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Capacity of Space-Time Block Codes in MIMO Rayleigh Fading Channels With Adaptive Transmission and Estimation Errors Amine Maaref, Student Member, IEEE, and Sonia Aïssa, Senior Member, IEEE
Abstract—Orthogonal space-time block coding (STBC) has recently raised a lot of research interest due to its inherent mathematical feature that enables simple linear decoding at the receiver and provides full diversity over the multiple-input multiple-output (MIMO) fading channel. In this paper, we derive general closed-form expressions for the Shannon capacity achieved by this transmit-diversity scheme over Rayleigh fading channels under adaptive transmission and channel-estimation errors. Adaptive transmission can be performed on a frame-byframe basis, provided that a channel state information (CSI), consisting of the signal-to-noise ratio (SNR) level as estimated by the receiver, is fed back to the transmitter, thereby allowing for different compromises between the achievable capacity and the corresponding implementation complexity. The closed-form capacity formulas, derived for different power- and rate-allocation policies, are expressed in terms of the number of transmit and receive antennas, the code rate of the STBC mapping, and a single parameter capturing Gaussian channel-estimation errors. Numerical results showing the effects of these parameters on the capacity of STBC subject to the adaptive-transmission policies under consideration are provided. Index Terms—Adaptive transmission, channel-estimation errors, multiple-input multiple-output (MIMO) systems, Shannon capacity, space-time block coding (STBC).
I. I NTRODUCTION
F
UTURE wireless systems are likely to be equipped with multiple antennas. Hence, adaptive-transmission techniques already implemented in single-input single-output (SISO) channels have to be extended to encompass the features of the multiple-input multiple-output (MIMO) fading channel. In an MIMO system, the use of multiple antennas adheres to one of two distinct approaches that seek to improve either the diversity gain or the information rate of the system. These two approaches are commonly referred to as MIMO diversity and spatial multiplexing, respectively [1]. Herein, we consider the former approach and investigate the Shannon capacity of MIMO Rayleigh fading channels using orthogonal space-time block coding (STBC). The Shannon capacity of a channel characterizes its maximum achievable rate, given no delay or Manuscript received February 25, 2004; revised July 8, 2004; accepted August 11, 2004. The editor coordinating the review of this paper and approving it for publication is R. Murch. This work was supported in part by the National Sciences and Engineering Research Council of Canada (NSERC) Discovery Research Grants Program. Parts of this work were presented at the IEEE Globecom’04 Conference, Dallas, TX, Nov. 29–Dec. 3, 2004. The authors are with INRS-EMT, University of Quebec, Montreal, QC H5A 1K6, Canada (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TWC.2005.853968
complexity constraints for an arbitrarily small bit error rate. In practice, different power- and rate-allocation policies that allow for different compromises between the achievable capacity and the corresponding implementation complexity can be performed when a channel state information (CSI), consisting of the signal-to-noise ratio (SNR) as estimated by the receiver, can be made available to the transmitter. In this paper, we derive closed-form expressions for the Shannon capacity of MIMO Rayleigh fading channels using STBC for three adaptivetransmission policies: 1) optimal power and rate adaptation (opra); 2) optimal rate adaptation (ora), given constant transmit power, and 3) channel inversion with fixed rate adaptation (cifr). Space-time coding is a transmit-diversity technique that consists of spreading the information across the transmit antennas in order to maximize the diversity gain in fading channels [1]. It applies to both MIMO and multiple-input single-output (MISO) systems and is especially attractive when the receiver cannot be equipped with more than one antenna. One family of space-time codes is the space-time trellis code (STTC), which provides both diversity and coding gains [2]. However, a major drawback of this technique lies in its encoding/decoding complexity, which grows exponentially as a function of the diversity order and the transmission rate [2]. A way of avoiding such a complexity is to use linear orthogonal STBC which is a transmit-diversity scheme first discovered by Alamouti in [3]. The Alamouti transmission paradigm already adopted in the wideband code division multiple access (WCDMA) and cdma2000 standards [4] was generalized by Tarokh et al. in [5] to an arbitrary number of antennas (MIMO systems) and further shown to outperform some of the best known STTCs when concatenated with outer additive white Gaussian noise (AWGN) trellis codes [6]. While the approach to STBC in [5] focused on the theory of orthogonal designs, a rather interesting approach from an SNR-maximization point of view [7] also led to the STBC diversity scheme. The latter approach was exploited by Sandhu and Paulraj in [8] to assess the gap in performance between the nonergodic STBC channel capacity and the true MIMO channel capacity. In this work, we draw upon the analysis in [8] to derive closed-form expressions for the Shannon capacity of MIMO Rayleigh fading channels using STBC under different power- and rate-adaptation policies. The Shannon capacity of an SISO frequency-nonselective slowly fading channel with CSI at both the transmitter and receiver was first derived by Wolfowitz [9, Th. 4.6.1]. This result, which is only applicable to fading channels with a finite set of
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MAAREF AND AÏSSA: CAPACITY OF SPACE-TIME BLOCK CODES IN MIMO RAYLEIGH FADING CHANNELS
states, was then generalized to an infinite set of states, under the consideration of different power- and rate-allocation policies, by Goldsmith and Varaiya in [10]. Therein, it was shown that when CSI is available at the transmitter, simultaneous adaptation of the transmit power and rate to the variations of the fading becomes an option. In [11], Alouini and Goldsmith capitalized on the seminal work [10] pertaining to SISO channels, in order to derive exact capacity expressions for maximal ratio combining (MRC) receive diversity under Rayleigh fading. Given the emergence of systems with multiple antennas at both sides of the communication link, it is of paramount importance to consider the problem in the more general framework of MIMO systems. In this contribution, we investigate the capacity of a particular MIMO setting, i.e., orthogonal STBC, under adaptive transmission in Rayleigh fading, by adopting an equivalent SISO-channel approach. Our capacity results may serve as upper bounds on the achievable spectral efficiency for any practical application of adaptive modulation using STBC in MIMO Rayleigh fading channels [12]. Furthermore, it can be shown that the results derived in [11] can be retrieved from our formulas when taking the special case of one transmit antenna, full-rate STBC, and perfect channel estimation, albeit STBC does not apply in the case of a single-transmit-antenna system. Also, important results pertaining to the cutoff SNR in the opra policy are derived. In particular, it is shown that this cutoff SNR strictly increases as a function of the average SNR per receive antenna. In our study, the effects of imperfect channel estimation are taken into consideration, and general expressions for the Shannon capacity under adaptive transmission are derived, from which those corresponding to perfect channel estimation can easily be extracted. The remainder of this paper is organized as follows. First, in Section II, a system-model description is presented. Section III considers an equivalent SISO-channel approach and derives the probability density function (pdf) of the output SNR resulting from employing the STBC diversity scheme subject to Gaussian channel-estimation errors at the receiver. Then, in Section IV, the Shannon capacity of STBC in MIMO Rayleigh fading channels, assuming channel knowledge at the receiver only, is derived. Section V considers the case where CSI is fed back to the transmitter and derives the Shannon capacity for the opra policy and for the suboptimal cifr policy. Finally, Section VI provides numerical results and comparisons, followed by concluding remarks in Section VII. II. S YSTEM AND C HANNEL M ODELS A block diagram of the MIMO transmission system, with nT > 1 transmit and nR receive antennas, is given in Fig. 1. We consider a discrete-time baseband channel model and assume quasi-static flat Rayleigh fading. The flat-fading assumption implies that the delay spread of the channel is much less than a symbol duration, whereas the quasi-stationarity accounts for the fact that the channel characteristics remain constant at least for the period of transmission of an entire frame, with a duration of T symbol periods. For a given channel realization, ∆ the MIMO system with diversity order K = nT nR can then be represented within a frame period by the channel matrix
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Fig. 1. System model.
