1
Decoding and Performance Bound of Demodulate-and-Forward Based Distributed Alamouti STBC Ankur Bansal, Member, IEEE, Manav R. Bhatnagar† , Member, IEEE, and Are Hjørungnes, Senior Member, IEEE
Abstract—In a demodulate-and-forward (DF) based cooperative communication system, erroneous relaying of the data leads to degradation in the performance of the destination receiver. However, a maximum likelihood (ML) decoder in the destination can improve the receiver performance. For achieving a diversity gain, the Alamouti space-time block code (STBC) can be used in the DF based cooperative system in a distributed manner. In this paper, we derive an ML decoder of the distributed Alamouti STBC for the DF based cooperative system with two imperfect relaying nodes. We also consider a DF cooperative communication system in which one out of two relays is in outage. A piece-wise linear (PL) decoder for the DF cooperative system with the distributed Alamouti code and one relay in outage is proposed. The PL decoder provides approximately the same performance as that of the ML decoder with reduced decoding complexity. We derive the pairwise error probability (PEP) of the proposed ML decoder with binary phase-shift keying constellation. An optimized transmit power allocation for the relays is performed by minimizing an upper bound of the PEP. It is shown by simulations that the proposed ML decoder enables the DF protocol based cooperative system to outperform the same rate amplify-and-forward protocol based cooperative system when both systems utilize the distributed Alamouti STBC.
I. I NTRODUCTION Cooperative communication is a potential technology for future wireless communication systems. It is very useful for the nodes aspiring for better throughput, despite their poor links to the destination. In cooperative communication, a source node transmits the data to the neighboring nodes (relays) which have better links to the destination and agree to cooperate with the source node. The cooperation among the distributed nodes provide benefits of a co-located multiple antenna system, such as diversity gain [1], [2]. The relays can simply demodulate-and-forward (DF) [3]–[5] or amplify-and-forward (AF) [6] the data of the source in a symbol-wise manner. The AF protocol is a non-regenerative protocol, where the relays merely scale the signals received Ankur Bansal and Manav R. Bhatnagar are with the Department of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India. (E-mail:
[email protected],
[email protected]). Are Hjørungnes was with UNIK - University Graduate Center, University of Oslo, NO-2027, Kjeller, Norway. (E-mail:
[email protected]). This work was supported in part by the Department of Science and Technology, Government of India under SERC scheme for the Project "Interference Cancelation in MAC Based Multiuser MIMO Communication Systems" (Project Ref. No. SR/S3/EECE/0089/2009). The material in this paper was presented in part at IEEE Vehicular Technology Conference (VTC) Fall, Sep. 2012, Quebec City, Canada. † Corresponding author.
from the source by an analog value. In the DF or regenerative protocol, the source broadcasts uncoded1 data to the relay and the destination. The relay demodulates the symbols transmitted by the source and forwards them to the destination in uncoded form; hence, the destination has two received replicas of the transmitted data. However, the relay cannot always demodulate the symbols, sent by the source, perfectly and it can relay erroneous data to the destination. Hence, the DF protocol cannot achieve diversity in its pure form [4]. The demodulate-and-forward protocol requires a low-level functionality of symbol-wise demodulation in the relay and reduces the complexity of the hardware and the energy consumption at the relay; nevertheless, it can be extended to combine with coding techniques [7], [8]. A more complicated form of the DF protocol is the decode-and-forward protocol [9], where the source and relays utilize forward error correction (FEC) coding. In the decode-and-forward protocol, the source transmits the coded data and the relay attempts to fully decode the source codeword; the relay re-encodes and transmits the decoded codeword to the destination upon successful decoding. The relays in the decode-and-forward protocol require more complex hardware and battery power. It is shown in [3], [5] that the performance of the DF based simple uncoded cooperative communication system with a single pair of the source and destination node and a single relay can be improved by using a maximum-likelihood (ML) decoder in the destination receiver. The ML decoder considers the possibility of erroneous transmission at the relay terminal and maximizes the probability density function (p.d.f.) of the received data in the destination terminal [3]. An ML decoder of the uncoded binary phase-shift keying (BPSK) data in a multiple antenna based DF cooperative system, where the relays utilize orthogonal transmissions, is given in [10]. In [11], an ML decoder and a low complexity piecewise linear (PL) decoder of the single antenna based DF cooperative system are derived, which require the average error probability of the source-relay links for decoding of the source’s data in the destination. In [12], [13], estimate-and-forward (EF) protocol is discussed for a cooperative system utilizing uncoded transmissions. In the EF protocol, the relay forwards an unconstrained minimum mean-square error estimate that maximizes the generalized signal-to-noise ratio (SNR) at the destination over additive white Gaussian noise (AWGN) channel [13, Eq. (12)]. It is shown in [13, Section IV-B and Fig. 4] that for M -point 1 Here from uncoded we mean that without forward error correction (FEC) coding.
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constellations, M > 2, the EF protocol tends to approach the DF protocol. In order to achieve better diversity in the cooperative communication system with multiple relays, distributed space-time block codes (STBCs) have been proposed mainly for the AF protocol in the literature [14]–[18]. In such systems, each relay transmits a single row of the STBC designed for a co-located multiple-input multiple-output (MIMO) system. Moreover, no orthogonal transmission is utilized by the relays. Therefore, the distributed STBC based cooperative communication system realizes a virtual MIMO system with distributed antennas. In [9], distributed STBC for DF protocol with relay selection is discussed by utilizing uncoded transmissions. In order to reduce the chances of wrong relaying, the relays, having high SNR links to the source, are selected for transmission of the distributed STBC in [9]. However, it is shown in [17] that the distributed STBC utilizing uncoded DF protocol with relay selection performs much worse than the AF based distributed STBC. In this paper, our main contributions are as follows. 1) We derive an ML decoder of the distributed Alamouti STBC [19] in a DF based cooperative communication system with two relays and M -ary constellation. 2) A low complexity PL decoder for the case when one of the two relays is in outage is derived. 3) An upper bound of the pair-wise error probability (PEP) is derived for the proposed ML decoder of the DF based distributed Alamouti code with BPSK constellation. 4) It is shown by simulation that the DF based distributed Alamouti STBC with the proposed ML decoder outperforms the distributed Alamouti code with AF protocol. The main differences between the present paper and [11] are as follows. In [11], a simple cooperative relay network with a single relay is considered; whereas, a cooperative setup with two relays and no direct link is presented in this paper. The proposed ML decoder in this paper is applicable to a DF cooperative system where the data of the source reaches the destination through two relays in the form of Alamouti STBC. A suboptimal PL decoder for a simple DF cooperative system is derived in [11], which closely approximates the ML decoder in [11]; whereas, in the present paper, a low-complexity PL decoder of the distributed Alamouti STBC is derived for the case when one of the two relays is in outage. The conditional symbol error rate (SER) of the PL decoder is derived in [11]; however, we find the conditional PEP of the proposed ML decoder with BPSK constellation. The rest of this paper is organized as follows: In Section II, the system model is introduced. The derivation of the ML decoder and an upper bound of the PEP of the DF based distributed Alamouti STBC is given in Section III. In Section IV, the ML and PL decoders of the distributed Alamouti code with a single relay in outage are derived. Simulation results are discussed in Section V. Section VI concludes the article. This article contains two appendices. II. S YSTEM M ODEL Let us consider a cooperative system containing one source S, one destination D, and two relays R1 and R2 as shown
Fig. 1. Block diagram of cooperative communication system with two relays and a single source-destination pair.
