Characterization of Relay Channels Using the Bhattacharyya Parameter

Transcription

1 Caracterization of Relay Cannels Using te Battacaryya Parameter Josepine P. K. Cu, Andrew W. Ecford, and Raviraj S. Adve Dept. of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada Dept. of Computer Science and Engineering, Yor University, Toronto, Ontario, Canada Abstract Relay systems ave large and complex parameter spaces, wic maes it difficult to determine te parameter region te system acieves a given performance criterion, suc as probability of frame error. In tis paper, we sow tat te union bound (UB) and te Battacaryya parameter (BP) can be used for fast analysis of te parameter space wen error-control coding is used. Tis is applicable wen amplify-and-forward (AF) or demodulate-and-forward (DemF) are used. For a given code ensemble, te associated UB tresold is found and can be used to define te signal-to-noise region a given frame error rate can be acieved. Using asymptotic results, te UB tresold can be used to specify te signal-to-noise ratio region successful decoding can be acieved for large bloclengt. In addition, te UB wit BP can be used wen fractional cooperation is used, eac relay only relays a fraction of te source codeword. Tis maes te UB wit BP a valuable tool in te system design of relay networs. I. INTRODUCTION In a relay cannel, te transmission of a message from a source to a destination is aided by one or more intermediary nodes, nown as relays [1]. Te use of relays is nown to improve performance in wireless fading cannels, as it is improbable tat all of te relays will simultaneously experience a deep fade. Some strategies for employing relays include amplifyand-forward (AF) [2], [3], in wic te relay only performs analog amplification on te source s signal; demodulate-andforward (DemF) [4], [5], in wic te source s transmission is demodulated by a relay prior to retransmission; and decodeand-forward (DF) [2], [6], in wic te relay decodes te source s codeword and re-encodes te information sequence. Te analysis of relay cannels is complicated by teir large parameter spaces. For instance, eac additional relay adds at least two parameters: te signal-to-noise ratio (SNR) on te source-to-relay lin, and te SNR on te relay-todestination lin (additional parameters may be introduced by te type of relaying in use). Tus, including te direct lin from source to destination, an m-relay system as at least a (2m + 1)-dimensional parameter space. Given a particular setting of all te parameters, it is not immediately clear weter a given performance criterion, suc as frame error rate, would be satisfied by tat setting. Furtermore, te large parameter space maes it difficult to caracterize te entire space by simulation, density evolution, or any oter metod tat examines te space at individual points. Previous wor as been done to caracterize parameter spaces in related scenarios. In [7], te Union Bound- Battacaryya parameter (UB-BP) metod was used to caracterize te parameter space of parallel cannels, a different segment of te same codeword was transmitted on eac cannel (suc a scenario may be found in a bloc fading cannel). In [8], te BP was used for relay selection and outage probability analysis in relay cannels employing DemF. In [9], te autors use te UB-BP metod to find te SNR tresold for incremental redundancy (IR) te decoding error becomes very small. It was also used to derive te diversity order of IR in fading cannels. Te contribution in tis present paper may be understood as an extension of tese wors, wit te use of te UB-BP metod, a parameter space caracterization of relay cannels wic employ AF and DemF is provided. Furter, te formulation ere provides a tool tat can be used to compare AF and DemF. We sow in Section IV tat te use of te BP gives rise to an easily calculated figure of merit to verify weter a particular relay system satisfies a given performance criterion, ere being te frame error rate (FER). We also give asymptotic results wic can be used as a guide for more detailed density evolution analysis. Hence, te UB can be used by system designers to twea te parameters of te relaying system, and ence determine te amount of relaying tat is required to acieve given system criteria. Tis paper is organized as follows. In Sec. II, we introduce te system model used trougout tis paper and briefly describe te relaying scemes AF and DemF. Some bacground information on te union bound and te Battacaryya parameter is given in Sec. III. Te application of UB and BP in relay cannels is illustrated in Sec. IV. Simulation results are sown in Sec. V, and conclusions drawn in Sec. VI. II. SYSTEM MODEL In tis paper te m-relay networ is used for te analysis, wic includes te source (S), destination (D) and m relay (R ) nodes. It is assumed tat eac relay can only transmit or receive at a time and tat symbol syncronization is not available. Ortogonal cannels are used to facilitate transmissions from different nodes. A quasi-static Rayleig fading cannel model is used to describe te lins between te nodes. It is assumed tat all receivers ave cannel state information, and te instantaneous SNR between S and all te relays are nown at te destination as well.

