellation; full

channel knowledge at the receiver and a total transmit power that is the same for all

configurations.

A. BER Simulation Results

[Fig. 2 here]Fig. 2 shows the BER simulation results for the ML detector, the proposed detector, the

decorrelator-based detector [14] and the OB-MMSE detector [17]. We see that the proposed

detector performs with a negligible difference to ML detection for all MASM configurations

considered down to the order of 10−6, whilst the OB-MMSE detector exhibits a ~1dB drop in

performance from ML detection for all configurations down to the order of 10−5. The

performance of the decorrelator-based detector is much worse than ML detection, with a gap

at a BER of 10−5 of approximately 14dB for the 6bits/s/Hz and 10bits/s/Hz schemes respectively

and very large for 15bits/s/Hz (assuming no error floor exists, in which case, the gap is infinite).

We can conclude from this that for the configurations visited, both the proposed detector and

the OB-MMSE detector perform aptly whilst the decorrelator-based detector performs badly.

17

Fig. 3 shows the analytical bounds and ML detection simulation BERs for each MASM

configuration. The analytical bounds predict well the ML detection BER of MASM over the

entire range of considered SNR for each of the three MASM configurations. Since the proposed

detector is ML based and exhibits an almost identical performance to ML detection, we can say

that the theoretical bounds apply to the proposed detector too, if we assume the candidate list

includes the transmitted symbol. We see from Fig. 2 that this assumption is valid.

[Fig. 3 here]

B. Complexity Analysis Results

In this section, we combine the results in (26-27), (28-29), (30-39), (53-54) to create figures of

merit measured in floating operations per second (FLOPS) for each of the detectors

respectively. We assume a complex multiplication and a complex addition require 6 and 2

FLOPS respectively as in [20]. The results are tabulated, plotted and compared for the decoding

schemes in question at the three spectral efficiencies simulated. The spectral efficiencies of

6bits/s/Hz, 10bits/s/Hz and 15bits/s/Hz are shown in Figures 4, 5 and 6.

From Figures 4, 5 and 6, we see that the decorrelator-based detector [14] exhibits significantly

lower complexity than the other detectors for all three configurations. However, Figure 1 shows

that the BER performance for said detector is significantly lacking in comparison to ML

detection. We also see for all configurations that ML detection has the highest complexity in all

but the 6bits/s/Hz configuration, where the proposed method has a slightly higher complexity.

The focus of comparison is thus between the OB-MMSE and proposed detectors.

For OB-MMSE, we derive two figures of merit: the first is the complexity plotted for each SNR

used in simulations and the second is the complexity averaged over all SNRs. For 6bits/s/Hz, the

complexity is relatively stable over the SNR range, however, for 10bits/s/Hz and 15bits/s/Hz, we

see a variation of about 10 times.

18

In the case of 6bits/s/Hz, we see that the OB-MMSE detector has a lower average complexity,

by about 10 times, than both the ML detector and the proposed detector. This fact, combined

with stability of complexity with SNR, makes OB-MMSE a good decoding candidate for the

6bits/s/Hz configuration. It is obvious that for the 4QAM (6bits/s/Hz) case, in which the

neighbour set is identical to the full MQAM set, no reduction in complexity can be achieved

using the proposed detector due to the SIC overhead.

At 10bits/s/Hz, the proposed detector has a slightly lower complexity than the average OB-

MMSE complexity, although, at high SNR, the OB-MMSE complexity is significantly lower than

the proposed method. Taking into account the instability of OB-MMSE over the SNR range and

the lower complexity of the proposed detector versus the average complexity of OB-MMSE, the

proposed detector is has merit.

At 15bits/s/Hz, the proposed decoder complexity is significantly lower than the average OB-

MMSE detector complexity and thus, notwithstanding the instability of OB-MMSE complexity in

this configuration, the proposed detector is advantageous in comparison.

Analysis of (53) and (54) shows that the SIC overhead of the proposed detector accounts for

25%, 12% and 4% of the total complexity for 6, 10 and 15bits/s/Hz respectively. Further, we see

that the modulation size is not included and that complexity increases linearly with the number

of antenna groups. Thus, the ML step is significant in all configurations. With the significance of

the ML step in mind, greater complexity savings can be expected for MQAM constellations

larger than the largest 16QAM constellation considered in this work if d , the radius encircling

neighbouring MQAM points, is kept constant.