in Fig. 1. Each node contains one antenna and it can either transmit or receive the data at a time. It is assumed that the channel between the source and the destination is very poor; therefore, direct transmission between the source and the destination is not possible. The source utilizes two relaying nodes for transmission of its uncoded data to the destination. The transmission from the source to the destination can be decoupled into two orthogonal phases. In Phase I, the source transmits the uncoded data sequentially to both relays. The relays symbol-wise demodulate the data of the source and transmit the Alamouti STBC in a distributed manner to the destination in Phase II. In this phase, the source remains silent. The destination decodes the data by utilizing an ML decoder. It is assumed that the channels of all links involved in the cooperation follow the Rayleigh block fading model [20] and stay constant for a block of at least two consecutive time intervals. Let s1 , s2 ∈ A, where A is a complex-valued M -point constellation, be two uncoded symbols to be transmitted by the source to the destination in a block2 . The transmission of s = [s1 , s2 ] to the destination is performed in two decoupled phases. In Phase I, the source sequentially transmits s1 and s2 to the relays; hence, the data received in the m-th relay, m ∈ {1, 2}, can be written as y m = hm s + e m ,
(1)
where y m = [ym,1 , ym,2 ] represents the 1 × 2 received data vector, ym,n , n = 1, 2, denotes the signal received in the m-th relay in n-th time interval, hm denotes the channel gain of the link between the source and the m-th relay, and em = [em,1 , em,2 ] is the 1 × 2 complex-valued additive white Gaussian noise vector with each element having zero mean and N0 variance. It is assumed that the channel hm is a circular 2 complex Gaussian random variable with zero mean and σm variance, and remains constant over transmission of s1 and s2 . The Alamouti STBC is given as [19] s1 −s∗2 . (2) X= s2 s∗1 2 Since each block is independent of other because of the block fading model, we skip the notation for a block in the article.
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In a co-located antenna system with two transmit antennas, each antenna transmits a distinct row of (2) in two consecutive time intervals. However, in the DF cooperative system with two relays, both relays act as distributed spatial dimensions; hence, they can transmit distinct rows of Alamouti STBC in two consecutive time intervals. Therefore, in Phase II, the relays demodulate the symbols transmitted by the source by using an ML demodulator and transmit the Alamouti STBC in two consecutive time intervals by using the the estimates of s1 and s2 . It is further assumed that the transmissions from the relays are perfectly synchronized. Let x ˆn,m be the estimated symbol by the m-th relay in the n-th time interval, then we can write the data received at the destination in the two consecutive time-intervals, due to transmission of Alamouti code from the relays, as y1 = f1 x ˆ1,1 + f2 x ˆ2,2 + z1 , y2 = −f1 x ˆ∗2,1 + f2 x ˆ∗1,2 + z2 ,
as ǫ1 (1 − ǫ2 )a1 + (1 − ǫ1 )ǫ2 a2 +ǫ1 ǫ2 a3 + (1 − ǫ1 )(1 − ǫ2 )a4 Λdp,q = ln ǫ1 (1 − ǫ2 )b1 + (1 − ǫ1 )ǫ2 b2 +ǫ1 ǫ2 b3 + (1 − ǫ1 )(1 − ǫ2 )b4 ǫ2 (1 − ǫ1 )c1 + (1 − ǫ2 )ǫ1 c2 +ǫ1 ǫ2 c3 + (1 − ǫ1 )(1 − ǫ2 )c4 +ln ǫ2 (1 − ǫ1 )d1 + (1 − ǫ2 )ǫ1 d2 ,
+ǫ1 ǫ2 d3 + (1 − ǫ1 )(1 − ǫ2 )d4
where ǫ1 and ǫ2 denote the uncoded instantaneous probability of errors in decoding a symbol belonging to A in R1 and R2 , respectively; p = [p1 , p2 ], q = [q1 , q2 ], p1 , p2 , q1 , q2 = 1, 2, ..., M , p 6= q; M X
a1 = (M − 1) (3)
where fm is the channel gain of the m-th relay-destination link and zm denotes the complex-valued zero mean AWGN noise with N1 variance. It is assumed that the channel fm is circular complex Gaussian with zero mean and Ω2m variance, and remains constant over the transmission period of one distributed Alamouti block. We can rewrite (3) as y d = [f1 , f2 ]
x ˆ1,1 x ˆ2,2
−ˆ x∗2,1 x ˆ∗1,2
S=
x ˆ1,1 x ˆ2,2
−ˆ x∗2,1 x ˆ∗1,2
1
M X
a2 = (M − 1)
2
e− N1 |y1 −f1 xp1 −f2 xl | , 1
l=1, l6=p2
a3 =
M M X X
2
1
e− N1 |y1 −f1 xl −f2 xk | ,
l=1, k=1, l6=p1 k6=p2 2
a4 = (M − 1) e− N1 |y1 −f1 xp1 −f2 xp2 | , + zd,
1
(7)
(4)
where y d = [y1 , y2 ] represents the 1 × 2 received data vector and z d = [z1 , z2 ] denotes the 1 × 2 AWGN noise vector. From (4), we can observe that the distributed Alamouti STBC, transmitted by both relays, will be
2
e− N1 |y1 −f1 xl −f2 xp2 | ,
l=1, l6=p1
2
(6)
c1 = (M − 1)
M X
∗ 2
e− N1 |y2 +f1 xp2 −f2 xl | , ∗
1
l=1, l6=p1
c2 = (M − 1)
M X
2
e− N1 |y2 +f1 xl −f2 xp1 | , ∗
1
∗
l=1, l6=p2
.
(5)
It can be seen from (5), that when x ˆn,i 6= x ˆn,j , where n, i, j ∈ {1, 2}, and i 6= j, then SS H is not necessarily proportional to the identity matrix. Therefore, S is not necessarily an orthogonal STBC when the relays commit error in demodulation of the symbols transmitted by the source.
III. ML D ECODER OF THE D ISTRIBUTED A LAMOUTI STBC AND ITS P ERFORMANCE B OUND
c3 =
M M X X
1
∗
∗ 2
e− N1 |y2 +f1 xk −f2 xl | ,
l=1, k=1, l6=p1 k6=p2 2
c4 = (M − 1)2 e− N1 |y2 +f1 xp2 −f2 xp1 | ; 1
∗
∗
(8)
and bi and di , i = 1, 2, 3, 4, can be obtained by substituting {q1 , q2 } in place of {p1 , p2 } in (7) and (8), respectively; and xp1 , xp2 , xq1 , xq2 ∈ A. Proof: Refer Appendix A for the proof. The proposed LLR decoder, given in (6), is used to decide between two symbol vectors xp = [xp1 , xp2 ] and xq = xp
We assume that the relays utilize an ML decoder for demodulation of the symbols transmitted by the source. The ML demodulator of an arbitrary constellation over co-located MIMO links is well known in the literature [21]. In this section, we will explain the ML decoding of the distributed Alamouti code in the destination node. Theorem 1: For the DF based cooperative relay network with no direct link and two relays, a log likelihood ratio (LLR) based ML decoder of the distributed Alamouti STBC using an arbitrary complex-valued M -point constellation (A) is given
[xq1 , xq2 ], xp 6= xq , as follows: Λdp,q ≷ 0. The proposed xq
decoder can be applied to all possible dissimilar vector pairs containing the symbols belonging to the M -point constellation, A, for a final decision of the transmitted symbol vector. Remark 1: It can be seen from (6) that the ML decoding of the two symbols transmitted by the source is performed jointly. Since the distributed Alamouti STBC S, in DF based cooperative system is not necessarily an orthogonal design under the estimation errors in relays, decoupled decoding of the symbols is not possible.
