6. Layered Space-Time Codesweb.xidian.edu.cn/lqzhao/files/20111019_222139.pdf · code blocks with...

State Key Lab. of ISN, Xidian University

6. Layered Space-Time Codes

6.1 Introduction

6.2 LST Transmitters: Types of Encoding

6.3 Layered Space-Time Coding: Design Criteria

6.4 LST Receivers

6.5 Iterative Receivers

6.6 The Effect of Imperfect Channel Estimation on Code Performance

6.7 Effect of Antenna Correlation on Performance

6.8 Diversity Performance of SM Receivers

6.9 Summary

6.10 Simulation Exercises

References


6.1 Introduction

In Chapter 4 and 5 we considered codes that has a spatial rate of

unity or less than unity. These codes did not have any multiplexing

gain with respect to a SISO channel, but, on the other hand, unlike

a SISO channel, they possessed a diversity order of ideally MTMR.

Hence, these codes were excellent for improving the link quality

by combating deep fades.

In this chapter, we shall examine codes that are expressly meant

for improving the multiplexing gain by transmitting MT independent

data streams. This yields a spatial rate and multiplexing gain of MT.


Foschini proposed a layered space-time (LST) architecture. In this landmark paper, he proposed to exploit the delay spreads existing in a wideband channel.

Basically, the reader should image the delay spreads as independent highways existing through the channel between the transmitter and receiver antennas. Depending on the frequency and bandwidth of the transmitted signals, they take a particular path through the channel. The time taken to traverse that path is delay spread for that path. These paths by their very nature are independent and each path does not “see” the other.

Foschini decided to exploit this property existing in nature by transmitting signals through these paths. Hence, we do not need to deliberately make the data streams orthogonal like we do in space-time block coding. The orthogonal property just exists in nature and we simply exploit it.


In view of the narrowband nature of the transmission, each data stream follows only one router to the receiver and, consequently, there are no multipath experienced by the individual data streams. This is a very important criterion in that the signals are narrowband. Indeed it is applicable to all MIMO transmission systems.

This technology is called spatial multiplexing (SM). However, the reality is far from this ideal. The data streams are not truly independent and they do possess a certain amount of interaction with each other, giving rise to a phenomenon called multistream interference (MSI) and the usual channel fading and additive noise.


6.2 LST Transmitters: Types of Encoding

6.2.1 Horizontal Encoding

6.2.2 Vertical Encoding


6.2.1 Horizontal Encoding (HE)

6.2.1.1 Diagonal Encoding

6.2.1.2 Threaded Space-Time Encoding


HLST: horizontal layered space-time code.

– In this method the bit stream is first demultiplexed

into MT data streams. Each data stream is thereafter

block encoded and interleaved. This is followed by

mapping into the chosen modulation scheme from a

constellation. The temporal encoding is independent

and each data stream is transmitted from its

individual antenna.


Figure 6.1 Horizontal encoding


– In this scheme each transmitted stream is received by MR antennas. Hence, the maximum diversity attainable is MR order diversity. This makes this system suboptimal, since we would have preferred MTMR order diversity. We have assumed no channel knowledge at the transmitter and perfect channel knowledge at the receiver, yielding an array gain of MR at the receiver.

– There is also a coding gain accruing from the encoder.

There are two variants of this scheme made with a view to increasing the diversity order as close to MTMR as possible. These are diagonal encoding (DE) and threaded encoding (TE).


6.2.1.1 Diagonal Encoding

The initial signal processing is just like that for HE. However, before going to the antenna, the signal is stream rotated in a round-robin fashion so that the bit-stream and antenna association is periodically recycled. If the code word is large enough we can ensure that it is transmitted over all MT antennas. This imparts a diversity of MT at the transmitter. This is the Diagonal Bell Laboratories Layered Space-Time (D-BLAST) scheme.


Figure 6.2 Diagonal encoding or D-BLAST transmission technique


Note the wasted space-time area in Figure 6-2, where

no transmission takes place. This initial wastage is

required to enable optimal decoding.

– D-BLAST architecture arranges the layers

diagonally in space and time. Each block represents

a transmitted symbol. Each layer is represented with

a different shade and runs diagonally through the

antenna elements as time progresses. Instead of

committing each data stream to a single antenna

like in Figure 6.1, the diagonal approach ensures

that none of the layers miss out because of a poor

transmission path.


– The BLAST receiver uses a multiuser detection strategy based

on a combination of interference, cancellation, and

suppression.

– In D-BLAST, each diagonal layer constitutes a complete code

word, so decoding is performed layer-by-layer. Consider the

code word matrix in Figure 6.2. The entries below the first

diagonal layer are zeros. To decode the first diagonal, the

receiver generates a soft-decision statistic for each entry in

that diagonal. In doing so, the interference from the upper

diagonals is suppressed by projecting the received signal onto

the null space of the upper interfaces. The soft statistics are

then used by the corresponding channel decoder to decode

this diagonal. The decoder output is then fed back to cancel

the first diagonal contribution in the interface while decoding

the next diagonal. The receiver then proceeds to decode the

next diagonal in the same manner.


D-BLAST can achieve MTMR order diversity if we use Gaussian code blocks with infinite block size. As usual, coding gain will depend on the encoder and array gain of MR is realizable at the receiver, since it has perfect knowledge of the channel.

In reality this is not possible and so a new type of SM coding called threaded layered space-time code (TLST) was proposed.

– This method mixes the signal more thoroughly across the antennas than does the D-BLAST diagonal system.

– The only requirement here is that during each symbol period any given layer is transmitting over, at the most, one antenna. As a result all spatial interference will come from other layer.

– However, unlike D-BLAST we cannot deal with the signal processing at the receiver one layer at a time but we need to carry out joint decoding of multiple threads. Hence, it is more complex to implement.


