Differential Amplify-and-Forward Relaying Using Linear Combining in Time-Varying Channels

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1 Differential Amplify-and-Forward Relaying Using Linear Comining in Time-Varying Channels M. R. Avendi, Ha H. Nguyen and Dac-Binh Ha University of California, Irvine, USA University of Saskatchewan, Saskatoon, Canada Duy Tan University, Danang, Vietnam arxiv:5.56v [cs.it] 4 Dec 5 Astract Differential encoding and decoding can e employed to circumvent channel estimation in wireless relay networks. This article studies differential amplify-and-forward relaying using linear comining with aritrary fixed comining weights in timevarying channels. An exact it error rate analysis is otained for this system using DBPSK modulation and over timevarying Rayleigh fading channels. The analysis is verified with simulation results for several sets of comining weights and in various fading scenarios. Index Terms Differential amplify-and-forward relaying, differential modulation, linear comining, time-varying channels. I. INTRODUCTION Differential amplify and forward D-AF relaying has een considered in [] [3] to circumvent channel estimation at oth Relay and Destination. In D-AF system, information its are differentially encoded at Source. The Relay s function is simply to multiply the received signals with a fixed amplification factor. At Destination, decision variales are computed for all links and then linearly comined. Computing the optimum comining weights requires the instantaneous CSI of Relay- Destination RD channels unknown and the amplification factor. Thus, the average fading powers of the RD channels have een used to define a set of fixed weights in [] [3]. For further reference, these weights are referred to as semi- OPT weights. It was also pointed out that the performance of a D-AF system using the semi-opt weights is difficult to derive [] [3]. Instead, the performance of a D-AF system, in slow-fading environments, ased on the optimum weights assuming instantaneous CSI of RD links are availale was derived for enchmarking the performance of the system using the semi-opt weights. On the other hand, channels may vary in time due to moility of users, and this would affect the system performance. In [4], the authors analysed D-AF relaying using a new set of comining weights over time-varying Rayleigh fading channels. The new comining weights were computed ased on the second-order statistics variance and auto-correlation of the transmission links and the amplification factor. For further reference, these weights are referred to as semi-opt. However, only a lower ound of the it-error-rate was derived in [4]. When there is no access to the aove information, Destination may simply use equal gain comining EGC aleit with some performance penalty. Therefore, it is necessary to consider a D-AF relaying system using aritrary fixed comining weights and analyse its performance. Motivated y the aove discussion, in this article, D-AF relaying using linear comining with aritrary fixed comining weights is analysed over time-varying Rayleigh-fading channels. An exact average it-error-rate ression of the system using DBPSK is otained. Specifically, the analysis is verified with simulation results for semi-opt and semi-opt weights in various fading scenarios. The outline of the paper is as follows. Section II descries the system model. In Section III, linear comining, differential detection and the performance of the system are considered. Simulation results are given in Section IV. Section V concludes the paper. Notation:,, R{ },I{ } denote conjugate, asolute value, the real part and the imaginary part of a complex numer, respectively. CN,σ stands for complex Gaussian distriution with mean zero and variance σ and χ stands for chi-squared distriution with two degrees of freedom. E{ } and Var{ } are ectation and variance operations, respectively. Both ande indicate onential function and E x = e t /tdt is the onential integral function. x II. SYSTEM MODEL The wireless relay model under consideration is shown in Figure. It has one source, one relay and one destination. Source communicates with Destination oth directly and via Relay. Each node has a single antenna, and the communication etween nodes is half duplex i.e., each node is ale to only send or receive in any given time. The channel coefficients at timek, from Source to Destination SD, from Source to Relay SR and from Relay to Destination RD are shown with h [k], h [k] and h [k], respectively. A Rayleigh flat-fading model is assumed for each channel, i.e., h i CN,σ i,i =,,. The channels are spatially uncorrelated and changing continuously in time. It is assumed that all the channels follow a moile-to-moile channel model [5]. Let, f i, i =,, are the maximum normalized Doppler frequencies induced y the motion of Source, Relay and Destination in the SD, SR and RD links. The time correlation etween two channel coefficients,