2 Te source node first forms a codeword d (S) {0, 1} l s of rate r s, and assuming binary pase-sift eying (BPSK) is used, te codeword is mapped to c (S) {+1, 1} ls. Tis mapping is denoted by te function ξ( ). Te source node broadcasts te codeword c (S) and te discrete-time received signal at relay and D are given by y (SR) = SR, c (S) + n (SR) = 1,..., m (1) y (SD) = SD c (S) + n (SD), (2) SR, and SD are te fading cannel coefficients between S and te t relay and between S and D respectively, and n (SR) and n (SD) are independent additive wite Gaussian noise wit variance N R, and N D respectively. After receiving y (SR), relay processes te received data, and forms a new symbol vector c (R), wic is ten transmitted to te destination node. Te lengt of te new signal formed by different relays can be different from tat of te source. Te discrete time signal transmitted by relay and received by D is given by y (RD) = RD, c (R) + n (RD), = 1,..., m, (3) RD, is te cannel coefficient between relay and D, and n (RD) is te additive wite Gaussian noise wit variance N D. Te average received signal-to-noise ratio (SNR) of te cannel between S and relay is given by γ SR, = E[γ SR, ] = E[ SR, 2 ]/N R,, (4) E[ ] denotes statistical expectation and γ SR, is te instantaneous SNR of te cannel between S and te t relay. Te average received SNR of te S-D and R -D cannel, γ SD and γ RD,, can be found in a similar fasion. A. Amplify-and-Forward Recall tat for AF, eac relay R amplifies te received te signal and transmits it to D. Te multiplication factor can be determined by eiter te amplitude limit imposed by te transmitting antenna, or te average transmitted power. In tis paper, we will assume tat te average transmitted power constraint is enforced, te amplification factor is set suc tat te SNR over te cannel between relay and D is constant for eac bloc. Instead of relaying te complete source codeword, eac relay can also relay only a fraction of te codeword. Wit fractional cooperation [10], eac relay cooses at random a fraction of te source codeword to relay, no centralized coordination is required. Here we assume tat relay cooses to relay eac bit wit probability ɛ. Let te it bit of te source codeword be relayed as te b,i t bit of te symbol vector transmitted by te t relay. Ten c (R) y (SR),i,b,i = SR, 2 + N R, At te destination node, te received signal from te relays are combined wit te received signal from te source node, eac of tem scaled accordingly to reflect cannel conditions and te noise variance. B. Demodulate-and-Forward Cooperative DemF scemes are devised wit sensor networs in mind, te battery power and ardware complexity of a source or relay node are limited, but te destination is assumed to possess relatively more complex ardware, and energy consumption is not an issue. DemF differs from DF as it only requires symbol-by-symbol ard detection, and te relay is not required to decode te complete codeword. In addition, wit DF, te signal is only relayed if correct decoding is acieved, as wit DemF, te signals are always forwarded. If DemF is used, after receiving te signal y SR,, relay maes ard decisions on te received signal to form d (R). Similar to AF, fractional cooperation can be used so tat te relay can select only part of te received signal to relay. In tis case te new codeword d (R) will only contain bits tat are selected. It ten generates c (R) = ξ(d (R) ), wic is ten transmitted to D. Te decoding process uses sum-product algoritm (SPA) [11] to account for te reliability of te S-R cannel. More information regarding te decoding process can be found in [10]. III. UNION BOUND AND BHATTACHARYYA PARAMETER Te union bound can be used to provide upper bounds on te maximum lieliood (ML) bit error rate (BER) and frame error rate (FER) for linear codes [12]. Two components are required for calculating te union bound: te weigt enumerator (WE) and te Battacaryya parameter (BP). Te WE is a vector of numbers A tat describes te number of codewords wit weigt. Te derivation of WE for turbolie codes was presented in [13], RA codes were also introduced. Te second component, BP, is associated wit te cannel condition. Te BP for binary input cannels wit inputs {0, 1} and output y is given by β y Y p(y 0)p(y 1) (5) Y is te alpabet of output y, and p(y 0) and p(y 1) are, respectively, te probability of y given 0 and 1 was sent. For te binary symmetric cannel (BSC) wit bit-flip probability p, β = 2 p(1 p), and for te additive wite Guassian noise cannel wit received SNR γ, β = e γ. After deriving te WE and BP, te FER bound for a point-to-point cannel is given by P f A β. (6) Tese results can be extended to scenarios te codeword is sent troug multiple parallel cannels. Te scenario te codeword is divided into subsets, wit eac subset is transmitted troug a different cannel was analyzed in [7], eac cannel can ave a different BP. By averaging over all possible cannel assignments for te codeword bits, te

