Performance Analysis of Bidirectional Relay Selection with Imperfect Channel State Information
aa r X i v : . [ c s . N I] D ec Performance Analysis of Bidirectional RelaySelection with Imperfect Channel StateInformation
Hongyu Cui, Rongqing Zhang, Lingyang Song, and Bingli JiaoSchool of Electronics Engineering and Computer SciencePeking University, Beijing, China,
Abstract
In this paper, we investigate the performance of bidirectional relay selection using amplify-and-forward protocol with imperfect channel state information, i.e., delay effect and channel estimation error.The asymptotic expression of end-to-end SER in high SNR regime is derived in a closed form, whichindicates that the delay effect causes the loss of both coding gain and diversity order, while the channelestimation error merely affects the coding gain. Finally, analytical results are verified by Monte-Carlosimulations.
Index Terms bidirectional relay selection, analog network coding, imperfect channel state information
I. I
NTRODUCTION
Bidirectional relay communications, in which two sources exchange information through intermediaterelays, have gained a lot of interest by now, and different transmission schemes have been proposed[1]. In [2], [3], an amplify-and-forward (AF) based network coding scheme, named as analog networkcoding (ANC), was introduced. With ANC, the data transmission process can be divided into two phases,and the spectral efficiency, which is restricted by half-duplex antennas, can get improved. Recently, relayselection (RS) for bidirectional relay networks has been intensively researched to achieve full spatialdiversity and better system performance, which requires fewer orthogonal resources in comparison ofall-participate relay approaches [4], [5]. Performing RS, the “best” relay is firstly selected before data transmission by the predefined criterion [6]–[11]. In [6], [7], the authors proposed the max-min sum rateselection criterion for AF bidirectional relay. In [8]–[11], selection criterions in minimizing the symbolerror rate (SER) were introduced and analyzed.To the authors’ best knowledge, most works about RS in bidirectional relay only consider perfectchannel state information (CSI). However, imperfect CSI, i.e., delay effect and channel estimation er-ror (CEE), has great impact on the performance of bidirectional relay selection. Specifically, the timedelay between relay selection and data transmission causes that the selected relay may not be optimal fordata transmission [12]–[14]. And similarly, channel estimation errors can not be ignored either [15]–[18].In [18], the authors analyzed the performance loss of bidirectional relay selection using decode-and-forward protocol with CEE, but the impact of imperfect CSI on a general bidirectional AF relay selectionwas not provided.In light of the aforementioned researches, we analyze the impact of imperfect CSI, including delayeffect and CEE, for bidirectional AF relay selection in this paper, which has not been studied previously.The asymptotic expression of end-to-end SER is derived in a closed form, and verified by computersimulations. Analytical and simulated results reveal that delay effect reduces both the diversity order andthe coding gain, while channel estimation error merely causes the coding gain loss. The main contributionof this paper can be summarized as follows:1) The asymptotic SER expression for bidirectional relay selection is provided in a closed form, whichmatches the simulated results in high SNR regime;2) Imperfect CSI, i.e., delay and channel estimation errors, is taken into account to derive the analyticalresults, and its therein impact is investigated.The remainder of this paper is organized as follows: In Section II, the system model of bidirectional AFrelay selection, and the imperfect CSI model are described in detail. Section III provides the analyticalexpression of bidirectional relay selection with imperfect CSI. Simulation results and performance analysisare presented in Section IV. Finally, section V concludes this paper.