2 replacements Destination Fig.. h [k] Relay h [k] h [k] The wireless relay model under consideration. Source n symols apart, follows the Akki and Haer s model [5] ϕ i n = E{h i [k]h i[k +n]} = σ ij πf i nj πf j n where i =,,, j =,,, J is the zeroth-order Bessel function of the first kind. The normalized Doppler frequency is a function of the velocity of the nodes. A higher velocity leads to a higher Doppler value and hence a lower time-correlation etween the channel coefficients. It should also e noted that the moile-to-moile channel model [5] is a more general model which includes the fix-to-moile model [6] as well. For instance, in case of fixed Destination f = and then ϕ n,ϕ n change to the Jakes fading model [6] defined with one Bessel function. LetV = {,+} e the set of BPSK symols. Information it at time k is transformed to an BPSK symol x[k] V. Before transmission, the symols are encoded differentially as s[k] = x[k]s[k ], s[] =. The transmission process is divided into two phases. Assume that, in phase I, symol s[k] is transmitted y Source to Relay and Destination. Let P e the average Source power per symol. The received signals at Destination and Relay are y [k] = P h [k]s[k]+z [k], 3 y [k] = P h [k]s[k]+z [k] 4 where z [k],z [k] CN,N are noise components at Destination and Relay, respectively. It is easy to see that, for given s[k], y [k] CN,N ρ +, where ρ is the averaged received SNR per symol from the direct path defined as ρ = Pσ N. Also, the average received SNR per symol at Relay is defined as ρ = Pσ N. The received signal at Relay is then multiplied y an amplification factor A, and re-transmitted to Destination. Here, A can e any aritrarily fixed value. The corresponding received signal at Destination is y [k] = A h [k]y [k]+z [k], 5 where z [k] CN,N is the noise at Destination. After some manuipulations, it yields that y [k] = A P h[k]s[k]+z[k], 6 where h[k] = h [k]h [k] is the equivalent doule-rayleigh channel with zero mean and variance σ σ [7] and z[k] = A h [k]z [k] + z [k] is the equivalent noise. It should e noted that for a given h [k], z[k] is a complex Gaussian random variale with zero mean and variance σz = N + A h [k] and hence y [k], conditioned on s[k] and h [k] is a complex Gaussian random variale with zero mean and variance ρ +σz. Here, ρ is the average received SNR per symol from the cascaded path at Destination, conditioned on h [k]. It is defined as ρ = A ρ h [k] +A h [k]. III. DETECTION AND PERFORMANCE ANALYSIS A. Linear Comining and Differential Detection Given two consecutive received symols at a time, noncoherent detection of the transmitted symols can e otained. Then, the decision variales for the direct and cascaded links are computed from the two latest symols as ζ = R{y [k ]y [k]}, 7 ζ = R{y [k ]y [k]}. 8 To achieve the cooperative diversity, the decision variales from the two phases are linearly comined as ζ = w ζ +w ζ, 9 where w,w are the comining weights. Finally, the output of the cominer is used to decode the transmitted signal as { if ζ < ˆx[k] = + if ζ >. In the next section, the performance of this system is analysed. B. Error Performance Analysis The transmitted symols are equally proale and, without loss of generality, assume that symol x[k] = + is transmitted and it is decoded erroneously as ˆx[k] = y the decoder. The can e ressed as P E = Prζ <,x[k] = + = Pr ζ + ζ < = f ζ rdrd = F ζ d, where ζ = w ζ, ζ = w ζ. Also, is the pdf of ζ and is the cdf of ζ F ζ. To compute these function, first, ζ and ζ are simplified. For time-varying channels, individual channels are ressed y an AR model as h i [k] = α i h i [k ]+ α i e i[k], i =,,

3 whereα i = ϕ i /σi is the auto-correlation of the ith channel and e i [k] CN,σi is independent of h i[k ]. Based on these ressions, a first-order time-series model has een derived in [4] to characterise the evolution of the cascaded channel in time. The time-series model of the cascaded channel is given as the reader is referred to [4] for the detailed derivations/verification h[k] = αh[k ]+ α h [k ]e [k], 3 where α = α α is the equivalent auto-correlation of the cascaded channel, which is equal to the product of the autocorrelations of individual channels, and e [k] CN,σ is independent of h[k ]. It should e noted that these timeseries models are used only for performance analysis purpose and not for generating channel coefficients in the simulation. Using and 3 into y [k],y [k] one has y [k] = α x[k]y [k ]+ z [k], 4 z [k] = z [k] α x[k]z [k ]+ α P s[k]e [k]. y [k] = αx[k]y [k ]+ z[k], 5 z[k] = z[k] αx[k]z[k ]+ α A P h [k ]s[k]e [k]. Then y sustituting 4 and 5 into 7 and 8, one has ζ = R { α x[k] y [k ] +y [k ] z [k] } 6 ζ = R { αx[k] y [k ] +y [k ] z[k]}. 7 It is seen that, for given y [k ], ζ is a comination of complex Gaussian random variales with conditional mean and variance computed as µ ζ = E{ζ y [k ]} = α ρ ρ + y [k ], 8 Σ ζ = Var{ζ y [k ]} = N + α ρ ρ + + α ρ y [k ]. 