R ,nT H = [hi,j ]ni,j=1 , where hi,j is the channel coefficient between the jth transmit and ith receive antennas. Since we are concerned with independent Rayleigh fading, the channel coefficients hi,j , i = 1, . . . , nR and j = 1, . . . , nT are modeled as independent identically distributed (i.i.d.) complex circular Gaussian random variables, each with a CN (0, 1) distribution. For the assumption of independent fading to hold, sufficient spacing between antenna elements should be provided at both ends of the transmission link. The input–output relationship can be expressed as
Y =HG+V
(1)
where the received signal Y is an nR × T matrix, G is the nT × T matrix of transmitted symbols, and the receiver noise V is an nR × T matrix with elements modeled as i.i.d. complex circular Gaussian random variables, each with a CN (0, σ 2 ) distribution. Channel knowledge is assumed at the receiver. We also consider the case where the channel state, which is the SNR level, is known at both the receiver and the transmitter as might be obtained through an error-free zerodelay feedback channel. Finally, let PT be the total transmit power per symbol duration, and define the average SNR per ∆ receive antenna as γ¯ = PT /σ 2 . III. E QUIVALENT SISO C HANNEL M ODEL W ITH I MPERFECT C HANNEL E STIMATION Transmit diversity over the wireless link using STBC is achieved by mapping each R ≤ T complex input symbols {s1 , s2 , . . . , sR }, belonging to a given signal set S, into nT orthogonal sequences of length T to be simultaneously transmitted through the nT transmit antennas. Since R input symbols from the signal set S are transmitted within T symbol durations (a frame period), the information code rate of the ∆ space-time block code is defined as Rc = R/T . Let Pframe designate the average transmit power over a frame period, i.e., Pframe = T PT . Then, given a mapping of the input symbols characterized by the matrix G, we have Pframe = E G2F = E trace (GG H ) R 2 |sr | = E trace c InT r=1
(2) (3) (4)
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where E[·] stands for expectation, · F denotes the matrix Frobenius norm,1 (·)H denotes the transpose conjugate operator, and InT is the nT × nT identity matrix. Equation (3) follows from the definition of the Frobenius norm, whereas (4) is due to the orthogonality of the rows of G. The constant c, which appears in (4), is dependent on the employed STBC mapping. For instance, taking the codes proposed in [13], c = 2 for the rate 1/2 space-time block codes G 3 and G 4 , and c = 1 for the rate 3/4 space-time block codes H3 and H4 .2 Hence, according to (4), the average total transmit power per frame is given by Pframe = c nT
R
E |sr |2 = c nT R Ps
(5)
r=1
where Ps = E[|si |2 ] is the average power per symbol. On the other hand, we also have Pframe = T PT . Therefore, the following relationship holds: Ps =
PT T PT = . c nT R c nT Rc
(6)
Due to the decoupling of signals transmitted from different antennas, it can be shown that STBC converts the matrix channel into a scalar channel [7], [8]. In fact, the effective induced channel is equivalent to two blocks of R subchannels corresponding to the real and imaginary parts of the transmitted symbols. Before maximum likelihood detection at the receiver, each subchannel can be described by the input–output relationship y = c H2F s + v
(7)
where s denotes the real or imaginary part of a transmitted symbol with power Ps /2, and v is the noise term after STBC decoding with a distribution N (0, cH2F σ 2 /2). Thus, the effective SNR at the receiver is given by γ STBC =
c2 H4F
Ps 2 2 cH2F σ2
= c H2F
Ps . σ2
(8)
pγ STBC (γ) =
γ H2F nT Rc
nT Rc γ
K
γ K−1 − nTγRc γ e , Γ(K)
γ>0
(11)
where Γ(·) is the Gamma function [15]. Assuming Gaussian distributed channel-estimation errors [16, App.], the pdf of the output SNR can be rewritten as follows: pγ STBC (γ) = (1 − ρ2 ) ×
K−1 nT Rc
K−1 k=0
γ K−1 k
e
−
nT Rc γ
γ
ρ2 γ nT Rc (1 − ρ2 ) γ
k
1 k!
(12)
where ρ denotes the correlation between the actual channel coefficients hi,j and their estimates hi,j defined as ∆ ρ2 = |E[hi,j h∗i,j ]|2 . By rearranging the terms [17], (12) can further be expressed as pγ STBC (γ) =
K B K−1 (ρ2 ) nT Rc k k−1
Γ(k)
k=1
where Bin (t) = [18], satisfying Bin (t) = 1
n i
γ
γ k−1 e
−
nT Rc γ
γ
(13)
ti (1 − t)n−i are Bernstein polynomials
∀ 0 ≤ t ≤ 1;
∀ n = 0, 1, 2, . . .
