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154 MIMO System Technology for Wireless Communications

TAS can even lead to “increased capacity” for low rank channels [11],

which arise due to keyhole effects [12] or due to spatial correlation between

the transmit and receive antenna elements [13]. This increase is a manifes-

tation of the additional CSI available at the transmitter about which antennas

to use; a transmitter without this additional information would have allo-

cated equal powers to all the N

elements.

6.3.3 Antenna Selection in Presence of Interference

So far, we have assumed that the noise is complex white circular Gaussian.

In cellular systems, the transmissions from other cells or from within the cell

give rise to co-channel interference, i. The received signal now takes the form

(6.9)

Let denote the covariance of the interference plus noise component.

The capacity formula for antenna selection changes to

(6.10)

It turns out that this case can be easily mapped to the white Gaussian noise

case that we have considered so far. By rearranging the terms in the deter-

minant above, it can be shown that that the scenario above is equivalent to

a system with an interference-free MIMO channel [14]. The equivalent chan-

nel takes the form where U

is the eigenspace of Q

and 1

a diagonal matrix of the eigenvalues of Q

. When the interference is not

Gaussian, the above formula provides a lower bound on the achievable

capacity as Gaussian interference is the worst form of interference [5].

We shall not delve into the case where the signaling of the interferers can

also be adjusted to maximize global system capacity [15].

FIGURE 6.2

Capacity of a spatial multiplexing MIMO system with receive antenna selection for different

= 3, N

= 2 L

= 3

= 8

cdf(C)

Capacity C(bits/s/Hz)

0.5

yHxni=++.

rXI

sel

=+.



max log

()

†

IHQHQ



HUH



†

4190_book.fm Page 154 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 155

6.4 Space–Time Codes

Space–time codes were presented in a series of seminal papers by Tarokh

et al. [16,17]. They provide a practical means of exploiting spatial diversity

when multiple transmit and receive antenna elements are used. We shall

study antenna selection for two different kinds of space–time codes: space–

time trellis codes and orthogonal space–time block codes, which offer differ-

ent trade-offs between performance and complexity. The reader is referred

to Chapter 4 for a review of space–time codes.

6.4.1 Space–Time Trellis Codes

Space–time trellis codes (STTC) [16,18] are designed to exploit the spatial

diversity present in MIMO systems. Compared to orthogonal space–time

block codes, which we discuss in the following section, STTCs also deliver

a coding gain. However, the receiver for STTCs is more complicated, with

the complexity increasing exponentially with the number of transmit

antenna elements.

The performance of STTCs with RAS and a maximum likelihood detector

(MLD) was studied in [19]. For STTCs, the expression for the pair-wise error

probability of codewords of a given length provides considerable insight

into the performance of antenna selection. Performance measures such as

the overall error probability of the STTC may then be obtained from the pair-

wise error probabilities using classical approaches for trellis codes [20].

Let the ith antenna element transmit a codeword of length denoted by

. In this notation, the vector (of size l × 1) is transmitted

from the ith antenna element. Given that the receiver knows the channel H

perfectly, the probability that a full complexity receiver erroneously decodes

C to the codeword depends on the Euclidean distance

between the codewords. It is upper bounded by [16]

(6.11)

where is the jth row of H and corresponds to the jth receive antenna

element. The matrix B (of size N

× N

) depends on the codewords, with its

(p, q)

element given by . Note that the non-negative

term

Cc c=,,[]

…

Ee e=,,[]

…

()exp ,

†

HhBhf

Bcece

pq p p q q

= ()()

†

hBh

†

4190_book.fm Page 155 Tuesday, February 7, 2006 12:44 PM

156 MIMO System Technology for Wireless Communications

corresponds to the distance between the codewords.

Ideally, the optimal antenna selection algorithm would choose the L

receive antenna elements with indices i

, …, i

that maximize the sum

However, the indices need not be the same for different pairs of codewords.

However, at high SNR, it is sufﬁcient to only consider codewords that are

separated by the minimum distance of the code because they largely deter-

mine the probability of decoding error. For such pairs, it can be argued that

the norm-based selection criterion that chooses the antenna elements based

on the received signal strength effectively achieves this selection [19]. We

shall discuss this further in Section 6.6.2.