3 autors derived te parallel-cannel UB on te FER. Assuming tat tere are J parallel cannels, ten J P f A α j β j (7) α j is te probability tat eac bit of te codeword transmitted troug cannel j and J α j = 1 and β j is te BP for cannel j. Te IR cooperative coding sceme, te source codeword is divided into subsets, wit eac subset relayed by at most one relay, is analyzed using UB-BP in [9]. For codewords wit large bloc lengts, asymptotic results can be used to estimate te SNR tresold above wic te error rate decreases very quicly. Let [C(n)] denote a binary code ensemble wit rate r and lengt n, and let A [C(n)] denote te average number of codewords wit Hamming weigt for te entire code ensemble [C(n)]. Ten te asymptotic normalized exponent of te weigt spectrum is defined as [9] r [C] (δ) lim sup n ln A [C(n)] n [C] denotes te binary code ensemble wit rate r. Using te same notation as [9], we define te asymptotic UB tresold of a code ensemble [C] as r(δ) c 0 sup 0<δ 1 δ, (8) Wit some manipulation of (6), it can be sown tat if J α j β j < exp( c 0 ) (9) ten te average FER approaces zero as n. Note tat tis section only provides a brief overview of te union bound and BP. A more detailed description, including te derivation of te union bound, can be found in [12], and discussions of tigter decoding error bounds can be found in [7], [14], [15]. IV. APPLICATION OF UB-BP TO RELAY CHANNELS Similar to te single-input single-output cannels, te UB and BP can be applied to relay cannels. We will first sow te BP corresponding to two different scemes repetition code is used: AF and DemF. In bot cases, te relay repeats data based on te received signals, and no coding is performed at te relay. We will ten sow ow te UB-BP can be used in relay cannels to determine te region in te parameter space a given performance criterion can be met wen AF or DemF is used. A. BP for AF Repetition Code Recall tat wit te use of fractional cooperation, eac bit can be relayed by any number of relays ranging from 0 to m. Let M i be te set of relays tat are relaying bit i of te source codeword. After combining te received signals from te source and all te relays, we ave te it bit of te combined signal y d y d,i = SD y(sd) i N D + M i SR, RD, y(rd),b,i N SRD,i SR, 2 + N R, (10) x is te conjugate of x and N SRD, = N D + RD, 2 N R, SR, 2 +N R,. After some manipulation, it can be sown tat y d,i is a Gaussian signal, wit SNR given by γ AF,i = γ SD + γ SR,γ RD, (11) γ SR, + γ RD, + 1 M i Recall tat for a Gaussian cannel wit SNR γ, te BP is given by β = exp{ γ}. Hence te BP of bit i for te AF coding sceme is given by β AF = exp { = γ SD M i γ SR,γ RD, γ SR, + γ RD, + 1 } (12) M i β AF, (13) = exp{ γ SD } { } γ SR, γ RD, β AF, = exp γ SR, + γ RD, + 1 (14) In te case of AF, te SNR of te equivalent cannel may be exactly calculated, as in (11). By placing it in te UB-BP expression (12), and noting tat tis expression can be used to verify tat a given performance criterion is satisfied, β AF, or te equivalent expression in any oter relaying system may be used as a figure of merit for tat system. B. BP for DemF Repetition Code For DemF, we ave te following lieliood equations for te it received signal from relay p(y (RD),i c (S) i = +1) = (1 p SR, )f(y (RD),i ; 1, 1/2γ RD, ) + p SR,i f(y (RD),i ; 1, 1/2γ RD, ) p(y (RD),i c (S) i = 1) = (1 p SR, )f(y (RD),i ; 1, 1/2γ RD, ) + p SR, f(y (RD),i ; 1, 1/2γ RD, ) p SR, = 1 2 erfc( γ SR, ) { } f(y; µ, σ 2 1 ) = exp (y µ)2 2πσ 2 2σ 2 After substituting tese equations into (5) and some simple manipulations, we ave β DemF, = γrd, exp { γ RD, (y 2 + 1) } π [(1 p SR, + p SR, exp { γ RD, y}) (1 p SR, + p SR, exp {γ RD, y})] 1/2 dy (15)

4 γ RD,i (db) γ SR,i (db) Fig. 1. Contours of β i for AF (solid) and DemF (das-dot). A closed form solution to te integration in (15) does not exist, but te integral can be approximated using te Taylor expansion of 1 + x for x < 1. Te received signal from te relays are combined wit te signal from S in te decoding process. Similar to te AF repetition code, if fractional cooperation is used, ten we ave te BP associated wit it bit of te equivalent received signal given by β DemF,i = β DemF,. (16) M i Tis quantity may be used as a figure-of-merit for a DemF system, wic is significant since te equivalent SNR of DemF cannot be calculated as easily as for AF. In bot AF and DemF, te BP contribution for eac relay can be isolated. Te β AF, and β DemF, values are sown in Fig. 1. Te contours of various β values are plotted for different γ SR, and γ RD, values. In te figure, te solid lines represent values of β AF, and te das-dot lines represent values of β DemF,. In most cases, β DemF, < β