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Sum of Squared Shadowed-Rice Random Variables and its Application to Communication Systems Performance Prediction Giuseppa Alfano and Antonio De Maio
Abstract— The distribution of the sum of non-negative random variables plays an essential role in the performance analysis of diversity schemes for wireless communications over fading channels. While for common fading models such as the Rayleigh, Rice, and Nakagami, the performance of diversity systems is well understood, a minor attention has been devoted to the Shadowed-Rice (SR) case, namely a Rice fading channel with fluctuating (e.g. random) Line of Sight (LOS) component. Indeed, the analytical performance evaluation of diversity systems on SR fading channels requires the availability of handy expressions for the distribution of the combined received power. To this end, the rationale of this paper is twofold: first, to evaluate the distribution of the sum of SR random variables, both for the case of independent as well as correlated LOS components, and then to carry out an extensive performance analysis of Maximal Ratio Combining (MRC) detection scheme on SR fading channels. Index Terms— Diversity, LMS, MRC, Shadowed-Rice.
I. I NTRODUCTION HE distribution of the sum of Gamma variates plays an essential role in the performance analysis of diversity schemes for wireless communications over Nakagami fading channels, since it characterizes the statistical properties of the received power at the output of a receiver combining scheme [1]. Series expressions for the pdf of the sum of Gamma variates are available in [2] and [3] for the case of independent, non identically distributed random variables, in [4], for the case of correlated variables sharing the same fading order (with possibly different scale parameters), and in [5] for the general case of correlated random variables with arbitrary scale parameters and fading orders1. The Nakagami distribution, while being adequate for the terrestrial wireless channel modeling, doesn’t accurately describe the Land-Mobile Satellite (LMS) communication channel, whose statistical modeling has seen a growing interest in recent years, due to its key role in third and fourth generation wireless communications systems. Among the different models proposed for the LMS channel in the open literature [8, and references therein], we focus here on the SR one, defined in [9], due to its attractive features, i.e. the accuracy in terms of statistical fitting and analytical tractability. Such features
T
Manuscript received April 1, 2006; revised October 17, 2006 and December 6, 2006; accepted December 17, 2006. The associate editor coordinating the review of this letter and approving it for publication was S. Aissa. G. Alfano is with the University of Sannio, Piazza Roma, Benevento 82100, BN Italy (e-mail:
[email protected]). A. De Maio is with Federico II University of Naples, Via Claudio 21, Napoli 80125, BN Italy (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2007.060202. 1 The particular case of exponentially correlated but identically distributed Gamma variates is handled in [6], while for integer fading orders the sum of correlated Gamma variates has been characterized through its characteristic function in [7].
are here capitalized, together with the results of [2] and [4], to obtain a single series expression for the distribution of the received, combined power for MRC detection on SR fading channels, both for the cases of independent as well as correlated LOS components. The new expressions are exploited to evaluate the outage probability and the channel capacity of two adaptive strategies, the Optimal Rate Allocation (ORA) with constant transmit power [10], [11], and the Channel Inversion with Fixed Rate (CIFR) [10], [11]. At the analysis stage, the obtained results are supported by a good match with computer generated statistics. The paper is organized as follows: Section II deals with the problem of evaluating the distribution of the sum of independent as well as arbitrarily correlated squared SR random variables. Section III contains the channel model and the performance analysis of the MRC diversity system. Conclusions are given in Section IV, together with some directions for future research. II. S UMS OF S HADOWED -R ICE R ANDOM VARIABLES In this section, we derive the pdf of the sum of both independent as well as correlated squared SR distributed random variables. Following [9], we define a SR random variable, α ∼ SR(Ω, b0 , m), as a Rice random variable with random LOS component b, i.e. 2 r br r + b2 fα (r) = Eb exp − I0 , (1) b0 2b0 b0 with Eb [·] the statistical expectation with respect to b, b0 > 0, and In (·) the n-th order modified Bessel function of the first kind. We assume that the random variable b2 is Gamma distributed with shape parameter m > 0 and scale parameter m−1 −x/β Ω e β=m > 0, i.e. fb2 (x) = x β m Γ(m) u(x), where Γ(·) denotes the Eulerian Gamma function [12] and u(·) is the unit step function. Performing the average in (1), we can write the SR pdf as [9] m −r2 /2b0 2b0 m re fα (r) = 2b0 m + Ω b0 Ωr2 (2b0 )−1 u(r), (2) 1 F1 m, 1; (2b0 m + Ω) where 1 F1 (·, ·; ·) is the confluent hypergeometric function [12]. In a wireless communication scenario, Ω is the average power of the LOS component, 2b0 the average power of the scattered component, and m is the fading order, defined in analogy to the Nakagami model2 . 2 Notice that, for m = 0, expression (2) reduces to the Rayleigh distribution, while for m → ∞, (2) gives the Rice pdf [9].
