MITSUBISHI ELECTRIC RESEARCH LABORATORIEShttp://www.merl.com

Capacity of MIMO Systems with AntennaSelection

Andreas F. Molisch, Moe Z. Win, Yang-seok Choi, Jack H. Winters

TR2005-144 July 2005

Abstract

We consider the capacity of multiple-input multiple-output systems with reduced complexity.One link-end uses all available antennas, while the other chooses the L out of N antennas thatmaximize capacity. We derive an upper bound on the capacity that can be expressed sa sthe sumof the logarithms of ordered chi-square-distributed variables. This bound is then evaluated ana-lytically and compared to the results obtained by Monte Carlo simulations. Our results show thatthe achieved capacity is close to the capacity of a full-complexity system provided that L is atleast as large as the number of antennas at the other link-end. For example, for L=3, N=8 anten-nas at the receiver and three antennas at the transmitter, the capacity of the reduced-complexityscheme is 20 bits/s/Hz compared to 23 bits/s/Hz of a full-complexity scheme. We also presenta suboptimum antenna subset selection algorithm that has a complexity of N2 compared to ehtoptimum algorithm with a complexity of (N L).

IEEE Transactions on Wireless Communications

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005 1759

Capacity of MIMO Systems With Antenna SelectionAndreas F. Molisch, Fellow, IEEE, Moe Z. Win, Fellow, IEEE,

Yang-Seok Choi, Member, IEEE, and Jack H. Winters, Fellow, IEEE

Abstract—We consider the capacity of multiple-input multiple-output systems with reduced complexity. One link-end uses allavailable antennas, while the other chooses the L out of N an-tennas that maximize capacity. We derive an upper bound on thecapacity that can be expressed as the sum of the logarithms ofordered chi-square-distributed variables. This bound is then eval-uated analytically and compared to the results obtained by MonteCarlo simulations. Our results show that the achieved capacity isclose to the capacity of a full-complexity system provided that Lis at least as large as the number of antennas at the other link-end. For example, for L = 3, N = 8 antennas at the receiverand three antennas at the transmitter, the capacity of the reduced-complexity scheme is 20 bits/s/Hz compared to 23 bits/s/Hz of afull-complexity scheme. We also present a suboptimum antennasubset selection algorithm that has a complexity of N 2 comparedto the optimum algorithm with a complexity of

(N

L

).

Index Terms—Antenna arrays, information rates, MIMOsystems.

I. INTRODUCTION

MULTIPLE-INPUT MULTIPLE-OUTPUT (MIMO)wireless systems are those that have antenna arrays at

both transmitter and receiver. Early simulation studies that re-vealed the potentially large capacities of those systems weredone in the 1980s [1], and subsequent papers explored thecapacity analytically [2], [3]. Since that time, interest in MIMOsystems has exploded. Layered space–time (ST) receiver struc-tures [4]–[6] and ST codes [7] make it possible to approach thecapacity limits revealed in [2]. Commercial products based onsuch codes are under development [8]. Most importantly, thestandard for third-generation cellular phones [3rd Generation

Manuscript received June 25, 2003; revised February 20, 2004; acceptedApril 23, 2004. The editor coordinating the review of this paper and approv-ing it for publication is G. Leus. This work was supported in part by anINGVAR grant of the Swedish Strategic Research Fund, a cooperation grantfrom the Swedish STINT, the Office of Naval Research Young InvestigatorAward N00014-03-1-0489, the National Science Foundation under Grant ANI-0335256, and the Charles Stark Draper Endowment. Parts of this work werepresented at ICC 2001 and VTC fall 2003.

A. F. Molisch was with AT&T Laboratories-Research, Middletown, NJ07748 USA. He is now with Mitsubishi Electric Research Laboratory (MERL),Cambridge, MA 02139 USA and also at the Department of Electroscience,Lund University, Lund, Sweden (e-mail: Andreas.Molisch@ieee.org).

M. Z. Win was with AT&T Laboratories-Research, Middletown, NJ 07748USA. He is now with the Laboratory for Information and Decision Systems,Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:win@ieee.org; moewin@mit.edu).

Y.-S. Choi was with AT&T Laboratories-Research, Middletown, NJ 07748USA. He is now with Intel, Inc., Hillsboro, OR 97229 USA (e-mail: yschoi@ieee.org).

J. H. Winters was with AT&T Labs-Research, Middletown, NJ 07748USA. He is now with Motia Inc., Middletown, NJ 07748 USA (email: jack.winters@ieee.org).

Digital Object Identifier 10.1109/TWC.2005.850307

Partnership Project (3GPP)] foresees the use of a simple STcode [9] with two transmit antennas and one or more receiveantennas for circuit-switched communications and spatialmultiplexing (multiple transmit data streams) for high-speeddownlink packet data access [10].

In an earlier work, it was shown that the incremental gain ofadditional receive antennas is negligible if the total number ofreceive antennas Nr is far larger than the number of transmitantennas Nt [4].1 This can be explained by the fact that ad-ditional antennas do not provide independent communicationchannels but just increase the diversity order. This motivatesresearchers to explore the possibility of replacing the maximalratio diversity that is normally achieved in a such a MIMOsystem with selection diversity (SD). Thus, in this paper, wepropose a reduced-complexity MIMO scheme that selects theLr “best” of the available Nr antennas. Such a scheme canprovide the full number of independent communication chan-nels, and additionally an SD gain. Compared to the use of allantennas, the antenna selection has the advantage that only Lr

instead of Nr receiver RF chains are required. We still requirethe full number of antenna elements, but these are usuallyinexpensive, as they are patch or dipole antennas that can beeasily produced and placed.

Antenna selection, or more precisely, the principle of usingL out of N antennas, was first studied in the context of antennaselection at one link-end, while only a single antenna is presentat the other link-end [11]–[14]. This is referred to as “hybridselection/maximum ratio combining (MRC)” in the literature.Therefore, we will employ the term “hybrid selection/MIMO”(H-S/MIMO) for the more general case studied in this paper,namely antenna selection at one link-end, and multiple anten-nas, all of which are used, at the other link-end.

