Amplify-and-Forward Two-Way Relay Channels: Error ... - IEEE Xplore

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

Amplify-and-Forward Two-Way Relay Channels: Error Exponents Hien Quoc Ngo

Tony Q. S. Quek

Hyundong Shin

School of Electronics and Information Kyung Hee University Yongin-si, Gyeonggi-do 446-701, Korea Email: [email protected]

Institute for Infocomm Research 1 Fusionopolis Way #21-01 Connexis, Singapore Email: [email protected]

School of Electronics and Information Kyung Hee University Yongin-si, Gyeonggi-do 446-701, Korea Email: [email protected]

Abstract— In a two-way relay network, two terminals exchange information over a shared wireless half-duplex channel with the help of a relay. Due to its fundamental and practical importance, there has been an increasing interest in this channel. However, surprisingly, there has been little work that characterizes the fundamental tradeoff between the communication reliability and transmission rate across all signal-to-noise ratio (SNR) ratios. In this paper, we consider amplify-and-forward (AF) two-way relaying due to its simplicity. We first derive the random coding error exponent for the link in each direction. From the exponent expression, the capacity and cutoff rate for each link are also deduced. We then put forth the notion of the bottleneck error exponent, which is the worst exponent decay between the two links, to give us insight into the fundamental tradeoff between the rate pair and information-exchange reliability of the two terminals.

I. I NTRODUCTION The two-way communication channel was first introduced by Shannon, showing how to efficiently design message structures to enable simultaneous bidirectional communication at the highest possible data rates [1]. Recently, this model has regained significant interest by introducing an additional relay to support the exchange of information between the two communicating terminals. The attractive feature of this two-way relay model is that it can compensate the spectral inefficiency of one-way relaying under a half-duplex constraint [2]–[5]. With one-way relaying, we should use four phases to exchange information between two terminals via a half-duplex relay, i.e., it takes two phases to send information from one terminal to the other terminal and two phases for the reverse direction. However, exploiting the knowledge of terminals’ own transmitted signals and the broadcast nature of the wireless medium, we can improve the spectral efficiency by using only two phases to exchange information in the two-way relay channel (TWRC) [2]. Due to the fundamental and practical importance of the TWRC, much work has investigated the sum rate and the achievable rate region of the TWRC with different relaying protocols [2]–[5]. However, there has been little work that characterizes that the fundamental tradeoff between the communication reliability and transmission rate in the TWRC across all signal-to-noise ratio (SNR) regimes.1 1 Throughout the paper, we shall use a rate measured in units of nats per second per Hz (nats/s/Hz).

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In this paper, we consider half-duplex two-way AF relaying due to its simplicity in practical implementation. To characterize the fundamental tradeoff between the communication reliability and rate, we first derive Gallager’s random coding error exponent (RCEE)—the classical lower bound to Shannon’s reliability function (see, e.g., [6]–[8] and references therein)— for the link of each direction in the AF TWRC. Instead of considering only the achievable rate or error probability as a performance measure, the RCEE results can reveal the inherent tradeoff between these measures to unveil the effectiveness of two-way relaying in redeeming a significant portion of the half-duplex loss in the information exchange. From the exponent expression, the capacity and cutoff rate for each link in the TWRC are further deduced. We then introduce the bottleneck error exponent, which is defined by the worst exponent decay between the links of two directions and captures the tradeoff between the rate pair of both links and the reliability of information exchange at such a rate pair. II. S YSTEM M ODEL We consider the TWRC, where a half-duplex relay node R bidirectionally communicates between two terminals Tk∈T ={1,2} with AF relaying. In the first multiple access phase, the terminals Tk∈T transmit their information to the relay and the received signal at the relay is given by yR = h 1 x 1 + h 2 x 2 + z R

(1)

where xk∈T is the transmitted signal from the terminal Tk∈T with E |xk |2 = pk , hk ∼ CN (0, Ωk ) is the channel coefficient from Tk to the relay, and zR ∼ CN (0, N0 ) is the complex AWGN.2 Note that |hk |2 ∼ E (1/Ωk ).3 At the relay, the received signal is scaled and broadcasted to both terminals in the second broadcast phase, while satisfying its power constraint pR . Then, the received signal at the terminal Tk∈T is given by yT,k = hk xR + zT,k

(2)

2 CN µ, σ 2
denotes a circularly symmetric complex Gaussian distribution with mean µ and variance σ 2 . 3 E (λ) denotes an exponential distribution with a hazard rate λ. As in [2], we assume the channel reciprocity for hk .