9 Also, for giveny [k ] andh [k ],ζ is a comination of complex Gaussian random variales and hence it is Gaussian as well with conditional mean and variance computed as µ ζ = E{ζ y [k ],h [k ]} = αρ ρ + y [k ], Σ ζ = Var{ζ y [k ],h [k ]} = σ z + α ρ ρ + + α ρ y [k ]. From now on, the time index [k ] is omitted for simplicity. From 8 and 9, ζ CNw µ ζ,wσ ζ and its conditional pdf is written as y = πw Σ ζ w µ ζ w Σ ζ. Since y CN,N ρ +, it follows that y.5n ρ +χ. By taking the ectation of over the distriution of y, the pdf of ζ is otained as = c, 3 d, = w N +ρ,c =.5w N + α ρ,d =.5w N ++α ρ. On the other hand, from and, ζ CNw µ ζ,w Σ ζ and its conditional pdf is written as f ζ y,h = πw Σ ζ w µ ζ w Σ ζ. 4 Since, conditioned on h, y CN,σz ρ +, one concludes that y.5σzρ +χ. By taking the ectation of 4 over the distriution of y, one has h f ζ = d c,, 5 = w σz ρ +,c =.5w σz + αρ,d =.5w σz ++αρ, are functions of random variale λ = h whose pdf is f λ λ = /σλ/σ. Thus, the cdf of ζ conditioned on h is otained as = c h F ζ = + d c d,,. Using 3 and 6, can e evaluated as P E h = e c + d F ζ h d e d d + e d c 6 e c d = c c d + d c + d c d c. 7 By taking the final average over the distriution of λ = h, one has P E = I α + I α,α + I 3 α,α. The first term is otained as I α = c = + αρ +ρ. Interestingly, the term I α, which corresponds to the direct link, is the same as the of DBPSK of point-to-point communications over fast-fading channels derived in [8, eq.8.3]. Also, I α,α and I 3 α,α are derived as I α,α = c = B 3 d d c f λλdλ + B B σ B B σ E σ 8

4 B = w +α+w +w α ρ ρ +w + α ρ +w A w +ρ ++αρ 9 I 3 α,α = d c d c f λλdλ = I α, α. 3 with B = /A + + αρ, B 3 = + α ρ / B A + ρ + ρ, B defined on the top of the next page. As can e seen, the otained ression depends on the comining weights through B and it gives the of the D-AF relaying system using DBPSK and aritrary fixed comining weights in time-varying Rayleigh fading channels. Moreover, it is seen that the otained ression depends on the channel auto-correlations through the defined parameters. This dependency on the channel auto-correlations leads to a performance degradation in fast-fading channels and eventually, at high transmit power, the reaches an error floor. The amount of error floor can e otained as lim P E, which is straightforward and not ressed P /N here due to space limit. IV. NUMERICAL RESULTS As mentioned efore, the analysis in the previous section works for aritrary fixed comining weights. In this section, the analysis is verified for two sets of comining weights, semi-opt [] [3] and semi-opt [4], with simulation results. The semi-opt weights are written as [] [3], [9], w semi OPT = N, w semi OPT = 3 N +A σ. Also, the semi-opt weights are written as [4] w semi OPT α = N [+α + α ρ ], w semi OPT = α N [+α +A σ + α A ρ σ ]. 3 Practically, the two sets of weights are the same for slowfading channels with α =,α =. The optimum power allocation values, reported in [] are considered for three scenarios: symmetric channels, [σ,σ,σ ] = [,,], non-symmetric channels with strong PSfrag replacements SR channel [σ,σ,σ ] = [,,] and non-symmetric channels with strong RD channel [σ,σ,σ ] = [,,]. The values are q opt =.66,.54,.8 for symmetric, strong SR and strong RD channels, respectively. To generate the channel coefficients, the simulation method in [] is used. Based on the moility of nodes, three fading cases, listed in Tale I are considered here. In, all nodes are assumed to e fixed or slowly moving such that f = f = f =.. In case II, only Source is moving and Relay and Destination are static such that f =.,f =.,f =.. Both SD and SR channels are time-varying and RD channel is static. In II, oth Source and Relay are moving and Destination is fixed such that f =.5,f =.3,f =.. All channels are time-varying in this case. TABLE I THREE FADING CASES. f f f Channels status... all are slow-fading I... SD and SR are fast-fading II.5.. all are fast-fading The simulated and theoretical values using the semi- OPT and semi-opt weights are computed for all cases and different channel variances and plotted versus P/N in Figs The theoretical values of the error floor for Cases II and III are computed and added to the figures. The error floor for is very small and hence not plotted. For comparison purposes, the theoretical lower ound derived in [4] for symmetric channels is also added to Fig.. From Figs. -4, it can e seen that the simulation and our analytical results in all fading and channel scenarios are tight to each other for oth semi-opt and semi-opt weights whereas the lower ound of [4] is relatively loose in Cases II and III of Fig.. Specifically, in of all figures slowfading channels, the desired diversity is achieved as ected. The comining weights of semi-opt and semi-opt are practically the same in this case and their results overlap. On the other hand, in Cases II and III fast-fading channels of all Simulation semi OPT 3 4 Simulation semi OPT Analysis semi OPT Analysis semi OPT Lower Bound [4] II I P/N db Fig.. Theoretical and simulation of the D-AF system with the semi- OPT and semi-opt weights in different fading rates and [σ,σ,σ 3 ] = [,,].