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Further, it can be noticed from (6) that for decoding the data transmitted by the source, the destination requires the knowledge of the uncoded instantaneous error probabilities of R1 and R2 . Since the instantaneous probability of error depends upon the instantaneous channel coefficients [22], in practice, the values of h1 and h2 can be forwarded by the relays to the destination for calculation of the values of ǫ1 and ǫ2 . In order to provide the channel state information (CSI) of the source-relay and the relay-destination links in the destination, we can use a three phase training protocol as follows. In the first phase, the source transmits training data to the relays such that the relays estimate the source-relay channels. In the second phase, the destination estimates the relay-destination channels by using the training data transmitted by the relays. The acquired CSI of the source-relay links is forwarded by the relays to the destination over some reliable and secure channels in the third phase. In Subsection V-A, we will discuss effect of imperfect values of ǫ1 and ǫ2 over the performance of the proposed ML decoder. Remark 2: Let us define γ¯1 , P0 σ12 /N0 , γ¯2 , P0 σ22 /N0 , γ¯3 , P1 Ω21 /N1 , and γ¯4 , P2 Ω22 /N1 as the average SNRs of S-R1 , S-R2 , R1 -D, and R2 -D links, respectively, where P0 and Pm are the average transmitted powers of the source and the m-th relay, respectively. If the channel between the source and the relays is very good, i.e., γ¯1 , γ¯2 → +∞, such that ǫ1 , ǫ2 → 0+ , then after some algebra we get a simplified decoder from (6) as Λdp,q= 2Re{(y1∗ f1 +y2 f2∗ )(xp1−xq1)+(y1∗ f2 −y2 f1∗ )(xp2−xq2)}. (9) The decoder of (9) is equivalent to the ML decoder of the Alamouti code in a co-located antenna system [19, Eq. (13)] which provides decoupled decoding of the transmitted symbols. For BPSK (i.e., M = 2), the ML decoder of (6) gets simplified into Λdp,q
= ln
Ap Aq
+ ln
Bp Bq
,
(10)
where 2
1 Ap = ǫ1 (1 − ǫ2 )e− N1 |y1 −f1 x¯p1 −f2 xp2 | + (1 − ǫ1 )ǫ2 2 2 1 1 × e− N1 |y1 −f1 xp1 −f2 x¯p2 | + ǫ ǫ e− N1 |y1 −f1 x¯p1 −f2 x¯p2 |
1 2
− N1 |y1 −f1 xp1 −f2 xp2 |
2
, + (1 − ǫ1 )(1 − ǫ2 )e 1 2 1 Bp = ǫ2 (1 − ǫ1 )e− N1 |y2 +f1 xp2 −f2 x¯p1 | + (1 − ǫ2 )ǫ1 2 2 1 1 × e− N1 |y2 +f1 x¯p2 −f2 xp1 | + ǫ ǫ e− N1 |y2 +f1 x¯p2 −f2 x¯p1 | 1 2
+ (1 − ǫ1 )(1 − ǫ2 )e
− N1 |y2 +f1 xp2 −f2 xp1 | 1
2
,
(11)
Aq and Bq can be obtained by replacing the variables {p1 , p2 } with {q1 , q2 } in the expressions of Ap and Bp , respectively; ¯k = p1 , p2 , q1 , q2 = 1, 2; xp1 , xp2 , xq1 , xq2 ∈ [1, −1]; and x −xk , for k ∈ {p1 , p2 , q1 , q2 }. Theorem 2: The uncoded average PEP of the proposed ML
TABLE I VALUES OF P R (El |h1 , h2 ).
El E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16
x ˆ1,1 x q1 x ¯ q1 x q1 x q1 x q1 x ¯ q1 x q1 x q1 x ¯ q1 x ¯ q1 x q1 x ¯ q1 x q1 x ¯ q1 x ¯ q1 x ¯ q1
x ˆ2,1 xq2 xq2 x ¯ q2 xq2 xq2 x ¯ q2 x ¯ q2 xq2 xq2 xq2 x ¯ q2 x ¯ q2 x ¯ q2 xq2 x ¯ q2 x ¯ q2
x ˆ1,2 x q1 x q1 x q1 x ¯ q1 x q1 x q1 x ¯ q1 x ¯ q1 x q1 x ¯ q1 x q1 x ¯ q1 x ¯ q1 x ¯ q1 x q1 x ¯ q1
x ˆ2,2 xq2 xq2 xq2 xq2 x ¯ q2 xq2 xq2 x ¯ q2 x ¯ q2 xq2 x ¯ q2 xq2 x ¯ q2 x ¯ q2 x ¯ q2 x ¯ q2
Pr (El |h1 , h2 ) (1 − ǫ1 )2 (1 − ǫ2 )2 ǫ1 (1 − ǫ1 )(1 − ǫ2 )2 ǫ1 (1 − ǫ1 )(1 − ǫ2 )2 ǫ2 (1 − ǫ1 )2 (1 − ǫ2 ) ǫ2 (1 − ǫ1 )2 (1 − ǫ2 ) ǫ21 (1 − ǫ2 )2 ǫ1 ǫ2 (1 − ǫ1 )(1 − ǫ2 ) ǫ22 (1 − ǫ1 )2 ǫ1 ǫ2 (1 − ǫ1 )(1 − ǫ2 ) ǫ1 ǫ2 (1 − ǫ1 )(1 − ǫ2 ) ǫ1 ǫ2 (1 − ǫ1 )(1 − ǫ2 ) ǫ21 ǫ2 (1 − ǫ2 ) ǫ1 ǫ22 (1 − ǫ1 ) ǫ1 ǫ22 (1 − ǫ1 ) ǫ21 ǫ2 (1 − ǫ2 ) ǫ21 ǫ22
decoder of the distributed Alamouti code is given by Pr{xq → xp |s = xq } = " 16 X µp +λp −µq −λq Eh1,h2,f1,f2 Qq σp2 +δp2 +σq2 +δq2 +2σp σq +2δp δq l=1 El # × Pr {El |h1, h2 } ,
(12)
where xp 6= xq , s denotes the transmitted symbol vector, Pr{·} represents the probability, Eh1 ,h2 ,f1 ,f2 [·] denotes the expectation over h1 , h2 , f1 , and f2 ; and Q (·) |El denotes qfunction [21, Eq. (2.3.10)] evaluated at El , where El , l = 1, 2, ..., 16 is the l-th mutually exclusive event depending upon the correct and erroneous decoding by both relays. For example, the event E1 will occur if both relays decode s1 and s2 correctly, and E2 corresponds to an event when R1 commits error in decoding s1 but it decodes s2 correct, and R2 decodes s1 and s2 correctly. All such events are listed in Table I. The probability Pr {El |h1 , h2 } is the conditional probability of occurrence of the event El given that the channel gains h1 , h2 are perfectly known. The terms µℓ , λℓ , σℓ2 , and δℓ2 , ℓ ∈ {p, q} are given in (33). Proof: Refer Appendix B for the proof. It is difficult to analytically find the average PEP from (12), therefore, we can numerically calculate the average PEP by using (12). From (12), an upper bound of PEP (UBPEP) for the proposed ML decoder of distributed Alamouti code can be given as Pr{xq → xp |s = xq} ≤
max
{xp1,xp2,xq1,xq2}∈A4
Pr{xq → xp |s = xq}.
