Tsoulos George (ред.) MIMO System Technology for Wireless Communications

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104 MIMO System Technology for Wireless Communications

certain set of sub-streams that will be treated differently. In SM, M

inde-

pendent symbols are transmitted from M

transmitting antennas. Therefore,

there are two main advantages of SM techniques. First, these techniques can

reach a closer bound to the available capacity of MIMO systems, since the

spatial rate is equal to the number of transmit antennas M

. On the other

hand, the complexity requirement, while heavy, remains constant and inde-

pendent of the modulation in use, which is not the case of STTCs. In general,

in order to perform SM, the number of receive antennas must be equal to

or greater than the number of transmit antennas.

The main idea behind SM encoding techniques relies on the use of pow-

erful decoding techniques on the receiver side. At the transmitter side, the

information sequence is subdivided into several sub-streams through a

demultiplexer. The sub-stream where the signal processing is conducted is

identiﬁed as “Layer.” Hence, SM is also called the Layer Space–Time (LST).

The demultiplexing operation can be applied to bits or symbols. Conse-

quently, the demultiplexing of the data information stream can be realized

in three different ways according to the demultiplexer position in the trans-

mitter chain and the directions of the layers. The encoding processes are

known as horizontal, vertical and diagonal SM and are schematically drawn

from top to bottom in Figure 4.7.

In the horizontal encoding case, the data bits are demultiplexed into M

sub-streams that are independently encoded, interleaved and modulated. In

vertical encoding SM, the data stream is at ﬁrst coded, interleaved and

FIGURE 4.7

Spatial Multiplexing Encoding Schemes.

Symbols

Demultiplexer

Bits

Encoder

interleaver

modulator

Bits

Encoder

interleaver

modulator

Encoder

interleaver

modulator

Demultiplexer

Stream

interleaver

Symbols

Bits

Encoder

interleaver

modulator

Encoder

interleaver

modulator

Demultiplexer

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Introduction to Space–Time Coding 105

modulated and the resulting symbols are then demultiplexed into M

sub-

streams. Diagonal SM encoding is similar to horizontal encoding with the

only difference that, after the ﬁnal stage, the frames of symbols undergo a

stream interleaver, which rotates the transmitted frames by padding with

zero some antennas. The forms of the layers for each case are shown in the

right side of Figure 4.7.

4.5.1 V-BLAST Algorithm

The vertical SM encoding is referred in [13] as V-BLAST. The detection in

vertical architecture is simpler than the detection in diagonal architecture

and is preferable to the sub-optimal horizontal case. The information data

are at ﬁrst coded, interleaved and mapped into their corresponding symbols.

Then, the symbols are demultiplexed to the M

transmit antennas. Hence,

the symbols transmitted are independent of each other. Each set of M

symbols composes a transmission vector as shown in the second scheme of

Figure 4.7.

Assuming that the receiver is aware of the modulation used, the decoding

at the receiver becomes possible through the V-BLAST algorithm. In vertical

SM, detection and estimation of the transmitted symbols is performed in a

vector-by-vector basis, since each vector represents the layer of the SM coding.

The algorithm is initialized and operates on each vector in a symbol-per-

symbol basis by iteratively detecting and estimating the transmitted sym-

bols. We assume that the inter-symbol interference is negligible and that the

system is quasi-stationary, which implies that the matrix of the channel

coefﬁcients remains constant. We also assume that the channel matrix is

perfectly known at the receiver side.

The algorithm is based on linear interference suppression in combination

with interference cancellation, methods commonly used in multi-user environ-

ments. The algorithm also performs a rearrangement of the symbol detection

in an optimal sense. In general, V-BLAST is characterized as an interference

avoidance technique. The received signals are perceived as a set of signals

issued from different users. Each signal is an interferer to the rest of the

signals. The best signal stream in terms of SNR is selected for detection and

is also removed from the remaining signals. This way, the remaining signals

have one interferer less. Optimal combining is repeated until all signals are

detected.

Three consecutive phases take place:

• Linear interference suppression through Minimum Mean Square

Estimation (MMSE) or Zero Forcing (ZF)

• Interference cancellation of the symbols detected

• Reordering of the detection process through SNR post-detection

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106 MIMO System Technology for Wireless Communications

The initial form of V-BLAST described in the literature uses ZF for the

suppression of the linear interference. This operation is realized in three

steps:

(4.22)

The ﬁrst step consist on ﬁnding the zero-forcing vector w

that satisﬁes:

(4.23)

In Equation 4.23, (H)

represents the jth column of H which is a vector

composed of all the channel coefﬁcients produced from transmit antenna j.

