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144 MIMO System Technology for Wireless Communications

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147

Antenna Selection in MIMO Systems

Neelesh B. Mehta and Andreas F. Molisch

CONTENTS

6.1 Introduction................................................................................................147

6.2 MIMO System Model................................................................................149

6.2.1 General Notation...........................................................................150

6.3 Spatial Multiplexing..................................................................................151

6.3.1 Optimum Signaling with Transmit Antenna Selection...........151

6.3.2 Performance of Antenna Selection.............................................152

6.3.3 Antenna Selection in Presence of Interference.........................154

6.4 Space–Time Codes .....................................................................................155

6.4.1 Space–Time Trellis Codes.............................................................155

6.4.2 Orthogonal Space–Time Block Codes........................................157

6.5 SIMO Systems ............................................................................................159

6.6 Implementing Antenna Selection: Criteria and Algorithms...............161

6.6.1 Capacity-Achieving Spatial Multiplexing.................................162

6.6.1.1 When Transmitter Has CSI ...........................................164

6.6.1.2 With Linear Receivers ....................................................164

6.6.2 Space–Time Codes.........................................................................165

6.7 Performance with Non-Idealities............................................................166

6.8 Antenna Selection with Spatial Correlation..........................................167

6.8.1 Alternate Antenna Selection Structures.....................................168

6.9 Summary.....................................................................................................170

References.............................................................................................................171

6.1 Introduction

MIMO systems, which employ multiple transmit and receive antenna ele-

ments, substantially improve the data rates that can be transmitted over the

channel and the reliability with which they can be received without any

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148 MIMO System Technology for Wireless Communications

additional bandwidth. Higher data rates are achieved by transmitting mul-

tiple data streams simultaneously using spatial multiplexing techniques. For

spatially uncorrelated channels, the data rates even increase linearly with

the minimum of the number of transmit and receive antenna elements [1].

Increased reliability is achieved by exploiting spatial diversity to signiﬁ-

cantly reduce the probability that the channel is in a deep fade. Orthogonal

space–time block codes and space–time trellis codes are examples of diver-

sity techniques tailored to MIMO systems. A single input multiple output

(SIMO) system, which combines the many received copies of the transmitted

signal to improve reliability, is another example of a spatial diversity system.

While MIMO systems perform impressively, they also increase the hard-

ware and signal processing complexity, power consumption, and component

size in the transmitter and the receiver [2]. One of the main culprits behind

this increase in complexity is that each receive antenna element requires a

dedicated radio frequency (RF) chain that comprises a low noise ampliﬁer,

a frequency down-converter, and an analog-to-digital converter, and each

transmit antenna element requires an RF chain that comprises a digital-to-

analog converter, a frequency up-converter, and a power ampliﬁer. More-

over, processing the signals received in spatial multiplexing schemes or with

STTCs calls for sophisticated receivers whose complexity increases, sometimes

exponentially, with the number of transmit and receive antenna elements.

This increase in complexity has inhibited the widespread adoption of

MIMO systems. For example, the third-generation cellular system speciﬁca-

tion (3GPP) currently supports only an optional two antenna space–time

transmit diversity scheme and does not require the handsets to have more

than one antenna element [3]. Sophisticated techniques that employ spatial

multiplexing or support more antenna elements have met with considerable

opposition in 3GPP. While the adoption of MIMO has made headway in the

next-generation wireless local area network (WLAN) standard IEEE 802.11n,

which aims to transfer raw information at rates greater than 100 Mbps over

a 20-MHz bandwidth, complexity considerations are likely to make the

adopted MIMO scheme an elementary one that supports a small number of

antenna elements.

Antenna selection is a solution that addresses some of the complexity

drawbacks associated with MIMO systems. It reduces the hardware com-

plexity of transmitters and receivers by using fewer RF chains than the

number of antenna elements. While the antenna elements are typically cheap,

and in some cases are just a patch of copper, the RF chains are considerably

more expensive. In antenna selection, a subset of the available antenna

elements is adaptively chosen by a switch, and only signals from the chosen

subset are processed further by the available RF chains.

Given its promise as a low-complexity solution, antenna selection has

received considerable attention recently. It has been considered at the trans-

mitter (transmit antenna selection [TAS]), at the receiver (receive antenna

selection [RAS]), and at both the transmitter and the receiver (transmit and

receive antenna selection [T-RAS]). Its performance has been explored from

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Antenna Selection in MIMO Systems 149

various angles such as capacity and outage for spatial multiplexing systems,

and diversity order and array gain for space–time coded systems. The diver-

sity order quantiﬁes the effectiveness in avoiding deep fades and is deﬁned

as the slope of average symbol error probability vs. input signal to noise

ratio (SNR) curve for high SNRs. The array gain quantiﬁes the improvement

in average SNR seen at the combiner output when signals received by the

multiple antenna elements are combined.