p6=q
(13) The UBPEP given in (13) provides the probability of the worst possible error, which contributes significantly in the
5
20 15
ε =0 1
Λd(1) p,q
−6
ε =10 1
10
−5
ε =10 1
ε =10 1
−3
ε =10
− N1 |y1 −f1 xq1 −f2 xp2 | ǫ1 1 (M −1) e 2 − N1 |y1 −f1 xp1 −f2 xp2 | − ǫ1 )e 1
2
− N1 |y1 −f1 xp1 −f2 xq2 | ǫ1 1 (M −1) e 2 − N1 |y1 −f1 xq1 −f2 xq2 | − ǫ1 )e 1
2
αp,q +
+(1 = ln α + p,q +(1
−4
5
,
(15)
φ(t1)
1
0
−2
ε =10 1
where αp,q =
−5 −10 −15 −20 −20
−15
−10
−5
0 t
5
10
15
20
1
Fig. 2. Plots of the function φ(t1 ) given in (16) versus the variable t1 for different values of the probability of error in the relay using QPSK constellation. TABLE II VALUES OF T1 FOR DIFFERENT VALUES OF ǫ1 USED IN F IG . 2.
ǫ 10−2 10−3 10−4 10−5 10−6 T1 ±5.6937 ±8.0054 ±10.3089 ±12.6115 ±14.9141
PEP; therefore, for finding an optimized power distribution, we minimize the UBPEP of (13). IV. ML AND PL D ECODERS IN D ESTINATION WITH A S INGLE R ELAY IN O UTAGE Let us assume that one of the two relays is in outage. This scenario exists when one of the two relays is very close to the source such that the channel between the source and this relay is very good; whereas, another relay is relatively very far from the source, hence, it experiences its channel from the source in outage. If R1 is the relay in outage, then ǫ2 = 0, and we get the following ML decoder for M -ary constellation from (6): 2 PM − N1 |y1 −f1 xl −f2 xp2 | ǫ1 1 l=1, e (M −1) l6=p1 2 − N1 |y1 −f1 xp1 −f2 xp2 | 1 +(1 − ǫ )e 1 d Λp,q = ln 2 1 P M y −f x −f x − | | ǫ 1 1 l 2 q2 1 N1 l=1, e (M −1) l6=q1
2
1 +(1 − ǫ1 )e− N1 |y1 −f1 xq1 −f2 xq2 | ∗ 2 PM − N1 |y2 +f1 x∗ ǫ1 l −f2 xp1 | 1 l=1, e (M −1) l6=p2 ∗ ∗ 2 1 +(1 − ǫ1 )e− N1 |y2 +f1 xp2 −f2 xp1 | . (14) +ln ∗ ∗ 2 1 P M − N |y2 +f1 xl −f2 xq1 | ǫ1 1 l=1, e (M −1) l6=q2
+(1 − ǫ1 )e− N1 |y2 +f1 xq2 −f2 xq1 | 1
∗
∗
2
If p 6= q such that xp1 6= xq1 and xp2 = xq2 , then we can rewrite the first term in the right hand side (R.H.S.) of the LLR decoder of (14) as
ǫ1 (M −1)
M P
2
e− N1 |y1 −f1 xl −f2 xp2 | . It is diffi1
l=1 l6=p1 ,q1
cult to simplify (15) further. However, let us neglect the term αp,q in the numerator and denominator in (15) to obtain a suboptimal decoder. It will be shown by using the quadrature phase shift keying (QPSK) signaling scheme in Fig. 8 that neglecting these terms does not degrade the performance of the decoder significantly at all SNRs. Therefore, we get the following approximate LLR from (15): ǫ1 + (M − 1)(1 − ǫ1 )et1 Λd(1) ≈ φ(t ) = ln , (16) 1 p,q ǫ1 et1 + (M − 1)(1 − ǫ1 )
where, for M -PSK constellation 2 ∗ (17) Re (y1 − f2 xp2 )f1∗ (xp1 −xq1 ) . t1 = N1 In [10], a PL combiner is derived for DF based MIMO relay network using the binary phase shift keying (BPSK) constellation. It can be seen from (16) that when ǫ1 = 0, φ(t1 ) = t1 ; and for very large and very small values of t1 , φ(t1 ) is clipped to T1 = ±ln [(M − 1)(1 − ǫ1 )/ǫ1 ]. We have plotted φ(t1 ) for different values of ǫ1 and t1 for QPSK constellation (M = 4) in Fig. 2. The values of T1 for different values of ǫ1 are also listed in Table II. It can be seen from Fig. 2 and Table II that when t1 > T1 , φ(t1 ) ≈ T1 and when t1 < −T1 , φ(t1 ) ≈ −T1 . Moreover, for −T1 ≤ t1 ≤ T1 , φ(t1 ) ≈ t1 . Therefore, we can approximate φ(t1 ) by a PL function as follows: t1 < −T1 , −T1 , if t1 , if −T1 ≤ t1 ≤ T1 , φ(t1 ) ≈ φPL (t1 ) , (18) T1 , if t1 > T 1 .
Following a similar procedure as stated above, we can get an approximation for the second term on the R.H.S. of (14) for xp2 6= xq2 and xp1 = xq1 as follows: ǫ1 + (M − 1)(1 − ǫ1 )et2 , (19) Λd(2) ≈ φ(t ) = ln 2 p,q ǫ1 et2 + (M − 1)(1 − ǫ1 )
where, for M -PSK constellation, t2 = ∗ 2 ∗ ∗ . We can approximate ) −x )f (x Re (y − f x p2 q2 2 2 p1 1 N1 φ(t2 ) by a PL function similar to (18) with t1 replaced by t2 . Therefore, we get a low complexity and approximate LLR decoder as follows: d(2) φPL (t1 ) + Λp,q , if xp1 6= xq1 and xp2 = xq2 , Λdp,q ≈ Λd(1) p,q + φPL (t2 ), if xp1 = xq1 and xp2 6= xq2 , (20) d(1) d(2) Λp,q + Λp,q , if xp1 6= xq1 and xp2 6= xq2 . V. S IMULATION R ESULTS
We have considered BPSK, QPSK, and 16-QAM constellation and Rayleigh fading channels in the simulations.