[Fig. 4 here]

[Fig. 5 here]

[Fig. 6 here]

19

VII. Conclusion

Higher spectral efficiency transmission schemes such as MASM perform well when the

exhaustive search method of optimal (ML) detection is used. A decorrelator-based detector [14]

and OB-MMSE detector [17] have been proposed to cater to this problem, however, when

applied to the MASM configurations in this paper, the decorrelator-based detector exhibits

poor BER performance and the OB-MMSE detector performs slightly worse than ML, with a

complexity that varies with SNR.

We propose a simplified ML detection scheme based on the symbol cancellation method in [1]

and multi-level subset searching method [18] that, in simulation, achieves the performance of

optimal detection down to bit error rates in the order of 10−6. The proposed detector makes an

estimate of a transmitted symbol for each antenna group by assuming, in turn, each antenna

group a priori. The estimates are expanded to include the adjacent neighbours in the MQAM

constellation and the resultant reduced set of possible transmitted MASM symbols is forwarded

to an ML detector. Complexity analysis shows that for the higher two spectral efficiencies

considered, the proposed detector is favourable in comparison to the sub-optimal methods,

whilst at the lowest spectral efficiency, OB-MMSE is favourable. In all configurations

considered, the decorrelator-based detector [14] exhibits poor BER performance.

In the work, we also found a bound for the ML detector performance, which was shown to

aptly predict the simulation BER.

References

[1] P. Wolniansky, G. Foschini, G. Golden, and R. Valenzuela, “V-blast: an architecture for

realizing very high data rates over the rich-scattering wireless channel,” Proc. ISSSE 98. Oct.

1998, pp. 295 –300.

[2] R. Mesleh, H. Haas, C. W. Ahn, and S. Yun, “Spatial modulation - a new low complexity

spectral efficiency enhancing technique,” Proc. ChinaCom ’06. Beijing, Oct. 2006, pp. 1 –5.

20

[3] M. Di Renzo, D. De Leonardis, F. Graziosi, and H. Haas, “Space shift keying (ssk x2014;) mimo

with practical channel estimates,” IEEE Trans. Commun, vol. 60, no. 4, pp. 998 –1012, April

2012.

[4] H.-W. Liang, R. Chang, W.-H. Chung, H. Zhang, and S.-Y. Kuo, “Bi-space shift keying

modulation for mimo systems,” IEEE Commun. Letters,, vol. 16, no. 8, pp. 1161–1164, 2012.

[5] A. ElKalagy and E. Alsusa, “A novel two-antenna spatial modulation technique with

simultaneous transmission,” Proc. SoftCOM 2009, 2009, pp. 230–234.

[6] R. Chang, S.-J. Lin, and W.-H. Chung, “New space shift keying modulation with hamming

code-aided constellation design,” IEEE Wireless Commun. Letters,, vol. 1, no. 1, pp. 2–5, 2012.

[7] S. Sugiura, C. Xu, and L. Hanzo, “Reduced-complexity qam-aided space-time shift keying,”

Proc. IEEE GLOBECOM 2011, 2011, pp. 1–6.

[8] C.-M. Yu, S.-H. Hsieh, H.-W. Liang, C.-S. Lu, W.-H. Chung, S.-Y. Kuo, and S.-c. Pei,

“Compressed sensing detector design for space shift keying in mimo systems,”, IEEE Commun.

Letters, vol. 16, no. 10, pp. 1556–1559, 2012.

[9] N. Naidoo, H. Xu, and T. Quazi, “Spatial modulation: optimal detector asymptotic

performance and multiple-stage detection,” IET Commun., vol. 5, no. 10, pp. 1368 –1376, 2011.

[10] R. Mesleh, M. D. Renzo, H. Haas, and P. M. Grant, “Trellis coded spatial modulation,”, IEEE

Trans. Wireless Commun., vol. 9, no. 7, pp. 2349 –2361, July 2010.

[11] N. Serafimovski, M. Di Renzo, S. Sinanovic, R. Mesleh, and H. Haas, “Fractional bit encoded

spatial modulation (fbe-sm),” IEEE Commun. Letters, vol. 14, no. 5, pp. 429 –431, May 2010.

