Antenna selection for multiple-antenna ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

2669

Correspondence________________________________________________________________________ Antenna Selection for Multiple-Antenna Transmission Systems: Performance Analysis and Code Construction Israfil Bahceci, Student Member, IEEE, Tolga M. Duman, Member, IEEE, and Yucel Altunbasak, Senior Member, IEEE

Abstract—This correspondence studies antenna selection for wireless communications systems that employ multiple transmit and receive antennas. We assume that 1) the channel is characterized by quasi-static Rayleigh flat fading, and the subchannels fade independently, 2) the channel state information (CSI) is exactly known at the receiver, 3) the selection is available only at the receiver, and it is based on the instantaneous signal-to-noise ratio (SNR) at each receive antenna, and 4) space–time codes are used at the transmitter. We analyze the performance of such systems by deriving explicit upper bounds on the pairwise error probability (PEP). This performance analysis shows that 1) by selecting the set of antennas that observe the largest instantaneous SNR, one can achieve the same diversity gain as the one obtained by using all the receive antennas, provided that the underlying space–time code has full spatial diversity, and 2) in the case of rank-deficient space–time codes, the diversity gain may be dramatically reduced when antenna selection is used. However, we emphasize that in both cases the coding gain is reduced with antenna selection compared to the full complexity system. Based on the upper bounds derived, we describe code design principles suitable for antenna selection. Specifically, for systems with two transmit antennas, we design space–time codes that perform better than the known ones when antenna selection is employed. Finally, we present numerical examples and simulation results that validate our analysis and code design principles. Index Terms—Antenna selection, diversity, multiple-input multipleoutput (MIMO) systems, multiple-antenna communications, pairwise error probability (PEP), space–time coding, wireless communications.

I. INTRODUCTION The capacity and error rate performance of a wireless communication system can be dramatically improved by employing multiple-antenna elements at the transmitter and/or at the receiver. Recently, it has been shown that under Rayleigh fading, the capacity of a multiple-antenna link increases almost linearly with the number of transmit antennas provided that there are at least as many receive antennas as transmit antennas and the channel gain between each transmit/receive antenna pair is known to the receiver [1], [2]. To achieve this promised capacity, various space–time coding schemes have been developed. See, for example, [3], [4]. A natural drawback of the multiple-antenna systems is the increased complexity due to the need for multiple radio-frequency (RF) chains. Therefore, there is a considerable effort in exploring multiple-input Manuscript received October 30, 2002; revised June 15, 2003. This work was supported in part by the National Science Foundation under Awards CCR0105654 and INT-0217549, and under the National Science Foundation CAREER Award CCR-9984237. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Yokohama, Japan, June/July 2003. I. Bahceci and Y. Altunbasak are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, 30332-0250 USA (e-mail: bahceci @ece.gatech.edu; [email protected]). T. M. Duman is with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287-5706 USA (e-mail: [email protected]). Communicated by G. Caire, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2003.817455

multiple-output (MIMO) systems that significantly reduce this complexity, but still provide similar capacity and performance improvements. A promising technique to achieve this goal is to select a subset of antennas at the transmitter and/or receiver [5]–[7]. For example, for the case of single antenna selection at the receiver, assuming that the fading is slow, the received signal power can be monitored periodically, and only the signal of the receive antenna observing the largest instantaneous signal-to-noise ratio (SNR) can be fed to the RF chain for processing. Thus, significant reduction in hardware costs can be attained while reaping the benefits of MIMO signaling. The capacity of MIMO systems with antenna selection (only at the receiver) is considered in [5]. The selection is based on the capacity, i.e., those antennas that achieve the largest capacity are selected. The authors evaluate upper bounds on the capacity of the system and conclude that one can achieveacapacity veryclosetothatofthefull-complexity system as long as the number of antennas selected is greater than or equal to the number of transmit antennas. In [6] and [7], transmit antenna selection is studied for systems where limited feedback on the channel state information (CSI) is available to the transmitter. In these systems, channel capacity is used as the optimality criterion and the selection is performed by an exhaustive search. Gorokhov proposes a suboptimal selection algorithm in [8] that decreases the computational complexity significantly. An antenna-selection method seeking the minimization of the error rate using linear receivers is considered in [9]. In [10], the authors considered the use of antenna selection in conjunction with orthogonal space–time block codes. They present antenna selection algorithms for cases when exact channel knowledge or statistical channel knowledge is available. For the case of exact channel knowledge, the expressions for the average SNR and the outage capacity improvement are derived assuming that the selection criterion used is the maximization of the channel Frobenius norm. This selection criterion is equivalent to minimizing the error probability for the case of space–time block codes. Using the outage probability analysis, the authors hint that the diversity gain is preserved for this system. However, they do not explicitly provide an analysis that includes the evaluation of the pairwise error probability (PEP) for the system with antenna selection. Furthermore, these results are valid only for orthogonal space–time codes and cannot be directly applied to the case of more general space–time codes. In [11], Ghrayeb and Duman present an approximate analysis for the PEP of the space–time coding system using antenna selection. They show that the diversity order available is maintained. However, this analysis is based on several approximations. Thus, it is not an explicit proof. Other work on antenna selection is also reported in [12]–[16]. In this correspondence, we present a comprehensive theoretical performance analysis for MIMO systems over quasi-static Rayleighfading channels that use antenna selection at the receiver. We assume that the CSI is known at the receiver, but not at the transmitter, and that the subchannels fade independently. Antenna selection is performed only at the receiver. We base our selection criterion on the maximization of the received signal power, that is, we select the subset of antennas having the largest instantaneous SNRs. Under certain cases, this selection criterion may be optimal in the sense that it may achieve the maximum channel capacity, e.g., for the case of single antenna selection at the receiver. The PEP is central in our approach. We calculate the diversity and coding gains by computing upper bounds on the PEP. For the case of

