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84 MIMO System Technology for Wireless Communications

3.7 Spatial Capacity of Waveguide Channels

While the maximum in Equation 3.9 is difﬁcult to ﬁnd in general since one

of the constraints is a partial differential equation with an arbitrary boundary

condition, it can be found in an explicit closed form for some waveguide

channels. Consider, for example, a lossless uniform waveguide. Using Equa-

tion 3.27, we conclude that (i) the optimizations over and can be car-

ried out separately (since they are independent of each other) and (ii) the

optimization over is equivalent to optimization over (since the expan-

sion coefﬁcients determine the ﬁeld uniquely). When the Tx does not know

the channel, is i.i.d. complex Gaussian because (i) the channel is AWGN

and Gaussian distribution maximizes the entropy, and (ii) the lack of the

channel knowledge at the Tx forces the covariance of to be the identity

matrix, i.e., no “preferred direction” in the eigenspace (see [2] for more

details), and the capacity is given by Equation 3.1 [2].

Further optimization of Equation 3.9 over subject to results

in and the channel correlation matrix , i.e., all the modes

are independent and carry the same power. Hence, all the capacity results

above give, in fact, the spatial capacity, i.e., the maximum capacity limited

by the laws of electromagnetics. One may say that the spatial capacity con-

cept encompasses the limits in information transmission due to both the

information theory and electromagnetics.

When there is mode coupling and, additionally, different modes experience

different attenuation (e.g., lossy waveguide with different L

), one has to

FIGURE 3.12

Capacity in a rectangular cavity for a = 4Q, b = 2Q and )f/f

= 0.2.

1 2 3 4 5 6 7 8 9 10 11

100

120

140

2D array

3D array

Exact

Approximate

Limit(36)

Capacity,

it/s/Hz

c/λ

p()xE

E F

pCNPn

() (, / )x0 I=

tr n

()GG

= 1

RGG I==

4190_book.fm Page 84 Tuesday, February 21, 2006 9:14 AM

Information Theory and Electromagnetism: Are They Related? 85

consider generic correlation matrix . The optimum power allocation in

the mode eigenspace can be found by applying the water-ﬁlling solution

[14] to the eigenvalues Q

of R,

, (3.45)

where and 0 otherwise, and is chosen to satisfy the power

constraint,

. (3.46)

is large for large eigenvalues, and small or even zero (i.e., no transmis-

sion on the eigenmode) for small Q

. Without loss of generality, we further

assume that Q

v Q

v … v Q

. Two important conclusions follow from

Equation 3.45 and Equation 3.46 [14]. In the large SNR mode, ,

all the eigenmodes carry approximately the same power (F

~ 1). We also

note that the same solution applies when all the eigenvalues are equal, Q

, regardless of the SNR. In the small SNR regime, W f 1/Q

–

1/Q

, all the power is allocated to the largest eigenmode, F

= n

, .

From this, we conclude that the traditional transmission strategy (i.e., using

a single dominant mode only) is optimum in the small SNR regime only.

For large SNR, the best strategy is to use all the modes. In all the other cases,

between these two extremes, several dominant modes should be used, the

exact number being determined by the available SNR and by the eigenvalues

Remarkably, this insight requires a joint application of electromagnetic and

information-theoretic techniques. It is expected that an efﬁcient design of

eigenmode modulators and demodulators (see Figure 3.7) will also call for

signiﬁcant uniﬁcation of these two presently unrelated areas of research.

3.8 Acknowledgments

The authors would like to acknowledge numerous insightful discussions

with C. Charalambous, A. Kouki, A. Molisch, I.E. Telatar, D. McNamara and

also the colleagues within the European projects COST 273 and 284.

RI|

= 





()xxx

= vif 0 S







WQ>> n

QFq = 

i= |01

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86 MIMO System Technology for Wireless Communications

References

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ciency of BLAST Architecture, AP2000 Millennium Conference on Antennas &

Propagation, Davos, Switzerland, April 9–14, 2000.

10. S. Loyka. 2002. MIMO Channel Capacity: Electromagnetic Wave Perspective,

27th General Assembly of the International Union of Radio Science, Maastricht,

the Netherlands, Aug. 17–24, 2002.

11. S.L. Loyka. 2003. On MIMO Channel Capacity, Spatial Sampling and the Laws

of Electromagnetism, the 3rd IASTED International Conference on Wireless

and Optical Communications (WOC 2003), Banff, Alberta, Canada, July 14–16,

2003, pp. 132–137.

12. S.L. Loyka. 2003. “Multi-antenna capacities of waveguide and cavity channels,”

IEEE CCECE’03, Montreal, May 2003.

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System Aspects in Wireless Communications, Chalmers University of Technol-

ogy, Gothenburg, Sweden, June 2004.

14. S.L. Loyka. 2005. “Multi-antenna capacities of waveguide and cavity channels,”

IEEE Trans. Vehicular Technology, Vol. 54, No. 3, May 2005, pp. 863–872.

15. E.D. Rothwell and M.J. Cloud. 2001. Electromagnetics, Boca Raton, FL: CRC

Press.

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17. J.D. Gibson, ed. 2002. The Communications Handbook, Boca Raton, FL: CRC Press.

18. S.R. Saunders. 1999. Antennas and Propagation for Wireless Communication Sys-

tems, Chichester, U.K.: Wiley.

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Information Theory and Electromagnetism: Are They Related? 87

19. A.J. Jerry. 1977. The Shannon sampling theorem — its various extensions and

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tial correlation matrix,” IEEE Communication Letters, Vol. 5, No. 9, Sep 2001,

pp. 369–371.

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capacity,” IEEE ICASSP 2003.

25. J.W. Wallace and M.A. Jensen. 2002. “Intrinsic capacity of the MIMO wireless

channel,” IEEE VTC (Fall), Sept. 24–28, 2002.

26. R.C. Johnson. 1993. Antenna Engineering Handbook, New York: McGraw Hill.

27. D.H. Johnson and D.E. Dudgeon. 1993. Array Signal Processing, Upper Saddle

River, NJ: Prentice Hall.

28. J.W. Wallace and M.A. Jensen. 2004. “Mutual coupling in MIMO wireless sys-

tems: a rigorous network theory analysis,” IEEE Trans. Wireless Commun., Vol. 3,

No. 4, July 2004, pp. 1317–1325.

29. V. Jungnickel, V. Pohl, and C. von Helmolt. 2003. “Capacity of MIMO systems

with closely spaced antennas,” IEEE Communications Letters, Vol. 7, No. 8, Aug.

2003, pp. 361–363.

30. R. Janaswamy. 2002. “Effect of element mutual coupling on the capacity of

ﬁxed length linear arrays,” IEEE Antennas and Wireless Propagation Letters, Vol. 1,

2002, pp. 157–160.

31. P.S. Kildal and K. Rosengren. 2004. “Correlation and capacity of MIMO systems

and mutual coupling, radiation efﬁciency, and diversity gain of their antennas:

simulations and measurements in a reverberation chamber,” IEEE Communica-

tions Magazine, Vol. 42, No. 12, Dec. 2004, pp. 104–112.

32. M.K. Ozdemir, H. Arslan and E. Arvas. 2003. “Mutual coupling effect in multi-

antenna wireless communication systems,” IEEE Globecom., pp. 829–833.

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4190_book.fm Page 87 Tuesday, February 21, 2006 9:14 AM

Introduction to Space–Time Coding

Antonis D. Valkanas and Alexander D. Poularikas

CONTENTS

4.1 Introduction..................................................................................................89

4.2 MIMO System and Space–Time Coding..................................................90

4.2.1 MIMO System and Capacity.........................................................90

4.2.2 Methodologies and Diversity........................................................92

4.3 Space–Time Block Codes............................................................................94

4.3.1 Alamouti’s Transmit Technique....................................................94

4.3.2 Orthogonal Space–Time Block Codes..........................................96

4.4 Space–Time Trellis Codes...........................................................................98

4.4.1 Encoding and Decoding.................................................................98

4.4.2 Performance Analysis and Code Design...................................101

4.5 Spatial Multiplexing..................................................................................103

4.5.1 V-BLAST Algorithm......................................................................105

4.6 Space–Time Coding with CSI Knowledge at the Transmitter ...........107

4.7 Other Space–Time Coding Schemes....................................................... 110

References............................................................................................................. 110

Acronyms............................................................................................................. 111