There has been considerable interest in H-S/MIMO in recentyears. The case of antenna selection at the transmitter is treatedin [15] using Monte Carlo simulations; this paper also developsa criterion for optimal antenna set selection for high signal-to-noise ratios (SNRs); [16] extended this to the low-SNR case.It has been shown that antenna selection is beneficial in a low-rank environment [17] and in interference-limited systems [18].A selection algorithm for minimizing the bit error probabilityof linear MIMO receivers is given in [19]. The use of STcodes in combination with antenna selection was investigated in[20] and [21]; the use of antenna selection in transmit–receivediversity systems with channel knowledge at both link-ends was

1Under certain circumstances, increasing that number can even lead toperformance degradation, as the channel estimation becomes more difficult andintroduces estimation errors.

1536-1276/$20.00 © 2005 IEEE

1760 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 1. Block diagram of the considered system.

treated in [22].2 A more detailed overview of the literature isgiven in [28].

In this paper, we derive analytical bounds for the capacitydistribution function of an H-S/MIMO system at one link-end.We show that an exact antenna selection algorithm requireshigh computational complexity and propose several alternativemethods that have much lower complexity while performingalmost as well as the exact selection criteria. The rest of thepaper is organized as follows: In Section II, we set up thesystem model. Analytical bounds for the capacity are derived inSection III. Next, we present a fast antenna selection algorithmin Section IV. Section V gives evaluations for the analyticalbounds of H-S/MIMO and compares them to numerical sim-ulation results. Conclusions and system design considerationsare given in Section VI.

II. SYSTEM MODEL

We consider the case where the transmitter uses all availableantennas while the receiver uses antenna selection. Fig. 1exhibits a block diagram. At the transmitter, the data streamenters an ST encoder, whose outputs are forwarded to the Nt

transmit antennas. The signals are subsequently upconvertedto passband, amplified by a power amplifier, and filtered. Forour model, we omit these stages, as well as their equivalentsat the receiver, which allows us to treat the whole problemin equivalent baseband. Note, however, that it is exactly theseparts that are most expensive and make the use of reduced-complexity systems desirable.

From the antennas, the signal is sent through the mobile radiochannel, which is assumed to be flat fading and quasi-static. Byquasi-static, we mean that the coherence time of the channelis so long that “a large number” of bits can be transmittedwithin this time. More specifically, we assume that the data areencoded with near Shannon limit achieving codes.3 It has beenshown that LDPC codes with a block length of 10 000 approachthe Shannon limit within less than 1 dB [30]. For a data rate of10 Mbits/s, such a block can be transmitted within 1 ms, whichis shorter than the typical 10 ms coherence time of wirelesschannels. Thus, each channel realization can be associated with

2Parallel to our work (see also [23] and [24]), an alternative algorithm for theselection of antenna subsets was presented and a lower bound of the capacitywas derived in [25]–[27]; this algorithm will also be discussed in Section IV.

3Such a code could be, e.g., the combination of ST processing [6] with alow-density parity check code [29].

a (Shannon-AWGN) capacity value. The capacity thus becomesa random variable (RV), rendering the concept of “capacitycumulative distribution function” and “outage capacity” mean-ingful performance measures [2].

We denote the Nr × Nt matrix of the channel as

H =

h11 h12 · · · h1Nt

h21 h22 · · · h2Nt

......

. . ....

hNr1 hNr2 · · · hNrNt

. (1)

If the channel is Rayleigh fading, the hij are independentidentically distributed (i.i.d.) zero-mean circularly symmetriccomplex Gaussian RVs with unit variance, i.e., the real andimaginary parts have a variance of 1/2 each. Consequently, thepower carried by each transmission channel hij is chi-squaredistributed with 2 degrees of freedom. The channel also addswhite Gaussian noise, which is assumed to be independentamong the Nr receiver antenna elements. Following [2], weconsider the case in which the hij are independently fading, asthis simplifies the theoretical analysis. More involved channelmodels are discussed, e.g., in [31]–[33].

The received signal, which is written as

y = Hs + n = x + n (2)

is received by Nr antenna elements, where s is the transmitsignal vector and n is the noise vector. A control algorithm (tobe discussed in Sections III and IV) selects the best Lr of theavailable Nr antenna elements and downconverts their signalsfor further processing (note that only Lr receiver chains arerequired). ST encoder and decoder are assumed to be ideal sothat the capacity can be achieved. We assume ideal knowledgeof the channel at the receiver so that it is always possibleto select the best antennas. However, we do not assume anyknowledge of the channel at the transmitter. This implies thatno waterfilling can be used and that the available transmitterpower is equally distributed among the transmit antennas.

III. THEORY

Let us first explore the scenarios that are suited forH-S/MIMO. As shown in [2], the capacity is linearly propor-tional to min(Nr, Nt). Any further increase of either Nr orNt while keeping the other fixed only increases the diversity

MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1761

order and possibly the mean SNR, possibly. Thus, if the num-ber of antennas at one link-end is limited, e.g., due to spacerestrictions, a further increase in the antenna number at theother link-end does not allow us to add statistically independenttransmission channels (which would imply linear increase insystem capacity), but only provides additional diversity. Sinceit is well known that SD has the same diversity order as that ofMRC [34], we can anticipate that a hybrid scheme with Nr >Lr = Nt will give a good performance. In the next subsections,we will give a quantitative confirmations of this conjecture.

A. Exact Expression for the Capacity

The capacity of MIMO system using all antenna elements isgiven by [2]

Cfull = log2

[det(INr +

ΓNt

HH†)]

(3)

where INr is the Nr × Nr identity matrix, Γ is the mean SNRper receiver branch, and superscript † denotes the Hermitiantranspose. The receiver now selects those antennas that allowa maximization of the capacity, so that

Cselect = maxS(H)

{log2

[det(ILr +

ΓNt

HH†)]}

(4)

where H is created by deleting Nr − Lr rows from H, andS(H) denotes the set of all possible H, whose cardinality is(NrLr

).