(14a)
i=0
(9)
Bin (1) = δi,n
∀ i = 0, . . . , n;
∀ n = 0, 1, 2, . . .
(14b)
where δi,n are Kronecker delta symbols. Note that for perfect channel estimation, each channel coefficient is perfectly correlated with its estimate, i.e., ρ2 = 1, and then, using (14b), (13) reduces to (11). We now use the pdf given by (13) to derive closed-form expressions for the Shannon capacity of STBC in Rayleigh fading under the aforementioned adaptive-transmission policies. We first derive the Shannon capacity expression assuming no CSI at the transmitter.
matrix Frobenius norm of an r × t matrix M = [mj,k ]r,t is given j,k=1
by MF =
r j=1
t
k=1
|mj,k |2 =
trace(MMH ).
matrices G and H in [13, eqs. (4)–(7)] designate specific STBC mappings and correspond to instances of G T according to our notation, where T (·) stands for matrix transpose. 2 The
n
where γ¯ is the average SNR per receive antenna. Equation (9) represents the effective SNR per symbol at the output of the STBC decoder. It is worthwhile to mention that the constants c and Rc , which are determined by the choice of the STBC mapping, are missing from the analysis in [6] and [8]. Although the code-dependent constant c does not appear in the expression of the SNR of the equivalent SISO channel (9), the code rate Rc does induce an SNR gain by a factor 1 The
∆
where Rc = Rc / ln(2), which corrects the corresponding expression given by [8, eq. (4)]. H2F is the sum of 2K i.i.d. χ2 random variables (due to the Rayleigh fading assumption); it is then χ2 distributed with 2K degrees of freedom [14], and hence, using a change of variables, it can be shown that the received SNR γ STBC follows a Gamma distribution with parameter K and mean ∆ γ = Kγ/nT Rc = nR γ¯ /Rc , according to the pdf pγ STBC (γ) given by
∆
Plugging (6) into (8) yields γ STBC =
1/Rc (Rc ≤ 1). As a result, the nonergodic capacity of orthogonal STBC is given (in bits per second per hertz) by
γ 2 HF C = Rc ln 1 + (10) nT Rc
IV. C APACITY W ITH R ECEIVER -S IDE I NFORMATION Based on the expression of the Shannon capacity for an SISO fading channel with CSI at the receiver [19], [20], the Shannon
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capacity of STBC subject to Rayleigh fading is given by CSTBC = Rc E ln(1 + γ STBC ) ora
[b/s/Hz]
(15)
where E[·] denotes the expectation over the pdf of the received SNR given by (13). Note that when i.i.d. fading is assumed, which corresponds to our case of study, (15) also denotes the achievable capacity when CSI is available at both the transmitter and the receiver but with no power adaptation [10]. It corresponds to the capacity with ora to the fading level given constant transmit power. Hence, we have the subscript ora in (15), referring to the ora policy [11]. Since the function γ → ln(1 + γ) is concave for γ > 0, we can apply Jensen’s inequality [15] to obtain an upper bound for as follows: CSTBC ora CSTBC ≤ Rc ln 1 + E[γ STBC ] ora K γ K−1 2 kBk−1 (ρ ) . (16) = Rc ln 1 + nT Rc
Fig. 2. Comparison between the capacity per unit bandwidth of a full-rate STBC with receiver-side information CSTBC and its asymptotic approxora imation for ρ2 = 1, Rc = 1, nT = 2, and (a) nR = 1, (b) nR = 2, and (c) nR = 4.
k=1
We now state the following result, giving the analytical expression of the Shannon capacity for an MIMO Rayleigh fading channel using STBC, with imperfect channel knowledge at the receiver, as a function of the code rate Rc , the correlation coefficient ρ2 , and the average SNR per receive antenna γ¯ . Theorem 1: Let nT ∈ {2, 3, . . .}, nR ∈ {1, 2, . . .}, γ > 0, and 0 ≤ ρ ≤ 1. The Shannon capacity (in bits per second per STBC , with CSI available at the receiver, is given by hertz) Cora
= Rc CSTBC ora
In the special case of perfect channel estimation, i.e., when ρ2 = 1, (17) reduces to
nT Rc nT Rc P = R CSTBC − E K 1 ora c γ γ
+
K−1
Pl
nT Rc γ
l=1
nT Rc nT Rc K−1 2 Bk−1 (ρ ) Pk − E1 γ γ k=1 nT Rc k−1 P Pk−l − nTγRc l γ (17) + l l=1
where E1 (·) denotes the exponential integral function of the first order, which is defined as [21]
∆
+∞
E1 (x) =
E1 (x) = −E − ln(x) −
x>0
(18)
x
and Pn (·) designates the Poisson distribution
+∞ (−x)k k=1
(20)
k.k!
,
x>0
(21)
of the exponential integral function, where E = 0.577215665 is the Euler–Mascheroni constant, which is an asymptotic approximation for CSTBC that is valid for large values of γ, ora and can be obtained as follows: ≈ Rc CSTBC ora
e−u du, u
PK−l − nTγRc . l
Using the series representation [15, eq. (8.214-2)]
K
K
K−1 2 Bk−1 (ρ )
k=1
nT Rc nT Rc nT Rc . Pk − −E − ln + γ γ γ nT Rc k−1 P Pk−l − nTγRc l γ . (22) + l l=1
∆
Pn (x) =
n−1 k x k=0
k!
e−x ,
n = 1, 2, . . .
(19)
Proof: The derivation of the above theorem is given in Appendix A.
Fig. 2 depicts the channel capacity with perfect channel estimation at the receiver (20) and its asymptotic approximation (22) for different antenna configurations. This figure shows that the approximate capacity expression closely matches the exact expression for γ¯ > 6 dB. Further numerical results will be discussed in Section VI.