6
0
10
γ¯1 = γ¯2 = γ¯3 = γ¯4
γ¯1 = γ¯2 = γ¯3 = γ¯4
−1
10
−1
10
−2
10
−2
10 −3
BER
SER
10
−3
−4
−5
10
γ¯1 = γ¯2 = 10¯ γ3 = 10¯ γ4
10
10
γ¯1 = γ¯2 = 10¯ γ3 = 10¯ γ4 −4
10 −6
10
0
5
10
15 20 SNR [dB]
25
30
0
35
Fig. 3. Performance of the proposed ML decoder , sub-optimal decoder ◦ [19, Eq. (13)], and distributed Alamouti code with no decoding error in the relays ∗ under BPSK modulation.
5
10
15
20
25
SNR [dB]
Fig. 4. Performance of the proposed ML decoder , sub-optimal decoder ◦ [19, Eq. (13)], distributed Alamouti code with no decoding error in the relays ∗ under QPSK modulation.
A. Performance of the Proposed ML Decoder and A Suboptimal Decoder of Distributed Alamouti Code −1
10
−2
SER
We have simulated the performance of the distributed Alamouti code with BPSK and QPSK constellations, and two erroneous relays. The simulations are performed under the following scenarios: 1) γ¯1 = γ¯2 = γ¯3 = γ¯4 , i.e., all links have same SNR. 2) γ¯1 = γ¯2 = 10¯ γ3 = 10¯ γ4 , i.e., the SNR of the S-Rm link is 10 dB higher than that of the Rm -D link. We apply the proposed ML decoder and existing ML decoder of the co-located Alamouti code [19, Eq. (13)] for decoding of the distributed Alamouti STBC in the destination node. The existing ML decoder of [19, Eq. (13)] is a sub-optimal decoder of the distributed Alamouti STBC which assumes that the relays are error free or perfectly know the data of source. It can be seen from Figs. 3 and 4 that the proposed ML decoder significantly outperforms the sub-optimal decoder of the distributed Alamouti code at all SNRs for BPSK and QPSK constellations. For example, a gain of approximately 5 dB is obtained at SER=10−2 by the proposed ML decoder as compared to the sub-optimal decoder [19, Eq. (13)] for BPSK and QPSK constellations when the SNRs of all links are same. Moreover, the proposed ML decoder provides better diversity than the sub-optimal decoder of the distributed Alamouti STBC. The performance of the distributed Alamouti STBC with no decoding error in the relays, i.e. that of the co-located Alamouti code, is also plotted in Figs. 3 and 4. It can be seen from Figs. 3 and 4 that if we increase the SNR of the sourcerelay links, the distributed Alamouti STBC with the proposed ML decoder performs close to the co-located Alamouti STBC at all SNRs considered in the figures. It can be observed from Fig 3 that for γ¯1 = γ¯2 = γ¯3 = γ¯4 , the proposed ML decoder provides a bit error rate (BER) of 1.4×10−4 and 1.4×10−6 at 20 dB and 30 dB SNR, respectively. Therefore, the proposed ML decoder achieves full diversity of two. In order to demonstrate the sensitivity of the proposed ML decoder with respect to the error in the values of ǫ1 and ǫ2
10
−3
10
Sub−optimal decoder MSE=100% MSE=50% MSE=40% MSE=30% MSE=20% MSE=10% MSE=0
−4
10
0
5
10 15 SNR [dB]
20
25
Fig. 5. Performance of the proposed ML decoder with perfect and imperfect values of hm known in the destination.
(the instantaneous probability of error of S-Rm links), we have plotted SER versus SNR plots for QPSK constellation in Fig. 5 under the following assumption: γ¯1 = γ¯2 = γ¯3 = γ¯4 and for different values of the mean square error (MSE) in the values of the channel gains forwarded by MSE the relays. The 2 ˆ is defined in percentage as MSE , E hm − hm /σ 2 × m
ˆ m denotes the estimate of hm available in the 100%, where h destination. It is assumed that the channel estimates of both source-relay links received by the destination from the relays lead to the same amount of MSE. Due to this estimation error, the values of ǫ1 and ǫ2 calculated by the destination will be erroneous. It can be seen from Fig. 5 that for small values of MSE, i.e., MSE=10%, the proposed ML decoder performs very close to the ML decoding with MSE=0. However, as the value of MSE increases, the performance of the proposed
7
−1
10
Algebraic code Golden code Alamouti code −1
10
−2
−2
10
PEP
SER
10
−3
10
−3
10
γ¯1 = γ¯2 = 32¯γ3 = 32¯γ4 −4
10
γ¯1 = γ¯2 = 100¯γ3 = 100¯γ4 PEP of co-located Alamouti Code
−4
10
5
10
15 SNR [dB]
20
25
0
30
Fig. 6. Comparison of the performance of the distributed Golden [23], the distributed algebraic [24], and the proposed distributed Alamouti STBC.
ML decoder approaches the performance of the sub-optimal decoder [19, Eq. (13)] of the distributed Alamouti STBC. Nevertheless, from low to moderate values of the MSE, i.e., 0 ≤MSE≤ 50%, the proposed ML decoder performs better than the sub-optimal decoder of the distributed Alamouti STBC. For example, for MSE=50% and SER=10−2 , an SNR gain of approximately 3 dB is achieved by the proposed ML decoder as compared to the sub-optimal decoder. For acquiring the partial/imperfect channel knowledge of the source-relay links, the destination needs less number of feed-forward bits as compared to those required for perfect CSI. This saves the useful bandwidth. We have shown the performance of the distributed Golden code [23] and distributed algebraic STBC [24] in an uncoded cooperative system utilizing the DF protocol in Fig. 6. It is assumed that γ¯1 = γ¯2 = 32¯ γ3 = 32¯ γ4 . Since both STBCs are non-orthogonal STBCs, the destination receiver utilizes joint decoding based ML decoder under the assumption that the relays are error free. We have also shown the performance of the distributed Alamouti STBC with the proposed ML decoder in Fig. 6. The constellations for all three schemes are chosen to provide a data-rate of 2 bits/sec/Hz in the DF based uncoded cooperative system with two relays. It can be seen from Fig. 6 that the proposed ML decoder based distributed Alamouti STBC significantly outperforms the other distributed non-orthogonal STBCs. For example, the proposed ML decoder for SER=10−3 achieves a SNR gain of approximately 3 dB and 6 dB as compared to the distributed Golden code and distributed algebraic code, respectively. Moreover, the distributed Alamouti STBC with the proposed ML decoder achieves full diversity of two in contrast to the other distributed non-orthogonal STBCs. In Fig. 7, we have plotted the PEP for the proposed ML decoder of the distributed Alamouti STBC with BPSK constellation under the following scenarios: 1) γ¯1 = γ¯2 = 32¯ γ3 = 32¯ γ4 . 2) γ¯1 = γ¯2 = 100¯ γ3 = 100¯ γ4 . The value of the PEP at each SNR is calculated by using (12). We have also plotted the exact PEP of the co-located Alamouti STBC
5
10 SNR [dB]
15
20
Fig. 7. PEP versus SNR performance of the proposed ML decoder with BPSK constellation. −1
10
−2
10
−3
10 SER
0
−4
10
Proposed ML decoder with γ¯1=¯ γ2 =¯ γ3 −5
10
Proposed PL decoder with γ¯1=¯ γ2 =¯ γ3 Perfect relaying at both relays
−6
10
0
5
10
15 20 SNR [dB]
25
30
35
Fig. 8. SER versus SNR Performance of the proposed ML and PL decoders with QPSK constellation.
in Fig. 7. It can be observed from Fig. 7 that the PEP of the distributed Alamouti code with the proposed ML decoder gets closer to the exact PEP of co-located Alamouti STBC with increasing value of the average SNR of S-Rm links. B. Performance of the Proposed PL Decoder In Fig. 8, we have shown the performance of the proposed ML and PL decoders for the uncoded cooperative communication system with one of the two relays in outage. The simulations are performed with the uncoded symbols based distributed Alamouti STBC and QPSK constellation. It is assumed that γ¯2 → ∞, i.e., R2 has perfect knowledge of the data transmitted by the source and γ¯1 = γ¯3 = γ¯4 . It can be seen from Fig. 8 that the proposed PL decoder works similar to the proposed ML decoder at all SNRs considered in the figure. Further, it can be seen from Fig. 8 that the proposed ML and PL decoders achieve full diversity of two under the considered scenario.