Therefore, the pair-wise error probability with RAS can be written as

(6.12)

(6.13)

The second step follows from the following simple inequality that holds for

any sequence of n numbers x

, x

, …, x

(6.14)

By averaging all the realizations of the channel, H, it can be shown that,

for large W, the average pair-wise error probability for a full rank STTC takes

the form

(6.15)

Notice that the exponent of W is N

and is independent of L

. This shows

that RAS retains the full diversity order N

. However, compared to a full

complexity system (with N

receive antenna elements and N

RF chains) this

is accompanied by an array gain loss of 10 log

). Tighter bounds on

the array gain loss are also derived in [19] for the case when only one receive

antenna element is selected (L

= 1).

hBh

†

sel

()exp ,

†

HhBhf

f .

exp

†

hBh

xx x mn

== =

¨¨ ¨

ff ff.

11 1

[]

,( )

EHB

sel

()



4190_book.fm Page 156 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 157

Figure 6.3 plots the pair-wise error probability as a function of SNR, W, for

2 transmit antennas for different numbers of receive antennas and RF chains.

It can be seen that when N

= 3, L

= 1, 2, and 3 cases have the same diversity

order, which is the slope at high SNR. Moreover, compared to the L

= N

3 case, the array gain losses equal 0.9 dB for L

= 2 and 3.4 dB for L

= 1.

Both the losses are considerably less than the respective upper bound values

of 1.76 dB and 4.77 dB.

TAS for STTCs was studied in [21]. The analysis turns out to be very similar

to the one given above for RAS. TAS also achieves the full diversity order

of the system, N

, and incurs an array gain loss that is upper bounded by

10 log

). That TAS achieves the full diversity order while transmitting

from only L

antenna elements is noteworthy. This implies that smaller STTCs

can be used in conjunction with antenna selection to achieve the full diversity

order. This circumvents the exponential increase in the number of trellis

states in STTCs that are designed for larger N

6.4.2 Orthogonal Space–Time Block Codes

Orthogonal space–time block codes (OSTBC), introduced in [17], achieve full

diversity and yet require only a simple linear receiver. They can be optimally

decoded by a low-complexity linear receiver.*

FIGURE 6.3

Pair-wise error probability of STTC with N

= 2 for different L

and N

as a function of SNR, W

* Strictly speaking, the receiver should be called a widely linear receiver as it involves a conju-

gation operation [22].

(dB)

= 3, N

= 3

= 2, N

= 3

= 1, N

= 3

= 2, N

= 2

= 1, N

= 1

Frame error rate

−3

−2

−1

6 8 10 12 14 16 18

4190_book.fm Page 157 Tuesday, February 7, 2006 12:44 PM

158 MIMO System Technology for Wireless Communications

Therefore, antenna selection with OSTBCs has received considerable atten-

tion in the literature. In OSTBCs, the receive SNR after linear processing can

be expressed in terms of the Frobenius norm of the matrix, which is simply

the square root of the sum of the squares of the amplitudes of all the elements

of the matrix. This simple form for the SNR greatly facilitates analysis. The

probability of bit error or codeword error can then be easily written in terms

of the SNR depending on the modulation and channel coding scheme under

consideration. TAS and RAS have been studied in [23]. The analysis for RAS

was generalized to cover several cases in [24]. We illustrate the analysis for

RAS below.

Given that the channel subset is selected, the SNR, L

sel

, of the received

data stream can be directly written in terms of the Frobenius norm, , of

[23] as follows:

(6.16)

where n

, …, n

and m

, …, m

denote the indices of the chosen receive and

transmit antenna elements, respectively.

Maximizing the instantaneous SNR is equivalent to maximizing the

mutual information or minimizing the BER. The optimal selection criterion

for all of these is therefore:

(6.17)

The above simple criterion is often referred to as the norm-based criterion.

In other words, in the case of TAS, the transmitter chooses the antenna

elements that correspond to the L

columns with the largest column-norms.

In the case of RAS, the receiver chooses antenna elements that correspond

to the L

rows with the largest row-norms. However, in the case of T-RAS,

transmit and receive antenna selection cannot be done independently.

As was the case for STTCs, the analyses for RAS and TAS are very similar.