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V. C APACITY W ITH T RANSMITTER - AND R ECEIVER -S IDE I NFORMATION Efficient power allocation can be performed on a frame-byframe basis subject to an average power constraint, provided that the CSI is available at the transmitter. Such a powerallocation policy may be done in conjunction with either a variable transmission rate, thus yielding the opra policy, or a fixed transmission rate, when the so-called channel-inversion policy is implemented [10], [22]. A. Optimal Power and Rate Adaptation (opra) When the channel state is known at the transmitter, both the transmit power and rate, can be simultaneously adapted to the variations of the fading channel. This may be achieved by allowing the transmitter to vary its transmit power and rate, subject to the average power constraint PT (γ) =1 (23) E PT where PT (γ) denotes the variable transmit power. Using methods from the calculus of variations, it can be shown that the capacity is maximized with the following power-allocation strategy [10], [22] 1 PT (γ) − γ1 , if γ ≥ γ0 = γ0 (24) PT 0, if γ < γ0
for the opra policy is 3) The Shannon capacity CSTBC opra given (in bits per second per hertz) by
nT Rc γ0 STBC Copra = Rc E1 γ
+
K k=1
γ0
1 1 − γ0 γ
pγ STBC (γ)dγ = 1.
(25)
k=2
=
+∞ γ ln pγ STBC (γ)dγ. γ0
γ0
Since γ → ln(γ/γ0 ) is a concave function, then, by applying Jensen’s inequality [15], an upper bound for CSTBC opra can be obtained as follows: K γ STBC K−1 2 Copra ≤ Rc ln k Bk−1 (ρ ) . (27) nT Rc γ0
(30)
K
K−1
[P1 (nT Rc x)−nT Rc x E1 (nT Rc x)] Pk−1 (nT Rc x) K−1 2 Bk−1 (ρ ) Pk (nT Rc x)−nT Rc x (k−1)
k=2
∀x > 0.
−x γ
(31)
Using the first-order derivative of PK (nT Rc x), given by (56) in Appendix A with nT Rc instead of µ, and differentiating f with respect to x over the interval ]0, +∞[, yields f (x) = − nT Rc (1 − ρ2 ) − nT Rc
k=1
We now state the following result, giving a closed-form expression for the Shannon capacity of an MIMO system using STBC subject to the opra policy, as a function of the code rate Rc , the correlation coefficient ρ2 , and the average SNR per receive antenna γ¯ . Theorem 2: Let nT ∈ {2, 3, . . .}, nR ∈ {1, 2, . . .}, and 0 ≤ ρ2 ≤ 1. 1) ∀ γ > 0, ∃ a unique γ0 ∈ ]0, 1[ satisfying (25). 2) γ0 strictly increases as function of γ.
. (28)
Denote by x the ratio γ0 /¯ γ , and consider the function f (x) defined as follows:
+ (26)
l
nT Rc γ0 γ nT Rc γ0 Pk−1 . − γ (k − 1)
f (x) = (1−ρ2 )
Rc
l=1
nT Rc γ0 γ
and further noting from the definition of the Bernstein polynomials that B0K−1 (ρ2 ) = (1 − ρ2 )K−1 , we find, after some manipulations, that γ0 must satisfy the following equation:
nT Rc γ0 nT Rc γ0 nT Rc γ0 2 K−1 E1 P1 γ0 = (1−ρ ) − γ γ γ
K nT Rc γ0 K−1 2 Bk−1 (ρ ) Pk + γ
Shannon capacity for the opra policy is then given by CSTBC opra
Proof: Let γ > 0 be the average SNR per receive antennas and suppose there exists a γ0 that satisfies (25). By inserting (13) into (25), using the fact that the Poisson distribution related to the upper incomplete Gamma function, Pn (·) can be ∆ +∞ Γ(n, µ) = µ un−1 e−u du, by means of [23, eq. (11.6)] +∞ n−1 −u u e du µ , n≥1 (29) Pn (µ) = Γ(n)
where γ0 represents a cutoff SNR value below which the channel is not used. Inserting (24) into (23) implies that the optimal cutoff level has to satisfy the following: +∞
K−1 2 Bk−1 (ρ )
k−1 P l
K
K−1
E1 (nT Rc x)
K−1 2 Bk−1 (ρ )
k=2
Pk−1 (nT Rc x) − γ. (32) (k − 1)
Hence, f (x) < 0 ∀x > 0, meaning that f is a strictly decreasing function of x. Moreover, observe that lim xE1 (x) = 0,
x→0+
(a)
lim Pk x = 1,
x→0+
lim xE1 (x) = 0
x→+∞
(b)
lim Pk x = 0,
x→+∞
(33a) ∀ k = 0, 1, 2, . . . (33b)
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Then, from (31) with (33), we deduce that along K−1 2 limx→0+ f (x) = K k=1 Bk−1 (ρ ) = 1 and limx→+∞ f (x) = −∞. Since f is a continuous function of x, then, according to Weierstrass’s intermediate value theorem, there exists a unique x0 ∈ ]0, +∞[ such that f (x0 ) = 0. Uniqueness is ensured owing to the strictly decreasing nature of f . We thereby proved that for each γ > 0, there exists a unique γ0 > 0 satisfying (25). Consider now the function g(·) : γ → (1/γ0 ) − (1/γ) defined over the interval ]0, +∞[. This function is concave since its second derivative with respect to γ is given by g (·) : γ → −(2/γ 3 ), which is < 0 for γ > 0. Then, applying Jensen’s inequality [15] to (25) yields the following inequality: nT Rc
1 γ . − 1 ≥ K K−1 2 γ0 kB k=1 k−1 (ρ )
(34)
Hence, γ0 is always less than 1. Additionally, using (21) and (33a) and (33b), an asymptotic development of (30), when γ → +∞, leads to the following approximation for the cutoff SNR:
nT Rc γ0 nT Rc γ0 2 K−1 − (1 − ρ ) 1 − γ0 ≈ E + ln γ γ K K−1 2 Bk−1 (ρ ) (35) + (k − 1) k=2
which implies that limγ→+∞ γ0 = 1. Next, in order to prove the strictly increasing nature of γ0 as a function of γ, we consider the following two functions, deduced from the function f and defined over the interval ]0, +∞[: γ 0 ∀ γ0 > 0 (36) fγ0 (·) : x → f x
x fγ¯ (·) : x → f (37) ∀ γ > 0. γ Due to the strictly decreasing nature of f , it can easily be seen that fγ¯ (·) is also a strictly decreasing function of x for each average SNR value γ, whereas fγ0 (·) is a strictly increasing function of x for each cutoff SNR γ0 . Now, consider two average SNR values γ 1 and γ 2 such that γ 1 < γ 2 , and let their corresponding cutoff SNRs be γ01 and γ02 , respectively. Then, we have
γ01 (b) (a) = 0 (38) fγ 1 (γ01 ) = f γ1 where (a) follows from the definition of fγ 1 (·) (37), and (b) results from the fact that γ01 /γ 1 is a zero of f. On the other hand, we also have
(d) γ01 (c) = fγ01 (γ 1 ) < fγ01 (γ 2 ) (39) f γ1 where (c) originates from the definition of fγ01 (·) (36) whereas (d) is due to the strictly increasing nature of fγ01 (·).