8
−1
10
10¯ γ1 = 10¯ γ2 = γ¯3 = γ¯4 SER
BER
−2
10
−2
10
γ¯1 = γ¯2 = γ¯3 = γ¯4
−3
10
Uniform Optimized
ML−AF with poor S−R links Proposed ML−DF with poor S−R links ML−AF with similar links Proposed ML−DF with similar links ML−DF with perfect knowledge at relays
−3
10
−4
10
0
5
10 SNR [dB]
15
20
Fig. 9. Performance of the proposed ML decoder with uniform and PEP optimized power distributions, and BPSK constellation.
0
5
10
15 SNR [dB]
20
25
30
Fig. 10. Comparison of the DF and AF based distributed Alamouti code with 16-QAM constellation and different SNR conditions of S-R links. 0
C. UBPEP Optimized Power Distribution
10
In this subsection, we find an optimized transmit power 2 P Pm = distribution of both relays under a constraint C1: m=1
D. Comparison of the DF and AF Based Distributed Alamouti STBC We compare the performance of the uncoded transmissions based two cooperative communication systems with a single source-destination pair and two relays for 16-QAM constellation by assuming that all links are similar, i.e., having the same average SNR. One of the cooperative communication systems utilizes distributed Alamouti STBC with uncoded DF protocol and the proposed ML decoder in the destination; whereas, another cooperative system uses uncoded transmissions based distributed Alamouti STBC with AF protocol [17].
−1
10
SER
2Pmax , where Pmax is the maximum power transmitted by both relays. Another distribution of the transmit powers of both relays is obtained under a different constraint (other than C1) defined as C2: 0 ≤ Pm ≤ Pmax . For the above two optimization problems, we consider 20 dB SNR of the relay-destination links and BPSK constellation. It is assumed that the S-R1 channel is poor with γ¯1 = 100.5 and SNR γ4 γ3 = 100¯ of the S-R2 link is given as follows: γ¯2 = 100¯ P1 P2 . Both optimized distributions are obtained by minimizing the UBPEP in (13). Under the constraint C1, the optimized distribution is P1 = 0.7Pmax and P2 = 1.3Pmax . However, the optimized power distribution results into uniform distribution under constraint C2. Intuitively, this optimized distribution makes sense because the destination has perfect knowledge of the channel gain h1 , and it utilizes it in an ML sense for decoding of the data. Therefore, it is not recommended to reduce the power of a poor relay if this reduction cannot be compensated by increment in the transmit power of other relay. It can be seen from Fig. 9 that the proposed ML decoder provides improvement in the performance of the distributed Alamouti STBC with optimized power distributions under constraint C1.
−2
10
−3
ML−AF with 80% estimation error ML−AF with 20% estimation error ML−AF with perfect CSI Proposed ML−DF with 80% estimation error Proposed ML−DF with 20% estimation error Proposed ML−DF with perfect CSI
10
0
5
10
15 SNR [dB]
20
25
30
Fig. 11. Comparison of the DF and AF based distributed Alamouti code with imperfect channel knowledge of the source-relay link in the destination utilizing 16-QAM constellation.
It is assumed that the DF and AF based distributed Alamouti STBCs use half-duplex relaying nodes. We have plotted the SER versus SNR performance of the proposed ML decoder (6) of the DF based distributed Alamouti STBC and an existing ML decoder [17, Section II] of the AF based distributed Alamouti STBC in Fig. 10 for 16-QAM constellation. Further, it is assumed that both decoders have perfect knowledge of the channel coefficients of all links involved in cooperation. It can be seen from Fig. 10 that the DF protocol based distributed Alamouti STBC with the proposed ML decoder significantly outperforms the AF protocol based distributed Alamouti STBC for SNR > 5 dB. For example, a SNR gain of approximately 5 dB is achieved at SER=10−2 by the DF based distributed Alamouti STBC with the proposed ML decoder as compared to the AF based distributed Alamouti STBC of [17]. Moreover, we have compared the SER versus SNR performance of the AF and DF systems with S-Rm links being ten times poorer
9
than the Rm -D links in Fig. 10. It can be noticed from Fig. 10 that the DF based Alamouti code outperforms the AF based Alamouti STBC in this case as well. We have also shown the performance of the proposed ML decoder when the relays are error free in Fig. 10. By comparing different plots in Fig. 10, it can be deduced that the proposed ML decoder enables the distributed Alamouti STBC based DF system to achieve the second order diversity. It can also be noticed from Fig. 10 that the SER of the DF based distributed Alamouti code decays at the same rate as regular co-located Alamouti code. Hence, unlike distributed STBC based AF system, no log factor is involved in the asymptotic behavior of the proposed distributed Alamouti code based DF cooperative system. Fig. 11 shows the comparison of the distributed Alamouti code utilizing DF and AF protocols with imperfect channel knowledge of the source-relay links in the destination. The SER versus SNR performance curves are plotted for MSE of 20% and 80% in the CSI of the source-relay links. It can be observed from Fig. 11 that the performance of the AF based Alamouti code [17, Section II] is more severely affected due to the imperfect CSI of the source-relay links as compared to the DF based distributed Alamouti code with the proposed ML decoder. Moreover, it can be noticed from Fig. 11 that the proposed ML decoder with imperfect CSI of the source-relay links outperforms the AF based distributed Alamouti code with perfect CSI of the source-relay links, for the MSE and SNR values considered in the figure.
destination. Since the AWGN noises z1 and z2 are independent of each other, it can be observed from (3) that y1 and y2 are independent when hm , fm , sn , and x ˆn,m are known for all m, n. Therefore, pyd |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,1 ,ˆx1,2 ,ˆx2,1 ,ˆx2,2 }∈A4 = py1 |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,1 ,ˆx2,2 }∈A2 ×py2 |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,2 ,ˆx2,1 }∈A2 .
(22)
Depending upon the channel quality of the S-R1 and SR2 links, there exist the following four possibilities: 1) R1 demodulates the data erroneously, and R2 takes a correct decision. 2) R2 demodulates the data erroneously, and R1 takes a correct decision. 3) R1 and R2 both take wrong decisions. 4) R1 and R2 both demodulate the data correctly. Considering these four cases, we can write py1 |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,1 ,ˆx2,2 }∈A2 = ǫ1 (1−ǫ2 ) py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 =xp2 + (1 − ǫ1 ) ǫ2 py1 |f1 ,f2 ,ˆx1,1 =xp1 ,ˆx2,2 6=xp2 + ǫ1 ǫ2 py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 6=xp2 + (1 − ǫ1 ) (1 − ǫ2 ) py1 |f1 ,f2 ,ˆx1,1 =xp1 ,ˆx2,2 =xp2 .