6.2.1.2 Threaded Space-Time Encoding (TSTE)

This is also called threaded layered space-time code

(TLST).

This is again a variant of HE. The last block is a spatial

interleaver, which interleaves the symbols as shown in

the space-time matrix. Each shade represents a thread.

We have one code word per thread. In the first column,

the symbols of each layer are not shifted. In the second

column they are shifted once in a cyclic manner. In the

third column they are shifted twice and so on.


Figure 6.3 Threaded layered space-time (TLST)


We define the space-time code matrix A. The MTxl matrix A contains the symbols transmitted over the MT transmit antennas for l symbol periods. We can describe each layer in general by specifying a set of elements from A.

– Let L={L1,L2,…,LMT} be a set of indices specifying the elements

of A. Mathematically Li is defined as

(6.1a) – To avoid spatial interference between layers, if we specify

element (a,t) as belonging to Li, then (b,t) cannot belong to Li since a≠b.

To take full advantage of the diversity available, each layer should use the full range of antenna elements over the full range of symbol periods. Putting it in another way, we require that each layer has a spatial span of MT and a temporal span of l.

ltlitLTMi

0:,11


Now consider the MT data streams that are demultiplexed and encode.

– If the length of the data stream before demultiplexing was k, then after demultiplexing its length would be k/MT. After block encoding, let each of the MT independent data streams have a length of N/MT where N>k. Finally, we modulate the encoded vectors using MT spatial modulators (mapping), which map the encoded data into MTxl matrix A. The relationship between l and N/MT is defined by the type of constellation being mapped.

– We are now in a position to analyze the new TLST architecture. This approach maximizes the diversity available through the encoding and interleaving of layers. To take full advantage of the resources, the TLST architecture requires that each layer transmit one symbol during each symbol interval and use on an average each transmission antenna equally often. Hence, all layers have equal use of the system resources.

– This approach helps impart full transmit diversity MT to the transmitted signal.


The TLST architecture is a combination of the coding process and the layering process.

– The reader should bear in mind that each symbol is a signal constellation point corresponding to b alphabet symbols (bits) that will be modulated into the l constellation points in Li.

– The design of this encoding process can be represented using a matrix construction. Let g be a mapping such that g(x) is the encoded vector of length bl. We can specify g(x) by

(6.1b)

where M1,M2,…,MMT are binary matrixes of dimension kxbl/MT.

Thus, for any arbitrary inputs x, g(x) is the associated code word for the block code specified by M1,M2,…,MMT

. This code word can be split up into MT segments corresponding to the encoded vectors to be modulated over each of the MT antennas. In other words, xMj is mapped by the spatial modulator fi into the l/MT symbols of Li to be transmitted over antenna j. The complete transmission matrix corresponding to layer i is given by ci=fi(g(x)).

TM

MxMxMxxg ...21


A space-time code specified by the encoding matrixes M1,M2,…,MMT

and the spatial modulator fi achieves spatial diversity equal to dMR in a quasi-static fading channel if and only if d is the largest integer such that the encoding matrixes M have the property that:

∀ a1,a2,…,aMT∈{0,1} such that a1+a2+…+aMT

=MT-d+1:

M=[a1M1 a2M2… aMTMMT

] has rank k over the binary field

We conclude that full spatial diversity MTMR is achieved iff M1,M2,…,MMT

are of rank k over the binary field. Thus, by designing the encoding matrixes as discussed, dMR level diversity can be obtained for a given layer within a system with MT transmit and MR receive antennas.


Suppose we code with a binary convolutional code of rate k/MT. Let xi(t)

and yi(t) denote the ith input and output sequences, respectively. The

convolutional code is defined by an “impluse response” gij(t) that relates

the ith input (xi(t)) to the jth output (yi(t)). If we deinfe the D-transform of

the sequence {x(t)} as

X(D) = x(0) + x(1)D + x(2)D2 + …

then the encoder action in the D-domain is given by the matrix equation:

Y(D) = X(D)G(D) (6.1c)

where Y(D) is an n-dimensional vector of the output sequences

Y1(D)Y2(D)…Yn(D), X(D) is a k-dimensional vector of the input sequences

X1(D)X2(D)…Xk(D), and G(D) is given by:

DGDGDG

DGDGDG

DGDGDG

DG

nkkk

n

n

,2,1,

,22,21,2

,12,11,1


We now define the jth column of G(D) as Fj(D) and write the jth output of the encoder as:

Yj(D) = X(D)Fj(D)

Bearing in mind the above code, the MT outputs of the encoder, Y1(D)Y2(D)…Yn(D), can be assigned directly to the MT transmission layers in a space-time layered architecture. In this case, the resulting space-time code, defined by the convolutional code and a spatial modulator f, will have a diversity level of dMR with d=MT-v+1 if v is the smallest integer such that whenever a1+a2+…+aMT

=v for all a1,a2,…,aMT∈{0,1},

then the matrix

has full rank.

DFaDFaDFaT

TMM

2211


Until now, we have designed the space-time code for a

given layer and its performance was considered

independently of other layers. But we need to take into

account the interference from other layers. The overall

performance of TSTE architecture will dependent

largely on the ability of the receiver to separate layers.

Thus, the signal processing scheme at the receiver is

an integral part of system design.

The signal processing for the space-time modulated

signal at the receiver can be regarded as a multiuser

detection problem. Each layer must be independently

detected and decoded. The detection and decoding

tasks can be combined into an efficient iterative

process.