As we shall see, antenna selection — for a variety of MIMO techniques —

achieves the full diversity inherent in the system at the expense of a small

loss in array gain compared to a full complexity system, i.e., a system that

can always allocate RF chains to each and every antenna element. Another

way of stating this is that for the same number of RF chains, using additional

antenna elements with antenna selection outperforms a system that lacks

additional antenna elements. Antenna selection has been found to modify,

on a fundamental level, the optimum Gaussian signaling required to transmit

information at the maximum possible rate. Considerable effort has also been

spent to develop various criteria, both optimal (but complicated) and sub-

optimal (but simple), to implement antenna selection algorithms.

This chapter studies various aspects of antenna selection in some depth,

and assumes basic knowledge about the operation of MIMO systems. The

reader is referred to the tutorial papers in [2,4] for a more basic introduction.

We ﬁrst analyze the performance of antenna selection for different MIMO

systems. We then consider the algorithms and criteria to implement antenna

selection. We also consider antenna selection performance in the presence

of non-idealities, such as imperfect channel estimates, multipath dispersion,

fast fading, etc. Finally, we discuss a class of antenna selection algorithms

that are based on knowledge of the statistical variations of the channel. Using

statistical information instead of the instantaneous channel state information

often signiﬁcantly reduces the demands on the transmitter and receiver. We

show that even a moderate spatial correlation, when properly exploited by

antenna selection, helps recover, to a great extent, the array gain loss that is

incurred by antenna selection.

In this chapter, we will focus most of our attention on the simplest setup

in which the receiver has full channel state information (CSI) while the

transmitter has no CSI, and knows only which antenna elements to use in

case of TAS and T-RAS. However, we do brieﬂy discuss how other scenarios,

such as the availability of CSI at the transmitter and the presence of co-

channel interference in addition to noise, can be handled.

6.2 MIMO System Model

Let and denote the number of antenna elements that are available (but

not necessarily used every time) at the transmitter and receiver, respectively.

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150 MIMO System Technology for Wireless Communications

Let and denote the number of RF chains at the transmitter and receiver,

respectively. We always have 1 f L

f N

and 1 f L

f N

. A block diagram

representation of antenna selection at the transmitter and the receiver is

given in Figure 6.1.

The transmitted signal, of size N

× 1, is denoted by x. The signal, y, of size

× 1, received by the antenna elements is given by

(6.1)

where n represents noise and the matrix H, of size N

× N

, denotes the

instantaneous channel state. Unless otherwise mentioned, the elements of H

are assumed to be independent of each other. In case of TAS, N

elements

of x, which correspond to the transmit antenna elements not chosen, will be

0. The noise vector, n, of size N

× 1, is a zero-mean complex white Gaussian

random vector. To simplify notation, we shall use x to also denote the L

× 1

transmit vector and y to denote the L

× 1 receive vector that correspond to

only the selected transmit and receive elements, respectively. The notation

shall be obvious from the context. The total power transmitted from all the

antennas is denoted by W. Without loss of generality, we shall assume that

each element of n and H has unit variance. Therefore, the average SNR of

the signal input to any receive antenna element equals W/L

when the power

is equally allocated among the L

transmit elements.

We shall assume that the receiver has perfect channel state information.

The receiver can estimate the channel from the training sequences or by

using blind estimation techniques. A frequency-ﬂat channel is assumed. The

channel is assumed to be block-fading, in that it does not change during

the transmission of the space–time codeword and decorrelates thereafter. We

shall comment on the impact of the breakdown of these assumptions toward

the end of the chapter.

6.2.1 General Notation

For a matrix A, the symbol A

†

denotes the Hermitian transpose of A, A

the

transpose, A

–1

the matrix inverse, the Frobenius norm, the determinant,

FIGURE 6.1

Block diagram of antenna selection with L

out of N

transmit antenna elements selected and

out of N

Data

source

Data

sink

Space/time

decoder

out

of N

switch

Mobile

radio

channel

out

of N

switch

Space/time

coder

− 1

yHxn=+,

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Antenna Selection in MIMO Systems 151

and the trace. The identity matrix of size shall be denoted by

. The symbol denotes the expectation with respect to the random

variable X.

As we shall see, order statistics, which deals with the statistics of ordered

random variables, is often used in the analysis of antenna selection. We shall

resort to the following commonly used order statistics notation. When n

numbers, x

, …, x

, are rearranged in descending order, they shall be denoted

by , where [i] denotes the index of the ith largest number.