(23)
Similarly, py2 |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,2 ,ˆx2,1 }∈A2 = ǫ1 (1 − ǫ2 ) py2 |f1 ,f2 ,ˆx1,2 =xp1 ,ˆx2,1 6=xp2 + (1 − ǫ1 ) ǫ2 py2 |f1 ,f2 ,ˆx1,2 6=xp1 ,ˆx2,1 =xp2
VI. C ONCLUSIONS We have derived an ML decoder of the DF based distributed Alamouti STBC in a cooperative system which works well for the complex-valued M -ary constellations. Moreover, the proposed ML decoder enables the DF based distributed Alamouti STBC to significantly outperform an existing AF based distributed Alamouti STBC. We have also derived the conditional PEP of the proposed ML decoder applied to the distributed Alamouti STBC with BPSK constellation. A PPENDIX A P ROOF OF T HEOREM 1 An ML decoder of the symbols s1 , s2 can be obtained by maximizing the conditional joint p.d.f. of received data vector y d (given in (4)) in the destination over two consecutive time intervals. It is equivalent to maximize a likelihood ratio for decoding the data [25]. It can be shown by using the analysis given in [25, Section 2.3] that the destination needs to find the following LLR to decide between two symbol vectors xp and xq : pyd |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,1 ,ˆx1,2 ,ˆx2,1 ,ˆx2,2 }∈A4 , Λdp,q = ln pyd |f1 ,f2 ,h1 ,h2 ,s=xq ,{ˆx1,1 ,ˆx1,2 ,ˆx2,1 ,ˆx2,2 }∈A4 (21)
+ ǫ1 ǫ2 py2 |f1 ,f2 ,ˆx1,2 6=xp1 ,ˆx2,1 6=xp2 + (1 − ǫ1 ) (1 − ǫ2 ) py2 |f1 ,f2 ,ˆx1,2 =xp1 ,ˆx2,1 =xp2 .
(24)
Since z1 and z2 ∼ CN (0, N1 ), we have 1 − N1 |y1 −f1 xp1 −f2 xp2 |2 e 1 , πN1 1 − N1 |y2 +f1 x∗p −f2 x∗p |2 2 1 . py2 |f1 ,f2 ,ˆx1,2 =xp1 ,ˆx2,1 =xp2 = e 1 πN1 (25)
py1 |f1 ,f2 ,ˆx1,1 =xp1 ,ˆx2,2 =xp2 =
From [26, Section III], it can be deduced that the p.d.f.s py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 =xp2 , py1 |f1 ,f2 ,ˆx1,1 =xp1 ,ˆx2,2 6=xp2 , and py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 6=xp2 denote the p.d.f.s of a Gaussian mixture random variable. With these observations, it follows that py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 =xp2 = k0
M X
2
e− N1 |y1 −f1 xl −f2 xp2 | , 1
l=1, l6=p1
py1 |f1 ,f2 ,ˆx1,1 =xp1 ,ˆx2,2 6=xp2 = k0
M X
2
e− N1 |y1 −f1 xp1 −f2 xl | , 1
l=1, l6=p2
py1 |f1 ,f2 ,ˆx1,1 6=xp1 ,ˆx2,2 6=xp2
M X M X k0 = (M − 1) l=1, k=1,
l6=p1 k6=p2
where pyd |f1 ,f2 ,h1 ,h2 ,s=xp ,{ˆx1,1 ,ˆx1,2 ,ˆx2,1 ,ˆx2,2 }∈A4 is the conditional joint p.d.f. of the received data vector y d given that channels of the S-R1 , S-R2 , R1 -D, and R2 -D links, the symbols transmitted by the source s1 , s2 , and the symbols transmitted by two relays x ˆ1,1 , x ˆ1,2 , x ˆ2,1 , x ˆ2,2 are perfectly known in the
×e
− N1 |y1 −f1 xl −f2 xk |2 1
,
(26)
where k0 = 1/[πN1 (M − 1)]. Similarly, conditional p.d.f.s of y2 can be obtained. Using (21)-(26) and the conditional p.d.f.s of y2 , we get (6).
10
and variances of wp,i , i = 1, 2, 3, 4 are given as
A PPENDIX B P ROOF OF T HEOREM 2 Let us state the following definition and results before providing a proof of Theorem 2. Definition 1: If X ∼ N µ, σ 2 , where N µ, σ 2 denotes the real-valued normal distribution with mean µ and variance σ 2 , then Y = exp {X} ∼ LogN µ, σ 2 , where LogN (·, ·) denotes the log-normal distribution [27, Eq. (5.30)]. Lemma 1: If Yi = exp {Xi } ∼ LogN µi , σi2 , i = 1, 2, .., N , where Xi ∼ N µi , σi2 are correlated normal random variables (RVs) with correlation coefficient ρi,j , i, j = N P Yi approximately3 follows 1, 2, ..., N, i 6= j, then Y = i=1
the distribution of a log-normal distributed random variable Z ∼ LogN µz , σz2 , where # "N X σ2 σ2 µi + 2i − z, (27) µz = ln e 2 i=1 and
N P
2µi +σi2
σi2
N NP −1 P −1 +2
e e i=1 i=1 j=i+1 2 2 µi +σi /2 µj +σj /2 ρi,j σi σj −1) ×e e (e . σz2 = ln +1 N 2 P µ + σi2 e i 2 i=1
(28)
Refer [28] for a proof. Lemma 2: If Z ∼ LogN µz , σz2 , then 1/Z ∼ LogN −µz , σz2 . Proof: Let Z = eX , where X ∼ N µz , σz2 , then ′ 1/Z = e−X = eX and X ′ ∼ N −µz , σz2 . Therefore, 1/Z ∼ LogN −µz , σz2 . X1 and Z2 = eX2 , where X1 ∼ Lemma 3: If Z1 = e 2 N µz1 , σz1 and X2 ∼ N µz2 , σz22 are correlated normal random variables with correlation coefficient ρ, then Z1 /Z2 ∼ LogN (µz1 −µz2 , σz21 + σz22 + 2ρσz1 σz2 . Proof: Since 1/Z2 ∼ LogN −µz2 , σz22 , From the properties of the sum of two correlated Normal random variables [21], we get Z1 /Z2 ∼ LogN (µz1 − µz2 , σz21 + σz22 + 2ρσz1 σz2 . Lemma 4: If Y = exp {X} ∼ LogN µ, σ 2 , then aY ∼ LogN ln(a) + µ, σ 2 , where a is a constant. Proof: We can write aY = eln(a) Y = eln(a)+X = ′ eX , where ln(·) denotes the natural logarithm. Since X ′ ∼ N ln(a) + µ, σ 2 , therefore, it proves the above result. Using (3) and Lemma 4 in (10), we get wp,1 + wp,2 + wp,3 + wp,4 Λdp,q = ln w + wq,2 + wq,3 + wq,4 q,1 vp,1 + vp,2 + vp,3 + vp,4 , (29) +ln vq,1 + vq,2 + vq,3 + vq,4 2 , and vℓ,i ∼ where wℓ,i ∼ LogN µℓ,i , σℓ,i 2 LogN λℓ,i , δℓ,i , ℓ ∈ {p, q}, i = 1, 2, 3, 4. The means 3 Only matches the mean and variance of the log-normal sum and the approximating log-normal RV.