For example, an MMSE receiver can be used to generate soft decisions estimating the symbols received from the MT transmission antennas. The MT streams of estimates can be unwoven into the MT layers, which can then be decoded. Feeding the results back to the MMSE receiver, the a prior probabilities can be updated and interference to other layers can be canceled. The estimate of the symbol sent by the ith antenna at time t can be written as:

(6.1d)

where wf is a vector of weighting coefficients suppressing interference from other layers, and wb is the feedback coefficient to cancel the effects from previously decoded symbols.

i

b

i

f

i wrwyT


When compared with a horizontal layering

approach, known as H-BLAST, we see that the

TST provides greater spatial diversity through

the efficient distribution of layers across

antenna elements.

To compare the performance of TST with D-

BLAST and a multilayered approach, the

simulation results demonstrated a gain in

performance for the TST architecture of 3-8 dB

over the D-BLAST and multilayering

architecture.


6.2.2 Vertical Encoding

In this method the bit stream is encoded, interleaved, and mapped before being fed to a demultiplexer. It is then split into MT streams. The implication in the design is that each information bit can be spread across all antennas. This kind of transmission, however, requires joint decoding at the receiver, making it very complex. The spatial rate is MT and the signaling rate is rmrcMT bits/transmission. Since the transmission bits are spread over MT antennas and each stream is received by MR antenna, the diversity order is MTMR.


Figure 6.4 Vertical encoding


6.2.2.1 V-BLAST

A variant of the virtual encoding (VE) is the

vertical BLAST (V-BLAST) algorithm, shown in

Figure 6.5.

The basic idea here is to usefully exploit the

multipath, rather than mitigate it, by

considering the multipath itself as a source of

diversity that allows the parallel transmission of

substreams from the same user.


Figure 6.5 V-BLAST configuration


The Bell Labs Layered Space-Time (BLAST)

architecture uses multielement antenna arrays at both

transmitter and receiver to provide high-capacity

wireless communications in a rich scattering

environment. It has been shown that the theoretical

capacity approximately increases linearly as the

number of antennas is increased. Two types of BLAST

realizations have been widely publicized: V-BLAST and

D-BLAST.

– The V-BLAST is a practical algorithm shown to achieve a large

fraction of the MIMO channel capacity in the case of

narrowband point-to-point communication scenarios.

– The V-BLAST algorithm implements a nonlinear detection

technique based on zero forcing (ZF) combined with symbol

cancellation to improve the performance.


The idea is to look at the signals from all the receive antennas simultaneously, first extracting the strongest substream from the received signals, then proceeding with the remaining weaker signals, which are easier to recover once the strongest signals have been removed as a source of interference. This is called successive interference cancellation (SIC) and is somewhat analogous to decision feedback equalization.

When symbol cancellation is used, the order in which the substreams are detected becomes important for the overall performance of the system. In fact, the transmitted symbol with the smallest postdetection SNR will dominate the error performance of the system. Postdetection SNR is determined by ordering. The optimal ordering is based on the result that simply choosing the best postdetection SNR at each stage of the detection process leads to the maximization of the worst SNR over all possible orderings.


Figure 6.6 V-BLAST system


For simplicity, we base our explanation on Figure 6.6.

– Flat fading is assumed and the matrix channel transfer functin is

HMTxMR, where hi,j is the complex transfer function from transmitter j to

receiver i and MT≤MR.

– We assume that the transmission is organized in bursts of L symbols

and that the channel time variation is negligible over the L symbol

periods, comprising a burst, ant that the channel is estimated

accurately using training symbols embedded in each burst.

Suppose the number of transmitter is MT and the number of

receivers is MR. QAM transmitter 1 to MT operate cochannel at

symbol rate 1/T symbols, with synchronized symbol timing. This

collection of transmitters constitutes a vector drawn from a QAM

constellation.

Receivers 1 to MR are individually conventional QAM receivers.

The receivers also operate cochannel, each receiving the signals

radiated from all MT transmit antennas.


Let a=(a1 a2 … aM)T denote the vector of transmit symbols. Then the corresponding received MR vector i

(6.2)

where v is a wide sense stationary (WSS) noise vector with i.i.d components.

Step 1: Using nulling vector wk1, form a linear combination of the

components of r1 to yield yk1:

(6.3)

Step 2: Slice yk1 to obtain âk1

:

(6.4)

where Q(.) denotes the quantization (slicing) operation appropriate to the constellation in use.

Step 3: Assuming that âk1=ak1

, cancel ak1 from the received vector r1; resulting in modified received vector r2:

(6.5)

where (H)k1 denotes the k1th column of H.

Steps 1-3 are then performed for components k2, k3,…,kMT by operation in

turn on the progression of modified received vectors r2, r3,…,rMT.

111

rwy T

kk

11

ˆkk

yQa

11

ˆ12 kk

Harr

vHar 1


The specifics of the detection process depend on the criterion chosen to compute the nulling vectors wki, the most common choices being minimum MMSE and ZF. The detection process used in this section is the latter and simpler. The kith ZF nulling vector is defined as the unique minimum norm vector satisfying

(6.6)

Thus, the kith ZF nulling vector is orthogonal to the subspace spanned by the contributions to ri, due to those symbols not yet estimated and canceled. It can be easily shown that the unique vector satifying (6.6) is just the kith row of Hkj-1

∓, where the notation Hkj-

denotes the matrix obtained by zeroing columns k1, k2,…,kj of H and + denotes the Moore-Penrose pseudoinverse.

ij

ijHw

jik

T

k

1

0


The full ZF detection algorithm is a recursive procedure, including determination of optimal ordering:

(6.7a)

(6.7b)

(6.7c)

where j ∉ {k1 k2 … ki-1}

(6.7d)

(6.7e)

(6.7f)

(6.7g)

(6.7h)

(6.7i)

where (Gi)j is the jth row of Gi.

HGtion:initializa1

1i

2

minarg:jij

Gkrecursion

ii kik

Gw

i

T

kkrwy

ii

ii kk

yQa ˆ

ii

kkiiHarr ˆ

1

ikiHG

1

1 ii


Equation (6.7c) determines the elements the

optimal row sequence that maximizes the

performance of the BER curve.