We illustrate the performance of RAS below assuming that uncoded BPSK

is used [24]. The bit error probability is given in terms of the Q-function as

(6.18)

(6.19)



sel

¨¨





opt

=.argmax

()SH

sel

()

[]

H =

¨¨

ºº

f .

¨¨

4190_book.fm Page 158 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 159

For large W, the average bit error probability simpliﬁes to

(6.20)

Therefore, receive antenna selection maintains the diversity order of N

with an array gain loss that is upper bounded by 10 log

As was the case with STTCs, tighter bounds on the array gain loss exist

for the case where L

= 1 [24]. When the Alamouti code (L

= 2) is used, exact

closed-form expressions for the average SNR and the outage probability have

also been derived [23].

6.5 SIMO Systems

In SIMO systems, which have only one transmit antenna, only one data

stream can be transmitted. Multiple copies of the transmitted signal arrive

at the receiver, but with different amplitudes and phases that depend on the

channel. The receiver uses maximum ratio combining (MRC) to combine the

multiple received signals and achieve the maximum possible SNR. In MRC,

each received signal is weighted with a complex conjugate of the correspond-

ing channel coefﬁcient.

With antenna selection, L

out of the N

received signals are selected and

maximum ratio combined. Such a receiver is commonly referred to as the

hybrid-selection MRC (H-S/MRC) receiver. The SNR at the output of the

H-S/MRC receiver is given by

. (6.21)

Setting L

= 1 corresponds to the special case of a selection combining (SC)

receiver that uses only the best receive antenna element.

Closed-form expressions and tight bounds are available for both the SNR

and the BER for this case. At ﬁrst glance, the analysis of HS-MRC receivers

seems complicated because the ordered random variables L

[1]

, …, L

]

are

correlated even though the underlying variables L

, …, L

are independent of

each other. However, the minimum value L

]

and the differences L

[1]

– L

[2]

, …,

are unconstrained, i.e., they can take any value between 0 and h.

This motivates the following linear mapping, called virtual branch analysis

[25], between L

[i]

s and the virtual variables V

sel

()



rrt

HSMRC /

[]i

[][]NN



4190_book.fm Page 159 Tuesday, February 7, 2006 12:44 PM

160 MIMO System Technology for Wireless Communications

(6.22)

where the V

s are just conveniently scaled versions of the differences. In case

of Rayleigh fading, the V

s turn out to be independent and identically dis-

tributed exponential random variables with unit mean.

The mean SNR at the output of the H-S/MRC receiver can then be easily

shown to be [25,26]

(6.23)

(6.24)

(6.25)

And the average symbol error probability (SEP) for MPSK can be shown

to be [27]

(6.26)

where 6 = U(M – 1)/M and c = (sin(U/M))

. For the large SNR case, the

formula reduces to:

(6.27)

WW W

[]



















HSMRC /



[]



¨¨

¾¾

W 1

()

(sin )







(sin )

() (sin)



22N

dV.

4190_book.fm Page 160 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 161

Therefore, H-S/MRC achieves a diversity order of N

, and compared to a

full complexity receiver with N

RF chains, incurs an array gain loss equal to

Upper and lower bounds for the array gain loss for HS-MRC are derived in

[28]. Figure 6.4 plots the SEP for N

= 8 as a function of the SNR, W, for

different values of L

. We again see that the SEP decreases as L

increases.

The gains diminish for larger L

The hybrid selection combining that we considered above can also be done

at the transmitter in a closed-loop MIMO diversity system, in which signals

input to the transmit antenna elements are weighted replicas of the same

data stream. The received streams are combined to maximize the instanta-

neous SNR at the output of the combiner. This is analyzed for the L

= 1 case

in [29] and for the general case in [30].

6.6 Implementing Antenna Selection:

Criteria and Algorithms

In TAS, the transmitter needs to examine possibilities to choose the best

subset of L

antenna elements out of the N

available. Similarly, in RAS, the

receiver needs to examine possibilities. In the case of T-RAS, the number

FIGURE 6.4

Symbol error probability as a function of SNR for different L

−1

−2

−3

−4

−5

−6

−2 0 2 4 6 8

ρ (SNR per branch, cB)

Symbol error probabiliy (SEP)

10 12 14 16 18 20

= 1

= 2,.....7

= N

= 8

log



v 1

()

4190_book.fm Page 161 Tuesday, February 7, 2006 12:44 PM

162 MIMO System Technology for Wireless Communications

of possibilities balloons to , and coordination between the transmitter

and receiver is required to choose the optimal transmit and receive antenna

subsets. The combinatorial increase in the number of possibilities makes an

exhaustive search impractical even for moderate values of N

and N

Therefore, a large number of sub-optimal selection algorithms and heuris-

tics with varying levels of complexity have been proposed in the literature.