Notation: ( · ) ∗ and |·| represent the conjugate and the absolute value, respectively. E is used for theexpectation and P r represents the probability. The probability density function and the cumulativeprobability function of variable x are denoted by f x ( · ) and F x ( · ) , respectively.II. S YSTEM M ODEL
The system investigated in this paper is a general bidirectional AF relay network with two sources S j , j = 1 , exchanging information through the intermediate N relays R i , i = 1 , . . . , N . The direct link between S and S does not exist, and each node is equipped with a single half-duplex antenna.The transmit power of the sources is assumed to be the same, denoted by p s , and all the relays havethe individual power constraint, denoted by p r . The channel coefficients between sources and relays arereciprocal, and these coefficients are constant over the duration of one data block.The whole procedure of bidirectional AF relay selection is divided into two parts periodically: relayselection process and data transmission process , which will be described concretely in the next section.Let h s,ji and ˆ h s,ji represent the actual and the estimated channel coefficients between S j and R i duringthe relay selection process, respectively; let h t,ji and ˆ h t,ji represent the actual and the estimated channelcoefficients between S j and R i during the data transmission process, respectively. All the actual channelcoefficients are independent identically distributed (i.i.d.) Rayleigh flat-fading with zero mean and unitvariance, i.e., E (cid:16) | h t,ji | (cid:17) = E (cid:16) | h s,ji | (cid:17) = 1 , and thus, | h s,ji | and | h t,ji | are both exponentiallydistributed with unit mean. Both the sources can know the global channel coefficients by estimating thetraining symbols, while each relay only has its local channel information. A. Model of Delay Effect
Due to the time delay between relay selection process and data transmission process, h s,ji is not thesame as h t,ji , which means the CSI is outdated . Their relationship can be modeled by the first-orderautoregressive model [14]: h t,ji = ρ f j h s,ji + q − ρ f j ε j , (1)where h t,ji is a zero mean complex-Gaussian RV with variance of σ h t,ji ; h s,ji and ε j are i.i.d. randomvariable (RVs) with zero mean and variance of σ h s,ji and σ ε j , respectively. In this paper, we assume σ h t,ji = σ h s,ji = σ ε j = 1 .The correlation coefficient ρ f j ( ≤ ρ f j ≤ , where ρ f j = 1 represents no delay effect, in other words,the CSI is not outdated) between S j and relays is defined by Jakes’ autocorrelation model [14]: ρ f j = J (cid:0) πf d j T (cid:1) , (2)where J ( · ) stands for the zeroth order Bessel function [23], f d j is the Doppler frequency, and T isthe time delay between the relay selection process and the data transmission process. In this paper,two variables ρ f j , j = 1 , are used to represent the correlation coefficients between S j and the relays,respectively, for f d and f d may be different. B. Model of Channel Estimation Error
Let h denote the actual channel coefficient and ˆ h represent the estimated channel coefficient, and thentheir relationship can be modeled as follows [15]: ˆ h = h + e , (3)and h = ρ e ˆ h + d , (4)where h and CEE e are independent complex-Gaussian RVs with zero mean and variances of σ h , σ e ,respectively. ˆ h and CEE d are also independent complex-Gaussian RVs with zero mean and variancesof σ h , σ D , respectively. The correlation coefficient ρ e = σ h /σ h ( ≤ ρ e ≤ , where ρ e = 1 meansno CEE) is determined by the concrete channel estimation method. In addition, ρ e can be modeled asan increasing function of the training symbols’ power P , i.e., ρ e → when P approaches infinity [19],[20]. In this paper, we assume ρ e = P/ ( P + N ) [21].According to the above relationship, the variances of CEE are given by : σ e = σ h − σ h = (1 − ρ e ) σ h = 1 − ρ e ρ e σ h , (5)and σ D = σ h − ρ e σ h = (1 − ρ e ) σ h = (cid:0) ρ e − ρ e (cid:1) σ h . (6)Assuming σ h = 1 in this paper, we have σ h = ρ − e and σ D = 1 − ρ e according to (5) and (6). C. Relationship between ˆ h s,ji and ˆ h t,ji For the bidirectional relay selection communications, ˆ h s,ji is used for relay selection, and ˆ h t,ji is usedfor data detection. According to the model of imperfect CSI, we have : Lemma 1: ˆ h t,ji and ˆ h s,ji can be related as : ˆ h t,ji = ρ j ˆ h s,ji + q − ρ j v j , (7)where v j and ˆ h s,ji are i.i.d. RVs, and ρ j = , if ρ f j = 1; ρ e ρ f j , if ρ f j < . (8)When the CSI is not outdated, i.e., ρ f j = 1 , ˆ γ t,ji = (cid:12)(cid:12)(cid:12) ˆ h t,ji (cid:12)(cid:12)(cid:12) and ˆ γ s,ji = (cid:12)(cid:12)(cid:12) ˆ h s,ji (cid:12)(cid:12)(cid:12) have the samedistribution. When the CSI is outdated, i.e., ρ f j < , the probability density function (PDF) of ˆ γ t,ji conditioned by ˆ γ s,ji can be expressed as : f ˆ γ t,ji | ˆ γ s,ji ( y | x ) = 1 (cid:16) − ρ j (cid:17) σ h exp − x + ρ j y (cid:16) − ρ j (cid:17) σ h I q ρ j xy (cid:16) − ρ j (cid:17) σ h , (9)where I ( · ) stands for the zeroth order modified Bessel function of the first kind [23], and σ h = σ h s,ji = σ h t,ji = ρ − e . Proof:
The proof of Lemma 1 can be found in Appendix A. (cid:4)
III. P
ERFORMANCE A NALYSIS OF B IDIRECTIONAL R ELAY S ELECTION WITH I MPERFECT
CSI
A. Instantaneous Received SNR at the Sources
As mentioned above, the whole procedure of bidirectional relay selection is divided into relay selectionprocess and data transmission process.In the relay selection process, the central unit (CU), i.e., S or S , estimates all the channel coefficients ˆ h s,ji . Then, based on the predefined selection criterion, CU selects the “best” relay from all the availablerelays for the subsequent data transmission and other relays keep idle until the next relay selection instantcomes. There are several selection criterions for bidirectional relay [6]–[11]. In this paper, we adopt the Best-Worse-Channel method for relay selection which has the best performance in minimizing the averageSER and is tractable for analysis [10], [11]. According to this criterion, the index k of the selected relaysatisfies : k = arg max i min (cid:26)(cid:12)(cid:12)(cid:12) ˆ h s, i (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ˆ h s, i (cid:12)(cid:12)(cid:12) (cid:27) , (10)and thus, min (cid:26)(cid:12)(cid:12)(cid:12) ˆ h s, k (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ˆ h s, k (cid:12)(cid:12)(cid:12) (cid:27) = max i min (cid:26)(cid:12)(cid:12)(cid:12) ˆ h s, i (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) ˆ h s, i (cid:12)(cid:12)(cid:12) (cid:27) . (11)The subsequent data transmission process can be divided into two phases. During the first phase,the sources simultaneously send their respective information to the intermediate relays where only theselected relay R k is active. The superimposed signal at R k is y k = √ p s h t, k s + √ p s h t, k s + n k , where s j denotes the modulated symbols transmitted by S j with the average power normalized, j = 1 , , and n k is additive white Gaussian noise (AWGN) at R k , which is a zero mean complex-Gaussian RV withtwo-sided power spectral density of N / per dimension. During the second phase, R k amplifies thereceived signal and forwards it back to the sources. Let x k be the signal generated by R k , then we have x k = √ p r β k y k , where β k is the amplification factor. In this paper, we analyze the variable-gain AF relay [16], then β k = (cid:18) p s (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) + p s (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) + N (cid:19) − / is decided by the estimated instantaneous channelcoefficients.The received signals by S and S are similar due to the symmetry of the network topology, and thus,we take S as an example for analysis. The signal y ,r received by S can be written as y ,r = h t, k x k + n ,where n is AWGN at S ; n and n k are i.i.d. RVs. According to (4), h t, k and h t, k can be rewrittenas h t, k = ρ e ˆ h t, k + d k and h t, k = ρ e ˆ h t, k + d k , where d k and d k are independent RVs due to theindependence of h t, k and h t, k . Therefore, y ,r can be expanded as : y ,r = √ p r p s β k ρ e ˆ h t, k ˆ h t, k s (12) + √ p r p s β k (cid:16) ρ e ˆ h t, k d k + ρ e ˆ h t, k d k + d k d k (cid:17) s (13) + √ p r p s β k ρ e ˆ h t, k ˆ h t, k s (14) + √ p r p s β k (cid:16) ρ e ˆ h t, k d k + d k (cid:17) s (15) + √ p r β k ρ e ˆ h t, k n k + √ p r β k d k n k + n , (16)where (12) represents the useful information from S ; (13) represents the inter-interference from S causedby CEE; (14) and (15) represent the self-interference from S itself which can be subtracted totally by self-canceling if CEE does not exist [9]. However, with CEE, S can only reconstruct √ p r p s β k ρ e ˆ h t, k ˆ h t, k s at the receiver. Thus, only (14) can be subtracted totally, whereas the self-interference of (15) is residual;(16) includes the amplified noise from R k and the noise at S .After self-canceling √ p r p s β k ρ e ˆ h t, k ˆ h t, k s from y ,r , and then multiplied by ˆ h ∗ t, k ˆ h ∗ t, k to compensatethe phase rotation, the processed signal y at S is : y = ˆ h ∗ t, k ˆ h ∗ t, k (cid:16) y ,r − √ p r p s β k ρ e ˆ h t, k ˆ h t, k s (cid:17) . (17)The transmitted information s can be recovered by maximum likelihood detection: ˜ s = arg min s ′ ∈A (cid:13)(cid:13)(cid:13)(cid:13) y − √ p r p s βρ e (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) s ′ (cid:13)(cid:13)(cid:13)(cid:13) , (18)where k·k represents the Euclid-distance, A is the alphabet of modulation symbols, and ˜ s is therecovered signal.According to (17), the instantaneous received SNR γ at S can be written as : γ = ψ r ψ s ρ e (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) (cid:0) ψ r ψ s ρ e σ D + ψ r ρ e + ψ s (cid:1) (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) + (cid:0) ψ r ψ s ρ e σ D + ψ s (cid:1) (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) + 3 ψ r ψ s σ D + ψ r σ D + 1 , (19)where ψ s = p s /N , ψ r = p r /N , ρ e is the CEE coefficient, and the CEE variance σ D = 1 − ρ e . In high SNR regime, ρ e → and σ D = (1 − ρ e ) → , then the item ψ r ψ s σ D + ψ r σ D + 1 in thedenominator of (19) approaches 1, which can also be ignored when SNR approaches infinity [9].Therefore, γ in high SNR regime can be simplified into : γ = ˜ a (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) ˜ b (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) ˜ a (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) + ˜ b (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) , (20)where ˜ a = ψ r ρ e ψ r ρ e σ D , ˜ b = ψ r ψ s ρ e ψ r ψ s ρ e σ D + ψ r ρ e + ψ s . (21) γ in (20) is greater than that in (19), whereas they match tightly in high SNR regime. Therefore, weuse γ in (20) for asymptotic analysis in the followings. B. Distribution Function of the Received SNR
The distribution of γ in (20) is decided by ˆ γ t, k = (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) and ˆ γ t, k = (cid:12)(cid:12)(cid:12) ˆ h t, k (cid:12)(cid:12)(cid:12) , which are determinedby ˆ γ s, k = (cid:12)(cid:12)(cid:12) ˆ h s, k (cid:12)(cid:12)(cid:12) and ˆ γ s, k = (cid:12)(cid:12)(cid:12) ˆ h s, k (cid:12)(cid:12)(cid:12) according to Lemma 1. Furthermore, the distribution of ˆ γ s, k and ˆ γ s, k can be obtained by the above selection criterion. After some manipulations, we have Theorem 1:
With the definition that : a ∆ = ρ e ˜ a = 1 + ψ r ρ e σ D ψ r ρ e , b ∆ = ρ e ˜ b = 5 ψ r ψ s ρ e σ D + ψ r ρ e + ψ s ψ r ψ s ρ e , (22)the cumulative distribution function (CDF) of γ is : F γ ( z ) = 1 − N N − X m =0 N − X n =0 (cid:18) N − m (cid:19)(cid:18) N − n (cid:19) ( − m m + 1 ( − n n + 1 ( f + f + f + f ) (23)where f = 2 √ abz exp ( − ( a + b ) z ) K (cid:16) z √ ab (cid:17) , (24) f = s n ab n + 1) [(2 n + 1) (1 − ρ ) + 1] z exp (cid:18) − (cid:18) n + 1) a (2 n + 1) (1 − ρ ) + 1 + b (cid:19) z (cid:19) (25) × K z s n + 1) ab (2 n + 1) (1 − ρ ) + 1 ! ,f = s m ab ( m + 1) [(2 m + 1) (1 − ρ ) + 1] z exp (cid:18) − (cid:18) a + 2 ( m + 1) b (2 m + 1) (1 − ρ ) + 1 (cid:19) z (cid:19) (26) × K z s ab ( m + 1)(2 m + 1) (1 − ρ ) + 1 ! , f = s m n ab ( m + 1) ( n + 1) [(2 n + 1) (1 − ρ ) + 1] [(2 m + 1) (1 − ρ ) + 1] z (27) × exp (cid:18) − (cid:18) n + 1) a (2 n + 1) (1 − ρ ) + 1 + 2 ( m + 1) b (2 m + 1) (1 − ρ ) + 1 (cid:19) z (cid:19) × K z s ab ( m + 1) ( n + 1)[(2 m + 1) (1 − ρ ) + 1] [(2 n + 1) (1 − ρ ) + 1] ! . (28)And K ( · ) is the first order modified Bessel of the second kind [23], (cid:0) Nk (cid:1) is the binomial coefficient,and ρ j , j = 1 , satifies (8) in Lemma 1: ρ j = 1 if ρ f j = 1 , and ρ j = ρ e ρ f j if ρ f j < . Proof:
The proof of Theorem 1 can be found in Appendix B. (cid:4)
Due to the symmetry, it can be proved similarly that the CDF of the received SNR γ at S have thesame form as γ , and their PDFs can be obtained by differentiating the CDFs. C. Asymptotic Performance of Average Symbol Error Rate
For many common modulation formats, the average SER can be obtained by [13]:
SER = α E h Q (cid:16)p βγ (cid:17)i = α √ π ∞ Z F γ (cid:18) t β (cid:19) e − t dt , (29)where γ is the instantaneous received SNR, Q ( · ) is Gaussian Q-Function [23], and α = 1 , β = 2 forBPSK, α = 1 , β = 1 for QPSK, α = 1 / log M , β = log M sin ( π/M ) for MPSK ( M > ).Applying Theorem 1 and (29), the exact average SER of S can be obtained by [24, (6.621.3)]: Z ∞ x µ − e − αx K ν ( βx ) dx = √ π (2 β ) ν ( α + β ) µ + ν Γ ( µ + ν ) Γ ( µ + ν )Γ ( µ + 1 / F (cid:18) µ + ν, ν + 12 ; µ + 12 ; α − βα + β (cid:19) , (30)where Γ ( · ) is Gamma function, and F ( · ) is Confluent Hypergeometric function [23]. However, the exactform is too complicated to analyze the performance, thus we resort to the high SNR analysis [22]. Theorem 2:
The asymptotic performance of SER in high SNR regime can be obtained in two differentcases according to whether the CSI is outdated or not. • When the CSI is not outdated, i.e., the delay coefficients satisfy ρ f = ρ f = 1 , and the CEEcoefficient ρ e is arbitrary, the average SER of S in high SNR regime is: SER ∞ = α β N (2 N )! N ! (cid:18) ψ r ρ e σ D ψ r ρ e (cid:19) N + (cid:18) ψ r ψ s ρ e σ D + ψ r ρ e + ψ s ψ r ψ s ρ e (cid:19) N ! , (31)where α and β are decided by the modulation format in (29); ψ s = p s /N and ψ r = p r /N ; N ! isthe factorial of N ; σ D = 1 − ρ e . • When the CSI is outdated, i.e., ρ f < or ρ f < , and ρ e is arbitrary, the average SER of S inhigh SNR regime is: SER ∞ = α β (cid:18) ψ r ρ e σ D ψ r ρ e (cid:19) N N − X n =0 ( − n (cid:18) N − n (cid:19) − ρ (2 n + 1) (1 − ρ ) + 1 (32) + α β (cid:18) ψ r ψ s ρ e σ D + ψ r ρ e + ψ s ψ r ψ s ρ e (cid:19) N N − X m =0 ( − m (cid:18) N − m (cid:19) − ρ (2 m + 1) (1 − ρ ) + 1 , where ρ j , j = 1 , satifies (8) in Lemma 1: ρ j = 1 if ρ f j = 1 , and ρ j = ρ e ρ f j if ρ f j < . Proof:
The proof of Theorem 2 can be found in Appendix C. (cid:4)
Similarly, the average SER in high SNR regime of S can be obtained from (31) and (32) by permuting a with b . D. Performance Analysis of Diversity Order and Coding Gain
Diversity order d = − lim ψ t →∞ (cid:0) log SER ∞ / log ψ t (cid:1) [22], where ψ t = (2 p s + p r ) /N = 2 ψ s + ψ r , isan useful metric to describe the asymptotic performance of SER, i.e., greater diversity order means thecurve of SER attenuates more quickly. Theorem 3:
According to the definition of diversity order, the diversity order is : d = N, if the CSI is not outdated, i.e., ρ f = ρ f = 1;1 , if the CSI is outdated, i.e., ρ f < or ρ f < . (33) Proof:
Assuming ψ t = (2 p s + p r ) /N = 2 ψ s + ψ r , ψ s = p s /N = λψ t , and ψ r = p r /N =(1 − λ ) ψ t , the diversity order can be obtained by Theorem 2, and the fact that ρ e → and σ D =1 − ρ e → when SNR approaches infinity. (cid:4) Theorem 3 reveals that the diversity order is N if and only if the CSI is not outdated. Once the CSIis outdated, i.e., the delay exists, the diversity order reduces to , whereas CEE has no impact on theperformance loss of diversity order.However, both delay effect and CEE can reduce the coding gain, which is the shift of SER curve, e.g.,different delay coefficients ρ f j and CEE coefficients ρ e will result in different ρ j is Theorem 2, and thusthe coding gain is different. IV. S IMULATION R ESULTS AND D ISCUSSION