Equation (6.7d-f) compute the ZF nulling vector,

the decision statistic, and the estimated

component of a, respectively.

Equation (6.7g) performs the cancellation of

the detected component from the received

vector and (6.7h) computes the new

pseudoinverse for the next iteration.


Conclusions:

HLST

D-BLAST

TLST

V-BLAST


HLST

Output from encoder 1 s31 s21 s11 s01 Output for antenna 1




3 2 1 0 Time t


D-BLAST

Output from

encoder 4

Output from

encoder 3

Output from

encoder 2

Output from

encoder 1

s04 s03 s02 s01 Output for antenna 1




6 5 4 3 2 1 0 Time t


TLST

Output from

encoder 2

Output from

encoder 3

Output from

encoder 4

Output from

encoder 1





3 2 1 0 Time t


V-BLAST

Output from

encoder 4

Output from

encoder 3

Output from

encoder 2

Output from

encoder 1





3 2 1 0 Time t


6.3. Layered Space-Time Coding: Design Criteria

6.3.1 Performance Analysis of an HLST

System

6.3.2 Performance Analysis of a DLST System

6.3.3 Code Design Criteria


Consider the HLST encoder in Figure 6.1. We will now derive the design criteria for LST systems using a procedure similar to the one discussed in Chapter 5 for STTC systems.

Assume that an any given time t a symbol sti,

i∈{1,2,…,MT} is transmitted out of transmit antenna i. If we assume a Rayleigh fading channel, receive antenna j receives symbol rt

j, which can be expressed as,

(6.8)

where hti,j is the channel coefficient between the ith

transmit and the jth receive antenna at time t. ntj is an

additive white Gaussian noise sample at receive antenna j at time t, which is an independent sample of a zero mean complex Gaussian random variable with variance σ2 per dimension.

j

t

M

i

i

t

ji

t

j

tnshr

T

1

,


The modulated signal estimate ŝtk of the kth layer at time t can be written

as

(6.9)

For some linear combination coefficients w an MR dimensional row vector whose values are determined by using a ZF function or MMSE criterion. The received column vector rt = [rt

1 rt2 … rt

MR]T contains the received symbols at the MR receive antennas. The MR-dimensional column vector ht

l = [htl,1 ht

l,2 … htl,MR]T contains the channel coefficients between the lth

layer and MR receive antennas at time t. We carry out QR decomposition on the MRxMT channel matrix Ht = [ht

1 ht2 … ht

MT] and decompose it into an unitary matrix Ut and an upper triangular matrix Rt such that

Ht = UtRt (6.10)

where

(6.11)

is an MRxMT upper triangular matrix and Ut is of size MRxMT.

TM

kl

k

t

l

tt

k

tshrws

1

~

t

M

M

t

M

t

t

M

tt

t

M

ttt

t

T

R

T

T

T

R

RR

RRR

RRRR

R

000

00

0

3

3

3

2

3

2

2

2

1

3

1

2

1

1

1


We define yt = [yt1 yt

2 … ytMR] as

(6.12)

where ňt = Utnt is an MR-dimensional column vector of

AWGN noise samples and st is an MT-dimensional

column vector of the transmitted modulated symbols at

time t. Since Rt is upper triangular,

(6.13)

Assuming that hard decisions from the constituent

decoders are correct, they can be used to completely

suppress the interference term in (6.13). Therefore, the

decision variable ŝtk can be expressed as

(6.14)

for k ∈ {1, 2, …, MT}.

ttttttnsRrUy ˆ

TM

t

k

t

k

t

k

t

k

tt

k

k

k

tsssnsRy ,...,, from ceinterferenˆ 21

k

t

k

tt

k

k

k

tnsRs ˆ~


6.3.1 Performance Analysis of an HLST System

6.3.1.1 Behavior in Slow Fading Channels

6.3.1.2 Behavior in Fast Fading Channels


If we assume code words of length L, the pairwise error probability of the kth layer P(sk, ek) is the probability that the decoder selects as its output the sequence ek = (e1

k, e2k, …, et

k, …, eL

k), when the transmitted sequence on the kth layer was in fact sk = (s1

k, s2k, …, st

k, …, sLk).

This occurs if

(6.15)

or equivalently

(6.16)

where Re{.} indicates the real part of a complex number.

L

t

k

tt

k

k

k

t

L

t

k

tt

k

k

k

teRssRs

1

2

1

2 ~~

L

t

k

tt

k

k

k

t

L

t

k

t

k

tt

k

k

k

teRsesRn

1

2

1

~ˆRe2


For a given realization of the channel matrix H or equivalently the matrix R, the right-hand side of (6.16) is a constant. We call this constant d2(sk, ek). The left-hand side of the expression is zero mean Gaussian random variable with variance 4σ2d2(sk, ek). Therefore, for a given H, the conditional pairwise error probability can be expressed as

(6.17)

By using the inequality

(6.19)

we obtain

(6.20)

where d2(sk, ek) is defined by

(6.21)

2

2

8

,|

kk

kk esdQHesP

02

1

0

22

xdtexQ t

2

2

8

,exp

2

1|

kk

kk esdHesP

L

t

k

tt

k

k

k

t

kk eRsesd1

22 ~,


6.3.1.1 Behavior in Slow Fading Channels

From (6.20) and given the channel coefficient matrix H, the probability that the decoder decides erroneously in favor of a modulated sequence ek can be expressed as

(6.22)

where the matrix R comes from the QR decomposition of H and σ2 is the noise variance per dimension.

If the channel is a slow fading one, the fading coefficients remain constant over a frame. This makes ||(Rk

k)t||2 = ||Rk

k||2 and

(6.23)

where ||Rkk||2 is a sum of 2(MR-k) zero mean Gaussian distributed random

variables, each with variance 1/2.