5 the figures the results gradually deviate from slow-fading results and reach to error floors as predicted.as ected, in Cases II and III of the figures, the results of semi-opt are slightly etter than semi-opt at high SNR region. The reason is that the semi-opt weights take channel auto-correlation values into account so that they are closer to optimum weights. V. CONCLUSION II PSfrag replacements 3 Differential amplify-and-forward relaying Simulation using linear semi-opt comining with aritrary fixed comining weights Analysis in time-varying semi-opt Rayleigh fading channels was considered. The of the 4 Simulation EGC I system employing DBPSK was derived and verified Analysis with simulation for two sets of comining weights. With the otained EGC ression, the effect of channel variation on the system performance was also studied. P/N db Fig. 4. Theoretical and simulation of the D-AF system with the semi- OPT and semi-opt weights in different fading rates and [σ,σ,σ 3 ] = [,,]. Simulation semi OPT Simulation semi OPT Simulation semi OPT Analysis semi OPT Analysis semi OPT placements semi-opt semi-opt lation EGC 3 4 Simulation semi OPT Analysis semi OPT Analysis semi OPT II I P/N db Fig. 3. Theoretical and simulation of the D-AF system with the semi- OPT and semi-opt weights in different fading rates and [σ,σ,σ 3 ] = [,,]. REFERENCES [] T. Himsoon, W.P. Siriwongpairat, Weifeng Su, and K.J.R. Liu, Differential modulations for multinode cooperative communications, IEEE Trans. Signal Process., vol. 56, no. 7, pp , Jul. 8. [] Q. Zhao and H. Li, Differential modulation for cooperative wireless systems, IEEE Trans. Signal Process., vol. 55, no. 5, pp , May 7. [3] Z. Fang, L. Li, X. Bao, and Z. Wang, Generalized differential modulation for amplify-and-forward wireless relay networks, IEEE Trans. Veh. Technol., vol. 58, no. 6, pp , July 9. [4] M.R. Avendi and H.H. Nguyen, Performance of differential amplifyand-forward relaying in multinode wireless communications, IEEE Trans. Veh. Technol., vol. 6, no. 8, pp , 3. [5] A.S. Akki and F. Haer, A statistical model of moile-to-moile land communication channel, IEEE Trans. Veh. Technol., vol. 35, no., pp. 7, Fe [6] W. C. Jakes, Microwave Moile Communications, Wiley-IEEE Press, Piscataway, NJ, 994. [7] C.S. Patel, G.L. Stuer, and T.G. Pratt, Statistical properties of amplify and forward relay fading channels, IEEE Trans. Veh. Technol., vol. 55, no., pp. 9, Jan. 6. [8] M. Alouini and M. Simon, Digital Communication over Fading Channels, Wiley-IEEE Press, 5. [9] T. Himsoon, W. Su, and K.J.R. Liu, Differential transmission for amplify-and-forward cooperative communications, IEEE Signal Process. Letters, vol., no. 9, pp , Sept. 5. [] C.S. Patel, G.L. Stuer, and T.G. Pratt, Simulation of rayleigh-faded moile-to-moile communication channels, IEEE Trans. Commun., vol. 53, no., pp , 5.