The optimum choice of antennas requires the knowledge ofthe complete channel matrix. This may seem to necessitatethe use of Nr RF chains, which is in contrast with a low-complexity system. However, in a sufficiently slowly changingenvironment, the Lr RF chains can be cycled through the Nr

antennas during the training bits. In other words, RF chainsare connected to the first Lr antennas during the first part ofthe training sequence, then to the second Lr antenna duringthe next part, and so on. At the end of the training sequence,we pick the best Lr antennas. Thus, we only need a few moretraining bits instead of additional RF chains and the decreasein the spectral efficiency due to those additional training bits isnegligible, especially in high-data-rate systems.

B. Capacity Bound for Lr ≤ Nt

An exact analytical solution for Cselect seems difficult. Thus,we derive analytical bounds in this subsection and verify themwith Monte Carlo simulations in Section V. Our starting point

is the upper capacity bound for the full-complexity system withNt ≤ Nr [2]

Cfull ≤Nt∑i=1

log2

(1 +

ΓNt

γi

)(5)

where the γi are independent chi-square-distributed RVs with2Nr degrees of freedom. The equality applies in the “unrealisticcase when each of the Nt transmitted components is receivedby a separate set of Nr antennas in a manner where eachsignal component is received with no interference from theothers” [2].

In our case, we select the best Lr out of Nr receive antennas,where Lr ≤ Nt. The upper bound can be obtained similar to(5), except for exchanging the role of transmitter and receiver,and selecting those antennas whose instantaneous realizationsof γi are the largest. Since this equation is a crucial startingpoint, let us elaborate on its physical interpretation. We considera system where each of the Nr receive antennas has its ownset (of size Nt) of transmit antennas. Naturally, this case is notfeasible in practice but must result in an upper bound of thecapacity. Each set of transmit antennas corresponding to eachof the Nr receive antennas can carry one data stream. The max-imum SNR (which also achieves maximum capacity) for thisdata stream can be obtained with maximal ratio transmission,which in turn results in chi-square-distributed SNR with 2Nt

degrees of freedom at the receiver output. Finally, we selectthose Lr (out of Nr) receive antennas that give the best SNR,and thus highest capacity. The capacity bound with antennaselection is thus

Cbound =Lr∑i=1

log2

(1 + ργ(i)

)(6)

where ρ = Γ/Nt, and the γ(i) are ordered chi-square-distributed variables with 2Nt degrees of freedom, out of a setof Nr.4

The joint statistics of the ordered SNRs γ(i) is shown in(7) at the bottom of the page [14], where Γ(·) is Euler’s Gammafunction [35].

Thus, the characteristic function of the capacity bound is

Φ(jν) =Nr!

Γ(Nt)Nr

∞∫0

dγ(1)

γ(1)∫0

dγ(2) · · ·γ(Nr−1)∫

0

dγ(Nr)

× exp

[−jν

Lr∑i=1

log2

(1 + ργ(i)

)] Nr∏i=1

γNt−1(i) exp

(−γ(i)

). (8)

4We use {γ(i)} to denote the order set of {γi}, i.e., γ(1) > γ(2) >· · · γ[Nr]. Note that the possibility of at least two equal γ(i)’s is excluded asγ(i) �= γ(j) almost surely for continuous RVs γi.

pγ(i)

(γ(1), γ(2), . . . , γ(Nr)

)=

Nr!Nr∏i=1

1Γ(Nt)

γNt−1(i) exp

(−γ(i)

), for γ(1) > γ(2) > · · · > γ(Nr)

0, otherwise(7)

1762 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

First, we perform the integrations over the Nr − Lr discardedantennas. As shown in the Appendix, these Nr − Lr result in anexpression of the form

d(Nr−Lr) +

[Nr−Lr∑

p=1

exp(−b(Nr−Lr)

p γ(Lr)

)

×(Nr−Lr−p+1)(Nt−1)∑

k=0

c(Nr−Lr)p,k γk

(Lr)

]. (9)

The values of the coefficients b, d, and c are computed via aniteration. We initialize with

b(0)p = 0 for all p

c(0)p,k = 0 for all p, k

d(0) = 1 (10)

and then perform Nr − Lr iterations

b(q+1)p = b(q)

p + 1, for 1 ≤ p ≤ q (11)

b(q+1)q+1 = 1 (12)

[see (13)–(15) at the bottom of the page].For the next step of the iteration, it is advantageous to rewrite

(9) as

Nr−Lr∑p=0

exp(−b(Nr−Lr)

p γ(Lr)

)

×(Nr−Lr−p+1)(Nt−1)∑

k=0

c(Nr−Lr)p,k γk+α(Nr−Lr)

(Lr)

so that

c(Nr−Lr)0,0 = d(Nr−Lr)

c(Nr−Lr)0,k = 0

b(Nr−Lr)0 = 0

α(Nr−Lr) = 0. (16)

We then perform the next Lr − 1 integrations, which yield (seeAppendix) an expression of the form

Nr−Lr∑p=0

exp(−b(Nr−1)

p γ(1)

) M∑k=0

c(Nr−1)p,k γk+α(Nr−1)

(1) (17)

where the parameters c(Nr)p,r , α(Nr), and b

(Nr)p are again com-

puted via a recursion. In each step, we first compute

b(q)p = b(q)

p + 1 (18)

α(q) =α(q) +jν

ln(2)(19)

c(q)p,k =

c(q)p,k−(Nt−1), k = M

c(q)p,k−(Nt−1) + jν

ρ ln(2)c(q)p,k−Nt

, Nt − 1 ≤ k < M

jνρ ln(2)c

(q)p,k−Nt

, k = Nt − 20, otherwise

.

(20)

Then we can perform the second step, which is obtainingcoefficients for the next iteration step

α(q+1) = α(q) (21)

b(q+1)p = b(q)

p (22)

c(q+1)p,r =

r−1∑k=0

c(q)p,k f

(q)p,r−1−k (23)

with

f (q)p,n =

[b(q)p

]nn∏

i=0

(k + α(q) + 1 + i

) . (24)

The final integration and incorporation of constant multi-plicative factors yields

Φ( jν) =ρ

jνLrln(2) Nr!

Γ(Nt)Nr

Nr−Lr∑p=0

M∑r=0

c (Nr−1)p,r

Γ(r + α(Nr−1) + 1

)[b

(Nr−1)p

]r+α(Nr−1)+1.