0018-9448/03$17.00 © 2003 IEEE

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single antenna selection, we present a performance analysis based on the PEP, and demonstrate that the diversity gain with antenna selection is preserved for space–time codes with full spatial diversity. Since it is essential to employ full-rank space–time codes to make sure the diversity order is not reduced with antenna selection, it may be beneficial to use full-rank full-rate space–time code designs recently proposed in [17]–[19]. We also study the performance bounds when the space–time codes do not achieve full spatial diversity, and show that the diversity gain degrades substantially when antenna selection is employed compared to the full-complexity system. Furthermore, we present the PEP analysis when more than one antenna is selected, and generalize our results. We also compute tighter upper bounds on the PEP for the case of double transmit diversity systems. An immediate consequence of the performance analysis is the development of code design principles for space–time codes suitable for the systems employing antenna selection. Based on the bounds on the PEP, we propose two simple design criteria. In particular, for double transmit and double receive diversity systems, we design space–time codes that perform better than the known ones when antenna selection is used. For example, we design a 2-bits/s/Hz eight-state space–time code for two transmit antennas employing 4-ary phase-shift keying (PSK) modulation. We show that, while achieving the same performance with eight-state code in [3] for the full-complexity system, the new code provides about 0.7-dB performance improvement when antenna selection is employed at the receiver. The correspondence is organized as follows: in the next section, we introduce the multiple-antenna channel model, and summarize important results on PEP. In Section III, we compute the upper bounds on PEP for space–time codes when antenna selection is employed for both full-rank and rank-deficient space–time codes. In Section IV, we derive a tighter upper bound for systems with double transmit antenna diversity. We consider the space–time code design with antenna selection in Section V. In Section VI, we present several numerical examples and simulation results that validate our analysis and the new code design principles. We provide the conclusions in Section VII. II. CHANNEL AND SIGNAL MODEL We consider a single-user communication system where the transmitter has M antennas and the receiver has N antennas. Each receive antenna observes a noisy superposition of the M transmitted signals corrupted by Rayleigh flat fading. The subchannels between the transmit/receive antenna pairs are assumed to be independent and identically distributed (i.i.d.). The signal at the nth-receive antenna element at time t, xtn , is given by

xtn =

=M

M m=1

hnm stm + wtn ;

t = 1; 2; 1 1 1 ; l

(1)

where hnm is the complex-valued channel gain from the mth transmit antenna to the nth receive antenna, and wtn is the additive noise at the nth antenna. Both hnm and wtn CN (0; 1). The transmitted signals stm can be chosen from any signal constellation. We assume that the average energy of the transmitted signal at time t is normalized to unity over M antennas so that is the expected SNR at each receive antenna. We can rewrite (1) in vector form as

X=

HS + W M

(2)

where X is the N 2 l received signal vector, S is the M 2 l transmitted signal vector, H is the N 2 M channel transfer matrix, and W is the N 2 l additive white Gaussian noise vector. We assume that the CSI, i.e., H , is known at the receiver, but not at the transmitter.

The PEP conditioned on H is given by [3]

1 erfc ^ 2 4M k HS 0 H S k exp 0 4M k HS 0 H S^ k2 :

P (S ! S^ jH ) =

(3) (4)

Defining the codeword difference matrix B = S 0 S^ and A = BB H , and denoting trf1g as the trace operator, we can write

k HB k2 = trfHBB H H H g = trfHAH H g N M m j nm j2 =

(5)

n=1 m=1

where the last equality follows by using the eigenvalue decomposition A = U 3U H with 3 being a diagonal matrix whose entries (m ) are the eigenvalues of A , and = HU where nm are independent Gaussian random variables. The average PEP for a Rayleigh-fading channel is obtained by averaging the conditional PEP over the statistics of H resulting in

P (S ! S^ ) = EH fP (S ! S^ jH )g

0Nr

4M

r

m=1

m

0N (6)

where r = rank(A) = rank(B ). From this expression, we see that the diversity gain of the coded system is Nr and that the coding gain 1=r r . is m=1 m III. UPPER BOUNDS ON PEP WITH ANTENNA SELECTION In this section, we derive upper bounds on the PEP for the Rayleighfading channel. We start with the case when only one antenna is selected, and then generalize the results to the selection of more than one antenna at the receiver. A. Pairwise Error Probability (PEP) We first consider the case when only one antenna is selected at the receiver. The upper bound on the conditional PEP in (4) is then given by

P (S ! S^ jr^) exp

0 4M r^B B H r^H

where r^ is the row of H having the maximum Frobenius-2 norm. In order to obtain the average PEP, we simply evaluate the expected value of this upper bound with respect to the distribution of r^. That is,

P (S ! S^ ) ER^ fexp(0

=

C

^ H ^H 4M RB B R )g

exp 0 4M rBB H r H

fR (r )drr

where C M is the M -dimensional complex space and fR ^ (r ) denotes ^ . In order to compute fR^ (r ), the probability density function (pdf) of R we introduce the auxiliary event F = fnth row has the largest normg. Then, we can write (7) as shown at the top of the following page, where we use Bayes’ rule, and the fact that the R i ’s are i.i.d. Since all rows have the same statistics, we have

P (F ) =

1:

N

(8)


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f^r (r) = fR (rjF ) = P (FjRnP=(Fr))fR (R) 2 2 2 2 2 2 2 2 n+1 k < kr k ; 1 1 1 ; kR N k < kr k )fR (r ) = P (kR1 k < kr k ; 1 1 1 ; kRn01 k < kr k ; PkR (F ) P (kR1 k2 < kr k2 )N 01 fR (r ) = P (F ) Furthermore, using the statistics of H , we can say that the squared norm of each row, i.e.,

kr i k =

M

2

jhim j ;

m=1

1) Full-Rank Space–Time Codes With Antenna Selection: Let us simplify (11) by the change of variables zi = i ej , which yields

1

is a Chi-square random variable of order 2M , 2 (2M ). Hence, we can write

P (kRi k2 < kr k2 ) = 1 0 e0krk

0 kr k

M

m

1

2

m=0

m!

:

(9)

Finally, the term, fR (r ) in (7) is the unconditional pdf of the nth row, which is simply

1 fR (r ) = M exp(0kr k ): 2

1 0 e0krk

fR^ (r) = N

M

0 kr k

N

m

1

2

m=0

m!

0

1)

C

e0

(

M

0 kr k

C

1 1 0 e0kzk

N

m

2

m=0

m!

0

1)

m

1

2

m=0

m!

N

(

1)

2 1

+

)

m

2

N

(

0

1)

m!

m=0

1 2 1 1 1 M d1 d2 1 1 1 dM :

M

g(v) = 1 0 e0v

1 e0krk :

g(v) =

(12)

0 vm : 1

m=0

m!

1

2M N (M !)N 0

1

v

uM 01 du

0

(13)

1 e0 111 +

+

1

)

0 2 12 + 1 1 1 + M

(

M (N

1 e0

0

(

0

v

+1)

1 dv dv 1 1 1 dvM : 1

v1 + 1 1 1 + vM

(

i=1 M i

111

=1

1)

0

vi

1)

=

0

MN M

M

(v + 1 1 1 + vM )M N 0 =

M (N

(15)

2

We note that 1

1)

1 2 1 1 1 M d1 d2 1 1 1 dM : (14)

= i , we then obtain 1 1 N P (S ! S^ ) 1 1 1 N 0 (M !)