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A. Sums of Independent Squared SR Random Variables
B. Sums of Correlated Squared SR Random Variables
The pdf of the sum of H squared, independently distributed, SR random variables can be computed via the following Theorem: Theorem 1: Given αi ∼ SR(Ωi , b0 , mi ), i = 1, . . . , H, independently distributed, the pdf of the random variable γ = H 2 α can be written as (3) (see top of next page), with i=1 i H Ωi βi = mi , β = min{βi , i = 1, . . . , H}, ms = i=1 mi , Ψk = β(ms + k), δ0 = 1, and ⎡ ⎤ i k+1 H 1 ⎣ β ⎦ δk+1 = mj 1 − δk+1−i k + 1 i=1 j=1 βi
In this subsection we evaluate the pdf of the sum of H squared SR random variables with the same fading order m, whose LOS components are assumed to be arbitrarily correlated. Theorem 2: Given αi ∼ SR(Ωi , b0 , m), i = 1, . . . , H, the pdf of the random variable γ can be written as (9) (see top of next page), where λi is the i-th eigenvalue of the matrix Λ = Ω1 ΩH DC, D = diag{ m ,..., m } and3 C is the matrix whose 1 H (i,j)-th entry is the square root of the correlation coefficient ρi,j between the squared LOS components, b2i and b2j , of αi and αj , i.e.
k = 0, 1, 2, . . . Proof: Since each αi can be viewed as a Rice random variable with random non-centrality parameter bi , we can write its conditional pdf as 2 bi r r r + b2i exp − f (r) = I0 u(r) αi bi b0 2b0 b0 i = 1, . . . , H , Ωi with b2i a Gamma random variable with parameters m and i mi . It follows that, given b1 , . . . , bH , γ is the sum of H √ independent squared Rice random variables, and thus γ is non-central chi distributed with H degrees of freedom [13, p. 176], i.e.
f√
γ b1 ,...,bH
(r) =
rH r 2 + b2 exp − H−1 b0 b 2b0
IH−1
br b0
u(r),
(4) H 2 where b2 = b . This last quantity is the sum of i=1 i H independent Gamma distributed random variables with arbitrary scale parameters and its pdf can be expressed as [2] mi H ∞
β δk y ms +k−1 exp(−y/β) fb2 (y) = u(y). (5) βi β ms +k Γ(ms + k) i=1 k=0 √ It follows that the unconditional distribution of γ can be obtained by averaging (4) over (5). The resulting integral can be solved exploiting [14, Formula 28], after noticing that [12, Equation 9.6.3] and [14, Equation 7] imply that ν 2 ν + 1, x F /4 . By doing so, we get (7) Iν (x) = (x/2) ν! 0 1 (see top of next page), from which, after the proper random variable transformation, we get (3). The obtained expression for the pdf of the sum of H squared SR distributed random variables is thus a series of weighted squared SR pdf’s, with parameters strictly related to those of the original variables. Moreover, under some more restrictive hypotheses, expression (3) strongly simplifies. Precisely, the following Corollary holds true. Corollary 1: Under the hypotheses of Theorem 1, when Ω1 ΩH m1 = . . . = mH , then ms y H−1 exp − 2by0 2b m 0 s fγ (y)= (7) (2b0 )H (H − 1)! 2b0 ms + Ω Ωy u(y), 1 F1 ms , H; 2b0 (2b0 ms + Ω) H with Ω = i=1 Ωi , since, due to the additional hypothesis, only the term for k = 0 survives in (3).