Fig. 3. Cutoff level for the opra policy with ρ2 = 1, Rc = 1, nT = 2, and (a) nR = 1, (b) nR = 2, and (c) nR = 4.
Thus, (38) in conjunction with (39) yields (a)
(b)
fγ01 (γ 2 ) > 0 ⇒ fγ 2 (γ01 ) > fγ 2 (γ02 ) ⇒ γ01 < γ02 where (a) follows from the fact that, similar to (38), fγ 2 (γ02 ) = 0, whereas (b) follows from the strictly decreasing nature of fγ 2 (·). Therefore, the optimal cutoff SNR γ0 strictly increases whenever the average SNR γ increases. Fig. 3 shows the behavior of γ0 as a function of γ for ρ2 = 1 and different antenna configurations, which confirms the precedent results. Finally, to complete the proof of Theorem 2, the derivation of (28) is provided in Appendix B. For perfect channel estimation, the capacity of the opra policy (28) reduces to nT Rc γ0
K−1 P l γ ¯ nT Rc γ0 . (40) + CSTBC opra = Rc E1 γ¯ l l=1
Since, according to the opra policy, no data are transmitted when the experienced SNR falls below γ0 , this adaptivetransmission technique suffers a probability of outage given by γ0 STBC (γ0 ) Pout
=
pγ STBC (γ) dγ.
(41)
0
Inserting (13) into (41) and using (29) yields the following expression for the experienced outage probability by the opra policy when using STBC:
K nT Rc γ0 STBC K−1 2 Pout (γ0 ) = 1 − Bk−1 (ρ )Pk . (42) γ¯ k=1
B. Channel Inversion With Fixed Rate (cifr) Channel inversion is an adaptive-transmission technique whereby the transmitter uses the CSI fed back by the receiver in order to invert the channel fading. This way, the channel appears to the encoder/decoder as a time-invariant AWGN
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channel [10], [22]. However, this suboptimal adaptation policy, which is commonly used in spread-spectrum systems with near–far interference imbalances [24], may suffer a large capacity penalty compared to the previously analyzed adaptation policies, although it is much less complex to implement, only requiring a fixed coding and modulation scheme. For instance, the capacity of the channel-inversion policy for an SISO Rayleigh fading channel is 0 [10], [11]. When channel inversion is applied, the transmitter adjusts its power to maintain a constant received power. The power allocation obeys the following rule: κ PT (γ) = PT γ
(43)
where κ denotes the average received SNR that can be maintained subject to the average power constraint given by (23). Hence, κ has to satisfy κ = 1/E[1/γ]. The channel capacity with this power-allocation policy can be simply derived from the capacity of an AWGN channel with a received SNR κ according to 1 STBC . (44) Ccifr = Rc ln 1 + p ∞ γ STBC (γ) dγ 0 γ Thus, inserting (13) into (44), we have CSTBC = Rc cifr
× ln1 +
limγ→0+ E1
γ nT Rc (1−ρ2 )K−1 nT Rc γ
γ +
K
K−1 (ρ2 ) Bk−1
.
k=2 (k−1)(1−ρ2 )K−1
(45) Note that when ρ2 = 1, corresponding to an imperfect channel estimation, the capacity of the cifr policy equals 0, owing to the fact that limγ→0+ E1 ((nT Rc /γ)γ) = ∞. However, when ρ2 = 1, using the Bernstein polynomial property (14b), the capacity of the channel-inversion policy reduces to
γ STBC Ccifr = Rc ln 1 + (K − 1) . (46) nT Rc The achievable rate by the cifr policy may further be enhanced by compensating for parts of the channel fading, that is, by inverting the channel only when the SNR level is larger than a given threshold γ0 . This is known as the truncated inversion with fixed rate (tifr) policy [10]. The power allocation with the truncated policy is the following: κ0 PT (γ) γ , if γ ≥ γ0 = (47) PT 0, if γ < γ0
where κ0 represents the received SNR that can be maintained at the receiver subject to the average power constraint given by (23). Hence, κ0 can be related to the threshold γ0 through κ0 = +∞ γ0
.
(48)
dγ
As can be seen from (48), the larger the cutoff level γ0 is, the larger the received SNR will be, and hence, a lower probability of error after symbol detection will result at the receiver. Nevertheless, the decrease in the symbol error probability comes at the cost of an increase in the probability of outage suffered by this policy. The channel capacity with the truncated-inversion policy is obtained by maximizing, over all possible cutoff levels γ0 [10], the following expression: 1 CSTBC = Rc max ln 1 + p tifr ∞ γ STBC (γ) γ0 dγ γ0
γ
STBC (γ0 ) . 1 − Pout (49)
STBC (·) is the outage probability provided in (42). By where Pout using (29) and operating some mathematical calculations, it can be shown that (49) reduces to (50), shown at the bottom of the page, which represents the capacity of STBC under MIMO Rayleigh fading subject to the truncated cifr policy. As γ) → 1 can be seen in (50), (33b) implies that Pk−1 (nT Rc γ0 /¯ γ ) → 1 when γ0 → 0. Thus, the capacity and Pk (nT Rc γ0 /¯ induced by the truncated channel-inversion policy converges towards that induced by the total channel inversion provided in (44). Finally, for perfect channel estimation, the capacity of the truncated cifr policy reduces to
K−1 γ nT Rc γ0 nT Rc = R ln 1 + CSTBC P . K tifr c nT Rc γ0 γ¯ P K−1
γ ¯
(51) VI. N UMERICAL R ESULTS AND C OMPARISONS In this section, we start by providing results and comparisons for the Shannon capacity with receiver-side information only. We then present channel capacity results for the adaptivetransmission policies under study and provide analysis and comparison of their corresponding performance. Without loss of generality, we consider that space-time block codes exist for any given value of nT and are of rate Rc = 1. We also
CSTBC = Rc ln 1 + tifr
1 pγ STBC (γ) γ
(1 −
ρ2 )K−1 E
1
nT Rc γ0 γ ¯
γ nT Rc K B K−1 (ρ2 ) k−1 + (k−1) Pk−1 k=2
K nT Rc γ0 K−1 2 B (ρ )P k k−1 γ¯ nT Rc γ0 γ ¯
k=1
(50)
MAAREF AND AÏSSA: CAPACITY OF SPACE-TIME BLOCK CODES IN MIMO RAYLEIGH FADING CHANNELS
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Fig. 4. Average channel capacity per unit bandwidth using STBC for a Rayleigh fading channel with receiver-side information versus the average SNR per receive antenna γ ¯ for ρ2 = 1, Rc = 1, and (a) nR = 1, (b) nR = 2, and (c) nR = 4.