Let vi

0

2

1

In this section, we will derive simple expressions for the upper bounds on the PEP. In Section III-B1, we will consider the case when the space–time codes achieve full-spatial diversity, i.e., the rank of the codeword difference matrices B is M for all codeword pairs. For such codes, we will show that the diversity order achieved with antenna selection is the same as that of the full-complexity system. Then, in Section III-B2, we perform approximate analysis of the PEP for rank-deficient space–time codes, and show that the diversity order is dramatically reduced with antenna selection.

uM 01 e0u du

111

0

1 e0

(11)

B. Simplified Upper Bounds on the PEP for Single Antenna Selection

0

v

Since the value of the integrand in (12) is always greater than zero, we can further upper bound the right-hand side of (11) by substituting 2 2 v M ! in place of g (v ) with v = 1 + 1 1 1 + M to obtain

P (S ! S^ )

It is not easy to evaluate the integral in (11) to obtain a closed form expression. However, it is possible to further upper bound this expression to simplify the analysis as we will show in the next section.

1

(M 0 1)! (M 10 1)! M = vM ! :

1 e0krk drr:

1 e0kzk dzz : M

1

Lemma 1: Define

M

0

0 + 111 + M

vM for v > 0: ; M! Proof: Observing that g (v ) is the incomplete Gamma function,

j j

0 kz k

)

g(v)

z M

+

M

+

the proof follows easily, i.e.,

1

e0

+

We would like to find a simpler expression or bound that directly provides information about the diversity order and coding gain with antenna selection. To this end, we need the following auxiliary lemma.

M

(

0

Then

We can further simplify this expression by using the singular value decomposition of BB H = U 3U H and by applying the change of variable z = r U as

P (S ! S^ ) N

(

rBB r

1 1 0 e0krk

0

1 e0 111

Hence, the upper bound on the average PEP follows as

P (S ! S^ ) N

1 0 e

111

1 0 e0 111

(10)

Substituting (8)–(10) into (7), we obtain the desired pdf as (

1

P (S ! S^ ) 2M N

i 2 f1; 1 1 1 ; N g

2

(7)

M i

vi =1

1 1 1 vi

(16)

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where the indexes ik in vi , k 2 f1; 1 1 1 ; MN 0 M g, take values from the set J = f1; 1 1 1 ; M g. Assume the subscript index j appears lj times among the subscripts of the term vi 1 1 1 vi in (16). Then MN 0M

1 1 1 vi

vi

=

k=1 M =

0

vi

vj )l

(17)

(

j =1

such that M M . Using (16) and (17) in (15), and j =1 lj = MN changing the order of integration and summation, we obtain

P (S ! S^ ) M (M !)NN 01 1 1 1 i =1 i

1

1 0

111

P (S ! S^ )

=1

(

e

M

+1)v

vi )l dv1 dv2 1 1 1 dvM :

Recall that

(

i=1

0

i

=1

P (S ! S^ )

M

M !)N 01 i

(

1

=1

(1 +

For high SNR, it follows that M

P (S ! S^ )

N

M !)N 01 i

(

1 Since

M (l l =1 i

P (S ! S^ )

+ 1) =

N

m!

am+1

l1 ! 1 1 1 lM ! +1

111

(1 +

)l 4M

+1

: (19)

M i

=1

l1 ! 1 1 1 lM !

(l +1)

: (20)

MN , we finally arrive at 1

M ii

i=1

M =1

111

M

l1 ! 1 1 1 lM ! l 1 1 1 l r =1 1

i

r

0 4M

l

(22)

0

N 1 M !)N 01 r i

(

MN 0M

1

=1

)l 4M

111

=1

M i

111

r i

li MN M: i=1 In (22), the term in the square brackets is a function of =4M . Note that there certainly exist i1 ; ; iMN 0M such that ri=1 li = 0. We can regroup the terms in (22) to arrive at P (S ! S^ )

l1 +1 1 1 1 lM +1 (=4M )

M !)N 01

(

111

i=1

0r 1 4M : M l = MN 0 M , and therefore, i=1 i

(18) simplifies to

N

1

111

Using

xm e0ax dx =

M

0

(18)

1

N M !)N 01

(

1

M

1 0 0

with antenna selection based on the largest SNR observed, the diversity gain degrades dramatically as we will demonstrate shortly. Assume that there are r nonzero eigenvalues 1 ; 1 1 1 ; r . The analysis for rank-deficient space–time codes follows the same lines as (12)–(19); thus, we will not repeat it. However, it differs following (19), since some of the eigenvalues vanish in this expression. When the SNR is high, with the assumption that i = 0 for i 2 fr + 1; 1 1 1 ; M g; we can write (from (19))

M i

l1 ! 1 1 1 lM ! l l =1 1 1 1 1 M

0MN : (21) M Inequality (21) clearly shows that a diversity advantage of MN can

1

4

be achieved when only one antenna is selected based on the instantaneous SNR at the receiver. This diversity gain is equal to the diversity order of the system that uses all the antenna elements in the decoding. However, we note again that this is the case only if the space–time code has full spatial diversity. Although the diversity order is preserved, there will be a loss in the amount of coding gain with antenna selection. We will consider this loss later in more detail when we specifically study the case with two transmit antennas. 2) Rank-Deficient Space–Time Codes With Antenna Selection: In the analysis of the previous section, we assumed that the eigenvalues i , i = 1; 2; 1 1 1 ; M , of the matrix B B H were all nonzero. In this case, rank(B ) = rank(BB H ) = M and the maximum diversity advantage, MN , is achieved. When the codeword difference matrix is rank deficient, i.e., rank(B ) = r < M , the diversity gain obtained for the system using all the antenna elements is Nr . On the other hand,

j =0

i=1

j

0j M

0r M

4

4

(23)

r l , and is the sum of the terms multiplying where j = j i=1 i l 0 with the same exponents. For sufficiently high SNR, 4M 0j goes to 0 whenever r l > 0. Thus, we get the term with 4M i=1 i N 1 0r P (S ! S^ ) 0 : (24) r N 0 1 (M !) i 4M i=1 This expression suggests a diversity order of r as opposed to MN . However, since this is only an upper bound on the PEP, we need further analysis. Recall that we have used the Chernoff bound to obtain these upper bounds. We now approximate the PEP with the help of a lower bound on the bound in (11). First, we note the following simple result.

Lemma 2: If g (v ) is as defined in Lemma 1 e0v vM

g(v)

Proof:

g(v) = 1 0 e0v =

M 01 m v

m=0 m! M 01 m v

e0v ev 0 e0v vM M!

:

M!

m=0

=

m!

e0v

1 vm m=M

:

(25)

Using this bound on g (v ) in (12) with v 1 MN 2 P (S ! S^ ) 111 N 01 (M !) 0

=

0

2 12 + 1 1 1 + M , we obtain

1 0 e

1 e0N 111 1 1 1 M 1 1 1 1 M d d 1 1 1 dM : (

1

m!