(25)

c(q)p,k =

{c(q)p,k−(Nt−1), for (q − p + 2)(Nt − 1) ≥ k ≥ (Nt − 1)

0, otherwise(13)

d(q+1) = d(q)(Nt − 1)! +q∑

p=1

(q−p+2)(Nt−1)∑t=0

c(q)p,t

t!(b(q+1)p

)t+1 (14)

c(q+1)p,k =

−(q−p+2)(Nt−1)−k∑

t=0

c(q)p,k+t(

b(q+1)p

)t+1(k+t)!

k! , for 1 ≤ p ≤ q

−d(q) (Nt−1)!k! , for p = q + 1

(15)

MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1763

The upper summation limit M is theoretically infinite, but thesum converges reasonably fast. In our computations, M = 50proved to be sufficient for Nr = 8. Details about the derivationof the recursion relations for the coefficients can be found in theAppendix.

The above equation yields the characteristic function ofthe capacity bound (note that we have omitted the functionaldependence of the parameters on ν for notational convenience).The probability density function (pdf) of the capacity boundis obtained by performing an inverse Fourier transformation,which can be accomplished by a fast Fourier transform.

C. Capacity Bound for Lr > Nt

The bound derived above is quite tight for Lr ≤ Nt but tendsto become rather loose for Lr > Nt. Especially, this boundsuggests an “almost” linear increase of the capacity with Lr.5

However, we have shown in Section III-A that we can onlyanticipate a logarithmic increase. We thus derive an alternativebound that reflects this fact.

We consider the situation where each of the Nt transmitantennas transmits an independent data stream. Furthermore,we assume the (practically impossible) situation where none ofthe data streams interferes with each other. This is equivalent tohaving Nt single input multiple output (SIMO) systems eachwith separate Nr receive antenna elements dedicated to thereception of one such data stream. In each of the SIMO systems,we perform H-S/MRC, so that the (normalized) SNR of the jthSIMO system is given by

Lr∑i=1

γ(i). (26)

Assuming that none of the data streams interferes with anyother, the total capacity is then

Cselect ≤Nt∑j=1

log2

[1 + ρ

Lr∑i=1

γ(i)

]=

Nt∑j=1

ξj = Ψ (27)

where the γ(i) are ordered chi-square-distributed variables with2 degrees of freedom, taken from a set of Nr available ones.Since the ξj are i.i.d., the characteristic function of Ψ is finally

Nt∏j=1

Cj(ν) = C1(ν)Nt . (28)

The computation of Cj(ν), i.e., the characteristic function ofξj , is similar to the method described in [22] and [36].

IV. FAST ANTENNA SELECTION ALGORITHMS

The optimum selection of the antennas requires(NrLr

)compu-

tations of determinants and is thus computationally intensive.

5Note that the increase is only “almost” linear because we are dealing withordered stochastic variables. Thus, including more terms in the summationtends to give terms that have a lower SNR and thus a lower capacity.

It seems thus worthwhile to investigate suboptimum algo-rithms with lower computational complexity. In this section, wepresent a family of such algorithms that result in a small SNRpenalty while drastically reducing computation time.

The determinant in (4) can be written as

det(ILr +

ΓNt

HH†)

=r∏

k=1

(1 +

ΓNt

λ2k

)(29)

where r is the rank of the channel matrix and λk is the singularvalue of H. Note that the rank and the singular values shouldbe maximized for the maximum capacity. Suppose there aretwo rows of H which are identical. Clearly, only one of theserows should be selected in H. Since these two rows carry thesame information about the signal components, any one of thesetwo rows may be deleted. In addition, if they have differentpowers (i.e., square of the norm of the row), we select the rowwith the higher power. When there are no identical rows, wechoose two rows for the possible deletion whose correlation isthe highest and delete the one with the lower power. In thismanner, we can have the channel matrix H whose rows areminimally correlated and have maximum powers. The aboveargument leads to the following algorithm.

1) The channel vector hk is defined as the kth row of H,with k being an element of the set X = {1, . . . Nr}.

2) For all k and l, k > l, in X , compute the correlationΞ(k, l) defined as Ξ(k, l) = |〈hk,hl〉|, where 〈a,b〉 rep-resents an inner product between vector a and b.

3) Loopa) Choose the k and l (with k, l ∈ X ,k > l) that give

the largest Ξ(k, l). If ‖hk‖2 ≥ ‖hl‖2, eliminate hl,otherwise, eliminate hk.

b) Delete l (or k) from X .c) Go to Loop until Nr − Lr rows are eliminated.

The method defined above shall be called the correlationbased method (CBM). It does not require the SNR value andit is based on the correlation of the rows of the channel matrix〈hk,hl〉, which can be approximated by the correlation of thenoisy estimates E{yky∗

l }.As an alternative method when the SNR is available, we

suggest to use the mutual information between yk and yl.The zero-valued mutual information means that the kth receiveantenna output yk and the lth output yl carry totally different in-formation. This occurs when the corresponding channel vectorshk and hl are orthogonal. On the other hand, when the mutualinformation between yk and yl has a maximum value, yk and yl

carry the same information so that we can delete one of them.The mutual information is defined as [37]

I(yk; yl) = G(yk) + G(yl) − G(yk, yl) (30)

where G(·) denotes the entropy.6

6We deviate from the usual entropy notation H to avoid confusion with thechannel matrix H.

1764 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

In the MIMO system, the mutual information can be writ-ten as

I(yk; yl)=log

(‖hk‖2 Γ

Nt+1)(

‖hl‖2 ΓNt

+1)

(‖hk‖2 Γ

Nt+1)(‖hl‖2 Γ

Nt+1)−|〈hk,hl〉|2 Γ

2

N2t

.

(31)

The above equation can be rewritten as

I(yk; yl)

= log(‖hk‖2 Γ

Nt+ 1)

− log

(‖hk‖2 Γ

Nt+ 1)(

‖hl‖2 ΓNt

+ 1)− |〈hk,hl〉|2 Γ

2

N2t(

‖hl‖2 ΓNt

+ 1)

= log(‖hk‖2 Γ

Nt+ 1)

− log

1 +‖hk‖2 Γ

Nt+ ‖hk‖2‖hl‖2 Γ

2

N2t− |〈hk,hl〉|2 Γ

2

N2t

‖hl‖2 ΓNt

+ 1

.