It turns out that the condition of Equation 4.23 is realized by taking w

to be

the equivalent jth column from the Moore-Penrose pseudo-inverse. Hence,

the algorithm ﬁnds the zero-forcing vector w

through consecutively calcu-

lating the Moore-Penrose pseudo-inverse. In the ﬁrst line of Equation 4.22,

i is the iteration number, k

is the index of the transmit antenna which symbol

is detected during iteration i and G

is the Moore-Penrose pseudo-inverse at

iteration stage i. With w

, the statistic decision y

is obtained from the

received vector R

at stage i, and is quantized (represented by the operator

Q()) in order to obtain the k

th detected symbol.

The interference of the detected symbol is cancelled in the second step of

the algorithm by removing its interference from the received signal:

(4.24)

A new received vector R

i+1

is formed by removing from R

the estimated

transmit signal of the detected symbol. The Moore-Penrose pseudo-inverse

(represented by the + sign as superscript) is recalculated from the previous

stage transfer matrix after annulling its k

th row. The operations realized in

Equation 4.24 can be seen as redeﬁning a new system where the antenna

that used to transmit the previous detected symbol is extinguished. This is

achieved by removing the signal produced by the detected symbol and by

annulling the channel coefﬁcients related to the estimated symbol’s antenna.

The last step of the process is charged to select the optimum order for

decoding. In other words, the layers are rearranged in the process in order

to minimize the probability of error. This means ﬁnding an index k

of stage

ywR

xQy

()

= 

()

RRxH

iik

= 

()

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Introduction to Space–Time Coding 107

i. The method proposed in the algorithm is to select the antenna with the

best post-detection SNR. The post-detection SNR of each transmit antenna

j is the absolute value of the jth column of the Moore-Penrose pseudo-inverse

at stage i. If S is the optimum ordering and S = {k

, k

, … k

}, the search

for the best SNR post-detection at stage i is performed for the set {k

, …, k

since the symbols transmitted from antennas k

to k

already have been

detected in previous stages. The best post-detection SNR is the minimum

absolute value calculated:

. (4.25)

The generalized algorithm is composed of two phases, an initialization

and a recursive phase. The initialization phase consists of calculating the

Moore-Penrose pseudo-inverse:

(4.26)

The recursive phase is composed of the previously described steps:

(4.27)

4.6 Space–Time Coding with CSI Knowledge

at the Transmitter

The coding schemes described in the previous sections involve space–time

processing when no channel state information (CSI) is available at the trans-

mitter side. However, in a duplex system, it is sometimes possible to provide

jSj k k

()



{}



, ...

argmin

jSj k k

()



{}



, ...

argmin

kkk

iik

xQy

RRxH

= 

()

= 

()

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108 MIMO System Technology for Wireless Communications

CSI at the transmitter. The optimum exploitation of the CSI at the transmitter

is linear pre-ﬁlter or pre-equalization realized with the Singular Value

Decomposition (SVD) theorem, [14]. The channel matrix can be written as:

. (4.28)

In this equation, matrix U and V are composed of the eigenvectors of the

channel and D is a diagonal matrix composed of the eigenvalues of H. It

becomes possible to left multiply the transmit matrix X with V and to left

multiply the received matrix R with U

. In this case, the received signals

become:

(4.29)

The MIMO system becomes equivalent to a set of parallel SISO systems

as shown in Figure 4.8. Each SISO is ampliﬁed by an eigenvalue of D. The

capacity of the MIMO system is then the sum of distinct parallel SISO channel

capacities. The number of parallel connections is equal to the number of non-

zero eigenvalues or, in other words to the rank of H. Each SISO sub-channel

forms an eigenmode and optimal energy allocation can be applied through

the water-pouring algorithm.

FIGURE 4.8

Exploitation of CSI knowledge at the transmitter.

HUDV

= 

URU HVXN

UUDVVXN

 = 

()

= 

()



()

=  + XU N

Receiver

Transmitter

· R

V · X

HSTC V STCU

Channel

rank(H)

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Introduction to Space–Time Coding 109

The SVD method requires at least the transfer of the matrix V back to the

transmitter, and complex calculation tasks, such as diagonalization, must be

performed. There are several other methods that can sub-optimally use CSI

at the transmitter [15]. Such a method is the antenna selection method

described more analytically in [16].

Antenna selection is a simple and efﬁcient technique to exploit CSI at the

transmitter. The technique is referred to as Transmit Selection Diversity (TSD)

in [17]. The purpose of the TSD algorithm is to select the “best” subset of

transmit antennas that optimizes reception according to some sort of criteria.