2

1

22

8exp|

L

t

k

t

k

tt

k

kkk

esRHesP

L

t

k

t

k

t

k

k

kk esRHesP1

22

28

1exp|


Let Dk = dEk2||Rk

k||2 where dEk2 = Σt||st

k-etk||2. If 2(MR-k) ≥ 4, then

based on the Central Limit Theorem Dk approaches a Gaussian distribution with mean μDk = (MR-k)dEk and variance σDk

2 = (MR-k)dEk

2.

Therefore, the conditional error probability given in (6.23) can be rewritten as

(6.24)

The PEP is obtained by averaging (6.24) over the probability density function of Dk, p(Dk) resulting in

(6.25)

If 2(MR-k) < 4, the Gaussian approximation on the probability density functions of Dk no longer holds. In such a case Dk is a chi-square distributed with 2(MR-k) degrees of freedom. Therefore, the PEP is obtained by averaging (6.23) over a chi-square distribution resulting in

(6.26)

k

kk DHesP28

1exp|

2

22

2

2

22

2

8

8

882exp

kkk ERERERkk

dkMQ

dkMdkMesP

kMkk

kk

R

esesP

2

2

81


6.3.1.2 Behavior in Fast Fading Channels

For a channel coefficient matrix H, the PEP of the kth layer P(sk,

ek) can be expressed as

(6.27)

where the matrix R comes from the QR decomposition of H.

We introduce a variable bt,k = ||(Rkk)t||

2||stk-et

k||2 and Dk = Σtbt,k.

This makes bt,k a chi-square distributed random variable with

2(MR-k) degrees of freedom. Let dH be the Hamming distance

between the two code words and dH ≥ 4. Then, by virtue of the

Central Limit Theorem, Dk is a Gaussian random variable with

mean μDk = (MR-k)Σt||stk-et

k||2 and variance σDk2 = (MR-k)Σt||st

k-etk||4.

2

1

22

8exp|

L

t

k

t

k

tt

k

kkk

esRHesP


Since we know the probability density function of Dk, the PEP can be obtained by averaging the conditional error probability over the PDF p(Dk) as

(6.28)

where d2 = Σt||stk-et

k||2 and D4 = Σt||stk-et

k||4.

If the Hamming distance, dH < 4, the Gaussian assumption on the PDF of Dk is no longer valid. In such a case, Dk is a chi-square distributed with 2(MR-k) degrees of freedom. Hence, the PEP becomes

(6.29)

where η(sk, ek) = {t|stk ≠ et

k}.

4

2

2

4

2

2

22

4

02

8882exp

8exp

D

kMd

DkMQ

dkMDkM

DdDpD

esP

RRRR

kk

kkk

kk

R

kk est

Mkk

t

k

t

est

k

t

k

tt

k

kkkesesR

EesP,

2

2

,2

22

81

8exp


6.3.2 Performance Analysis of a DLST System

Given a channel coefficient matrix H, the probability that the receiver decides in favor of a distinct diagonal e = (e1

1, e22, …,

etk, …, eMT

MT), while the transmitted code word is s = (s11, s2

2, …, st

k, …, sMT

MT), can be expressed as,

(6.30)

where the matrix R comes from the QR decomposition of H.

Remember that because each code word occupies a diagonal in the transmission resource array, k = t, st

k = stt, and et

k = ett.

Similarly, (Rkk)t = (Rt

t)t. Equation (6.30) is valid on both fast and slow fading channels because each subchannel undergoes independent fading (they are transmitted one diagonal at a time and orthogonal in time).

2

1

22

8exp|

TM

t

k

t

k

tt

k

kkk

esRHesP


Let D = Σt+1L||(Rt

t)t||2||st

t-ett||2, where ||(Rt

t)t||2 is a chi-square distributed

random variable with 2(MR-k) degrees of freedom. If dH is the Hamming distance between the two code words and dH ≥ 4, then by virtue of the Central Limit Theorem, D is a Gaussian random variable with mean μD = Σt(MR-k)||st

t-ett||2 and variance σD

2 = Σt(MR-k)||stt-et

t||4.

If we know the PDF of D, the PEP can be obtained by averaging the conditional error probability over the PDF p(D) as

(6.33)

If dH < 4, the PDF of D is no longer approaching Gaussian distribution and instead is chi-square distributed. The error probability is obtained by taking the expectation of conditional PEP over a chi-square distribution as

(6.34)

where η(s, e) = {t|stt ≠ et

t}.

t

t

t

t

tR

t

t

t

t

t

t

t

t

tRt

t

t

t

tR

kk

eskM

eseskMQ

eskM

DdDpD

esP

42

22

2

2

2

02

8

8

82

18

1

exp

8exp

est

tM

tMt

t

t

tt

t

t

t

tt

t

t

kkest

R

R

esesREesP,

2

222

2

,

8

1

8

1exp


6.3.3 Code Design Criteria

6.3.3.1 HLST on Fast Fading Channels

6.3.3.2 HLST on Slow Fading Channels

6.3.3.3 DLST on Slow and Fast Fading

Channels

6.3.3.4 Performance Evaluation


6.3.3.1 HLST on Fast Fading Channels

Using (6.19) and assuming a high SNR, the PEP when dH ≥ 4 as given in (6.28) can be further approximated as

(6.35)

Hence, to minimize the PEP when dH ≥ 4, the minimum value of d2 = Σt||stk-

etk||2 has to be maximized.

Again, at high SNRs, the PEP when dH < 4 as given in (6.29) can be further approximated as

(6.36)

To minimize the PEP, the minimum of η has to be maximized over all code word pairs. This is equivalent to maximizing the Hamming distance, dH, of the code. Second, the minimum product distance πt∈η(s

k,e

k)||st

k-etk||2 over

all code word pairs that are dH Hamming distance apart has to be maximized.