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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

where  xR = GyR is the transmitted signal from the relay with E |xR |2 = pR , zT,k ∼ CN (0, N0 ) is the AWGN, and G is the relaying gain given by r pR . (3) G= 2 p1 |h1 | + p2 |h2 |2 + N0

We impose a total transmit power constraint P such that p1 + p2 ≤ P . As in [2], we further assume that the terminal Tk∈T knows its own transmitted signal and has perfect CSI to remove self-interference prior to decoding. For convenience, we shall refer to the communication link T1 → R → T2 as the link L1 and T2 → R → T1 as the link L2 , respectively. With the self-interference cancellation, the effective SNR of the link Lk∈T is given by pk pR α 1 α 2 (4) γkeff = pk αk + (pR + p1 p2 /pk ) α1 α2 /αk + 1 where αk , |hk |2 /N0 , k = 1, 2. III. E RROR E XPONENT A NALYSIS The TWRC consists of two communication links and the achievable rate can be characterized by the sum rate of two parallel relay channels under perfect self-interference cancellation [2]. As such, we need to first consider the RCEE for each link and subsequently introduce a notion of the bottleneck error exponent for the TWRC to effectively capture the tradeoff between the individual rates and the reliability. Using the Gaussian input distribution [6]–[8], we obtain the following proposition for the RCEE of each link in the TWRC. Proposition 1: With the Gaussian input distribution, the RCEE for the link Lk∈T of the TWRC with AF relaying is given by (5)

Er,k (R) = max {E0,k (ρ) − 2ρR} ρ∈[0,1]

where E0,k (ρ) = − ln Eγkeff

(

γ eff 1+ k 1+ρ

−ρ )

.

(6)

Proof: It follows directly from the results of [7] along with self-interference cancellation at the terminal Tk∈T . Remark 1: It is difficult to obtain a closed-form solution for (6) in Proposition 1 due to an analytically intractable form eff of the effective SNR γk∈T given in (4). In what follows, to alleviate such difficulty and render (6) more amenable to

further analysis, we use the upper bound γkub on the effective SNR γkeff by ignoring the term 1 in the denominator: pk pR α 1 α 2 γkub = (7) pk αk + (pR + p1 p2 /pk ) α1 α2 /αk which corresponds to the ideal/hypothetical AF relaying [9]. Remark 2: The factor 2 of ρR in (5) is due to the use of two phases for the exchange of information in the TWRC. In contrast, with one-way relaying, the information exchange occurs over four phases and hence, this factor should be 4, leading the RCEE for each link to Er,k (R) = maxρ∈[0,1] {E0,k (ρ) − 4ρR}.4 Theorem 1: With the Gaussian input distribution, the RCEE for the link Lk∈T of the TWRC with ideal/hypothetical AF relaying is given by n o ˜r,k (R) = max E ˜0,k (ρ) − 2ρR E (8) ρ∈[0,1]

˜0,k (ρ) given by (9) for 0 < ρ ≤ 1, shown at bottom of with E ˜0,k (ρ) = 0 for ρ = 0, where Γ (·) is Euler’s the page, and E K,N,N 0 ,M,M 0 gamma function, HE,(A:C),F,(B:D) [·] is the generalized Fox H-function [10, eq (2.2.1)], and ηk =

N0 N0 Ω k pR  . , λk = , µk =  p1 p2 pR + pp1 kp2 pk Ω k + p Ω Ω R 1 2 pk

Proof: Let Vk = pk αk and Wk = (p1 p2 /pk ) α1 α2 /αk . ub Then, Vk ∼ E (λk ), Wk ∼ E (µk ), γk∈T = ηk Vk Wk /(Vk + Wk ). Using the probability density function (PDF) of the Harmonic mean of the two exponential random variables [9], we obtain the PDF of γkub as  √  k )γ 2γ λk µk 4λk µk γ − (λk +µ ηk e K0 pγkub (γ) = ηk2 ηk   √ λk +µk )γ ( p 2 (λk + µk ) 2γ λk µk − ηk λk µk γe (10) + K1 ηk2 ηk where γ ≥ 0 and Kν (·) is the νth order modified Bessel function of the second kind [11, eq. (8.432.6)].

4 In the one-way relay channel (OWRC), if the total relaying power for information exchange is again constrained to pR , then the ideal/hypothetical AF relaying yields the upper bound on the effective SNR for the link Lk∈T

γkup =

pk (pR /2) α1 α2 pk αk + (pR /2) α1 α2 /αk

which is slightly different from (7) but makes no deviation in the analysis.

  √  4 λ k µk  √ √  2 π (λ + µ ) λ µ  (√λk −√µk )2 k k k k 1,0,1,2,1 ˜  E0,k (ρ) = − ln √ √
4 H1,(1:1),0,(2:1)  ηk  √  Γ (ρ) λ − µ √ 2 k k  (1+ρ)( λk − µk )  √ 4 λ k µk √ √ √ 2  4 πλk µk 1,0,1,2,1  ( λ k − µk ) + √ √
4 H1,(1:1),0,(2:1)  ηk √ Γ (ρ) λk − µk √ 2 (1+ρ)( λk − µk ) 2029

(2, 1) (1/2, 1) ; (1 − ρ, 1) — (1, 1) , (−1, 1) ; (0, 1) (2, 1) (1/2, 1) ; (1 − ρ, 1) — (0, 1) , (0, 1) ; (0, 1)

   

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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

Using (10), we have (Z  ∞ ˜0,k (ρ) = − ln E 1+ 0

γ 1+ρ

−ρ

pγkub (γ) dγ

)

. (11)

˜0,k (ρ) = 0 for ρ = 0, we define Since it is obvious that E Z ∞ −ρ I (ρ) , x (1 + ax) e−bx Kν (cx) dx (12) 0

˜0,k (ρ) in (11) for 0 < ρ ≤ 1. To evaluate the integral to find E −ρ I (ρ), we first express (1 + ax) and e−x Kν (x) in terms of the Fox H-functions with the help of [12, eqs. (8.3.2.21), (8.4.2.5), and (8.4.23.3)] as follows:   (1 − ρ, 1) 1 −ρ 1,1 H ax (13) (1 + ax) = (0, 1) Γ (ρ) 1,1   (1/2, 1) √ 2,0 e−x Kν (x) = πH1,2 2x (14) (ν, 1) , (−ν, 1) m,n where Hp,q [·] is the Fox H-function [12, eq. (8.3.1.1)]. Then, substituting (13) and (14) into (12), we have √ −2 π (b − c) I (ρ) = Γ (ρ)   2c (2, 1)  b−c (1/2, 1) ; (1 − ρ, 1)  1,0,1,2,1   . (15) × H1,(1:1),0,(2:1)   a — b−c (ν, 1) , (−ν, 1) ; (0, 1)

where the last equality follows from [10, eq. (2.6.2)]. From (10)–(12) and (15), we get (9) and complete the proof. ˜r,k (R) over ρ Remark 3: The maximum of  the exponent E 1 ˜ occurs at R = 2 ∂ E0,k (ρ) /∂ρ ρ=ρ and hence, the slope opt of the exponent–rate curve at a rate R is equal to −2ρopt . The maximizing ρopt lies in [0, 1] if " # " # ˜0,k (ρ) ˜0,k (ρ) 1 ∂E 1 ∂E Rcr,k = ≤R≤ = hCk i 2 ∂ρ 2 ∂ρ ρ=1