(32)

Since ‖hk‖2‖hl‖2 ≥ |〈hk,hl〉|2, the mutual information isupper bounded as follows:

I(yk; yl) ≤ log(‖hk‖2 Γ

Nt+ 1)

. (33)

Similarly, we have

I(yk; yl) ≤ log(‖hl‖2 Γ

Nt+ 1)

. (34)

Finally, the mutual information is upper bounded by

I(yk; yl) ≤ min{log(‖hk‖2 Γ

Nt+ 1)

, log(‖hl‖2 Γ

Nt+ 1)}

.

(35)

We therefore define the normalized mutual information

I0(yk; yl) =I(yk; yl)

min{

log(‖hk‖2 Γ

Nt+ 1), log(‖hl‖2 Γ

Nt+ 1)}(36)

as a measure of how close the two RVs are.We can also apply the mutual-information-based technique

to xk, which is the signal component of yk, in order to avoidrequiring the SNR value. Then the mutual information betweenthe data components xk and xl is

I(xk;xl) = log‖hk‖2‖hl‖2

‖hk‖2‖hl‖2 − |〈hk,hl〉|2. (37)

Similarly, we define the normalized mutual information as

I0(xk;xl) =I(xk;xl)

min{|log‖hk‖2|, |log‖hl‖2|} . (38)

The antenna selection algorithms based on mutual informa-tion then have a similar program structure as the one based oncorrelation (CBM). All that is required is to replace Ξ by I0

as defined in (36) (henceforth referred to as MIBM) or (38)(MIBM2).

V. RESULTS

In this section, we evaluate the bounds derived in previoussections and compare them to Monte Carlo simulations. Wefirst generate random realizations of mobile radio channelswith transfer function hij , which is an i.i.d. circularly complexGaussian RV with zero mean and a variance of 1/2 for the realand imaginary parts. From each realization of the matrix H, acomplete set S(H) of

(NrLr

)possible matrices H are obtained

by eliminating all possible permutations of Nr − Lr rows fromthe matrix H. For each of the H , we computed the capacity by(4), and selected the largest capacity from the set.

Fig. 2 shows the cumulative distribution function of capacityfor Nr = 8, Nt = 3, and various Lr. The SNR is 20 dB, andin the following, we consider the 10% outage capacity. Withfull exploitation of all available elements, 21.8 bits/s/Hz can betransmitted over the channel. This number decreases graduallyas the number of selected elements Lr is decreased, reaching18.2 bits/s/Hz at Lr = 3. For Lr < Nt, the capacity decreasesdrastically, since a sufficient number of antennas to provideNt independent transmission channels is no longer available.These trends are well reflected in the bounds: the bound forthe full-complexity system is 22.7 bits/s/Hz, decreasing to20.0 bits/s/Hz at Lr = 3. We also find that the bounds are tightfor Lr < Nt, become looser for Lr Nt, and become tighteragain for Lr Nt. This fact can be explained as follows. Aswe have noted in Section III, the bound reflects the situationthat each of the received signals has its own set (of size Nt)of transmit antennas. This is fulfilled perfectly for Lr = 1 andbecomes a progressively worse approximation as Lr increases.Note that this bound is used only up to Lr ≤ Nt. For largerLr, we bound the capacity by the case where we have Nt

independent data streams, none of which interferes with eachother. Now it is well known [38] that N receive antennas cansuppress K interfering data streams while retaining a diversityorder of N − K for the remaining data streams. The boundis thus approximately equivalent to a situation where we haveLr + Nt receive chains (instead of the Lr that are actuallyexisting). The relative error thus becomes progressively smalleras Lr increases. Finally, Fig. 2 also shows the capacity of anNt × Lr full complexity system. This shows us how muchperformance we would lose when using (for a fixed number ofRF chains) only the minimum number of antenna elements.

Fig. 3 shows the influence of the SNR on the capacity. Weplot the improvement of the 10% outage capacity of an H-S/MIMO system over a single-antenna system. We see that thecapacity increase is very large at low SNRs (factor of 25at SNR = 0 dB), while for high SNRs, it tends to a fixed

MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1765

Fig. 2. Exact capacity (solid curves) and bound (dashed curves) for Nr = 8, Nt = 3, SNR = 20 dB. Dotted lines show capacity of Nt = 3, Nr = L system.Note that for L = 1, the solid and the dashed lines coincide, while for L = 8, the solid and the dotted lines coincide.

Fig. 3. Ratio of 10% outage capacity of a system with Nr = 8, Lr = 6, Nt =3, over that of a single-antenna system: bound (dashed); exact (dotted); andsystem with Nt = Lr = 3 (solid).

value of about 4. A factor of 3 in the capacity increase canbe attributed to the number of independent communicationchannels between the transmitter and receiver. The remainderof the capacity increase is due to the diversity effect. Note alsothat Fig. 3 plots the improvement in 10% outage capacity. Ifwe were to consider the mean capacity, the influence of theSNR on the relative capacity increase would be significantlyreduced. For standard Nr = Lr = Nt systems, the relativemean capacity increase (compared to a SISO system) ispractically independent of the SNR.

Another interesting point is the comparison between antennaselection criteria based on capacity and antenna selection based

on the suboptimum algorithm that selects antennas with thehighest powers. In our MC simulations, we also recorded foreach channel realization the indices of those antennas thathave the highest SNR. The indices of those antennas werethen compared to those of the antennas that were chosen tomaximize capacity. We found that only in about 50% of allchannel realizations did the two selections agree with eachother. The geometric interpretation of this behavior is thatfor the deterministic case (corresponding to one channelrealization), the phase shifts between the antenna elements arethe decisive factors for capacity, and are far more importantthan instantaneous SNR [39]. Fig. 4 gives the capacities that areobtained by antenna selection based on an SNR criterion. Wesee that for Lr � Nr, the 10% outage capacity decreases from18.2 to 14.3 bits/s/Hz at 20 dB SNR when the SNR- (insteadof capacity-) based criterion is used for antenna selection. Thisloss gets smaller as Lr approaches Nr.