2

2

82

1exp

2

1

dkMesP Rkk

kk

R

est

kMk

t

k

tkkes

esP,

2

2

8


Design criteria for HLST fast fading channels

– If the Hamming distance of the code is greater than

or equal to four, the minimum value of

d2 = Σt||stk-et

k||2 has to be maximized.

– If the Hamming distance of the code is less than

four, first the Hamming distance of the code has to

be maximized. Second, the minimum product

distance πt∈η(sk,e

k)||st

k-etk||2 has to be maximized.


6.3.3.2 HLST on Slow Fading Channels

At high SNRs, the PEP on the kth layer given in (6.25) can be further approximated as

(6.37)

which suggests that the performance of HLST on a slow fading channel when 2(MR-k) ≥ 4 is dominated by the minimum value of dEk

2. Hence, to minimize the PEP requires that the minimum value of dEk

2 be maximized.

When 2(MR-k) < 4, however, the code performance is bounded by (6.26). It therefore suggests that to minimize the PEP the minimum squared Euclidian distance ||sk-ek||2 has to be maximized.

2exp

2

82

2 kM

dkMesP R

ER

kk

k


Hence, in slow fading channels, regardless of

the value of 2(MR-k) the design criterion of an

HLST is guided by maximizing the minimum

value of dEk

2 = Σt||stk-et

k||2.


6.3.3.3 DLST on Slow and Fast Fading Channels

The equation (6.33) upper-bounds the error performance of a diagonally layered

code when the Hamming distance dH ≥ 4. Using (6.19), this can be further

approximated as

(6.38)

where η(s, e) = {t|stt ≠ et

t}, which suggests that the PEP of DLST in slow and fast

fading channels is minimized when the value of Σt(MR-t)||stt-et

t||2 is maximized.

In the case when dH < 4 the code performance in fast and slow fading channels is

bounded by (6.34). We now define the truncated multidimensional effective length

(TMEL) and the truncated multidimensional product distance (TMPD) between two

distinct code words s and e as TMEL = Σt∈η(s,e)MR-t and TMPD = πt∈η(s,e)||stt-

ett||2(M

R-t). To minimize the PEP, the minimum value of TMEL over all code word

pairs has to be maximized. Second, of all the code word pairs with the maximum

minimum TMEL, the minimum value of TMPD ought to be maximized.

est

t

t

t

tR

t

t

t

t

tR

kkestM

estMesP,

2

2

2

2 8exp

2

1

8

1exp

2

1


Design criteria for DLST codes in fast and slow

fading channels

– If the Hamming distance of the code is greater than

or equal to four, the minimum value of

Σt(MR-t)||stt-et

t||2 has to be maximized.

– If the Hamming distance of the code is less than

four, first the minimum value of

TMEL = Σt∈η(s,e)MR-t has to be maximized. Second,

the minimum value of TMPD = πt∈η(s,e)||stt-et

t||2(MR

-t)

has to be maximized.


6.3.3.4 Performance Evaluation

Assumed that the receiver has perfect

knowledge of the channel.

The modulation adopted is QPSK.

Ring codes were used in the encoder.


Figure 6.7 FER comparison for HLST codes in a slow fading channel


The channel is a slow fading Rayleigh channel. The

performance of HLST with three different (8, 4) ring

codes are examined. Each code has a Hammng

distance of 3 and minimum squared Euclidian distance

of 6.0, 8.0 and 10.0. Six transmit and 10 receiver

antenna are assumed. This yielded a bandwidth

efficiency of 6 bit/s/Hz. A frame is assumed to consist

of a code word of eight 4PSK symbols.

The best performance is achieved when the minimum

squared Euclidian distance is the largest. In this case,

it is the ring code dEk2 = 10.0 which offers the 2 dB gain

at a frame error rate of 10-4.


Figure 6.8 FER comparison for HLST codes in a fast fading channel


Figure 6.8 shows the case of a fast fading

channel using (9, 3) ring codes with 8PSK

modulation. The codes have a Euclidian

distance of 5.515 and 4.343, respectively.

Their product distances are 0.081 and 0.235,

respectively. Once again, the code with the

highest Euclidian distance performs best

despite its much lower value of product

distance.


6.4 LST Receivers

6.4.1 ML Receiver

6.4.2 Zero-Forcing Receiver

6.4.3 MMSE Receiver

6.4.4 Successive Cancellation Receiver

6.4.5 Zero Forcing V-BLAST Receiver

6.4.6 MMSE V-BLAST Receiver

6.4.7 Simulation Results

6.4.8 Receivers for HLST and DLST Systems


The reality of MIMO receivers is that we need to contend with MSI, since the transmitted streams interfere with each other. In addition to this, we have the usual problem of channel fading and additive noise.

Initially we assume uncoded SM (i.e., the data stream comprises uncoded data, where no temporal coding has been employed, but only mapping). The three most common receivers for uncoded SM are ML, ZF, and MMSE.

We assume a flat fading environment following the law for the received signal

r = Hs + n (6.39)

where r is the received MRx1 vector, H is the channel matrix of size MRxMT, s is the MTx1 transmitted signal and n is the MRx1 ZMCSCG noise vector with covariance matrix N0IMR.


6.4.1 ML Receiver

This is an optimum receiver.

If the data stream is temporally uncoded, the ML receiver solves

(6.40)

where ŝ is the estimated symbol vector.

The ML receiver searches through all the vector constellation for the most probable transmitted signal vector. This implies investigating SMT combinations, a very difficult task. Hence, these receivers are difficult to implement, but provide full MR diversity and zero power losses as a consequence of the detection process. In this sense it is optimal.