In this section, the average SER of bidirectional relay selection with imperfect CSI is studied by Monte-Carlo simulations, and the analytical performance provided by Theorem 2 is verified by these simulation results. Due to the symmetry of the network, the following results only concern about the average SERof S . All the simulations are performed with BPSK modulation over the normalized Rayleigh fadingchannels. For simplicity, we assume that sources and relays have the same power, i.e., p s = p r = P ,and the x-axis of the following figures is SNR = P /N in decibel. To better understand the impact ofimperfect CSI, we discuss four different situation, i.e., perfect CSI, only delay effect, only CEE, andboth delay effect and CEE.In Fig. 1, we compare the simulated and the analytical SER of bidirectional relay selection withperfect CSI for N relays, i.e., ρ f = ρ f = 1 and ρ e = 1 . This figure shows that increasing the numberof available relays can reduce the average SER, because the diversity order is N when the CSI is perfect.This figure also shows that the asymptotic analytical SER given by Theorem 2 is the lower bound of thesimulated results due to the fact that γ in (20) is greater than that in (19), whereas both the analyticaland the simulated results match tightly in high SNR regime.In Fig. 2, we analyze the impact of delay on the SER performance without CEE , i.e., ρ e = 1 . Forsimplicity, we assume ρ f = ρ f = ρ f and N = 4 . The figure reveals that the diversity order degradesto 1 once ρ f < regardless of N . Although the diversity order is 1 once ρ f < , yet the coding gain isdifferent for different ρ f . Comparing the curves of ρ f = 0 . and ρ f = 0 , the coding gain gap betweenthem is approximately 6dB in high SNR regime. Besides, the performance at moderate SNR is differentfor different ρ f , i.e., greater ρ f has better performance at moderate SNR. For example, at moderate SNR,i.e., range from 8dB to 16dB, the slope of the SER curve of ρ f = 0 . is greater than , while the slopeof ρ f = 0 at the same range is . The performance at moderate SNR can be analyzed by the exactexpression of SER and Maclaurian Series [22].In Fig. 3, we study the impact of CEE on the SER performance without delay, i.e., ρ f = 1 and ρ e = P/ ( P + N ) , where P is the power of the training symbols [21]. P can be greater than the powerof the data symbols P to obtain better performance of channel estimation, thus we simulate the situationof P = P , P , P and ∞ ( P = ∞ means no CEE), respectively. With CEE, the diversity order isinvariant, which is the same as the number of relays. However, compared with the curve of P = ∞ ,there exists coding gain loss caused by CEE, and the loss could be reduced by increasing the power oftraining symbols P . As Fig. 3 illustrated, the coding gain loss in high SNR regime is about 5dB when P = P , but it reduces to 2dB when P = 4 P .In Fig. 4, the joint effect of delay and channel estimation error is considered and compared with thecases of only delay effect, only CEE, and perfect CSI. The results also indicate that delay will result inthe diversity order loss and the coding gain loss, and CEE will merely result in the coding gain loss. With both delay and CEE existing, the SER performance is the worst, which matches tightly with theanalytical result in high SNR regime. V. C
ONCLUSIONS
In this paper, we analyzed the performance of bidirectional AF relay selection with imperfect CSI, i.e.,delay effect and channel estimation error, and the asymptotic analytical expression of end-to-end SERwas derived and verified by the computer simulation. Both analytical and simulated results indicate thatdelay effect results in the coding gain loss and the diversity order loss, and channel estimation error willmerely cause the coding gain loss. A