ρ=0

where Rcr,k and hCk i are the critical rate and the (ergodic) capacity for the link Lk∈T , respectively. For R < Rcr,k , we have ρopt = 1, yielding the slope of the exponent–rate curve ˜r,k (R) = E ˜0,k (1)−2R. Furthermore, the is equal to −2 and E ˜0,k (1) /2. cutoff rate for the link Lk∈T is given by R0,k = E This quantity is equal to the value of R at which the exponent becomes zero by setting ρ = 1. While the capacity determines the maximum achievable rate, the cutoff rate determines the

hCk i =

√

maximum practical transmission rate for possible sequential decoding strategies and indicates both the value of the zerorate exponent and the rate regime in which the error probability can be made arbitrarily small by increasing the codeword length. Corollary 1: The ergodic capacity for the link Lk∈T of the TWRC with ideal/hypothetical AF relaying is given by (16), shown at bottom of the page. Proof: It follows from Theorem 1 that " # Z ˜0,k (ρ) 1 ∂E 1 ∞ hCk i = ln (1 + γ) pγkub (γ) dγ. (17) = 2 ∂ρ 2 0 ρ=0

˜0,k (ρ), we first express Similar to the derivation of E ln (1 + γ) in terms of the Fox H-function with the help of [12, eq. (8.4.6.5)] as   (1, 1) , (1, 1) 1,2 ln (1 + γ) = H2,3 γ . (18) (1, 1) , (0, 1)

Then, again using (14) and [10, eq. (2.6.2)], we evaluate (17) as (16) and complete the proof. Corollary 2: The cutoff rate for the link Lk∈T of the TWRC with ideal/hypothetical AF relaying is given by (19) shown at bottom of the next page. Proof: It follows directly from (9) by setting ρ = 1. Remark 4: It is insufficient to characterize the information exchange in the TWRC by only investigating the RCEE for each link individually, as it just reflects the tradeoff between the communication rate and reliability for the information transmission in one direction. Therefore, we introduce a notion of the bottleneck exponent for the TWRC to capture the tradeoff between the rate pair of both links and the reliability of information exchange at such a rate pair, enabling us to optimize the resource allocation in the TWRC. Definition 1 (Bottleneck Error Probability): For a TWRC
with the terminal Tk∈T transmitting a code eN Rk , N of rate Rk , the bottleneck error probability is defined as Pemax , max Pe(k) k∈T

(20)

(k)

where Pe is the error probability of the link Lk . Note that Definition 1 can be applicable for a general TWRC, regardless of relaying protocols. From the random coding bound ˜

Pe(k) ≤ e−N Er,k (Rk )

(21)

√ 4 λ k µk √  ( λk −√µk )2 π (λk + µk ) λk µk 1,0,2,2,1  √ √
4 H1,(1:2),0,(2:2)  ηk √ λk − µk √ 2 ( λ k − µk )

 (2, 1) (1/2, 1) ; (1, 1) , (1, 1)    — (1, 1) , (−1, 1) ; (1, 1) , (0, 1)   √ (2, 1) 4 λ k µk √ √  ( λk −√µk )2 (1/2, 1) ; (1, 1) , (1, 1)  2 πλk µk 1,0,2,2,1   + √ √
4 H1,(1:2),0,(2:2)   ηk — √ λk − µk √ 2 ( λk − µk ) (0, 1) , (0, 1) ; (1, 1) , (0, 1) √



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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009 5.0

TWRC: link L1 TWRC: link L2 O WRC

1.5

Capacity and sum rate (nats/s/Hz)

Random coding error exponent

1.8

1.2

0.9

0.6

0.3

0.0 0.0

0.3

0.6

0.9

1.2

R ( na t s / s / H z )

4.0

 

3.0

T W R C : lin k L T W R C : lin k L

2.0

2 1

O W R C 1.0

0.0

1.5

sum rate c ap ac i ty

0

5

10

15

SNR ( d B )

20

25

30

Fig. 1. Random coding error exponent for the link Lk∈T of the TWRC and OWRC with ideal/hypothetical AF relaying. Ω1 = 0.5, Ω2 = 2, and SNR = 20 dB.

Fig. 2. Capacity and sum rate versus SNR for for the link Lk∈T of the TWRC and OWRC with ideal/hypothetical AF relaying. Ω1 = 0.5 and Ω2 = 2.

the bottleneck error probability of the TWRC is bounded by

the rate pair (R1 , R2 ) and the bottleneck error exponent (i.e., information-exchange reliability).

Pemax ≤ max e k∈T

˜r,k (Rk ) −N E

.