The performance of our fast antenna selection algorithmsis detailed in Figs. 5 and 6. Again, the number of transmitand receive antennas is 3 and 8, respectively. For comparison,the ILM technique [25] is also evaluated. Each algorithmselects three receive antennas out of eight receive antennas.Among the proposed algorithms, the MIB methods outperformthe CB technique. The ILM is shown to have a performancethat is very close to the exhaustive search. However, itrequires Gram–Schmidt orthogonalization and thus matrixinversion/multiplications. The complexity thus goes likeNrLrN

3t or Nr(Nr − Lr)N3

t , whichever is the smaller [27]. Incontrast, the main computational burden of our fast algorithmscomes from the calculation of vector multiplications 〈hk,hl〉.

1766 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 4. CDF of the capacity of a system with Nr = 8, Nt = 3. Selection ofantenna by capacity criterion (solid) and by SNR criterion (dotted).

Each of those has a complexity of Nt, and we need Nr(Nr +1)/2 of them. The complexity thus goes as NtNr(Nr + 1)/2.The choice between ILM and MIBM2 (or a similar algorithm)is a tradeoff between performance and complexity.

Assuming that ideal coding is employed, the outageprobability when the bandwidth efficiency is 15 bits/s/Hz7 isshown in Fig. 7. The worst selection has 10 dB loss at 10−3

outage probability. The MIBM has about 2 dB loss whilethe correlation-based method exhibits around 6 dB loss. Theperformance of the fast algorithm MIBM2 is comparable tothat of the MIBM at high outage probability. The MIBM2 has agood performance overall while (similarly to the CB methods)it does not require the SNR value. Figs. 8 and 9 illustratethe outage capacities at 1% outage rate versus the number ofreceive antennas, Nr, under fixed Nt = Lr = 3 at 10 dB and30 dB SNR, respectively.

VI. SUMMARY AND CONCLUSION

We have investigated the behavior of MIMO systems thatselect a subset of available antennas at one link-end. In partic-ular, we have derived upper bounds for the capacity of antennaselection, and we have also derived several algorithms thatallow the selection of the antennas without an exhaustive searchover all possible antenna combinations. We compared the up-per bounds to computer simulation results and also comparedthe reduced-complexity selection algorithms. For Lr ≥ Nt,selecting the best Lr antennas gives almost the same capacityas the full-complexity system. Capacity losses are less than3.5 bits/s/Hz for Nr = 8, Nt = 3, Lr = 3 at 20 dB SNR. This

7That is, the probability that the capacity is smaller than 15 bits/s/Hz.

slight performance loss is offset by a considerable reductionin hardware costs. Instead of a full Nr receiver chains, onlyLr receiver chains, plus an RF switch are required. We havealso derived and compared several algorithms that allow theselection of the antennas without an exhaustive search over allpossible antenna combinations. These algorithms have a com-plexity proportional to N2

r , instead of the(NrLr

)complexity

of optimum algorithms, while resulting in a capacity loss ofless than 1 bit/s/Hz at 10 dB and 4 bits/s/Hz at 30 dB.

Important applications for such systems are cellular andwireless local area network systems with MIMO capability. Thenecessity of selecting antennas at one link-end (instead of usingall of them) stems from either complexity or cost considera-tions. For example, the number of transmit antennas foreseenfor the space–time coder could be limited, as is already thecase in the 3GPP standard. Furthermore, antenna selection canbe especially beneficial in low-rank and interference-limitedsystems. Thus, the results of this paper can serve as a guidelinefor designing reduced-complexity MIMO cellular systems forthird- and fourth-generation communications.

APPENDIX

DERIVATION OF THE RECURSION RELATION

The starting point for the derivation is (8). We first solve theNr − Lr innermost integrals.8 These integrals have the form

y∫0

[d(q) +

q∑p=1

exp(−b(q)

p x)

×(q−p+1)(Nt−1)∑

k=0

c(q)p,k xk

xNt−1 exp(−x) dx (39)

where for readability we have substituted γ(q) → x, γ(q−1)

→ y.The first part of the integral can be solved as [35]

y∫0

d(q)xNt−1 exp(−x) dx

= d(q)

[(Nt − 1)! − exp(−y)

Nt−1∑k=0

(Nt − 1)!k!

yk

]. (40)

Next, we pull out the summation over p from the integral andconsider the integrals

J (q)p =

y∫0

exp(−b(q)

p x) (q−p+1)(Nt−1)∑

k=0

c(q)p,kxk

xNt−1

× exp(−x) dx. (41)

8Integrals of a similar form are also solved by the authors in [22]. Forconvenience of the reader, we give here a short outline of the derivation.

MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1767

Fig. 5. Outage probabilities of fast algorithms, Nr = 8, Nt = Lr = 3, SNR = 10 dB.

Fig. 6. Outage probabilities of fast algorithms, Nr = 8, Nt = Lr = 3, SNR = 30 dB.

By introducing

b(q)p = b(q)

p + 1 for 1 ≤ p ≤ q (42)

M = (q − p + 2)(Nt − 1) (43)

c(q)p,k =

{c(q)p,k−(Nt−1), for (Nt − 1) ≤ k ≤ M

0, otherwise(44)

this integral can be written as

y∫0

exp(−b(q)

p x) M∑

k=0

c(q)p,k xk dx. (45)

Employing [40]∫ M∑k=0

c(q)p,kxke−b

(q)p x dx =

e−b(q)p x

−b(q)p

M∑l=0

(−1)l(−b

(q)p

)l

dl

dxl

M∑k=0

c(q)p,kxk

(46)

1768 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Fig. 7. Outage probability comparison, Nr = 8, Nt = Lr = 3.