Fig. 5. Average channel capacity per unit bandwidth using STBC for Rayleigh fading channels versus the average SNR per receive antenna γ ¯ for ρ2 = 1, Rc = 1, nT = 2, and (a) nR = 1, (b) nR = 2, and (c) nR = 4.
consider the case of perfect channel estimation, i.e., ρ2 = 1, unless otherwise mentioned. Consider the capacity of an MIMO Rayleigh fading channel using STBC with channel side information at the receiver only. The Shannon capacity in this case, which corresponds to the achievable capacity under the ora policy, is given by (17). When a single receive antenna is considered, the capacity of a full-rate space-time block code coincides with that of the underlying MIMO channel capacity as given by [25]
γ¯ H MIMO = E log2 det InR + HH . (52) C nT
of nR independent Rayleigh fading subchannels and the upper bound on the achievable capacity (53) corresponding to the number of receive antennas nR . Note, however, the diminishing returns as the number of receive antennas increases [case (c)] where the curves almost coincide. Here, it is worth noting that the STBC payoff, in terms of capacity improvement, which results from incrementing the number of transmit antennas, gets smaller as nT increases. The largest capacity improvement is the one corresponding to the use of STBC with nT = 2. As expected, for any value of nR , the achievable capacity through STBC improves as the number of transmit antennas increases. However, this capacity increase diminishes as nT increases, the largest improvement being the one obtained with two transmit antennas only. We now consider the capacity of STBC with transmitterside information. In particular, Fig. 5 shows the capacity per unit bandwidth for the opra CSTBC opra , the ora with constant , and the suboptimal channel-inversion transmit power CSTBC ora , with its truncated variant CSTBC . This policy CSTBC cifr tifr figure indicates how the opra policy achieves the highest capacity for any average receive SNR γ and for all antenna configurations. As can be observed, with no power adaptation at the transmitter, the ora policy achieves approximately the same capacity as the opra policy, especially for high SNR values. As expected, results of Fig. 5 also show that the channel-inversion policy yields a capacity decrease compared to the other adaptation policies: a decrease that is compensated somewhat by a reduction in the implementation complexity. Nevertheless, this decrease in capacity diminishes as the diversity order increases and/or the average received SNR increases. As for the effect of estimation errors, Fig. 6 compares the capacity performance of the different adaptive-transmission strategies for three values of ρ2 , namely, ρ2 = 1, ρ2 = 0.5, and ρ2 = 0.1. As can be seen, the Shannon capacity decreases as ρ2 decreases, which is expected. Besides, the total channel inversion is the most sensitive to channel-estimation errors. This adaptive-transmission
Indeed, when nR = 1, HHH = H2F and InR = 1, therefore, (52) reduces to (15) with Rc = 1 and γ STBC given by (9). This is the unique single case where the use of STBC does not incur a capacity loss compared to the capacity of the underlying MIMO channel, as proven in [8]. for Fig. 4 shows the capacity per unit bandwidth CSTBC ora nT = 2 and nT = 4 and different numbers of receive antennas [(a) nR = 1, (b) nR = 2, and (c) nR = 4]. For each case, we also display the corresponding upper bound on the achievable capacity provided in (16). Note that for perfect channel estimation, this upper bound only depends on the number of receive antennas. Indeed, when ρ2 = 1, (16) reduces to
nR CSTBC ≤ R ln 1 + γ (53) ora c Rc which represents the capacity of an AWGN channel with SNR scaled by nR /Rc . Fig. 4 also shows the capacity per unit bandwidth of an array of nR independent Rayleigh fading subchannels with MRC based on the expression given in [26, eq. (7)]. As can be observed from these results, the use of transmit diversity through STBC helps close the gap, in the achievable capacity, between the use of receive diversity (MRC) with an array
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Fig. 6. Average channel capacity per unit bandwidth using STBC with nT = 2 and nR = 2 versus the average SNR per receive antenna γ ¯ for (a) ρ2 = 1, (b) ρ2 = 0.5, and (c) ρ2 = 0.1.
Fig. 8. Probability of outage for the opra and the truncated channel-inversion policy versus the average SNR per receive antenna γ ¯ for ρ2 = 1, Rc = 1, nT = 2, and (a) nR = 1, (b) nR = 2, and (c) nR = 4.
VII. C ONCLUDING R EMARKS
Fig. 7. Optimal cutoff level for the truncated channel-inversion policy with ρ2 = 1, Rc = 1, nT = 2, and different numbers of receive antennas.
policy indeed yields zero capacity for ρ2 = 0.5 and ρ2 = 0.1. The capacity results obtained through total channel inversion are further increased by performing a truncated channel inversion, whereby the threshold level is chosen to maximize the corresponding capacity expression (50). Fig. 7 shows the behavior of the optimal threshold for different antenna configurations as a function of the average SNR per receive antenna. As can be seen, this threshold increases as the average SNR increases. However, since no data are transmitted when the SNR falls below this threshold, the truncated channel-inversion policy suffers an outage probability, which is always larger than the outage probability suffered by the opra policy for the same antenna configuration. This can be observed in Fig. 8, comparing the outage probabilities obtained when using these two adaptive-transmission policies with nT = 2 and different numbers of receive antennas.