There have been developments based on fast algorithms empolying sphere decoding.

2

minargˆ Hsrss


6.4.2 Zero-Forcing Receiver

The ZF receiver is a linear receiver. It behaves like a

linear filter and separates the data streams and

thereafter independently decodes each stream.

We assume that the channel matrix H is invertible and

estimate the transmitted data symbol vector as

(6.41)

where + represents pseudoinverse.

sHHsHHs H

1

ˆ


Since an inverse of H can only exist if the columns of H are independent, it is assumed that H=Hw (i.e., the entries are i.i.d). The noise in the separated streams is correlated and consequently the SNRs are not independent.

The SNR on any one channel averaged over all channel instances is upper-bounded by

(6.42)

Equation (6.42) shows that the diversity order of each stream is MR-MT+1. The ZF receiver decomposes the link into MT parallel streams, each with diversity gain and array gain proportional to MR-MT+1. Hence, it is suboptimum.

12

min

2

TR MM

T

ee

M

dNP


6.4.3 MMSE Receiver

We examine another linear detection algorithm to the

problem of estimating a random vector s on the basis

of observations y is to choose a matrix B that

minimizes the mean square error

(6.43)

The solution of the linear MMSE is given by

(6.44)

where the superscript H denotes the complex conjugate

transpose.

BysBysssssEeTT

ˆˆ2

rHHHISNR

rBs HH

M R

1

1ˆ


The ZF receiver perfectly separates the

cochannels’ signals at the cost of noise

enhancement. The MMSE receiver, on the

other hand, can minimize the overall error

caused by noise and mutual interference

between the cochannel signals, but this is at

the cost of separation quality of the signals.


6.4.4 Successive Cancellation Receiver

The SUC algorithm is usually combined with V-BLAST receivers. This provides improved performance at the cost of increased computational complexity. Rather than jointly decoding the transmitted signals, this nonlinear detection scheme first detects the first row of the signal and then cancels its effect from the overall received signal vector. It then proceeds to the next row. The reduced channel matrix now has dimension MRx(MT-1) and the signal vector has dimension (MT-1)x1. It then does the same operation on the next row. The channel matrix now reduces to MRx(MT-2) and the signal vector reduces to (MT-2)x1 and so on. If we assume that all the decisions at each layer are correct, then there is no error propagation. Otherwise, the error rate performance is dominated by the weakest stream, which is the first stream decoded by the receiver. Hence, the improved diversity performance of the succeeding layers does not help.


To get around this problem the ordered successive cancellation (OSUC) receiver is introduced. In the case, the signal with strongest signal-to-interference-noise (SINR) ratio is selected for processing. This improves the quality of the decision and reduces the channels of error propagation. This is like an inherent form of selection diversity wherein the signal with the strongest SNR is selected.

The OSUC algorithm is as following: – Ordering: Determine the optimal detection order by choosing the row

with minimum Euclidian norm (strongest SINR).

– Nulling: Estimate the strongest transmit signal by nulling out all the weaker transmit signals.

– Slicing: Detect the value of the strongest transmit signal by slicing to the nearest signal constellation value.

– Cancellation: Cancel the effect of the detected signal from the received signal vector to reduce the detection complexity for the remaining signals.

The OSUC algorithm is usually combined with the ZF or MMSE receiver, as in the V-BLAST algorithm discussed in Section 6.2.2.1.


6.4.5 Zero Forcing V-BLAST Receiver

This has already been discussed in Section

6.2.2.1.


6.4.6 MMSE V-BLAST Receiver

The OSUC is combined with the MMSE algorithm to yield a performance superior to the ZF with OSUC combination. The MMSE receiver suppresses both the interference and noise components, whereas the ZF receiver removes only the interference components. This implies that the mean square error between the transmitted symbols and the estimate of the receiver is minimized. Hence, MMSE is superior to ZF in the presence of noise.


Initialization

Recursion

where σ2 is the variance of I.I.d. complex Gaussian random noise with zero mean.

1i

rr 1

H

M

H HIHHGT

12

1

2

1minarg

jijGk

ikik

Gw 1

i

T

kkrwy

ii

ii kk

yQa ˆ

ii

kkiiHarr ˆ

1

H

iMi

H

iiHIHHG

T

12

1

2

1,...,,

1

21

minargji

kkkji

Gki

1 ii


6.4.7 Simulation Results

Simulation Model

– This was a 2x2 system with QPSK modulation.

– The power is normalized across the transmit

antennas and the channel is a slow Rayleigh fading

channel.


Figure 6.9 Comparison of ML, MMSE (OSUC), and ZF (OSUC) for a 2x2 using QPSK modulation


The results show that ML is the best in

performance followed by MMSE (OSUC) and

ZF (OSUC).


6.4.8 Receivers for HLST and DLST Systems

Receiver for such systems are the same as previously discussed. The main difference is that they should decode on block basis instead of symbol basis.

In a block, the information and parity bits are included in the same block. Therefore, the diversity order will remain as MR-MT+1 for ZF and MMSE receivers. Coding improves the quality of reception and will mitigate the error propagation phenomenon. The ML receiver achieves MR order diversity.

In addition, temporal coding provides coding gain.


6.5 Iterative Receivers

Ideally, in order to achieve high capacities, we would like to use very large code words followed by ML decoders at the receiver. Reality precludes us from these goals, because the block sizes are finite in length and ML receivers are too complex. In encoded systems the complexity of ML receivers is further enhanced because it has to perform joint detection and decoding on an overall trellis obtained by combining the trellises of the layered space-time coded and the channel code. The complexity of the receiver is an exponential function of the product of the number of transmit antennas and the code memory order.

The complexity of the ML receiver is an exponential function of the product of the number of transmit antennas and the code memory order.