[12] M. Di Renzo and H. Haas, “Transmit-diversity for spatial modulation (SM): Towards the

design of high-rate spatially-modulated space-time block codes,” Proc. IEEE ICC’2011, June

2011, pp. 1 –6.

[13] S. Alamouti, “A simple transmit diversity technique for wireless communications,”, IEEE J.

Sel. Areas Commun., vol. 16, no. 8, pp. 1451 –1458, Oct 1998.

[14] J. Wang, S. Jia, and J. Song, “Generalised spatial modulation system with multiple active

transmit antennas and low complexity detection scheme,” IEEE Trans. Wireless Commun.,, vol.

11, no. 4, pp. 1605 –1615, April 2012.

21

[15] Younis, A.; Di Renzo, M.; Mesleh, R.; Haas, H., "Sphere Decoding for Spatial Modulation,"

Communications (ICC), 2011 IEEE International Conference on , vol., no., pp.1,6, 5-9 June 2011

[16] Younis, A.; Sinanovic, S.; Di Renzo, M.; Mesleh, R.; Haas, H., "Generalised Sphere Decoding

for Spatial Modulation," Communications, IEEE Transactions on , vol.61, no.7, pp.2805,2815,

July 2013

[17] Xiao, Yue; Yang, Zongfei; Dan, Lilin; Yang, Ping; Yin, Lu; Xiang, Wei, "Low-Complexity Signal

Detection for Generalized Spatial Modulation," Communications Letters, IEEE , vol.18, no.3,

pp.403,406, March 2014

[18] H. Xu, “Simplified maximum likelihood-based detection schemes for M-ary quadrature

amplitude modulation spatial modulation,” IET Commun., vol. 6, no. 11, pp. 1356 –1363, 2012.

[19] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005.

[20] R. Chang and W.-H. Chung, “Low-complexity MIMO detection based on post-equalization

subspace search,” IEEE Trans. Vehicular Technol., vol. 61, no. 1, pp. 375–380, 2012.

22

FIGURE CAPTIONS

Fig. 1 Block diagram of MASM transmission with pseudo-example

Fig. 2 Performance of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and

ML detectors for 3 different MASM configurations

Fig. 3 Performance of ML detection and theoretical performance bound for 3 different MASM

configurations

Fig. 4 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and

ML detectors at 6bits/s/Hz over the simulated SNR range

Fig. 5 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and

ML detectors at 10bits/s/Hz over the simulated SNR range

Fig. 6 Complexities of OB-MMSE [17], average OB-MMSE [17], decorrelator [14], proposed and

ML detectors at 15bits/s/Hz over the simulated SNR range

23

FIGURES

Fig. 1

Fig. 2

24

Fig. 3

0 5 10 15 20 2510

-6

10-5

10-4

10-3

10-2

10-1

100

SNR (dB)

BE

R (b

its/b

it)

6bits/s/Hz ML10bits/s/Hz ML15bits/s/Hz ML15 bits/s/Hz Theory10 bits/s/Hz Theory6 bits/s/Hz Theory

Fig. 4

0 2 4 6 8 10 12 14 161E+02.

1E+03.

1E+04. 6 bits/s/Hz

OB-MMSE [17]Average OB-MMSE [17]Decorrelator [14]ProposedML

Signal to Noise Ratio (dB)

FLO

PS

25

Fig. 5

0 2 4 6 8 10 12 14 16 18 20 221E+03.

1E+04.

1E+05.

10 bits/s/Hz

OB-MMSE [17]Average OB-MMSE [17]Decorrelator [14]ProposedML

Signal to Noise Ratio (dB)

FLO

PS

Fig.6

0 2 4 6 8 10 12 14 16 18 20 22 241E+03.

1E+04.

1E+05.

1E+06.

1E+07.15 bits/s/Hz

OB-MMSE [17]Average OB-MMSE [17]Decorrelator [14]ProposedML

Signal to Noise Ratio (dB)

FLO

PS

26

TABLE CAPTIONS

Table 1: Simulation Parameters

Table 1

6bits/s/Hz 10bits/s/Hz 15bits/s/Hz

N p 2 2 3

N t 4 4 5

N r 5 5 5

N γ 4 4 8

M 4 16 16

27