This paper provided a thorough investigation of the capacity achieved by orthogonal space-time block codes over frequency-nonselective MIMO Rayleigh fading channels. In particular, closed-form expressions for the Shannon capacity of STBC were derived under various adaptive-transmission policies, taking into consideration the effect of channelestimation errors at the receiver. To be more specific, exact expressions for the channel capacity of STBC under: 1) the opra policy; 2) the ora policy given constant transmit power; and 3) the suboptimal channel-inversion policy were derived. Our numerical results showed that opra provides a slight capacity improvement over rate adaptation only given constant transmit power. As to the channel-inversion policy, though it decreases the achievable capacity, it represents a good alternative to the other adaptation policies when perfect channel estimation can be performed, as it is much less complex to implement. The achievable capacity by the latter policy can further be increased by applying a truncated channel inversion at the cost of an increase in the outage probability, which is always larger than that suffered by the opra policy. Finally, it is worthwhile to mention that by taking into account Gaussian estimation errors in the channel-estimation process, our derived capacity expressions represent general formulas from which those corresponding to perfect channel estimation can easily be retrieved. A PPENDIX A P ROOF OF T HEOREM 1 Expressing the expectation operator in (15) as an average over the pdf of the received SNR, given by (13), yields the following expression for the capacity per unit bandwidth (in bits per second per hertz): = Rc CSTBC ora
+∞ ln(1 + γ) pγ STBC (γ) dγ. 0
(54)
MAAREF AND AÏSSA: CAPACITY OF SPACE-TIME BLOCK CODES IN MIMO RAYLEIGH FADING CHANNELS
Inserting (13) into (54) and singling out the integral term yields CSTBC ora = Rc
+∞ K 0
K−1 2 Bk−1 (ρ )µ
k=1
(µγ)k−1 −µγ e ln(1 + γ)dγ Γ(k) (55)
where µ = nT Rc /γ. For x > 0, a primitive function of x → −µ
(µx)k−1 −µx e Γ(k)
K
k=1
+∞
l=1
Pk (µγ) dγ. 1+γ
(57)
0
Using (19) and operating the change of variable t = 1 + γ within the integral terms, (57) can further be written as CSTBC ora = Rc
K
K−1 2 Bk−1 (ρ )
k=1
k−1
µn −µ e n! n=0
+∞
Then, operating the change of variable u = µt within the integral terms and using (29), we get K CSTBC ora K−1 2 = Bk−1 (ρ ) Pk (−µ)E1 (µ) Rc k=1 k−1 n (−µ)n µ n −l + e (−µ) (l − 1)!Pl (µ) . (62) n! l n=1
(56)
is given by x → Pk (µx), where Pk (·) is the Poisson distribution function, as defined in (19).3 The summation in (55) is of finite order, and thus, one can invert the integral and summation orders. Then, integrating by parts the resulting elementary integrals yields CSTBC ora K−1 2 = Bk−1 (ρ ) Rc
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(t − 1)n −µt e dt. t
Inverting the summation order and replacing the binomial coefficients by their corresponding values yield K CSTBC ora K−1 2 = Bk−1 (ρ ) Pk (−µ)E1 (µ) Rc k=1 k−1 k−1 Pl (µ) (−µ)n−l µ + e . (63) l (n − l)! l=1
Operating the change of variable n = n − l in the second summation, and thereby recognizing that this summation reduces to Pk−l (−µ), yields the following: K CSTBC ora K−1 2 = Bk−1 (ρ ) Pk (−µ)E1 (µ) Rc k=1
1
(58) Furthermore, using the binomial expansion of (t − 1) given by
+
n
(t − 1)n =
n l=0
n (−1)n−l tl l
K k−1 n µ µ CSTBC ora K−1 2 e = B (ρ ) k−1 Rc n! n=0
×
l=0
+∞ tl−1 e−µt dt. (60)
n (−1)n−l l
.
(64)
Finally, replacing µ by nT Rc /γ in (64) yields the expression of (17), which concludes the proof of Theorem 1. A PPENDIX B P ROOF OF T HEOREM 2 Inserting (13) into (26) and operating the change of variable t = γ0 /γ within the integral term, then singling this term out yields CSTBC opra = Rc
+∞ K 1
K−1 2 Bk−1 (ρ )µ
k=1
(µt)k−1 ln(t)e−µt dt Γ(k)
(65)
1
By singling out the first terms in the summation over l, i.e., those corresponding to l = 0, and using the definitions of the exponential integral function of the first order (18) and that of the Poisson distribution function (19), we obtain K CSTBC ora K−1 2 = Bk−1 (ρ ) Pk (−µ)E1 (µ) Rc k=1
+∞ k−1 n n µ µ n n−l l−1 −µt e + t e dt . (61) (−1) l n! n=1 l=1
l
l=1
k=1
k−1 Pl (µ)Pk−l (−µ)
(59)
where nl = n!/(n − l)! l! are the binomial coefficients and inserting (59) into (58), we get
n
n=l
where µ = nT Rc γ0 /¯ γ . Following the same argument as in Appendix A, one can invert the integral and summation orders. Using the derivative of t → Pk (µt) given by (56) to integrate by parts the right-hand side of (65) yields K CSTBC opra K−1 2 = Bk−1 (ρ ) Rc k=1
Pk (µt) dt. t
(66)
1
Then, using the definition of the Poisson distribution given by (19), (66) can be written as follows:
1
3 Note that similar steps are used to derive the capacity of an array of independent Rayleigh subchannels with MRC at the receiver in [26].