2

+

+

)

2 1 +

1

2

+

2

M (N 01)

(26)


Evaluation of the above integration results in M M

P (S ! S^ )

N

(M !)N 01 i =1 1 1 1 i

Assuming that i can write

P (S ! S^ )

=

1

N

(M !)N 01 N MN

1

=1

l1 ! 1 1 1 lM ! 1 : (27) l +1 1 1 1 (N + )l +1 (N + ) 4M 4M 0 for i 2 fr + 1; 1 1 1 ; M g; for high SNR, we

M i

=1

111

1 4MN

i=1

M i

0r

r i

l1 ! 1 1 1 lM ! l 1 1 1 l r =1 1

all the rowsg. We also define another auxiliary event Al = fil th row has the minimum norm among r i ; 1 1 1 ; r i g. Then, we can obtain the joint pdf for the rows having the largest norms as follows in (30) at the bottom of the paage, where (a) follows because of the total probability theorem, (b) follows because of Bayes’ rule, and (c) follows becuase of the facts that P (F ; Al ) = 1=C (N; L) and P (Al ) = 1=L. IR (r 1 ; 1 1 1 r L ) is the indicator function 1; if (r 1 ; . . . ; r L ) 2 Rl (31) IR (r 1 ; . . . ; r L ) = 0; else where the region Rl is defined as

Rl = fr ; 1 1 1 ; r L : kr l k < kr k k; k = 1; 1 1 1 ; l 0 1; l + 1; 1 1 1 ; Lg : 1

0

4MN

2673

l

:

(28)

The PEP can thus be obtained by averaging the conditional PEP over this pdf L ~ N! P (S ! S^ ) e0 kHB k

With similar arguments used to obtain (24), we arrive at

P (S ! S^ )

N

(M !)N 01 N MN 0r

1

r i

00

0r 4M :

2

C. Upper Bound for Any M and N When L > Selected

=

(a)

=

(b)

L

kr l k m

m=0

2

(

+

N 0L

m!

+

)

drr1 1 1 1 drrL :

(32)

~ is the L 2 M channel transfer matrix generated by using L where H corresponding rows of H . However, the exact evaluation of (32) over this region is quite difficult. Instead, for its analytic tractability, we will evaluate the integral throughout the whole space which results in a looser upper bound. We also note that becuase of the symmetry of the pdf, the integral over Rl for each l will have the same value. We now consider the evaluation of the integral in the lth term. Before we proceed further, we note that ~ 2 ~ H ~H

kH B k = trfH BB H g = trfH~ U 33(H~ U )H g M = i kci k 2

(33)

i=1

where we used the eigenvalue decomposition of BB H . Here, c i is the ~ U . Using standard integration techniques, the lth term ith column of H of (32) can be rewritten as

1

Il = (N 0NL!)!L!L 0

0

111+u

(u

1e 1 1 0 e0 u 111 (

1 e0(u

111

+

+

111+u

+

+u

)

1

0

111+ (u +111+u ) M 01 (ul1 + 1 1 1 + ulM )k ) )+

k=0

du11 1 1 1 duLM :

N 0L

k!

(34)

(r 1 ; 1 1 1 ; r L jF 0 )

fR ; 111;R R (r 1 ; l=1 L P ( 0; l Ri l=1

L) = C (N; L

(c)

M 01

(N 0 L)!L!L

1 e0 kr k 111 kr k 1 ML

1 Antennas Are

In this subsection, we will extend the performance analysis presented in the previous sections to the more general case of L > 1. For the full spatial diversity system, since selecting a single receive antenna results in full diversity, we expect that the diversity obtained by selecting L out of N antennas will be the same. However, the coding gain may be different. Also, for a rank-deficient system, it will be interesting to observe the effect of the number of antennas selected at the receiver on the overall diversity order achieved. Let us denote the rows of H with the largest L norms by r^1 ; r^2 ; . . . ; r^L . Similar to the case of single antenna selection, let us introduce an event F 0 = fr i ; . . . ; r i have the largest norms among

fR^ ; 111; R^ (r 1 ; 1 1 1 ; r L ) = fR ; 111;R R

1 0 e0kr k

1

(29)

i=1 From (24) and (29), we observe that the Chernoff bound on the PEP is squeezed between two curves that have the same order in the exponent of , which is r = rank(B ). Since the Chernoff bound is tight in the exponential sense, i.e., P EP = K Chernoff bound + O(1), where K is a constant that does not depend on the SNR, the slopes of the exact PEP plot and the Chernoff bound plot will have the same slopes on a log-log scale. Hence, we conclude that the diversity gain of the system with antenna selection is only r , contrary to the case of the full-complexity system, where the diversity gain is Nr . Therefore, to exploit the diversity gain promised by MIMO systems when antenna selection is employed, space–time codes with full spatial diversity should be employed.

R

l=1

F Aj L l=1

= (N 0NL!)!L!L

P (kRi L l=1

1 1 1 ; r L jF 0 ; Al )P (Al ) = r1 ; 1 1 1 ; Ri = rL )fR ; 111;RR (r1 ; 1 1 1 ; rL )P (Al ) P (F 0 ; Al )

k

2

< kr l k2 ; 1 1 1 ; kRi

1 0 e0kR k

M 01 m=0

kr l k m 2

m!

k

2

< kr l k2 )

N 0L

L

j =1

fR (Rj )IR (r 1 ;

IR (r 1 ; 1 1 1 ; r L )

111

; rL)

e0(kr k +111+kr k ML

)

(30)

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will not repeat these steps. If we have r < M nonzero eigenvalues, one can obtain the upper bound on the PEP as

After some manipulations, we obtain

Il

=

1

N! N 0 L)!L!L

(

0 L

dudi

1

0 e0 u 111 (

1

+u

+

1 e0 u 111 (

+u

+

0

)

e

111

0

P (S ! S^ )

0

u

i=1 d=1; d6=l

1

111

0

1e M 1

1

)

0

M 01 k=0

u

1 0 e

ul1 + 1 1 1 + ulM )k k!

(

N 0L

i=1

(35)

1 1+

:

4M

(36)

The second integral is very similar to the one that we obtained for the case of single antenna selection and can be evaluated as described in (2) Section III-B. Using Lemma 1 in (35), we can upper-bound Il , as

Il (2)

M

1

N 0L (M !)

i

=1

111

M i

l1 ! 1 1 1 lM ! l +1 1 1 1 l +1 M =1 1 0M (N 0L+1) 1 4M : (37)

Using these results, we obtain

Il (N 0 L)!LN!L! (M !)N 0L M

1

i

=1

111

M i

L01 1

i=1

M 1 + 4M

l1 ! 1 1 1 lM ! l +1 1 1 1 l +1 M =1 1

0M (N 0L+1) : 4M

1

(38)

Note that this resulting bound is independent of l. Hence, substituting (38) into (32), and performing simple algebraic manipulations for the high-SNR region, we finally arrive at

P (S ! S^ )

1

N! N 0 L)!L!(M !)N 0L

(

M i

=1

111

M i

1

M i=1

i

l1 ! 1 1 1 lM ! l l =1 1 1 1 1 M

4

Similarly, the lower bound on the Chernoff bound (an approximation to the PEP) is given by (using (29))