Iterative receivers can approach optimal performance with a tolerable receiver complexity.


Figure 6.10 Generic block diagram of an iterative receiver


The receiver can be applied only to coded LST

systems. The decoders are soft input/soft

output decoders. The outer coded bits are

subtracted from the input and interleaved. The

interleaved output is canceled a posteriori fro

the preceding received signal. Interleaving

helps receiver convergence. This is called soft

iterative interference cancellation. The decoder

can apply the maximum a posteriori algorithm

(MAP) among many possible. This algorithm is

optimum in the sense that it minimizes the bit

error probability at the decoder output.


6.6 The Effect of Imperfect Channel Estimation Code Performance

We carry out imperfect estimation using the

MMSE technique discussed in the previous

chapters.


Simulation model

– QPSK code with one transmit and two receive

antennas (SIMO channel) and imperfect channel

estimation in a slow Rayleigh fading channel.

– In the simulation, 10 orthogonal signals in each data

frame of 130 symbols are used as pilot sequence

(preamble) to estimate the channel state information

at the receiver.


Figure 6.11 Imperfect channel estimation for QPSK SIMO channel


From the figure, we can see that the

deterioration due to imperfect channel

estimation is about 0.7 dB throughout.


6.7 Effect of Antenna Correlation on Performance

Simulation model

– The channel is a slow Rayleigh fading channel and

it is assumed that there is no correlation at the

transmitter.

– The modulation is QPSK.


Figure 6.12 Receiver correlation effect on a 2x2 system with QPSK modulation


There is degradation in bit error rates with

rising correlation across the receiver antennas.


6.8 Diversity Performance of SM Receivers

The ML receiver is the optimum receiver and

show the full diversity order of MR.

Both the ZF and MMSE receivers in the OSUC

mode show a diversity order of more than MR-

MT+1, but less than MR.


Table 6.1 Diversity order and SNR performance of SM receivers

Receiver Diversity Order SNR Loss

ZF (OSUC) ≥ MR-MT+1 ≤ MR Low

MMSE (OSUC) ≥ MR-MT+1 ≤ MR Low

ML MR Zero


6.9 Summary

LST transmitter

– Horizontal encoding

Diagonal encoding (D-BLAST)

Threaded encoding (TLST), which yielded the maximum

transmit diversity.

– Vertical encoding

Uncoded scheme (V-BLAST), which has gained a lot of

popularity because of its simplicity.


Criteria for HLST and DLST codes

– The best performance is determined by the code

with the highest Euclidian distance in both slow and

fast fading channels.

LST receivers

– ML is the best in performance, followed by MMSE

(OSUC) and ZF (OSUC).

– Iterative receivers

SM systems in the presence of channel

imperfections and antenna correlation


6.10 Simulation Exercises

Recover the curves given in the chapter for V-BLAST systems.

Study the coding.

Plot the MMSE curves for V-BLAST with a correlation of 0.6. How

does this curve compare with a MMSE curve with no correlation.


References

Foschini, G., “Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multielement Antennas,” Bell Labs Technical Journal, Autumn 1996, pp. 41-59.

El Gamal, H., and A. R. Hammons, “A New Approach to Layered Space-Time Coding and Signal Processing,” IEEE Trans. Inf. Theory, Vol. 47, No. 6, September 2001, pp. 2321-2334.

Tarokh, V., et al., “Combined Array Processing and Space-Time Coding,” IEEE Trans. on Inform. Theory, Vol. 45, May 1999, pp. 1121-1128.


Golden, G. D., et al., “Detection Algorithm and Initial Laboratory Results Using the V-BLAST Space-Time Communication,” Electronic Letters, Vol. 35, No. 1, January 7, 1999, pp. 14-16.

Strang, G., Linear Algebra and Its Applications, Fort Worth, TX: Saunders College Publishing, Brace Jovanovich College Publishers, 3rd edition, 1988.

Firmanto, W., et al., “Layered Space-Time Coding: Performance Analysis and Design Criteria,” Globecom 2001, Vol. 2, November 2001, pp.1083-1087.

Massey, J., and T. Mitterholzer, “Convolutional Codes over Rings,” Proc. Of the Fourth Joint Sweden-USSR Int. Workshop on Inform. Theory, Gotland, Sweden, 1989.


Viterbo, E., and J. Boutros, “A Universal Lattice Code Decoder for Fading Channels,” IEEE Trans. Inf. Theory, Vol. 45, July 1999, pp. 1639-1642.

Damen, G., A. Chkeif, and J. Belfiore, “Lattice Code Decoder for Space-Time Codes,” IEEE Commun. Letters, Vol. 4, No. 5, May 2000, pp. 161-163.

Paulraj, A., R. Nabar, and D. Gore, An Introduction to Space-Time Wireless Communications, Cambridge, UK: Cambridge University Press, 2003.

Bolcskei, H., and A. Paulraj, “Multiple-Input Multiple-Output (MIMO) Wireless Systems,” Communications Handbook, CRC Press, 2001.

Berrou, C., A. Glavieux, and P. Thitimajshima, “Near Shannon Limit Error Correcting Coding and Decoding: Turbo-code,” Proc. IEEE ICC, Switzerland, May 1993, pp. 1064-1070.


Wang, X., and V. Poor, “Iterative (Turbo) Soft

Interference Cancellation and Decoding for

Coded CDMA,” IEEE Trans. Commn., Vol. 47,

No. 7, July 1999, pp. 1046-1067.

Ten Brink, S., J. Speidel, and R. H. Yan,

“Iterative Demapping for QPSK Modulation,”

Electronic Letters, Vol. 34, No. 15, July 1998,

pp. 1459-1460.

Vucetic, B., and J. Yuan, Space-Time Coding,

Chichester, UK: John Wiley & Sons Ltd., 2003.

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