Cov(b2i , b2j ) , 0 ≤ ρi,j ≤ 1 ρi,j = ρj,i = Var(b2i )Var(b2j ) i, j = 1, . . . , H. Finally, λ = min{λi , i = 1, . . . , H}, ξk = λ(Hm+k), δ0 = 1 and ⎡ ⎤ i k+1 H m ⎣ λj ⎦ δk+1−i δk+1 = 1− k + 1 i=1 j=1 λ k = 0, 1, 2, . . . Proof: The case of correlation between the LOS components can be encompassed by assuming for b2 the following distribution [4, Corollary 1] m H ∞
λ δk y Hm+k−1 exp(−y/λ) u(y). (9) fb2 (y) = λi λHm+k Γ(Hm + k) i=1 k=0
As in Theorem 1, by averaging (4) over (9) [14, Formula 28] we get (8). III. P ERFORMANCE A NALYSIS OF THE MRC D IVERSITY S YSTEM A. System and Channel Model Assuming an H-branches MRC at the receiver, the output Signal to Noise Ratio (SNR) can be expressed as SNR = Eb H Eb α2i = N γ, where αi ∼ SR(Ωi , b0 , mi ) is the k=1 N0 0 Eb fading amplitude on the i-th branch and N is the energy per 0 bit divided by twice the power spectral density of the noise. The statistical characterization of the output SNR can be obtained both in absence as well as in presence of correlation between the branches, by virtue of Theorems 1 and 2. This allows for the evaluation of several performance indexes for a communication on a SR faded channel. B. Outage Probability To evaluate the outage probability Pout , we assume that the transmission is suspended when the received SNR falls below a prescribed threshold γ0 , and define Pout γ0 / NEb 0 Pout = fγ (y)dy, (10) 0
3 With diag{a , . . . , a } we denote a p-dimensional diagonal matrix with p 1 diagonal elements a1 , . . . , ap .
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fγ (y) =
y mi H ∞
β δk y H−1 e− 2b0
i=1
f√
γ (r)
=
k=0
(2b0 )H Γ(H)
r2 mi H ∞
β δk 2r2H−1 e− 2b0
i=1
fγ (y) =
βi
2b0 (ms + k) 2b0 (ms + k) + Ψk
βi
k=0
(2b0 )H Γ(H)
λi
k=0
2b0 (ms + k) 2b0 (ms + k) + Ψk
(2b0 )H Γ(H)
2b0 (Hm + k) 2b0 (Hm + k) + ξk
∞ (a)n n x . 1 F1 (a, b; x) = n!(b) n n=0
(11)
After some algebra, the outage probability can be thus written as mi H ∞
β
ms +k 2b0 (ms + k) βi 2b0 (ms + k) + Ψk i=1 k=0 n ∞ δk (ms + k)n Ψk n!Γ(n + H) 2b0 (ms + k) + Ψk n=0 ⎛ ⎞ ⎜ γ⎜ ⎝n + H,
γ0 ⎟ ⎟, Eb ⎠ 2b0 N0
x with γ(a, x) = 0 e−t ta−1 dt the unnormalized incomplete Gamma function [12] and (a)n the Pochhammer symbol [12]. The obtained expression involves two series; however, when the mi ’s are integers, it reduces to a finite sum. In fact, for H integer mi ’s, i=1 mi + k ≥ H, and the hypergeometric function in (3) can be represented via Kummer’s transform [12] as a polynomial, i.e. 1 F1 (a, b; x)
ms +k
ms + k, H;
Ψk y 2b0 (2b0 (ms + k) + Ψk )
1 F1 ms + k, H;
u(y) (3)
Ψk r 2 2b0 (2b0 (ms + k) + Ψk )
u(r)
Hm+k
1 F1 Hm + k, H;
ξk y 2b0 (2b0 (Hm + k) + ξk )
u(y) (8)
with fγ (y) given by (3) for the cases of independent branches, and by (8) if the branches are correlated. Evaluating the integral (10), under the hypotheses of Theorem 1, requires the series expansion of the confluent hypergeometric function [14], namely
Pout =
1 F1
(6)
m H ∞ − y
λ δk y H−1 e 2b0 i=1
ms +k
= ex 1 F1 (b − a, b; −x) = ex
a−b n=0
(a − b)!xn . (a − b − n)!n!(b)n (12)
Moreover, the series in (6) disappears exploiting the results of [15]. Expression (12) further simplifies under the assumptions of Corollary 1. Indeed, in this case, it becomes (13) (see top of next page). Finally, with reference to the correlated case of Theorem 2, the outage probability can be expressed as (14) (see next page).