P (S ! S^ )

L01

M

i

0Lr : M

000

(40)

dul1 1 1 1 dulM :

=

L

u

Denote the first integral in this expression by Il and the second inte(2) (1) gral by Il . The result of the first integral Il is (1)

1

r i=1

(1)

Il

N! N 0 L)!L!(M !)N 0L

(

L

0MN : (39) 4M

The inequality in (39) clearly indicates that the diversity order is MN when a selection of L antennas out of N antennas is made available at the receiver and the space–time code achieve full spatial diversity. Note also that this expression reduces to the expression in (21) when L = 1, and takes the same value as the full-complexity system for L = N . Clearly, (21) is also an upper bound on the PEP for the case under consideration. However, the bound in (39) is tighter than (21) and provides a better assessment of the coding gain of the system. For rank-deficient space–time codes, the analysis is very similar to the case when L = 1, which is described in Section III-B2. Hence, we

N0

(

1 000

N!

L)!L!(M !)N 0L 0Lr

0L

r L (N i=1 i

MN 0r

+ 1)

: (41) M In (40) and (41), 000 and 0000 are the coefficients that can be computed in a similar fashion to that used to compute 0 and 00 in (24) and (29). 0

4

The upper bound in (40) and the approximation in (41) obtained for the rank-deficient space–time codes indicate that the diversity order is Lr . To verify this approximation, we have conducted extensive simulations, and we have observed that the diversity gain is exactly Lr for this case as the expression suggests. These results indicate that, with antenna selection, the diversity order that can be obtained when the underlying space–time code is rank deficient is only Lr while the diversity order of the full-complexity system is Nr . Such a degradation in the achievable diversity order may be somewhat counterintuitive since one might have expected the results to be Nr as opposed to Lr as an extension of the full-rank space–time case. To explain this further, let us consider a space–time code with rank r < M . Such a code will achieve full spatial diversity for a system with r transmit antennas. Assume this code is used for a system having M transmit antennas, which can be trivially done by transmitting the same (dummy) symbols from the additional M 0 r antennas. Hence, these antennas will not give the receiver any useful information that will improve the diversity gain of the system. However, when we perform antenna selection, the channel gains from these antennas may enforce the selection of a “bad” subset of receive antennas, which will degrade the performance of the system severely. Even if this is the case for a fraction of time, asymptotic behavior of the PEP will depend on these falsified selections resulting in the reduced diversity order. Clearly, the performance with antenna selection will improve if we do not use the channel coefficients from the last M 0 r antennas for selection, in which case a diversity order of Nr will be achieved. However, this means that the spatial structure of the underlying space–time code has to be used in the selection process. For this example, this is simple. However, in general, it is not an easy task even for simple non-full-rank space–time codes. IV. TIGHTER UPPER BOUNDS FOR SYSTEMS WITH TWO TRANSMIT ANTENNAS Although the analysis in the previous section accurately predicts the diversity order, the upper bounds derived are not very tight. In this section, we will evaluate the exact value of the bound given in (11) to obtain a tighter bound for the case of double transmit antenna diversity. For M = 2, (11) reduces to

P (S ! S^ ) N

C

e0 ( jz^ j + jz^ j

)

1 0 e0 jz j jz j jz j 1 e0 jz j jz j dz dz : ( ^

1

1

2

( ^

+ ^

+ ^

)

)

(1 + ( ^1

^1

^2

2

+

jz j ^1

2

(N ))

01) (42)


z^1

Replacing the complex-valued integration variables z^1 and z^2 with = 1 ej and z^2 = 2 ej , we arrive at

1

P (S ! S^ ) 4N

0

0

1 0 ( e

+

P (S ! S^ ) 2

(N 01) (43)

since d2 d2 = 42 and the integrand does not depend on 1 or 2 . By a change of variables in (43) with 1 = y cos and 2 = y sin , we obtain =2

0

0

1 0 e

( cos + sin ) (N 01)

1 1 0 e0y 0 e0y y2 1 e0y y2 cos sin y dy d:

Assuming 1

1

1 0 e0y 0 e0y

0

1e0y

y

e0=8y 0 e0=8y =4(2 0 1 )

1

8N (2 0 1 )

0

When the eigenvalues are equal, i.e., 1 is given by

1

P (S ! S^ ) N

0

dy: (45)

= x to obtain

(1 + x log(x=e))(N 01)

1

x =8 0 x =8 dx: (46)

= 2 = , the upper bound

(1 + x log(x=e))(N 01)

x=8

log(x)dx: (47)

We can simplify (46) or (47) by using a binomial expansion and integrating each term separately to obtain the following closed-form expression (see Appendix):

P (S ! S^) N

N

01

n=0

C (N 0 1; n)(01)n

Pe; no

1 + n (an + bn ) a2 b2

an bn

n(n 0 1) a2n + an bn + bn2 + a3n bn3 n! (ann + 1 1 1 + bnn ) + ann+1 bnn+1

n n

+ 111

P (S ! S^ ) 2

P (S ! S^ )

6

sel

(51)

1 2

2

1

:

(52)

It is clear that (52) reaches its minimum value when the eigenvalues = 1; 2, associated with the codeword difference matrices, are equal to each other. This result can be used as an additional criterion to optimize the design of space–time codes for systems using antenna selection. When we have N = 3, we obtain the upper bound as shown in (53) at the bottom of the page, where a = 2 + 1 =8 and b = 2 + 2 =8:. When the SNR is high, the bound is given by

4 + 31 2 + 1 1 1 + 42 P (S ! S^) 18 1 (=8)06 : 51 52 Similarly, for N = 4, the upper bound for high SNRs (keeping only the highest order terms) is given as

6 + 51 2 + 1 1 1 + 62 P (S ! S^) 360 1 (=8)08 : 71 72 Example 2: M = 2, N = 3, and L = 2 Antennas Selected: Assume that there are two transmit and three receive antennas and two are selected based on the SNRs observed. The expression (32) can be rewritten as

(48)

1

~ ~ e0 trfHU 3(HU ) g

kr k 0, we can write

)

1 1 0 e0( + ) (1 + (12 + 22)) 1 e0( + ) 1 2 d1 d2

P (S ! S^ ) 4N

2675

= U 3U H . Let ~ U = 1 = H 2

(54)

11 12 : 21 22

Then kr i k = k i k; i = 1; 2. Transforming the integral into polar

(3a + 1)b5 + (3a2 + 7a + 2)b4 + (3a3 + 7a2 + 2a)b3 4 3 2 +(3a + 7a + 2a 0 6a 0 2)b2 + (3a5 + 7a4 + 2a3 0 6a2 0 5a 0 1)b + a5 + 2a4 0 2a2 0 a (a 0 1)(b 0 1)a2 b2 (a + 1)3 (b + 1)3

(53)

2676


COMPARISON

OF

KNOWN

TABLE I NEW 4-PSK SPACE–TIME CODES WITH THEORETICAL CODING GAINS. GM: GENERATOR MATRIX, FC: FULL-COMPLEXITY SYSTEM, S: SYSTEM WITH ANTENNA SELECTION

AND

coordinates with ij = ij e , we get

e0

P (S ! S^ ) 3

( (

+

)+ (

+

1 1 0 e0 (1 + + 1 e0 1 d d d d 1 d : (

(

+

11

12 2

+

+

21

22 4

)

+

11

2 21

2 22 )

)

12

21

• To achieve the maximum coding gain, the value of the coefficient in (21) must be minimized for all codeword pairs. We can also refer to (39) for the coding gain coefficient.