Performance results of different schemes including TSD for Orthogonal

Frequency Division Multiplexing (OFDM) systems are provided in Figure

4.9. In OFDM, diversity is generally applied in the spatial and frequency

domain. In the ﬁgure, Symbol Error Rate (SER) is shown in terms of SNR

when Quadrature Amplitude Modulation with 16 constellation points

(QAM-16) is used. The lines marked with stars are Space–Frequency Block

Codes while the circles are used for the performances of Space–Frequency

Trellis Coding schemes. Plain lines represent performances when CSI is

unknown at the transmitter side for a MIMO system of 2 transmit and 2

receive antennas, i.e., (2, 2). The dotted curves shows the SER when TSD is

used and 2 transmit antennas are selected out of 3 transmit antennas. Sim-

ilarly, the dashed lines are representing the case where 2 transmit antennas

are selected out of 4 transmit antennas.

FIGURE 4.9

Performance results for different SF schemes in an OFDM system for QAM-16.

−1

−2

Symbol error rate

(

SER)

−3

−4

0 5 10

SNR

(

)

15 20 25

Space-frequency block code(2, 2)

Space-frequency trellis code(2, 2)

No transmit selection diversity(TSD)

TSD: (2, 2) out of (3, 2)

TSD: (2, 2) out of (4, 2)

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110 MIMO System Technology for Wireless Communications

4.7 Other Space–Time Coding Schemes

The space–time coding schemes presented in the previous sections are just

some of the best-known schemes for MIMO systems. It is obvious that there

are several other possible schemes. The description of all these STC tech-

niques goes beyond the scope of this chapter. In this section, we provide

some general but not exhaustive directions of STC found in the literature.

A major open research direction is the design of STC in the case where the

channel matrix cannot be determined at the receiver. In this case, differential

schemes and techniques can be used in combination with classic ST coding

methods such as STBCs.

Another class of STCs not mentioned in the previous subsections is the

extension of STTCs with serial or parallel concatenate convolutional schemes.

In particular, recursive codes that perform iterative decoding, such as turbo

codes, can be applied to MIMO systems forming a new category of STC.

Some research has also been carried out for Low Density Parity Check

(LDPC) codes.

The most common form of STC is the combination of two distinct codes,

namely an inner and an outer code. The outer codes are usually well-known

bit-encoding schemes, such as reed-solomon or bit interleaved convolutional

codes, and they are placed before the modulator in the transmission chain.

Their task is to provide coding gain to the system. The inner codes are

positioned after the modulator and operate on a symbol basis. The inner

codes such as STBCs are mainly charged to provide diversity to the system

and to perform the serial-to-parallel signal conversion. It is evident that the

names “inner” and “outer” comes from the position of the codes in the

transmission chain. This combined scheme is considered in several emerging

standards and technologies such as the IEEE 802.16 standard, which is

known to the general public by the certiﬁcation mark WiMAX.

References

1. G.J. Foschini and M.J. Gans. 1998.“On limits of wireless communications in a

fading environment when using multiple antennas,” IEEE Wireless Personal

Communications, Vol. 6, March 1998, pp. 311–335.

2. L. Zheng and D.N.C. Tse. 2003. “Diversity and multiplexing: a fundamental

tradeoff in mulitple antenna channels,” IEEE Transactions on Information Theory,

Vol. 49, No. 5, May 2003, pp. 1073–1096.

3. S.M. Alamouti. 1998. “A simple transmit diversity technique for wireless com-

munications,” IEEE Journal on Selected Areas in Communications, Vol. 16, No. 8,

October 1998, pp. 1451–1458.

4. V. Tarok, H. Jafarkhami, and A.R. Calderbank. 1999. “Space-time block codes

from orthogonal designs,” IEEE Transactions on Information Theory, Vol. 45,

No. 5, July 1999, pp. 1456–1467.

4190_book.fm Page 110 Tuesday, February 7, 2006 12:44 PM

Introduction to Space–Time Coding 111

5. V. Tarok, H. Jafarkhami, and A.R. Calderbank. 1999. “Space-time block codes

for wireless communications: performance results,” IEEE Journal on Selected

Areas in Communications, Vol. 17, No. 3, March 1999, pp. 451–460.

6. G. Ganesan and P. Stoica. 2001. “Space-time block codes: a maximum SNR

approach,” IEEE Transactions on Information Theory, Vol. 47, No. 4, May 2001,

pp. 1650–1656.

7. X.B. Liang. 2003. “Orthogonal designs with maximal rates,” IEEE Transactions

on Information Theory, Vol. 49, No. 10, October 2003, pp. 2468–2503.

8. O. Tirkkonen and A. Hottinen. 2002. “Square-matrix embeddable space-time

block codes for complex signal constellations,” IEEE Transactions on Information

Theory, Vol. 48, No. 2, February 2002, pp. 384–395.

9. A. Boariu and D.M. Ionescu. 2003. “A class of nonorthogonal rate-one space-

time block codes with controlled interference,” IEEE Transactions on Wireless

Communications, Vol. 2, No. 2, March 2003, pp. 270–395.