(22)

Using (22), we define the bottleneck error exponent of the TWRC as follows: Definition 2 (Bottleneck Error Exponent): For a
TWRC with the terminal Tk∈T transmitting a code eN Rk , N of rate Rk , the bottleneck error exponent at the information-exchange rate pair (R1 , R2 ) is defined as ˜r,k (Rk ) . Er? (R1 , R2 ) , min E (23) k∈T Remark 5: Using the RCEE of the link Lk∈T in Theorem 1, we can readily obtain the bottleneck error exponent Er? (R1 , R2 ). From Definition 2, we can see that the bottleneck error exponent captures the behavior of the worst exponent decay between the two links in the TWRC and reflects the reliability of the information exchange at a rate pair (R1 , R2 ). Using the worst exponent decay as the reliability measure, we can design a two-way relay network such that both links can communicate reliably, and connect a bottleneck exponent value to each rate pair (R1 , R2 ). Therefore, besides the achievable rate region, we can also characterize the bottleneck exponent plane from the set of all possible rate pairs. This plane could provide us with further understanding of the tradeoff between

R0,k

IV. N UMERICAL R ESULTS In all examples, we choose Ω1 = 0.5, Ω2 = 2, pR = P , and define SNR , P/N0 . We further consider equal power allocation between two terminals Tk∈T (p1 = p2 = P/2).5 To ascertain the effectiveness of two-way relaying in terms of the error exponent, Fig. 1 shows the RCEE for the link Lk∈T of the TWRC and OWRC at SNR = 20 dB. To calculate the RCEE, we use Theorem 1 for two-way relaying, whereas we modify Theorem 1 for one-way relaying in such a manner as described in Remark 2. In the regime below the critical rate, the exponent of the TWRC decreases with the rate twice as slow as in the OWRC and hence, we require to increase the codeword length slowly with two-way relaying to achieve the same level of reliable information exchange as the rate increases. This is due to the spectral efficiency of two-way relaying that requires only half the time duration of one-way relaying to exchange the information. Figs. 2 and 3 demonstrate the effectiveness of two-way relaying on the achievable rates, where the capacity (or achiev5 For one-way relaying, the RCEE, capacity, and cutoff rate are symmetric and equal for two links in the case of equal power allocation.

  √ (2, 1)  4 λ k µk  √ √  (√λk −√µk )2 (1/2, 1) ; (0, 1) 1  2 π (λk + µk ) λk µk 1,0,1,2,1  = − ln H1,(1:1),0,(2:1)  √ √
4 ηk — 2  √  λ − µ √ 2 k k  2( λ k − µ k ) (1, 1) , (−1, 1) ; (0, 1)  √ (2, 1) 4 λ k µk √ √ √ 2  4 πλk µk ( λk − µk ) (1/2, 1) ; (0, 1) 1,0,1,2,1  H + √ √
4 1,(1:1),0,(2:1)  ηk — √ λ k − µk √ 2 2( λ k − µ k ) (0, 1) , (0, 1) ; (0, 1) 2031

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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009 1.5

TWRC: link L1 TWRC: link L2 O WRC

1.0

0.5

R2=0.2 0.9

R2=0.5 0.6

0.3

0

5

10

}

1.2

1.5

0.0

{

min G R1 ∈ R : E
 1 ( R1 ) ≤ E
 2 ( R2 )

R2 = 0

Minimal error exponent

Cutoff rate (nats/s/Hz)

2.0

15

SNR ( d B )

20

25

0.0 0.0

30

R2=0.7

R2=1.1 0.2

0.4

0.6

R1 ( n a t s / s / H z )

0.8

1.0

1.2

Fig. 3. Cutoff rate versus SNR for the link Lk∈T of the TWRC and OWRC with ideal/hypothetical AF relaying. Ω1 = 0.5 and Ω2 = 2.