))

1

4

22

(55)

0

Unfortunately, exact evaluation of the integral over the region

R=

2 2 2 2 11 > 0; 12 > 0; 21 > 0; 22 > 0 : 11 + 12 >21 + 22

as indicated previously, is quite difficult. However, we can evaluate the integral over the whole space. After some manipulations, we obtain (assuming 2 > 1 )

P (S ! S^ )

24

(2 0 1 )(1 + 1 =8)(1 + 2 =8)

1

1

0

(1 + x log x=e)

x

=8

0 x =

8

dx: (56)

Note that the integral in this expression is very similar to the integral obtained for the case M = N = 2 and L = 1. Using the previous results, we arrive at the following upper bound for high SNRs:

P (S ! S^ ) 3

21 + 22 + 1 2 41 42

06 :

(=8)

(57)

V. SPACE–TIME CODE DESIGN WITH ANTENNA SELECTION For the full-complexity system, two design criteria based on the upper bound on PEP have been proposed in [3]: to maximize the diversity gain, the minimum of the ranks of the codeword difference matrices B , and to maximize the coding gain, the minimum of the determinants of the matrices B B H should be maximized. The bounds developed in the previous sections can be used to develop similar criteria for the design of space–time codes for use with antenna selection at the receiver. We propose the following: • To achieve the maximum diversity gain MN , the underlying space–time code should be full-rank, i.e., rank(B ) = M:

The rank criterion imposes a significant constraint on the design of space–time codes when they will be used on MIMO systems employing antenna selection. The tradeoff between the diversity advantage and the transmission rate becomes more essential for such an application. Hence, the recently proposed full-rank and full-rate space–time code design techniques (i.e., [17]–[19] may be viable alternatives for systems with antenna selection. Clearly, instead of (21), we can also use the tighter bounds developed for specific M; N; and L. For example, for the case of double transmit diversity, (50) can be used. In this case, we should generate the code that maximizes the coding gain that is defined as the minimum of

Gain =

31 32 2 ( + 22 + 1 2 ) 2 1

over all codeword pairs for the case of M = N = 2. Let us give several examples using a systematic code search technique similar to the method presented in [20]. We first transform the trellis representation of the code into generator matrix form, and then perform an exhaustive search systematically using the generator matrix. For brevity, we refer the reader to [20] for the details of the method. Here, we briefly present our search results. Some of the known codes (the generator matrix forms) together with new codes using 4-, 8-, and 16-state trellises are shown in Table I. In Table I, we included the coding gains for both the full-complexity system and the system with antenna selection. Two notes are in order: first, the codes designed (by the proposed method) for the system with antenna selection performs better than known codes when they are used with antenna selection. This is because the already existing codes are designed for the full-complexity system, not for systems with antenna selection. Second, the proposed codes, which are designed for the system with antenna selection, also provide improved (or, at least the same) coding gains for the full-complexity system compared to the known codes. That is, the new design criteria do not conflict with those of full-complexity systems, they only impose additional constraints. VI. EXAMPLES In this section, we present several numerical examples and simulation results to clarify the theoretical analysis performed. We consider


2677

Fig. 1. PEP comparison between the full-complexity system and the system using single receive antenna observing maximum average SNR, L = 1. The codeword pairs are from 2-bits/s/Hz space–time trellis code using 4-PSK, eight-state trellis [3].

3;

M

=2

; N

=

three cases, i.e., upper bounds on the PEP, space–time code design with antenna selection, and the case of rank-deficient space–time codes separately.

tems achieve the same diversity advantage. There is about a 2-dB loss in coding gain as a result of not fully exploiting the receive antenna elements. However, the cost is significantly reduced.

A. Theoretical Upper Bounds on PEP

B. Space–Time Codes With Antenna Selection

We now evaluate the bounds given in Sections III-B1 and III-B2 for several codeword pairs that are selected from the codes developed in [3]. We also provide the actual frame error rates (FERs) for the space–time codes considered. In Figs. 1 and 2, we present the PEP bounds for the system with M = 2, N = 3, and L = 1; 2. We select the codeword pairs from the 2-bits/s/Hz eight-state space–time trellis codes using 4-PSK modulation (with M = 2). This code provides a diversity advantage of 6 [3], i.e., full spatial diversity. The two codewords considered differ in three consecutive symbols. The following observations are in order: 1) the simulated PEPs and the PEPs obtained numerically exactly match for both the full-complexity system and the one using antenna selection, 2) for L = 2, the Chernoff bound obtained by integration of the bound over the whole space (evaluated analytically) rather than over the support of the actual pdf (evaluated numerically) are very close to each other (differing by 0.8 dB for this example), 3) for both the cases of L = 1 and L = 2, the upper bound and the approximations to the Chernoff bound are very tight at high SNR, and, finally, 4) the performance of the system with optimal selection (that maximizes capacity) is only slightly superior to the selection we considered, i.e., optimal selection performs only slightly better for L = 2. The actual FER comparisons for M = N = 2 antennas are provided in Fig. Fig. 3 (see the solid lines). In this example, we present the simulation results (as opposed to the theoretical results) for the space–time trellis code considered in the previous example. The channel is assumed to be constant for a period of 130 transmissions. We observe that the slopes of the FER plots are the same, implying that both sys-

In Fig. 3, we also present a performance comparison between the space–time code designed in [3] and the code we designed for the system with antenna selection. Both space–time codes achieve 2 bits/s/Hz and they use an eight-state trellis with 4-PSK modulation. The coding gains of both codes for the full-complexity system are the same, i.e., the gains are 12, and, therefore, they have the same performance. However, for the system with antenna selection, we observe a significant improvement with the new space–time code. The improvement predicted by the theoretical coding gains shown in Table I is 0.88 dB which is very close to the 0.7 dB improvement observed by the simulations.

p

C. Effect of Rank-Deficiency on the Performance With antenna selection, we showed that the diversity order is preserved provided that the underlying space–time code achieves full spatial diversity. However, the upper bound analysis for rank deficient space–time codes has revealed that the diversity order is expected to degrade dramatically. To illustrate this point further, in Fig. 4, we depict the bounds on PEP together with the exact values of PEP obtained by simulations when M = N = 2 and the space–time code is rank deficient, i.e., the codeword difference matrix has rank 1. We observe from the slopes of the exact PEP that the diversity gain is only r = 1for antenna selection system while it is r = 2 for the full-complexity system. The bounds obtained for the system employing antenna selection also have the same asymptotic slopes as the exact value of the PEP. However, these bounds are not as tight as the ones for the full-rank space–time code presented earlier.