10. V. Tarok, N. Seshadri, and A.R. Calderbank. 1998. “Space-time codes for high

data rate wireless communication: performance criterion and code construc-

tion,” IEEE Transaction on Information Theory, Vol. 44, No. 2, March 1998,

pp. 744–765.

11. R.S. Blum. 2002. “Some analytical tools for the design of space-time convolu-

tional codes,” IEEE Transactions on Communications, Vol. 50, No. 10, October

2002, pp. 1593–1599.

12. Y. Liu, M.P. Fitz, and O.Y. Takeshita. 2002. “A rank criterion for QAM space-

time codes,” IEEE Transaction on Information Theory, Vol. 48, No. 12, December

2002, pp. 3062–3079.

13. G.D. Golden, G.J. Foschini, R.A. Valenzuela, and P.W. Wolniansky. 1999.

“Detection algorithm and initial laboratory results using the V-BLAST space-

time communication architecture,” Electronics Letters, Vol. 35, No. 1, January 7,

1999, pp. 14–15.

14. R. Horn and C. Johnson. 1985. Matrix Analysis, Cambridge, U.K.: Cambridge

University Press.

15. A.D. Valkanas and P.I. Dallas. 2003. “Adaptive modulation and coding in space-

time-frequency block and trellis for (2, 2) MIMO OFDM systems,” International

OFDM Workshop (InOWo), 2003.

16. I. Bahceci, T.M. Duman, and Y. Altunbasak. 2003. “Antenna selection for

multiple-antenna transmission systems: performance analysis and code con-

struction,” IEEE Transaction on Information Theory, Vol. 49, No. 10, October 2003,

pp. 2669–2681.

17. A.D. Valkanas. 2004. “Adaptive Space-Frequency Coding for Multiple-Input

Multiple-Output Orthogonal Frequency Division Multiplexing Systems,” Ph.D.

Dissertation, University of Alabama in Huntsville.

Acronyms

AWGN Additive White Gaussian Noise

BS Base Station

CSI Channel State Information

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112 MIMO System Technology for Wireless Communications

LDPC Low Density Parity Check

LOS Line of Sight

LST Layer Space Time

MEA Multiple Element Array

MIMO Multiple Input Multiple Output

MISO Multiple Input Single Output

ML Maximum Likelihood

MMSE Minimum Mean Square Estimation

MTMR Multiple Transmitter Multiple Receiver

NLOS Non Line of Sight

OFDM Orthogonal Frequency Division Multiplexing

OSTBC Orthogonal Space–Time Block Code

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

SER Symbol Error Rate

SF Space–Frequency

SIMO Single Input Multiple Output

SISO Single Input Single Output

SM Spatial Multiplexing

SNR Signal to Noise Ratio

ST Space–Time

STBC Space–Time Block Code

STC Space–Time Code/Coding

STF Space–Time-Frequency

STP Space–Time Process/Processing

STTC Space–Time Trellis Code

SVD Singular Value Decomposition

TSD Transmit Selection Diversity

V-BLAST Vertical Bell Laboratories Architecture Space Time

WiMAX Worldwide Interoperability for Microwave Access

ZF Zero Forcing

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113

Feedback Techniques for MIMO Channels

David J. Love and Robert W. Heath, Jr.

CONTENTS

5.1 Introduction................................................................................................ 113

5.2 Prior Work and Motivation...................................................................... 115

5.3 Limited Feedback MIMO .........................................................................119

5.3.1 System Description ....................................................................... 119

5.3.2 Channel Quantization .................................................................. 119

5.3.3 Quantized Signal Adaptation......................................................120

5.4 Quantized Signal Adaptation Algorithms.............................................121

5.4.1 Beamforming Example and Summary of Approach .................121

5.4.2 Precoded Orthogonal Space–Time Block Codes......................123

5.4.3 Precoded Spatial Multiplexing....................................................127

5.5 Effect of Spatial Correlation.....................................................................133

5.6 Feedback in Broadband Channels with MIMO-OFDM......................135

5.6.1 MIMO-OFDM System Model and Limited Feedback ............135

5.6.2 Interpolated Beamforming in MIMO-OFDM...........................136

5.6.3 Interpolated Precoding in MIMO-OFDM .................................138

5.6.4 Performance Example...................................................................139

5.7 Conclusions.................................................................................................139

References.............................................................................................................140

5.1 Introduction

Recent wireless innovations are improving bandwidth efﬁciency and reducing

the deleterious effects of fading in wireless communication channels. Mul-

tiple antennas offer high capacity, enhanced resistance to interference, and

reductions in fading thanks to diversity when arrays are used at both trans-

mitter and receiver in a MIMO (multiple-input multiple-output) conﬁguration.

Broadband signaling, such as orthogonal frequency division multiplexing

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