Fig. 4. Bottleneck error exponent Er? (R1 , R2 ) versus R1 for the TWRC with ideal/hypothetical AF relaying at R2 = 0, 0.2, 0.5, 0.7, and 1.1 nats/s/Hz: Ω1 = 0.5, Ω2 = 2, and SNR = 20 dB.

able sum rate) and cutoff rate versus SNR are depicted for the link Lk∈T of the TWRC and OWRC, respectively. We can see from the figures that the slopes of the capacity, achievable sum rate, and cutoff rate curves at high SNR are twice as large in the TWRC as in the OWRC due to again the fact that two-way relaying for the information exchange can reduce the spectral efficiency loss of half-duplex signaling by half in the TWRC. Hence, as can be seen in Fig. 2, the high-SNR slope of the capacity for the link Lk∈T of the TWRC is identical to that of the achievable sum rate in the OWRC. Fig. 4 shows the bottleneck error exponent Er? (R1 , R2 ) versus R1 for the TWRC at SNR = 20 dB when R2 = 0, 0.2, 0.5, 0.7, and 1.1. For fixed R2 , the bottleneck exponent Er? (R1 , R2 ) as a function of R1 behaves  identically ˜r,1 (R1 ) at R1 ≥ R, where R = min R1 ∈ R+ : to E ˜r,1 (R1 ) ≤ E ˜r,2 (R2 ) , whereas Er? (R1 , R2 ) is limited to E ˜r,1 (R) for all R1 ≤ R. In this example, the values of E R are equal to 0, 0.16, 0.36, 0.54, 0.92 for R2 = 0, 0.2, 0.5, 0.7, and 1.1, respectively. As can be seen, the bad link in terms of the exponent is a bottleneck limiting reliable information exchange, and the bottleneck exponent at large R1 or R2 becomes small, indicating the achievable reliability of information exchange would be low at such rate pairs.

information-exchange reliability at such a rate pair for the design of two-way relay networks such that both links can communicate reliably.

V. C ONCLUSIONS In this paper, we have derived Gallager’s random coding exponent to analyze the fundamental tradeoff between the communication reliability and transmission rate in two-way AF relay channels. The exponent has been expressed in terms of the generalized Fox H-function, from which the capacity and cutoff rate were also deduced for the link of each direction in the TWRC. Using the worst exponent decay between two links as the reliability measure for the information exchange, we defined the bottleneck error exponent to effectively capture the tradeoff between the rate pair of the two links and the

ACKNOWLEDGMENT This research was supported, in part, by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2007-000-11202-0). R EFERENCES [1] C. E. Shannon, “Two-way communication channels,” in Proc. 4th Berkeley Symp. Probability and Statistics, vol. 1, Berkeley, CA, 1961, pp. 611–644. [2] B. Rankov and A. Wittneben, “Spectral efficient protocols for halfduplex fading relay channels,” IEEE J. Select. Areas Commun., vol. 25, no. 2, pp. 379–389, Feb. 2007. [3] S. J. Kim, P. Mitran, and V. Tarokh, “Performance bounds for bidirectional coded cooperation protocols,” IEEE Trans. Inform. Theory, vol. 54, no. 11, pp. 5235–5241, Nov. 2008. [4] T. J. Oechtering, I. Bjelakovic, C. Schnurr, and H. Boche, “Broadcast capacity region of two-phase bidirectional relaying,” IEEE Trans. Inform. Theory, vol. 54, no. 1, pp. 454–458, Jan. 2008. [5] C. Schnurr, S. Stanczak, and T. J. Oechtering, “Achievable rates for the restricted half-duplex two-way relay channel under a partial-decodeand-forward protocol,” in Proc. IEEE Information Theory Workshop (ITW’08), Porto, Portugal, May 2008, pp. 134–138. [6] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968. [7] W. K. M. Ahmed and P. J. McLane, “Random coding error exponents for two-dimensional flat fading channels with complete channel state information,” IEEE Trans. Inform. Theory, vol. 45, no. 4, pp. 1338– 1346, May 1999. [8] H. Shin and M. Z. Win, “Gallager’s exponent for MIMO channels: A reliability–rate tradeoff,” IEEE Trans. Commun., vol. 57, no. 4, pp. 972– 985, Apr. 2009. [9] M. O. Hasna and M. S. Alouini, “End-to-end performance of transmission systems with relays over Rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1126–1131, Nov. 2003. [10] A. M. Mathai and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines. New York: Wiley, 1978. [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000. [12] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series. New York: Gordon and Breach Science, 1990, vol. 3.

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