Fig. 8. Outage probability comparison (1%) as a function of the number of receive antennas Nt = Lr = 3, SNR = 10 dB.

we get

J (q)p =

e−b(q)p x

−b(q)p

M∑k=0

c(q)p,k

k∑l=0

1(b(q)p

)l

k!(k − l)!

xk−l

∣∣∣∣∣∣∣y

0

. (47)

Introducing r = k − l, we can write this as

J (q)p =

e−b(q)p x

−b(q)p

M∑r=0

xrM−r∑t=0

c(q)p,r+t

1(b(q)p

)t

(r + t)!r!

∣∣∣∣∣∣∣y

0

. (48)

MOLISCH et al.: CAPACITY OF MIMO SYSTEMS WITH ANTENNA SELECTION 1769

Fig. 9. Outage probability comparison (1%) as a function of the number of receive antennas Nt = Lr = 3, SNR = 30 dB.

The total integral thus is

d(q)

[(Nt − 1)! − exp(−y)

Nt−1∑k=0

(Nt − 1)!k!

yk

]

+q∑

p=1

1

b(q)p

M∑t=0

c(q)p,t

t!(b(q)p

)t − e−b(q)p x

b(q)p

×M∑

r=0

yrM−r∑t=0

c(q)p,r+t

1(b(q)p

)t

(r + t)!r!

. (49)

Comparing this expression with the generic expression for theresult of the (q + 1)th integration

d(q+1) +q+1∑p=1

exp(−b(q+1)

p y) (q−p+2)(Nt−1)∑

k=0

c(q+1)p,k yk

(50)

and matching coefficients, we get the recursion relations givenin (11)–(15).

As mentioned in Section III-C, we perform this iterationNr − Lr times and write the result in the form

Nr−Lr∑p=0

exp(−b(Nr−Lr)

p γ(Lr)

)

×(Nr−Lr−p+1)(Nt−1)∑

k=0

c(Nr−Lr)p,k γk+α(Nr−Lr)

(Lr). (51)

The integrals we have to solve for the next Lr iteration steps arethus of the generic type

J (q) =

y∫0

dx

[Nr−Lr∑

p=0

exp(−b(q)

p x) M ′∑

k=0

c(q)p,k xk

]xNt−1

× exp(−x)[1 + ρx]jν

ln(2) (52)

where M ′= (Nr − Lr − p + 1)(Nt − 1) = M − (Nt − 1) forthe first iteration step and ∞ for the further steps (note thatsince the series converges well, a finite number of terms issufficient for the numerical computations), and x = γ(q), y =γ(q−1). Since ρ usually has reasonably large values, and thebehavior of the pdf is also mainly determined by the behaviorof the characteristic function near ν = 0 (the nth moment isthe nth derivative of the characteristic function at ν = 0), weapproximate

[1 + ρx]jν

ln(2) ≈ (ρx)jν

ln(2)

(1 +

jνln(2)

ρx

). (53)

This approximation was validated (for the parameters used inSection V) by computing Cbound by Monte Carlo simulationsand comparing it to the analytical results based on the approx-imation (53). Even for 1000 Monte Carlo runs, the differencebetween analytical and numerical results was smaller than theuncertainty of the MC results.

The integral J (q) can now be written as q = Nr −Lr, . . . . . . , Nr

J (q) = ρjν

ln(2)

y∫0

dx

[Nr−Lr∑

p=0

exp(−b(q)

p x) M∑

k=0

c(q)p,k xk+α(q)

](54)

1770 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

where

b(q)p = b(q)

p + 1 (55)

α(q) = α(q) +jν

ln(2)(56)

α(Nr−Lr) = 0 (57)

and

c(q)p,k =

c(q)p,k−(Nt−1), for k = M

c(q)p,k−(Nt−1) + jν

ρ ln(2)c(q)p,k−Nt

, for Nt − 1 ≤ k < M

jνρ ln(2)c

(q)p,k−Nt

, for k = Nt − 20, otherwise

.

(58)

Note that Nt ≥ 2.Now from [40]

y∫0

xk+α exp(−ax)dx = a−(k+α+1)γEuler(k + α + 1, ay)

(59)

where γEuler denotes here Euler’s Gamma function of thesecond kind. Using its series expansion [35], the above integralbecomes

a−(k+α+1) exp(−ay)∞∑

n=0

(ay)k+α+1+n

n∏i=0

(k + α + 1 + i)(60)

= exp(−ay)∞∑

n=0

yk+α+1+nfn (61)

with

fn =an

n∏i=0

(k + α + 1 + i). (62)

Applying this result now to (54), we get

J (q) = ρjν

ln(2)

[Nr−Lr∑

p=0

exp(−b(q)

p y)

×M∑

k=0

c(q)p,k

∞∑n=0

yk+α(q)+1+nfn

]. (63)

Comparing this to the generic form[Nr−Lr∑

p=0

exp(−b(q+1)

p x) M∑

k=0

c(q+1)p,k xα(q+1)+k

](64)

we find by comparison the coefficients in expressions(21)–(24).

By making use of the parameters b, c, the final integral is ofthe form

J (Nr) =

∞∫0

[Nr−Lr∑

p=0

exp(−b(Nr−1)

p x)

×M∑

k=0

c(Nr−1)p,k xk+α(Nr−1)

]dx (65)

which yields [40]

Nr−Lr∑p=0

∞∑r=0

c (Nr−1)p,r

Γ(r + α(Nr−1) + 1

)(b(Nr−1)p

)r+α(Nr−1)+1. (66)

ACKNOWLEDGMENT

The helpful suggestions of the reviewers are gratefully ac-knowledged. The authors also would like to thank H. Wangand J. L. Craig for critical reading of the manuscript.

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Andreas F. Molisch (S’89–M’95–SM’00–F’05)received the Dipl. Ing., Dr. techn., and habil-itation degrees from the Technical University (TU)Vienna, Vienna, Austria, in 1990, 1994, and 1999,respectively.

From 1991 to 2000, he was with the TU Vienna,becoming an Associate Professor there in 1999.From 2000 to 2002, he was with the Wireless Sys-tems Research Department at AT&T Laboratories-Research, Middletown, NJ. Since then, he has beena Senior Principal Member of Technical Staff with

Mitsubishi Electric Research Laboratories, Cambridge, MA. He is also Pro-fessor and Chairholder for radio systems at Lund University, Lund, Sweden.He has done research in the areas of SAW filters, radiative transfer in atomicvapors, atomic line filters, smart antennas, and wideband systems. His currentresearch interests are MIMO systems, measurement and modeling of mobileradio channels, and ultrawide bandwidth (UWB). He has authored, coauthored,or edited four books, eight book chapters, some 85 journal papers, and numer-ous conference contributions.