6.3 Spatial Multiplexing

As mentioned, we shall ﬁrst focus on the case where the transmitter has no

knowledge of the channel state H. (In the case of TAS, the transmitter knows,

in addition, which antenna elements to use.) This is typically achieved by

means of feedback from the receiver in frequency division duplex systems.

In time division duplex systems with slowly varying channels, the transmit-

ter can monitor the transmissions from the receiver to determine the channel

state without any help from the receiver.

The maximum data transmission rate of any communication system is

given by the Shannon capacity [5]. For MIMO systems, the Shannon capacity

given the channel state H takes the form [1]

(6.2)

where is the covariance matrix (of size N

× N

) of the transmitted signal.

The ergodic capacity is deﬁned as the average of over all the channel

realizations. We ﬁrst discuss the optimal form of for antenna selection.

6.3.1 Optimum Signaling with Transmit Antenna Selection

When no CSI is available at the transmitter, it is well known that in a full

complexity system it is optimal to allocate the transmit powers uniformly

across all the antenna elements and to keep the signals transmitted from

different antenna elements uncorrelated, i.e., [6]. In antenna

selection, however, this is no longer optimal. This was ﬁrst highlighted by

an example in [7], which showed that at low SNR, a covariance matrix of

the form can outperform . Here, 1 is a vector of size

× 1 and is given by 1 = [11

…

. . Notice that a covariance of the form 11

†

also treats all the transmit antenna elements equally and requires no CSI at

the transmitter. That is not optimal is because the channel is no

longer circularly symmetric Gaussian in the presence of antenna selection,

Tr A

{}

nn×

xx x

n[] [] [ ]12

vvv

() log ,

†

HIHQH=+

C()H

†

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152 MIMO System Technology for Wireless Communications

which, depending on the channel realization, changes which transmit

antenna elements are chosen.

In fact, the optimum signaling takes the following simple form for all SNR

[8]:

(6.3)

where F is a real number that lies between 0 and L

/(L

– 1). However, closed-

form expressions are not available for F, and a one-dimensional numerical

search is required to determine its optimal value.

A large number of papers in the literature do not take the above optimal

signaling into account and assume that . This is equivalent to

setting F = 1 in the above equation. We shall also make this assumption

henceforth.

6.3.2 Performance of Antenna Selection

When antenna selection is employed at the transmitter or the receiver or

both, a subset of the transmit and/or receive antenna elements is chosen.

The channel seen by the subset is the sub-matrix, , that is obtained by

selecting only the rows and columns of H that correspond to the selected

receive and transmit antenna elements. The optimal subset is one that leads

to the largest mutual information between the antenna elements used for

transmission and reception. The capacity with antenna selection is given by

(6.4)

where is an L

× L

matrix obtained by removing columns and N

–

rows from H and denotes the set of all possible sub-matrices

of H. The optimal channel subset,

opt

, is given by

(6.5)

An exact analytical solution for is quite difﬁcult. However, lower and

upper bounds are available for for RAS. An upper bound was derived

for in [9] by assuming that the signal components are received without

any interference from the other components. The receive antenna elements

that receive the largest signal components are then selected. Clearly,

neglecting the interference from other components is an unrealistic assump-

tion, which is why the formula below is an upper bound:

=+

FF(),

†

111



sel

=+max log ,

()

†

IHH





H NL



S()



H LL



HIHH

opt

=+.argmax log

()

†

sel

4190_book.fm Page 152 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 153

(6.6)

where are ordered chi-square random variables with 2N

degrees of

freedom such that . The bound is tight for L

f N

, but is

rather loose for . For the latter case, the following bound is better:

(6.7)

where

and, for each j,

are N

ordered chi-square random variables (ordered with respect to index

i) with 2 degrees of freedom.

Using matrix analysis, the following lower bound for the case L

= N

was derived in [10]:

(6.8)

where is the orthonormal basis for the column space of and

is the block of corresponding to receive antenna elements chosen.

As is a sub-matrix of the unitary matrix U, we always have .

The capacity loss term has been characterized in detail in [10].

The performance of RAS as a function of L

for N

= 3 and N

= 8 for W =

20 dB is shown in Figure 6.2. The ﬁgure plots the cumulative distribution

function (CDF) of the capacity (in bits/sec/Hz). The CDF is a useful repre-

sentation because it contains information about not just the ﬁrst moment

(expected value), but also its higher moments. It can be seen that the capacity

increases as the number of RF chains, L

, increases. However, the gains

diminish as L

increases. For L

= 5, the capacity CDF is already very close

to that of a full complexity receiver.

sel

f +

1log ,

[]

{}

[]

LL L

[] [] [ ]12

>>>…

sel

] ,

]

log

[]

()



[]

()

{}

sel

v ++log log ,

††

IHHUU

U NN

× U

log

†

0UU

log ||

†

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