PPENDIX AP ROOF OF L EMMA ρ f j = 1 , we have h t,ji = h s,ji by (1), and thus ˆ h t,ji = ˆ h s,ji , which is a special case of(7) when ρ j = 1 in Lemma 1.At the case of ρ f j < , by (1), (3) and (4), we have : ˆ h t,ji = h t,ji + e = ρ f j h s,ji + q − ρ f j ε j + e = ρ f j ρ e ˆ h s,ji + ρ f j d + q − ρ f j ε j + e , (34)where d , ε j , and e are independent zero mean complex-Gaussian RVs with variance of σ D , σ ε j , and σ e , respectively. Thus, ρ f j d + q − ρ f j ε j + e is a zero mean complex-Gaussian RV with varianceof ρ f j σ D + (cid:16) − ρ f j (cid:17) σ ε j + σ e , which can be simplified into (cid:16) − ρ f j ρ e (cid:17) σ h t,ji by the relationshipof variances (5),(6). Then, ρ f j d + q − ρ f j ε j + e can be written as q − ρ f j ρ e v j , where v j is anindependent RV with zero mean and variance of σ h t,ji . Defining ρ j = ρ e ρ f j , formula (7) in Lemma 1is proved. Thus, ˆ h t,ji and ˆ h s,ji are jointly complex-Gaussian, and ˆ γ t,ji = (cid:12)(cid:12)(cid:12) ˆ h t,ji (cid:12)(cid:12)(cid:12) and ˆ γ s,ji = (cid:12)(cid:12)(cid:12) ˆ h s,ji (cid:12)(cid:12)(cid:12) arecorrelated exponential distributions, then the joint PDF f ˆ γ t,ji , ˆ γ s,ji ( y, x ) is given by [25]: f ˆ γ t,ji , ˆ γ s,ji ( y, x ) = 1 (cid:16) − ρ j (cid:17) σ h exp − x + y (cid:16) − ρ j (cid:17) σ h I q ρ j xy (cid:16) − ρ j (cid:17) σ h . (35)And now, the conditional probability of (9) in Lemma 1 can be proved by [26]: f ˆ γ t,ji | ˆ γ s,ji ( y | x ) = f ˆ γ t,ji , ˆ γ s,ji ( y, x ) f ˆ γ s,ji ( x ) , (36)where f ˆ γ s,ji ( x ) = exp (cid:16) − x/σ h (cid:17) /σ h . A PPENDIX BP ROOF OF T HEOREM A. distribution of ˆ γ s, k and ˆ γ s, k Following the similar steps of [14] , the CDF of ˆ γ s, k can be expressed as : F ˆ γ s, k ( x ) ( a ) = N Pr { ˆ γ s, i < x, k = i } (37) ( b ) = N x Z f ˆ γ s, i ( y ) Pr { ˆ γ s, i ≤ ˆ γ s, i | ˆ γ s, i = y } Pr { k = i | ˆ γ s, i ≤ ˆ γ s, i , ˆ γ s, i = y } dy + N x Z f ˆ γ s, i ( y ) Pr { ˆ γ s, i > ˆ γ s, i | ˆ γ s, i = y } Pr { k = i | ˆ γ s, i > ˆ γ s, i , ˆ γ s, i = y } dy where (a) in (37) is satisfied due to the symmetry among the N end-to-end paths, and (b) is satisfiedby dividing the union event into two disjoint events, i.e., ˆ γ s, i > ˆ γ s, i and ˆ γ s, i ≤ ˆ γ s, i . Accordingto the selection criterion (11) and order statistics of independent RVs [27]: Pr (cid:8) min (cid:0) x , x (cid:1) ≤ z (cid:9) =1 − (1 − F x ( z )) (1 − F x ( z )) , and the fact that F ˆ γ s, i ( z ) = F ˆ γ s, i ( z ) = 1 − exp (cid:16) − z/σ h (cid:17) , we have Pr { k = i | ˆ γ s, i ≤ ˆ γ s, i , ˆ γ s, i = y } = Y p = i Pr { min (ˆ γ s, i , ˆ γ s, i ) ≤ y } = − exp − yσ h !! N − . (38)Similarly, the conditional probability Pr { k = i | ˆ γ s, i > ˆ γ s, i , ˆ γ s, i = y } can be achieved. Therefore, sub-stituting (38) into (37), F ˆ γ s, k ( x ) can be written as : F ˆ γ s, k ( x ) = N Z x σ h exp − yσ h ! Z y σ h exp − zσ h ! − exp − zσ h !! N − dz dy (39) + N Z x σ h exp − yσ h ! Z ∞ y σ h exp − zσ h ! dz ! − exp − yσ h !! N − dy . Applying binomial expansion (1 − x ) N = P Nk =0 (cid:0) Nk (cid:1) ( − k x k and N P N − n =0 (cid:0) N − n (cid:1) ( − n / ( n + 1) = 1 in [24, (0.155.1)], F ˆ γ s, k ( x ) can be rewritten as : F ˆ γ s, k ( x ) = 1 − N N − X n =0 (cid:18) N − n (cid:19) ( − n n + 1 " exp − xσ h ! + nn + 1 exp − n + 1) xσ h ! , (40)where σ h = ρ − e , and it can be proved similarly that the CDF of ˆ γ s, k have the same form, and theirPDFs can be obtained by differentiating the CDFs. B. distribution of γ At the case of ρ f < and ρ f < , and by Lemma 1 and ∞ R exp ( − αx ) I ( β √ x ) dx = (1 /α ) exp (cid:0) β / (4 α ) (cid:1) in [24, (6.614.3)], we have : f ˆ γ t, k ( x ) = N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 exp (cid:16) − x/σ h (cid:17) σ h (41) + 2 n/σ h (2 n + 1) (cid:0) − ρ (cid:1) + 1 exp − n + 1) x/σ h (2 n + 1) (cid:0) − ρ (cid:1) + 1 ! . The CDF of ˆ γ t, k can be obtained by integrating the PDF, and the distribution of ˆ γ t, k can be obtained bysubstituting ρ with ρ . Let Ω and Ω represent ˜ a ˆ γ t, k , and ˜ b ˆ γ t, k respectively, and the distribution of Ω and Ω can be obtained by f Y ( y ) = (1 /m ) f X ( x/m ) , and F Y ( y ) = F X ( x/m ) when Y = mX ( m > [26]. Thus, the CDF of γ can be written as : F γ ( z ) = Pr (cid:26) Ω Ω Ω + Ω < z (cid:27) (42) = Pr { (Ω − z ) Ω < z Ω , Ω > z } + Pr { (Ω − z ) Ω < z Ω , Ω ≤ z } = Z ∞ z F Ω (cid:18) zxx − z (cid:19) f Ω ( x ) dx + Z z (cid:20) − F Ω (cid:18) zxx − z (cid:19)(cid:21) f Ω ( x ) dx = 1 − ∞ Z f Ω ( x + z ) (cid:20) − F Ω (cid:18) z + z x (cid:19)(cid:21) dx Substituting ∞ R exp (cid:0) − mx − nx − (cid:1) dx = 2 p n/mK (2 √ mn ) in [24, (3.324)] into (42), Theorem 2 canbe proved when using N P N − n =0 (cid:0) N − n (cid:1) ( − n / ( n + 1) = 1 in [24, (0.155.1)].At the case of ρ f = 1 or ρ f = 1 , it can be proved in a similar way that the CDF of γ can also beexpressed as the formula (23) in Theorem 1. A PPENDIX CP ROOF OF T HEOREM (cid:0) ψ r , ψ s → ∞ (cid:1) , ρ e → and a, b → . By applying the Bessel functionapproximation for small x → , K ( x ) ≈ /x [23] in Theorem 1, we have : F γ ( z ) ≈ − N N − X m =0 N − X n =0 ( − m + n (cid:18) N − m (cid:19)(cid:18) N − n (cid:19) m + 1 12 n + 1 (43) × (cid:20) exp ( − ( a + b ) z ) + nn + 1 exp (cid:18) − (cid:18) n + 1) a (2 n + 1) (1 − ρ ) + 1 + b (cid:19) z (cid:19) + mm + 1 exp (cid:18) − (cid:18) a + 2 ( m + 1) b (2 m + 1) (1 − ρ ) + 1 (cid:19) z (cid:19) + mm + 1 nn + 1 exp (cid:18) − (cid:18) n + 1) a (2 n + 1) (1 − ρ ) + 1 + 2 ( m + 1) b (2 m + 1) (1 − ρ ) + 1 (cid:19) z (cid:19) (cid:21) . At the case of ρ f = ρ f = 1 , F γ ( z ) can be rewritten as : F γ ( z ) = 1 − N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 (cid:20) exp ( − az ) + nn + 1 exp ( − n + 1) az ) (cid:21) (44) × N N − X m =0 ( − m (cid:18) N − m (cid:19) m + 1 (cid:20) exp ( − bz ) + mm + 1 exp ( − m + 1) bz ) (cid:21) . Furthermore, we have : N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 (cid:20) exp ( − az ) + nn + 1 exp ( − n + 1) az ) (cid:21) (45) = N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 exp ( − n + 1) az )+ N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 [exp ( − az ) − exp ( − n + 1) az )] ( a ) = 1 − [1 − exp ( − az )] N − exp ( − az ) " N N − X n =0 ( − n (cid:18) N − n (cid:19) n + 1 (1 − exp ( − (2 n + 1) az )) ( b ) = 1 − − ∞ X p =1 ( − az ) p p ! N − ∞ X p =0 ( − az ) p p ! N N − X n =0 ( − n (cid:18) N − n (cid:19) ∞ X p =1 (2 n + 1) p − ( − az ) p p ! ( c ) = 1 −
12 (2 az ) N + o (( az ) N ) , where (a) in (45) is achieved by the fact that (cid:0) N − n (cid:1) Nn +1 = (cid:0) Nn +1 (cid:1) and (1 − x ) N = P Nk =0 (cid:0) Nk (cid:1) ( − k x k ; (b) isachieved by Maclaurian Series of exp ( x ) = P ∞ p =0 ( x p ) / ( p !) ; (c) is achieved by P Nk =0 ( − k (cid:0) Nn (cid:1) k n − =0 , (1 ≤ n ≤ N ) in [24, (0.154.3)]. Therefore, F γ ( z ) ≈ [ 12 (2 a ) N + 12 (2 b ) N ] z N . (46)Finally, (31) in Theorem 2 is proved by (22), (29), (46), and ∞ R t N exp (cid:0) − t / (cid:1) dt = 2 ( N − / Γ (1 / N ) = p π/ N )! / (cid:0) N N ! (cid:1) in [24, (3.326.2)], where Γ ( · ) is Gamma function [23].At the case of ρ f < or ρ f < , by exp ( x ) ≈ x , we similarly have : F γ ( z ) ≈ azN N − X n =0 ( − n (cid:18) N − n (cid:19) − ρ (2 n + 1) (1 − ρ ) + 1 (47) + bzN N − X m =0 ( − m (cid:18) N − m (cid:19) − ρ (2 m + 1) (1 − ρ ) + 1 . Then, (32) in Theorem 2 is proved by (22) and (29). R EFERENCES [1] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft, “Xors in the air: Practical wireless network coding,”
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Order Statistics . Jonh Wiley & Sons, Inc., 1970. −6 −5 −4 −3 −2 −1 SNR[dB] SE R Simulated,N=1Analytical,N=1Simulated,N=2Analytical,N=2Simulated,N=3Analytical,N=3Simulated,N=4Analytical,N=4
Fig. 1. Analytical and simulated SER with perfect CSI, with different N , ρ f = ρ f = 1 and ρ e = 1 . −5 −4 −3 −2 −1 SNR[dB] SE R Simulated, ρ f1 = ρ f2 =0Analytical, ρ f1 = ρ f2 =0Simulated, ρ f1 = ρ f2 =0.7Analytical, ρ f1 = ρ f2 =0.7Simulated, ρ f1 = ρ f2 =0.9Analytical, ρ f1 = ρ f2 =0.9Simulated, ρ f1 = ρ f2 =1Analytical, ρ f1 = ρ f2 =1 Fig. 2. Analytical and simulated SER of S with delay effect, with different ρ f j and ρ e = 1 , N = 4 . −5 −4 −3 −2 −1 SNR[dB] SE R Simulated,P=P Analytical,P=P Simulated,P=2P Analytical,P=2P Simulated,P=4P Analytical,P=4P Simulated,P= ∞ Analytical,P= ∞ Fig. 3. Analytical and simulated SER of S with estimation error, with different ρ e and ρ f = ρ f = 1 , N = 4 . −5 −4 −3 −2 −1 SNR[dB] SE R Simulated, ρ f = ρ f =0.7,P=P Analytical, ρ f = ρ f =0.7,P=P Simulated, ρ f = ρ f =0.7,P= ∞ Analytical, ρ f = ρ f =0.7,P= ∞ Simulated, ρ f = ρ f =1,P=P Analytical, ρ f = ρ f =1,P=P Simulated, ρ f = ρ f =1,P= ∞ Analytical, ρ f = ρ f =1,P= ∞ Fig. 4. Analytical and simulated SER of S with delay effect and estimation error, with different ρ e and ρ f i , N = 4= 4