Dr. Molisch is an editor of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, co-editor of a recent special issue on MIMO andsmart antennas in Journal on Wireless Communications Mobile Computing,and coeditor of an upcoming IEEE JOURNAL ON SELECTED AREAS IN

COMMUNICATONS special issue on UWB. He has participated in the Europeanresearch initiatives “COST 231,” “COST 259,” and “COST273,” where he ischairman of the MIMO channel working group. He is also vice chairman ofCommission C (signals and systems) of URSI (International Union of RadioScientists), chairman of the IEEE 802.15.4a channel modeling group, andrecipient of several awards.

Moe Z. Win (S’85–M’87–SM’97–F’04) receivedthe B.S. degree (magna cum laude) from Texas A&MUniversity, College Station, in 1987, and the M.S.degree from the University of Southern California(USC), Los Angeles, in 1989, both in electrical engi-neering. As a Presidential Fellow at USC, he receivedboth the M.S. degree in applied mathematics and thePh.D. degree in electrical engineering in 1998.

Dr. Win is an Associate Professor at the Labora-tory for Information and Decision Systems (LIDS),Massachusetts Institute of Technology, Cambridge.

Prior to joining LIDS, he spent 5 years at AT&T Research Laboratories and7 years at the Jet Propulsion Laboratory. His main research interests arethe application of mathematical and statistical theories to communication,detection, and estimation problems. Specific current research topics includemeasurement and modeling of time-varying channels, design and analysisof multiple antenna systems, ultrawide bandwidth (UWB) communicationssystems, optical communications systems, and space communications systems.

Dr. Win has been involved actively in organizing and chairing sessions, andhas served as a member of the Technical Program Committee in a number ofinternational conferences. He served as the Technical Program Chair for theIEEE Communication Theory Symposia of ICC-2004 and Globecom-2000, aswell as for the IEEE Conference on Ultra Wideband Systems and Technologiesin 2002, the Technical Program Vice Chair for the IEEE International Confer-ence on Communications in 2002, and the Tutorial Chair for the IEEE Semi-annual International Vehicular Technology Conference in Fall 2001. He is thecurrent Chair and past Secretary (2002–2004) for the Radio CommunicationsCommittee of the IEEE Communications Society. He currently serves as AreaEditor for Modulation and Signal Design and Editor for Wideband Wireless andDiversity, both for IEEE TRANSACTIONS ON COMMUNICATIONS. He servedas the Editor for Equalization and Diversity from July 1998 to June 2003 for theIEEE TRANSACTIONS ON COMMUNICATIONS, and as a Guest-Editor for the2002 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS SpecialIssue on Ultra-Wideband Radio in Multiaccess Wireless Communications. Hereceived the International Telecommunications Innovation Award from KoreaElectronics Technology Institute in 2002, the Young Investigator Award fromthe Office of Naval Research in 2003, and the IEEE Antennas and PropagationSociety Sergei A. Schelkunoff Transactions Prize Paper Award in 2003. In2004, he was named Young Aerospace Engineer of the Year by the AIAAand received the Fulbright Foundation Senior Scholar Lecturing and ResearchFellowship, the Institute of Advanced Study Natural Sciences and TechnologyFellowship, the Outstanding International Collaboration Award from the Indus-trial Technology Research Institute of Taiwan, and the Presidential Early CareerAward for Scientists and Engineers from the White House. He is an IEEEDistinguished Lecturer and elected Fellow of the IEEE, cited “for contributionsto wideband wireless transmission.”

1772 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 4, JULY 2005

Yang-Seok Choi (S’91–M’01) received the B.S.degree from Korea University, Seoul, South Korea,in 1990, the M.S.E.E. degree from the Korea Ad-vanced Institute of Science and Technology, Taejon,South Korea, in 1992, and the Ph.D. degree fromPolytechnic University, Brooklyn, NY, in 2000, allin electrical engineering.

From 1992 to 1996, he was with Samsung Elec-tronics, Co., Ltd., Suwon, Korea, where he developed32 QAM modem for HDTV and QPSK ASIC forDBS. During 2000 summer he held a Summer intern

position at AT&T Labs-Research Shannon Lab, Florham Park, NJ. In 2000, hejoined National Semiconductor, East Brunswick, NJ, where he was involved inthe development of W-CDMA. During 2001–2002, he was a Senior TechnicalStaff Member at AT&T Labs-Research, Middletown, NJ where he researchedon MIMO systems, OFDM systems and information theory. From 2002 to 2004he had been with ViVATO, Inc., Spokane, WA, working on MIMO OFDMsystems, smart antenna systems, and antenna/beam selection techniques. Heresearched on Smart antenna applications to CSMA protocol and co-inventedComplementary Beamforming. In 2004, he joined Intel Corporation, Hillsboro,OR where he studies on Broadband Wireless communications systems. Hisresearch interests include MIMO, OFDM, MC-CDMA, smart antenna, blindidentification/equalizer, carrier/timing recovery, space-time coding, cross-layerdesign and capacity of time-varying multipath channel. He holds seven U.S.patents.

Jack H. Winters (S’77–M’81–SM’88–F’96) re-ceived the Ph.D. degree in electrical engineeringfrom The Ohio State University, Columbus, in 1981.

He is the Chief Scientist at Motia, Inc., wherehe is involved with smart antennas for wireless sys-tems. In 1981, he joined AT&T and worked in theresearch area for more than 20 years. At AT&T,he was Division Manager of the Wireless SystemsResearch Division of AT&T Labs-Research, Middle-town, NJ. He is an IEEE Distinguished Lecturer forboth the IEEE Communications and the Vehicular

Technology Societies, Area Editor for Transmission Systems for the IEEETRANSACTIONS ON COMMUNICATIONS, and a New Jersey Inventor of theYear for 2001.