The criteria depend on whether OSTBCs, STTCs, or ideal capacity-achieving

space–time codes are assumed, and can also depend on the receiver archi-

tecture. We illustrate and compare a few of the proposed approaches for the

following cases:

• Capacity-achieving spatial multiplexing with an optimal receiver,

• Spatial multiplexing with linear receivers, and

• Space–time codes with an MLD receiver.

6.6.1 Capacity-Achieving Spatial Multiplexing

In Equation 6.4, the optimal choice of the selection subset is given by a non-

linear equation. However, for low SNR, the optimal selection criterion sim-

pliﬁes as follows:

(6.28)

Therefore, at low SNR, norm-based selection is optimal [2,31]. Moreover, this

is the same as the criterion in Equation 6.17 for antenna selection with

OSTBCs.

However, the situation is not as simple for high SNR because of the

form of the capacity formula in Equation 6.4. Therefore, several iterative and

computationally simpler criteria have been proposed that exploit basic

results from matrix theory [32].

The effect of adding an additional receive antenna element to the capacity

formula was considered in [10,31]. A greedy criterion for selecting the next

receive antenna element was proposed based on the observation that the

capacity with n + 1 antenna elements can be written in terms of the capacity

with n antenna elements as follows:

(6.29)

where the matrix depends on the channel subset chosen so far, and equals

()()

sel

~ +.

max log

()H



log

jnj

()()log ,

†



HH hBh

=++

IHH

()

1



†

4190_book.fm Page 162 Tuesday, February 7, 2006 12:44 PM

Antenna Selection in MIMO Systems 163

The criterion for choosing the next antenna element then becomes: choose

the receive antenna element j corresponding to the row h

of the channel

matrix, H, that maximizes the following quadratic form:

(6.30)

Calculating the matrix inverse is simpliﬁed by

using the following iteration:

(6.31)

The overall complexity of the algorithm is .

A decremental iterative search algorithm that successively removes

antenna elements has also been proposed [10]. The recursion formulae are

similar to the ones above, though with the + and – signs reversed. The

decremental approach is efﬁcient when L

is close to N

, while the incremental

approach is efﬁcient for large antenna arrays from which a small subset of

antenna elements needs to be chosen (L

 N

). Both approaches achieve

performance very close to the optimal selection criterion.

A faster decremental algorithm that avoids matrix inversion and determi-

nants was proposed in [33]. For RAS, it is based on the intuition that a row

(receive antenna element) of the channel matrix that is highly correlated with

another row adds little additional information about the channel and may

be removed. The algorithm calculates the correlation between all the rows

of the channel matrix and selects the two rows with the highest correlation.

Between the two, it eliminates the row with the lower power. Given that the

MIMO transmission rate is better measured by mutual information, an alter-

nate approach is to calculate the mutual information between two rows

instead of their correlation. Of the two rows that result in the highest pair-

wise mutual information, the one with a lower power is eliminated.

Transmit antenna selection criteria have also been motivated by the water-

ﬁlling principle [6], which optimally allocates more power to the channel

modes with better channel quality. For example, [34] proposed ﬁnding the

subset, represented by the transmit covariance K

sel

, that is close (in mean

square error sense) to the ideal water-ﬁlling transmit covariance K

(6.32)

Here, is a diagonal matrix (of size N

× N

) in which the diagonal entries

that correspond to the selected antenna elements are set as , while the

remaining elements are all set to 0.

jnj

=.arg max

†

hB h

IHH



()



†

hBh

Bh h B

JnJ

nJ J n

= 

†

ONNNL

trtr

(max{ } ),



HKK

=  .argmin

()S

sel

4190_book.fm Page 163 Tuesday, February 7, 2006 12:44 PM