1 2 µp,1 = ln (ǫ1 (1 − ǫ2 )) − |f1 x ¯ p1 + f 2 x p2 | N 1 ∗ − 2Re (f1 x ˆ1,1 + f2 x ˆ2,2 ) (f1 x ¯ p1 + f 2 x p2 ) , 1 2 ¯ p2 | |f1 xp1 + f2 x µp,2 = ln (ǫ2 (1 − ǫ1 )) − N 1 ∗ ¯ p2 ) , − 2Re (f1 x ˆ1,1 + f2 x ˆ2,2 ) (f1 xp1 + f2 x 1 2 ¯ p2 | |f1 x ¯ p1 + f 2 x µp,3 = ln (ǫ1 ǫ2 ) − N 1 ∗ ¯ p2 ) , − 2Re (f1 x ˆ1,1 + f2 x ˆ2,2 ) (f1 x ¯ p1 + f 2 x 1 2 µp,4 = ln ((1− ǫ1 )(1− ǫ2 ))− |f1 xp1 +f2 xp2 | N 1 ∗ − 2Re (f1 x ˆ1,1 + f2 x ˆ2,2 ) (f1 xp1 +f2 xp2 ) , 2 2 2 2 2 = ¯ p2 | , |f1 x ¯p1 +f2 xp2 | , σp,2 |f1 xp1 +f2 x N1 N1 2 2 2 2 2 2 = ¯p2 | , σp,4 |f1 x ¯p1 +f2 x |f1 xp1 +f2 xp2 | , (30) σp,3 = N1 N1 2 σp,1 =
and the means and variances of vp,i , i = 1, 2, 3, 4 are given as 1 h 2 λp,1 = ln (ǫ1 (1 − ǫ2 ))− |f1 x ¯ p2 − f 2 x p1 | N1 i + 2Re {(f2∗ x ˆ1,2 − f1∗ x ˆ2,1 ) (f1 x ¯p2 − f2 xp1 )} , 1 h 2 ¯ p1 | |f1 xp2 − f2 x λp,2 = ln (ǫ2 (1 − ǫ1 ))− N1 i ¯p1 )} , + 2Re {(f2∗ x ˆ1,2 − f1∗ x ˆ2,1 ) (f1 xp2 − f2 x 1 h 2 ¯ p1 | |f1 x ¯ p2 − f 2 x λp,3 = ln (ǫ1 ǫ2 )− N1 i ¯p1 )} , + 2Re {(f2∗ x ˆ1,2 − f1∗ x ˆ2,1 ) (f1 x ¯ p2 − f 2 x 1 h 2 λp,4 = ln ((1 − ǫ1 )(1 − ǫ2 ))− |f1 xp2 −f2 xp1 | N1 i + 2Re{(f2∗ x ˆ1,2 − f1∗ x ˆ2,1 ) (f1 xp2 −f2 xp1 )} , 2 2 2 2 2 = ¯ p1 | , |f1 x ¯p2 −f2 xp1 | , δp,2 |f1 xp2 −f2 x N1 N1 2 2 2 2 2 2 = ¯p1 | , δp,4 |f1 x ¯p2 −f2 x |f1 xp2 −f2 xp1 | , (31) δp,3 = N1 N1
2 δp,1 =
and the statistics (mean and variance) of wq,i and vq,i , i = 1, 2, 3, 4 can be found by substituting {xq1 , xq2 } in place of {xp1 , xp2 } in (30) and (31), respectively. It can be observed from (29) that the random variables wp,i , i = 1, 2, 3, 4, are fully correlated with unity correlation coefficient. Hence, by using Lemma 1 in (29), we can approximate the LLR decoder as Wp Vp d Λp,q = ln + ln , (32) Wq Vq where Wℓ =
4 P
i=1
wℓ,i and Vℓ =
4 P
i=1
vℓ,i , ℓ ∈ {p, q}. Following
Lemma 1, it can be observed that Wℓ and Vℓ , ℓ ∈ {p, q} are log-normal distributed random variables. The mean and
11
variance of Wℓ are given as # " 4 2 2 σℓ,i X σℓ,i , µℓ = ln eµℓ,i + 2 − 2 i=1 4 3 X 2 X 4 P 2 σℓ,i 2µℓ,i +σℓ,i e −1 +2 e i=1 i=1 j=i+1 1 2 1 2 ×eµℓ,i +µℓ,j + 2 σℓ,i + 2 σℓ,j(eσℓ,i σℓ,j −1) 2 +1 σℓ = ln , 2 4 σ2 P ℓ,i µ + ℓ,i 2 e i=1
(33)
2 µℓ,i , σℓ,i
where are given in (30). Similarly for Vℓ , ℓ ∈ {p, q}, the mean (λℓ ) and variance (δℓ2 ) can be obtained by chang2 2 ing the parameters {µℓ,i , σℓ,i } in (33) with {λℓ,i , δℓ,i } given in (31). Moreover, we can notice from (32) that Wp and Wq , and Vp and Vq , are fully correlated RVs with unity correlation coefficient. Applying Lemmas 1- 3, and the above results in (32), it can be concluded that the LLR decoder of (10) approximately follows the distribution of a real-valued normal distributed RV Z with mean (µp + λp − µq − λq ) and variance (σp2 + δp2 + σq2 + δq2 + 2σp σq + 2δp δq ). Therefore, the conditional probability of error of decoding xp in place of xq , such that p 6= q, using the LLR decoder (10) for BPSK constellation will be given as Pr{xq → xp |h1 , h2 , f1 , f2 , s = xq ,ˆ x1,1 ,ˆ x2,1 ,ˆ x1,2 ,ˆ x2,2} = Pr {Z > 0|h1 ,h2 ,f1 ,f2 , s = xq ,ˆ x1,1 ,ˆ x2,1 ,ˆ x1,2 ,ˆ x2,2} , µp + λp − µq − λq . (34) = Q q 2 2 2 2 σp + δp + σq + δq + 2σp σq + 2δp δq
The PEP of the ML decoder given that the channel gains of all involved links and the data transmitted by the source are known in the destination will depend upon sixteen mutually exclusive events of correct or erroneous decoding by both relays, i.e., El , given in Table I. Hence, the PEP of the proposed ML decoder conditioned on El can be expressed as Pr {xq → xp |h1 , h2 , f1 , f2 , s = xq } = 16 X
Pr {Z > 0|f1 , f2 , s = xq , El } Pr {El |h1 , h2 } .
(35)
l=1
The values of Pr {El |h1 , h2 } for each El are also listed in Table I. The average PEP given in (12) can be obtained by using (34) and (35). R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity Part-I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [2] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74–80, Oct. 2004. [3] J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” IEEE Wireless Communications and Networking Conference (WCNC), pp. 7–12, Sep. 2000, Chicago, IL, USA.
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