+∞
K k−1 l CSTBC µ opra K−1 2 = B (ρ ) k−1 Rc l! k=1
l=0
+∞ tl−1 e−µt dt. 1
(67)
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By operating the change of variable u = µt within the integral terms, then singling out the summation terms corresponding to l = 0, we obtain K CSTBC opra K−1 2 = Bk−1 (ρ ) Rc k=1 +∞ +∞ l−1 −u −u k−1 e 1 µ u e du du + . × u l Γ(l)
(68)
l=1
µ
Recognizing that the first term in the summation within the brackets corresponds to the exponential integral function of the first order (18), whereas the remaining terms can be written using the Poisson distribution through (29), yields CSTBC opra
=
Rc
E1 (µ) +
K
K−1 2 Bk−1 (ρ )
k=1
k−1 Pl (µ) l=1
l
(69)
where we made use of (14a). Finally, replacing µ by its corγ , yields (28), and responding value, which is nT Rc γ0 /¯ concludes the proof of Theorem 2. R EFERENCES [1] R. W. Heath, Jr. and A. J. Paulraj, “Switching between diversity and multiplexing in MIMO systems,” IEEE Trans. Commun., vol. 53, no. 6, pp. 962–968, Jun. 2005. [2] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data rate wireless communication: Performance criteria and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 1744–1765, Mar. 1998. [3] S. M. Alamouti, “A simple transmitter diversity scheme for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [4] Technical Specification Group Radio Access Network, Physical Layer Procedures (FDD) (Release 1999), 3rd Generation Partnership Project, 3GPP TS 25.214 V3.7.0, 2001. [5] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [6] S. Sandhu, R. W. Heath, Jr., and A. Paulraj, “Space–time block codes versus space–time trellis codes,” in Proc. IEEE Int. Conf. Communications, Helsinki, Finland, Jun. 2001, pp. 1132–1136. [7] G. Ganesan and P. Stoica, “Space–time block codes: A maximum SNR approach,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1650–1656, May 2001. [8] S. Sandhu and A. Paulraj, “Space time block codes: A capacity perspective,” IEEE Commun. Lett., vol. 4, no. 12, pp. 384–386, Dec. 2000. [9] J. Wolfowitz, Coding Theorems of Information Theory, 2nd ed. New York: Springer-Verlag, 1964. [10] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1986–1992, Nov. 1997. [11] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165–1181, Jul. 1999. [12] A. Maaref and S. Aïssa, “Rate-adaptive M-QAM in MIMO diversity systems using space–time block codes,” in Proc. IEEE Int. Symp. Personal, Indoor Mobile Radio Communication, Barcelona, Spain, Sep. 2004, pp. 2294–2298. [13] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block coding for wireless communications: Performance results,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. [14] H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
[15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000. [16] M. J. Gans, “The effect of Gaussian error in maximal ratio combiners,” IEEE Trans. Commun. Technol., vol. COM-19, no. 4, pp. 492–500, Aug. 1971. [17] B. R. Tomiuk, N. C. Beaulieu, and A. A. Abu-Dayya, “General forms for maximal ratio diversity with weighting errors,” IEEE Trans. Commun., vol. 47, no. 4, pp. 488–492, Apr. 1999. [18] G. G. Lorentz, Bernstein Polynomials. Toronto, ON, Canada: Toronto Univ. Press, 1953. [19] T. Ericson, “A Gaussian channel with slow fading,” IEEE Trans. Inf. Theory, vol. IT-16, no. 3, pp. 353–355, May 1970. [20] R. J. McEliece and W. E. Stark, “Channels with block interference,” IEEE Trans. Inf. Theory, vol. IT-30, no. 1, pp. 44–53, Jan. 1984. [21] J. Spanier and K. B. Oldham, An Atlas of Functions. Washington, DC: Hemisphere, 1987, ch. 37, pp. 351–360. [22] E. Biglieri, G. Caire, and G. Taricco, “Coding and modulation under power constraints,” IEEE Pers. Commun. Mag., vol. 5, no. 3, pp. 32–39, Jun. 1998. [23] N. M. Temme, Special Functions: An introduction to the Classical Functions of Mathematical Physics. New York: Wiley, 1996. [24] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, Jr., and C. E. Wheatley, III, “On the capacity of a cellular CDMA system,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 303–312, May 1991. [25] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [26] C. G. Günther, “Comment on: Estimate of channel capacity in Rayleigh fading environment,” IEEE Trans. Veh. Technol., vol. 45, no. 2, pp. 401–403, May 1996.
Amine Maaref (S’03) received the Diplôme National d’Ingénieur degree in telecommunications with distinction from the École Supérieure des Communications (Sup’Com), Tunis, Tunisia, in 2001, the M.Sc. degree in telecommunications from INRSEMT, University of Quebec, Montreal, QC, Canada, in 2003, and is currently working toward the Ph.D. degree in telecommunications at INRS-EMT. His current research interests lie in the area of radio resource management and multiuser adaptive transmission in multiple-input multiple-output (MIMO) systems. Mr. Maaref has been holding the Tunisian government Postgraduate Scholarship of Excellence at INRS-EMT since 2001, and he was awarded the Canadian Wireless Telecommunications Association Graduate Scholarship in 2004.
Sonia Aïssa (S’93–M’00–SM’03) received the Ph.D. degree in electrical and computer engineering from McGill University, Montreal, QC, Canada, in 1998. She is now an Associate Professor at INRS-EMT, University of Quebec, Montreal, where she holds the Quebec government FQRNT fellowship “Strategic Program for Professors-Researchers,” and an Adjunct Professor at Concordia University, Montreal. From 1996 to 1997, she was a Visiting Researcher at the department of electronics and communications, Kyoto University, Kyoto, Japan, and at the wireless systems laboratories of NTT, Kanagawa, Japan. From 1998 to 2000, she was a Research Associate at INRS-Telecommunications, Montreal. From 2000 to 2002, she was a Principal Investigator in the major program of personal and mobile communications of the Canadian Institute for Telecommunications Research, Montreal, conducting research in radio resource management in code division multiple access (CDMA) systems. Her research interest includes radio resource management and crosslayer design for MIMO wireless networks, with sponsors that include the federal NSERC and provincial FQRNT funding agencies. Dr. Aïssa is currently serving as Editor for the IEEE TRANSACTIONS ON W IRELESS C OMMUNICATIONS and Associate Editor for the IEEE Communications Magazine. She is the Chair of the Montreal Chapter IEEE Women In Engineering society and Cochair of the IEEE Wireless Communication Symposium of the International Conference on Communications (ICC’2006).