C. Channel Capacity This section is devoted to the characterization of a SR fading channel from an information-theoretical point of view, under the hypothesis of MRC detection. We investigate two adaptive strategies and for both of them evaluate the channel capacity. In particular, we consider the case of Optimal Rate Allocation (ORA) and Channel Inversion with Fixed Rate (CIFR) [16], [17]. 1) Optimal Rate Allocation With Constant Transmit Power: The first strategy to be analyzed is the ORA, obtained averaging the capacity of an Additive White Gaussian Noise (AWGN) channel over the distribution of the received SNR, which is tied up to the assumed fading law. The expression of the conditional channel capacity per unit bandwidth in the case of ORA and flat-flat fading is [10], [16] Eb γ . Cu (γ) = log2 1 + N0
(15)
In order to compute the average of (15) over fγ (y), under the hypotheses of Theorem 1, we use (11) and [16, Formula 78] in (3), thus obtaining (16) (see top of next page), where Γ(·, ·) is the Complementary Incomplete Gamma Function [12]. Notice that the two series disappear for the case of integer mi ’s following the same arguments outlined in previous subsection. Under the assumptions of Corollary 1, moreover, (16) becomes (17) (see top of the next page). Finally, under the hypotheses of Theorem 2, we get (18) (see next page). 2) Channel Inversion With Fixed Rate: We derive here the average channel capacity per unit bandwidth for the CIFR transmission strategy4 [4], [10], i.e. ⎛
⎞
⎜ Cu = log2 ⎜ ⎝1 + 0
∞
Eb N0
dyfγ (y)y −1
⎟ ⎟. ⎠
4 We highlight that the obtained expressions hold only for H>1, namely when an effective receive diversity is in force.
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Pout =
Pout =
H
i=1
(ms )n n!Γ(n + H) n=0
Ω 2b0 ms + Ω
n
⎛
⎞
⎜ γ⎜ ⎝n + H,
γ0 ⎟ ⎟. Eb ⎠ 2b0 N0
k=0
i=1
βi
N0
e 2b0 Eb
H Eb 2b0 N ln 2 0
N0
(ln 2)−1 e 2b0 Eb Cu = Eb H (2b0 N ) 0
∞ ∞
δk
k=0 n=0
2b0 ms 2b0 ms + Ω
2b0 (ms + k) 2b0 (ms + k) + Ωk
ms ∞
(ms )n n! n=0
⎛ ⎜ ⎜ Cu = log2 ⎜ ⎜1 + ⎝
(H − 1)
mi H
βi i=1
β
2b0
A
⎞ Eb N0 ⎟ ⎟ ⎟, ⎟ ⎠
ms +k
(ms + k)n n!
Eb −1 Ω(2b0 N ) 0
n n+H
(2b0 ms + Ω)
Exploiting the expression of fγ (y) obtained under the hypotheses of Theorem 1, we get via [14, Formula 28]
(19)
Γ − n − H,
=1
1 Eb 2b0 N 0
m H
λi
(H − 1) ⎜ λ ⎜ i=1 ⎜ Cu = log2 ⎜1 + S ⎝ A= 2 F1
δk
k=0
2b0 (ms + k) 2b0 (ms + k) + Ωk
Cu =
i=1
λi
Eb H (2b0 N ) ln 2 0
e
1 E 2b0 b N0
∞ k,n=0
(16)
(17)
⎞ Eb 2b0 N0 ⎟ ⎟ ⎟, ⎟ ⎠
(21)
with
δk
Eb 2b0 N0
(14)
ms +k
Ωk ms + k, H − 1, H; 2b0 (ms + k) + Ωk H λ m
γ0 ⎟ ⎟ Eb ⎠ 2b0 N0
If we make the stronger assumptions of Corollary 1, we can write the CIFR capacity as ⎛ −ms ⎞ 2b0 ms 2b0 Eb (H − 1) ⎜ ⎟ N0 2b0 ms + Ω ⎜ ⎟ Cu = log2 ⎜1 + ⎟. Ω ⎝ ⎠ 2 F1 ms , H − 1, H; 2b0 ms + Ω (20) For the correlated scenario of Theorem 2,
with
∞
⎞
n Ωk Eb 2b0 N (2b0 (ms + k) + Ωk ) 0 1 Γ
− n − H, n+H E 2b0 Nb 0 , − E b =1 2b0 N0
⎛
(13)
⎛ m∞ ∞ mH+k n ⎜ λ 2b0 (mH + k) ξk (mH + k)n δk × γ⎜ ⎝n + H, λi 2b0 (mH + k) + ξk n!Γ(n + H) 2b (mH + k) + ξ 0 k n=0
mi H
β Cu =
2b0 ms 2b0 ms + Ω
ms ∞
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S=
.