4.1 Introduction

The explosion of wired and wireless telecommunication systems marked the

end of the second millennium. The number of subscribers to the Internet or

to a second-generation mobile network has increased exponentially. Hence,

the users of both systems have augmented expectations in services and

capacity. For wireless communication systems, a novel direction proposed

to resolve capacity requirements in the challenging radio environment is the

exploitation of Multiple Element Array (MEA) at both transmitter and

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90 MIMO System Technology for Wireless Communications

receiver sides. Wireless systems with antenna elements at both edges are

referred to in the literature as Multiple Input Multiple Output (MIMO)

systems in contrast to Single Input Single Output (SISO) antenna systems

that have one transmit and one receive antenna. Following this classiﬁcation,

a system with one transmit antenna and several receive antennas is denoted

as Single Input Multiple Output (SIMO) system while several transmit anten-

nas and one receive antennas designate a Multiple Input Single Output

(MISO) system. In order to include the case where the transmitter and/or

receiver antenna elements are not located in the same devices, MIMO sys-

tems are referred to as Multiple Transmitters Multiple Receivers (MTMR).

In this chapter, we introduce some general concepts about the signal pro-

cessing used in MIMO systems, which is commonly called Space–Time Coding

(STC) or Space–Time Processing (STP). In the ﬁrst part of this chapter, we

describe the MIMO system model and formulate the general expression of

the capacity of a MIMO system. We also present exploitation methodologies

of this capacity and a classiﬁcation of STCs. The next subsection describes

the simplest class of STCs, i.e., Space–Time Block Codes (STBCs). Space–Time

Trellis Codes (STTCs) that provide diversity and coding gain at the same

time are analyzed next. Then, spatial multiplexing methods are tackled

through the description of the Vertical Bell Laboratories Architecture

Space–Time (V-BLAST) algorithm. Exploitation of the channel capacity in

the case where channel state information is available at the transmitter side

is brieﬂy introduced in the following subsection. We conclude this chapter

by mentioning different interesting space–time coding schemes that appear

in the literature but which detailed description surpasses the scope of this

chapter.

4.2 MIMO System and Space–Time Coding

4.2.1 MIMO System and Capacity

A MIMO system with M

transmit antennas and M

receive antenna is

shown in Figure 4.1. From the ﬁgure, it seems that space–time coding is

charged to perform serial-to-parallel and parallel-to-serial conversion of the

transmitted information data. The block responsible for the serial-to-parallel

transformation at the transmitter is the Space–Time Encoder. The opposite

operation, i.e., the parallel-to-serial conversion, is performed in the Space–

Time Decoder at the receiver side. However, space–time coding is more than

these transformations, since these codes can realize at the same time several

communication processes such as channel encoding/decoding, modulation/

demodulation, multiplexing/demultiplexing or equalization.

The transmitted symbols are noted by x

, with the subscript index i

taking

integers with non-zero values up to M

. Similarly, the received signals are

4190_book.fm Page 90 Tuesday, February 7, 2006 12:44 PM

Introduction to Space–Time Coding

marked by r

with j

taking values from 1 to M

. Based on this notation, the

channel transfer function from the i

th transmit antenna to the j

th receive

antenna are denoted by h

j,i

. In order to have a MIMO system comparable to

a SISO system, the sum of the transmitted powers of all antennas M

must

be equal to the transmitted power of a SISO system denoted P

. Hence, the

transmitted power from each antenna is P/M

. Additionally, the normaliza-

tion of the channel coefﬁcient must be such that at each receive antenna j

the sum of channel gains of all transmit antennas to receive antenna j

equal to the number of transmit antennas, i.e., M

. The Additive White

Gaussian Noise (AWGN) components are added to each receive antenna and

are designated by n

, with j

being an index taking values from 1 to M

. The

AWGN components are assumed identically and independently distributed.

The MIMO system can be described in a matrix form, using the previous

notations, as follows:

(4.1)

FIGURE 4.1

Block diagram of a MIMO system.