2678


Fig. 2. PEP comparison between the full-complexity system and the system using single receive antenna observing maximum average SNR, L = 2. The codeword pairs are from 2-bits/s/Hz space–time trellis code using 4-PSK, eight-state trellis [3].

3;

M

=2

; N

=

Fig. 3. FER comparison between 1) (solid lines) the new code and the known space–time trellis code in [3] when M = 2, N = 2, the 2-bits/s/Hz space–time trellis code using 4-PSK, eight-state trellis, and 2) (dashed lines) the full-complexity system and the system using antenna selection when M = 3 and N = 2, rank= 2.


2679

Fig. 4. Bounds for rank-deficient space–time code. PEP comparison between the full-complexity system and the system using single receive antenna observing maximum average SNR, M 2; N = 2. For rank-deficient code, rank(B ) = 1:

=

For M = 3 and N = 3, Fig. 5 depicts the PEP bounds for a codeword pair from a rank-deficient space–time code. We present two cases in this figure: 1) selection of a single antenna 2) selection of two antennas. We present exact PEP (obtained through simulations) as well as the bounds we derived for all cases. We observe from these plots that: 1) the diversity order with antenna selection is given by Lr , i.e., the achievable diversity orders are 2 and 4 with L = 1 and L = 2, respectively; 2) the optimal selection (that maximizes the capacity) performs slightly better than the selection based on energy for L = 2; and finally, 3) the bounds become tighter as L increases (i.e., as we select more antennas), and become looser as r decreases (i.e., the rank of the codeword difference matrix reduces). The FERs for a rank-deficient space–time code whose codeword difference matrices of rank-2 are shown in Fig. 3 (see dashed lines). In this example, M = 3 and the maximum diversity order for the full-complexity system with N = 2 receive antennas is 2 2 N = 4, which can be seen from the solid line in the figure. The dotted line shows the FER when L = 1. We observe that diversity gain in this case is the same as that would be obtained if we had used only one receive antenna, i.e., it is 2. However, there is a 3-dB improvement in the coding gain over the system with only one receive antenna.

we computed tighter upper bounds in closed form. We have also determined guidelines for optimal space–time code design with antenna selection, and presented several simple codes forthe case of two transmit antennas. We have also provided extensive numerical examples, and simulation results, and observed that the results are in agreement with the theoretical analysis. APPENDIX A. Upper Bound in Closed Form for M = 2 and Any N When L = 1 Applying the binomial expansion in (46), we obtain

P (S ! S^ )

C (N 0 1; n)

0 ) n 1 logn (x=e)

(2

1

=0

xn+ =8

1 0

0 xn

+

=8

dx: (58)

Letting x=e = u and then log (u) = t, we obtain

P (S ! S^ )

(2

1

VII. CONCLUSION We investigated antenna selection for a MIMO wireless system using space–time coding. We considered the case when only the receiver knows the CSI and antenna selection (based on received signal power) is performed only at the receiver. We have analytically shown that the diversity gain does not change and hence, we can exploit the full diversity advantage promised by the MIMO system that uses all available antenna elements, provided that the space–time code employed has full spatial diversity. For rank-deficient systems, we have shown that the diversity gain with antenna selection deteriorates significantly compared to the full-complexity one. Furthermore, for double transmit diversity,

N 01

8N

N 01

8N

0 ) n 1

=0

en+1+ =8

0

en+1+ =8

C (N 0 1; n)

01 01 01 01

tn et(n+1+ =8) tn et(n+1+ =8) dt :

Now, consider

In =

01 01

Integration by parts gives us

In = (01)n

tn eat dt:

e0a a

0 na In0 : 1

(59)

2680


Fig. 5. PEP for rank-deficient space–time code: M

= 3,

N

= 3, and

L

= 1 or 2. For the codeword pairs, rank(B ) = 2:

Solving this recurrent equation for In with the initial value of I0 e , we get a

In = (01)n

e0a a

1+

n a

+

n! n(n 0 1) + 111 + : a2 an

=

(60)

Using a = n + 1 + 1 =8 and b = n + 1 + 2 =8 in (59) along with (60), the closed-form expression is obtained as

P (S ! S^)

N01 N C (N 0 1; n)(01)n (2 0 1 ) n=0 n n(n 0 1) ! 1 1 an + a2 + a3 + 1 1 1 ann+1 n n n n n(n 0 1) n! 1 0 bn + b2 + b3 + 1 1 1 bn+1 n n n 8

:

(61)

Further simplification can be made after regrouping the terms in the summation and then using xn 0 y n = (x 0 y )(xn01 + xn02 y + 1 1 1 +

yn01 )

P (S ! S^)

N 1

N01 n=0 1

C (N 0 1; n)(01)n

an bn

+

n (an + bn ) n(n 0 1) a2n + an bn + bn2 + a2n bn2 a3n bn3 +

111

+

n! (ann + 1 1 1 + bnn ) : ann+1 bnn+1

(62)

ACKNOWLEDGMENT The authors would like to thank anonymous reviewers for their insightful comments and especially suggestions on Lemma 1 that helped us make our bounds tighter.

REFERENCES [1] E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell Labs. Internal Tech. Memo, June 1995. [2] G. J. Foschini Jr. and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, no. 2, pp. 311–335, March 1998. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [4] A. Stefanov and T. M. Duman, “Turbo coded modulation for systems with transmit and receive antenna diversity over block fading channels: System model, decoding approaches and practical considerations,” IEEE J. Select. Areas Commun., vol. 19, pp. 958–968, May 2001. [5] A. F. Molisch, M. Z. Win, and J. H. Winters, “Capacity of MIMO systems with antenna selection,” in Proc. Int. Conf. Communications, 2001, pp. 570–574. [6] D. A. Gore, R. U. Nabar, and A. Paulraj, “Selecting an optimal set of transmit antennas for a low rank matrix channel,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing, 2000, pp. 2785–2788. [7] S. Sandhu, R. Nabar, D. A. Gore, and A. Paulraj, “Near optimal antenna selection of transmit antennas for a MIMO channel based on Shannon capacity,” in Proc. 34th Asilomar Conf., Nov. 1999, pp. 567–571. [8] A. Gorokhov, “Antenna selection algorithms for MEA transmissions,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Sisgnal Processing, vol. 3, 2002, pp. 2857–2860. [9] R. Heath and A. Paulraj, “Antenna selection for spatial multiplexing systems based on minimum error rate,” in Proc. IEEE Int. Control Conf., 2001, pp. 2276–2280. [10] D. A. Gore and A. J. Paulraj, “MIMO antenna subset selection with space-time coding,” IEEE Trans. Signal Processing, vol. 50, pp. 2580–2588, Oct. 2002. [11] A. Ghrayeb and T. M. Duman, “Performance analysis of MIMO systems with antenna selection over quasi-static fading channels,” IEEE Trans. Vehic. Technol., vol. 52, pp. 281–288, Mar. 2003. [12] N. Kong and L. B. Milstein, “Combined average SNR of a generalized diversity selection combining scheme,” in IEEE Int. Conf. Communications, June 1998, pp. 1556–1560. [13] M. Z. Win and J. H. Winters, “Analysis of hybrid selection-maximal ratio combining in Rayleigh fading,” IEEE Trans. Commun., vol. 47, pp. 1773–1776, Dec. 1999.