2b0 (mH + k) 2b0 (mH + k) + ξk
∞ k=0
mH+k
δk
2b0 (mH + k) 2b0 (mH + k) + ξk
mH+k
n Eb −1 ξk (2b0 N ) (mH + k)n 0 n! (2b0 (mH + k) + ξk ) −1 Eb Γ
− n −H, 2b n+H 0 N0
=1
Eb 2b0 N 0
−
(18)
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5
4.5 0.3
4 0.25
3.5
H=4
0.2
Cora
fλ(λ)
3
0.15
H=2
2.5
2
0.1
H=1 1.5
0.05
1 0
0
Fig. 1. H = 4.
2
4
6 λ
8
10
12
Pdf (3) (solid curves) and simulated statistics (hystogram) with
0.5
0
1
2
3
4
5 Eb/N0
6
7
8
9
10
Fig. 3. Channel capacity (16) (solid curves) and simulated channel capacity Eb with H as a parameter. (+ marked points) versus N 0
0
10
5 −1
10
4.5
H=1
4
−2
10
3.5
H=4
3
H=3
Ccifr
Pout
H=2 −3
10
2.5
H=3
−4
10
H=2 2 −5
10
1.5
−6
10
1 0
1
2
3
4
5 E /N b
6
7
8
9
10
0
0.5
Fig. 2. Outage probability (12) (solid curves) and simulated outage E probability (+ marked points) versus N , for γ0 = 0.3, and H as a parameter.
2 F1
ξk mH + k, H − 1, H; 2b0 (mH + k) + ξk
Finally, if mi ’s are integers, the series in (19) and (21) reduce to finite sums while the Gauss series becomes a polynomial via [12, Formula 13.1.27]. IV. N UMERICAL R ESULTS We set H = 4 and consider independent SR random variables αi ’s whose parameters are given = (0.278, 0.27, 0.3, 0.277), by (Ω1 , Ω2 , Ω3 , Ω4 ) (m1 , m2 , m3 , m4 ) = (5.21, 5.2, 5.25, 5.3), and b0 = 0.251. In Fig. 1, we plot the pdf of γ given by (3) truncated at the 10-th term. For comparison purposes, the hystogram obtained through Montecarlo simulation with 106 independent runs is plotted too. The figure shows that 10 terms of the series in (3) are sufficient in order to achieve a good agreement between the theoretical distribution and the hystogram. In Fig. 2, we fix γ0 = 0.3 and, under the hypotheses of Eb , Theorem 1, plot the probability of channel outage versus N 0 for different values of H. Specifically, we adopt the parameter
0
1
2
3
4
5 E /N b
6
7
8
9
10
0
Fig. 4. Channel capacity (19) (solid curves) and simulated channel capacity Eb (+ marked points) versus N , with H as a parameter. 0
set (m1 , Ω1 ) for the case of H = 1, the set (m1 , m2 , Ω1 , Ω2 ) for the case of H = 2, and the set (m1 , m2 , m3 , Ω1 , Ω2 , Ω3 ) for the case H = 3. The theoretical curves, obtained truncating the external series in (12) at the 10-th term and the inner one to 20 terms, are plotted together with the results of Montecarlo simulations, where 106 independent realizations are averaged. A good match is achieved between the theoretical expressions and the Montecarlo simulations. Assuming the same parameters of the previous figure, we consider a communication over a SR fading channel with ORA at the transmitter. The attainable communication rate, given by (16), is plotted in Fig. 3 together with the plot of Montecarlo simulations where 10000 independent realizations are averaged to evaluate the channel capacity. Formula (16) is truncated at the 10-th term of the external series and at the 20-th term of the inner one, for all the values of H. The plots show that the truncated analytical expression matches well with the curve of the simulated capacity for all considered parameter values. Finally, under the same simulation setup