1,1

2,1

2,2

1,M

2,M

Space-time

encoder

Space-time

decoder

rhxn

=  +



hhxn

RHXN

Mi i

 +

¡ =  +

hh h

11 12 1

21 2







,, ,

,,222







hh h

MM MM

RR RT

,, ,









½½

4190_book.fm Page 91 Tuesday, February 7, 2006 12:44 PM

92 MIMO System Technology for Wireless Communications

In Equation 4.1, R

is a column vector M

composed of the received

signals r

, X

is a column vector M

composed of the transmit constellation

points x

, N

is a column vector M

composed of the noise components

, and H

is a M

matrix with ji

th component being the channel coef-

ﬁcient h

j,i

It has been demonstrated in [1] that MIMO systems provide tremendous

capacity. When Channel State Information (CSI) is not available at the trans-

mitter, the capacity of a MIMO system expressed in bits per second per hertz

(bps/Hz) can be written as:

(4.2)

In Equation 4.2, I

is the identity matrix of size M

×M

, H

is the channel

matrix of size M

with H

being its transpose conjugate, and W

gives

the average Signal-to-Noise Ratio (SNR) per receiver branch independent of

the number of transmitting antennas M

4.2.2 Methodologies and Diversity

Two methodologies have been developed to exploit the capacity offered by

the presence of several transmit and receive antennas in a telecommunication

system.

The ﬁrst method exploits the additional diversity of multiple antennas,

namely spatial diversity, to combat channel fading. This method is perfor-

mance oriented. It targets improving the reliability of the link, which can be

achieved by the transmission and reception of several replicas of the same

information through independent fading paths and, hence, reduces the prob-

ability of simultaneous signal fades. The provision of replicas of the same

information at the receiver is referred to as diversity. The number of inde-

pendent receptions of the same information at the receiver is deﬁned as the

“diversity order” or the “diversity gain” of the system. In a MIMO system,

the transmit and receiver antennas form M

×M

independent radio links

and, hence, can provide a maximum or full diversity gain equal to this

product. Diversity techniques are particularly interesting in the case of a

severely rich fading environment, but the targeted transmitted rate is equiv-

alent to that of a SISO system. In other words, the additional antennas of

the MIMO system are used to support the transmission of a SISO system.

Diversity methods are traditionally used in Base Stations (BS). In the down-

link, the BS transmits from two or more antennas, while in the uplink the

BS receives information via several receive antennas. The diversity approach

is particularly important for systems having a relatively small number of

transmit antennas that operate at low SNR values. A major drawback of a

=+



log det bps .

/Hz

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Introduction to Space–Time Coding

MIMO system is that the transmitted signals from distinct antennas must be

decorellated, and hence, the antenna elements must be sufﬁciently separated.

It has been shown in the literature that the spacing between antenna elements

must exceed half the wavelength of the transmitted signals. In practice, the

spading exceeds by three and even ten times the signal’s wavelength. There-

fore, the diversity schemes are popular to mobile/portable devices that have

size limitations.

In MIMO systems that target maximizing the transmission and reception

diversity, the channel coding is named according to the domains where

diversity is applied. Usually, diversity is applied to two or more domains.

Hence, we have Space–Time (ST) coding, Space–Frequency (SF) coding and

Space–Time–Frequency (STF) coding. Those coding techniques can be visu-

alized in the three-dimensional plane as shown in Figure 4.2.

The methodology described previously is suboptimal for MIMO systems

with a large number of transmit antennas when operating at high SNR

regimes. Since the capacity of a MIMO system increases according to the

number of antennas, this capacity augmentation can be exploited differently

for MIMO systems that have no limitations on the number of antennas. There

are other methods to exploit the capacity of a MIMO system, which are

throughput oriented. In this case, the target of the system is to transfer the

maximum possible information data. These techniques are particularly inter-

esting for the case of a Line of Sight (LOS) environment, where the channels

have practically no fading. This approach is known as Spatial Multiplexing

(SM) or Layer Space–Time (LST). The number of extra degrees of freedom

available for communication in a MIMO system is then equal to the mini-

mum between the number of transmit antennas and the number of receive

antennas.

Several efforts have been made recently in the literature to combine the

two methods described in this section. Some hybrid encoding schemes have

been suggested. A tradeoff between diversity and multiplexing has been

presented in [2].

FIGURE 4.2

Diversity exploitation in MIMO systems in three domains.

Time

Frequency

Spatial

Time

Frequency

Spatial

Time

Frequency

Spatial

4190_book.fm Page 93 Tuesday, February 7, 2006 12:44 PM