[14] R. Nabar, D. Gore, and A. Paulraj, “Optimal selection and use of transmit antennas in wireless systems,” in Proc. Int. Conf. Telecommunications (ICT’00), Acapulco, Mexico, 2000. [15] D. A. Gore and A. Paulraj, “Space-time block coding with optimal antenna selection,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing, 2001, pp. 2441–2444. , “Statistical MIMO antenna sub-set selection with space-time [16] coding,” in Proc. Int. Conf. Communications (ICC’2002), 2002, pp. 641–645. [17] H. El Gamal and M. O. Damen, “An algebraic number theoric framework for space-time coding,” in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, June 2002, p. 132. [18] S. Galliou and J.-C. Belfiore, “A new family of full rate, fully diverse space-time codes based on Galois theory,” in Proc. IEEE Int. Symp. Information Theory, Lausanne, Switzerland, June 2002, p. 419. [19] X. Ma and G. B. Giannakis, “Layered space-time complex field coding: Full diversity with full-rate, and tradeoffs,” in Proc. Sensor Array and Multichannel Signal Processing Workshop, 2002, pp. 442–446. [20] S. Baro, G. Bauch, and A. Hansmann, “Improved codes for space-time trellis codes,” IEEE Commun. Lett., vol. 41, pp. 20–22, Jan. 2000.

On Performance Limits of Space–Time Codes: A Sphere-Packing Bound Approach Majid Fozunbal, Student Member, IEEE, Steven W. McLaughlin, Senior Member, IEEE, and Ronald W. Schafer, Fellow, IEEE

Abstract—A sphere-packing bound (SPB) on average word-error probability (WEP) is derived to determine the performance limits of space–time codes on Rayleigh block-fading channels under delay and maximum energy constraints. Two other explicit bounds, a looser bound and a tight approximate bound, are also derived to provide more intuition on how the system parameters affect the performance limits. Moreover, it is shown that as the block length grows to infinity, the SPB converges to the outage probability, and the asymptotic behavior of performance limits is determined by the outage probability. Index Terms—Fading channels, multiple antennas, outage probability, performance limits, space–time codes.

I. INTRODUCTION In this correspondence, a sphere-packing bound (SPB) is derived on the average word-error probability (WEP) of codes over Rayleigh block-fading multiple-input–multiple-output (MIMO) channels. The SPB defines a bound on the performance (in an average WEP sense) of all possible codes satisfying the design constraints. Thus, it can be used to specify the merit of a code in a sense that how far the code performs from the performance limits. Moreover, it can be used to obtain some intuition on how system parameters affect the performance limits. Because the SPB does not have a closed form, inequalities are

2681

used to obtain two other bounds with explicit expressions, namely, a looser bound and a tight approximate bound. The results presented here show that for data rates less than the ergodic capacity [1], the performance limits improve significantly if a code spans a larger number of fading blocks. On the other hand, increasing the block length (length of portion of the code within one block) improves the performance limits marginally. In fact, as block length grows to infinity, the SPB converges to the outage probability (see [2] and [3] for definition); and the asymptotic behavior of performance limits of space–time codes are determined by the outage probability. Hence, for large block lengths, the SPB determines the outage probability. To show how effective the SPB is in determining the performance, it is shown that for a 2 2 2 MIMO system, the 64-state space–time trelliscoded modulation developed in [4], performs 2.5 and 1.4 dB away from the performance limits for data rates of 3 and 2 bits/s/Hz, respectively. In Section II, the system model is introduced. In Section III, it is shown that for sufficiently large dimension, the received signal space is bounded within a hyperellipsoid with an arbitrarily high probability. The derivation of the three sphere-packing lower bounds is given in Section IV. In Section V, it is shown that outage probability determines the asymptotic behavior of the performance limits. Finally, we conclude in Section VI. II. SYSTEM MODEL We assume a wireless communication system employing n transmit and m receive antennas. The channel is assumed to be memoryless, MIMO block fading. The channel gains remain constant throughout the duration of each fading block, but they change independently from one block to another. The fading is assumed to be Rayleigh, where the channel gains are independent and identically distributed (i.i.d.) complex normal with zero mean and unit variance. It is assumed that the delay constraint is equal to the duration of K fading blocks (fading intervals), and the codewords (words) span K fading blocks. In each fading block, the channel is used L times, which is called the block length of the code. At each channel use, all antennas are used simultaneously, and n complex symbols are transmitted through n transmit antennas. Let Xk be an n 2 L matrix denoting the nL symbols that are transmitted over the k th fading block. The channel output is described as

k = 0; . . . ; K 0 1

Yk = Hk Xk + Zk ;

variate i.i.d. complex-normal m 2 L noise matrices. To simplify the notation of the channel equation, we define

1 X=

X0 X1 .. .

Z0 Z1

1 ; Z=

ZK 01 H0

1 H=

0

.. .

1 ; Y =

.. .

XK 01 Manuscript received September 7, 2002; revised June 7, 2003. This work was supported in part by Texas Instruments. The material in this correspondence was presented in part at the IEEE International Conference on Communications, Anchorage, AK, May 2003. The authors are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; swm @ece.gatech.edu; [email protected]). Communicated by B. Hassibi, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2003.817453

(1)

01 where fHk CN (0m2n ; Imn )gK k=0 is a sequence of multivariate i.i.d. complex-normal m 2 n channel matrices over K fading blocks. 01 The noise fZk CN (0m2L ; 2 ImL )gK k=0 is a sequence of multi-

0

H1 .. .

111 111 ..

.

111

0

0

=

HX + Z

Y0 Y1 .. .

YK 01 0 0

.. .

:

HK 01

Thus, (1) can be rewritten as

0018-9448/03$17.00 © 2003 IEEE

Y

(2)