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of Fig. 3, we analyze the attainable communication rate with CIFR strategy (19). We consider the cases of H = 2, H = 3, and H = 4, and plot expression (19), truncated to the 10th term, together with the Montecarlo simulation where 106 independent realizations are averaged to estimate the channel capacity. Again, the figure shows a good agreement between the simulated and the theoretical curves. V. C ONCLUSIONS The sum of independent as well as correlated ShadowedRice distributed random variables has been completely characterized by giving the expression of the multivariate pdf. The knowledge of this distribution is crucial for the performance analysis of diversity systems for wireless communications on LMS channels. Under the hypothesis of MRC at the receiver, the outage probability and the ergodic channel capacity for ORA and CIFR, have been evaluated either in closed form or by means of series expressions, both for correlated as well as for uncorrelated LOS components of the signals on each receiver branch. The newly obtained statistical characterization is not only useful in the context of wireless communications, but also in the field of radar detectors performance analysis. Indeed, possible future research tracks might concern both the application of the model in the framework of radar detection as well as the extension of the Shadowed-Rice channel model to Multiple-Input-Multiple-Output (MIMO) LMS channels. R EFERENCES [1] G. L. Stuber, Principles of Mobile Communications. Boston, MA: Kluwer, 1996. [2] P. G. Moschopoulos, “The distribution of the sum of independent gamma random variables,” Annals Inst. Stat. Math. Part A, vol. 37, pp. 541-544, 1985.
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[3] C. H. Sim, “Point processes with correlated gamma interarrival times,” Stat. Prob. Lett., vol. 15, pp. 135-141, 1992. [4] M. S. Alouini, A. Abdi, and M. Kaveh, “Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1471-1480, Nov. 2001. [5] T. A. Tran and A. B. Sesay, “Sum of arbitrarily correlated gamma variates and performance of wireless communication systems over Nakagami-m fading channels,” IEE Proc. Commun., to be published. [6] S. Kotz and J. Adams, “Distribution of the sum of identically distributed exponentially correlated gamma random variables,” Annals Math. Stats., vol. 35, no. 2, pp. 277-283, June 1964. [7] R. K. Mallik, “On multivariate Rayleigh and exponential distribution,” IEEE Trans. Inform. Theory, vol. 49, no. 6, pp. 1499-1515, June 2003. [8] C. Loo, “A statistical model for a land mobile satellite link,” IEEE Trans. Veh. Technol., vol. 34, pp. 122-125, 1985. [9] A. Abdi, W. C. Lau, M.-S. Alouini, and M. Kaveh, “A new simple model for land mobile satellite channels: First and second order statistics,” IEEE Trans. Wireless Commun., vol. 2, no. 3, pp. 519-528, May 2003. [10] A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inform. Theory, vol. 43, no. 11, pp. 18961992, Nov. 1997. [11] R. K. Mallik, M. Z. Win, J. W. Shao, M.-S Alouini, and A. J. Goldsmith, “Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1124-1133, July 2004. [12] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover Publications, 1994. [13] L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, Addison-Wesley, 1991. [14] A. T. James, “Distribution of matrix variates and latent roots derived from normal samples,” Annals Math. Stats., vol. 35, no. 2, pp. 474-501, June 1964. [15] G. K. Karagiannidis, N. C. Sagias, and T. A. Tsiftsis, “Closed-form statistics for the sum of squared Nakagami-m variates and its applications,” IEEE Trans. Commun., vol. 54, no. 8, pp. 1353-1359, Aug. 2006. [16] M. S. Alouini and A. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining tgechniques,” IEEE Trans. Veh. Technol., vol. 48, no. 4, pp. 1165-1181, July 1999. [17] A. Maaref and S. Aissa, “Capacity of space-time block codes in MIMO Rayleigh fading channels with adaptive transmission and estimation errors,” IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2568–2578, Sep. 2005.