AF,, except wen γ RD, is muc larger tan γ SR,, as indicated by te crossover points in te plot. Te BP can be used to compare te performance te AF and DmF, and elps us estimate under wat condition it is more advantageous to use AF or DmF. If a relay is capable of bot AF and DmF, ten it can coose te relaying sceme tat will provide te best error performance given te cannel conditions. C. Decodable region Te upper bounds on te FER can be used to specify te region in te (2m + 1)-dimensional SNR space a given error rate can be acieved. Depending on te relaying sceme, we can ten substitute (14) or (15) into (7) to obtain te upper bound on te error performance. Let I = {1, 2,..., m} be te set of relays, and I j be all disjoint subsets of I, including te empty subset, j = 1,..., 2 m. Ten te UB is given by 2 m P f (17) A α j = I j ɛ α j I j β 1 ɛ I\I j and β can be β AF, or β DemF,, depending te type of relaying sceme cosen by relay. Wit some simple manipulations, (17) can be simplified to ( m P f A (1 ɛ (1 β ))) (18) =1 To find te region of SNR te FER is below P f,t, we need to first find te value of β t suc tat P f,t = A βt. Ten it is easy to see tat as long as te condition m =1 (1 ɛ (1 β )) β t (19) is satisfied, te FER is below P f,t. Te set of SNRs te condition in (19) is satisfied defines te region in te (2m + 1)-dimensional SNR space te FER requirement is fulfilled. For source codewords wit asymptotically large bloclengt, if te asymptotic UB tresold c 0 exists and is finite, it can be used to estimate te SNR region te FER approaces zero [7]. It can be assumed tat successful decoding can be acieved if te condition m =1 (1 ɛ (1 β )) exp( c 0 ) (20) is satisfied. As a result, te expression in (20) can be used to predict te outcomes of density evolution. V. SIMULATION RESULTS Tis section compares Monte Carlo simulation results and te associated union bounds. For all te results sown ere, a rate-1/2 systematic RA code is used, and te bloclengt of te source codeword, n is It is assumed tat te information bits are repeated q = 3 times and punctured to form te parity bits [16]. Te WE can be obtained using te formulation in [17]. In Fig. 2, only one relay is used, and te SNR over all cannels are te same. Simulation results for bot AF and DemF are sown, and te cases ɛ = 1 and ɛ = 1/2 are illustrated. Te SNR tresolds corresponding to te asymptotic UB tresold c 0 of te rate-1/2 systematic RA code for various cases are also illustrated. In all te cases,

5 FER AF ε i = 1 DmF ε i = 1 AF ε i = 1/2 DmF ε i = 1/ SNR (db) Fig. 2. FER for AF and DemF wit bloclengt n = and teir corresponding SNR tresolds derived from c 0 of te rate-1/2 RA code. FER/P out AF FER DmF FER AF P out DF P out SNR (db) Fig. 3. FER for AF and DemF wit bloclengt n = and P out of te rate-1/2 RA code wit m = 6 and ɛ i =. te waterfall of te FER is about db away from te SNR tresold corresponding to c 0. In Fig. 3, te FER for multiple relays in fading cannels are sown. In tis case, m = 6, ɛ i = and te average SNR is te same over all cannels. Te probability of (20) not satisfied, or equivalently, te outage probability P out for bot AF and DemF are also sown in te plot. As sown in te plot, P out follows te FER closely. Altoug te outage probability only provides an upper bound on te FER, and does not provide an exact expression for te FER, it can be used to gain an understanding of te diversity order of te system. Similar to te previous scenario, DmF performs better tan AF wen te average SNR over all te cannels are te same. code. We ave also sown ow te UB wit BP can be used to determine te SNR region a given error performance can be acieved. By applying asymptotic results, it can be used to determine correct decoding is acievable for large bloclengts. Finally, we ave sown tat tis is not only applicable to relay cannels wit one relay, but also ones wit multiple relays wit fractional cooperation. REFERENCES [1] T. M. Cover and A. A. E. Gamal, Capacity teorems for te relay cannel, IEEE Trans. on Inform. Teory, vol. IT-25, pp , Sept [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative diversity in wireless networs: Efficient protocols and outage beavior, IEEE Trans. Inform. Teory, vol. 50, pp , Dec [3] Y. Zao, R. Adve, and T. Lim, Improving amplify-and-forward relay networs: Optimal power allocation versus selection, IEEE Trans. on Wireless Comm., vol. 6, pp , Aug [4] J. P. K. Cu and R. S. Adve, Implementation of co-operative diversity using message-passing in wireless sensor networs, in Proc. IEEE Globecom, Dec [5] T. Wang, A. Cano, G. B. Giannais, and J. N. Laneman, Higperformance cooperative demodulation wit decode-and-forward relays, IEEE Trans. on Comm., vol. 55, pp , July [6] P. Razagi and W. Yu, Bilayer low-density parity-cec codes for decode-and-forward in relay cannels, IEEE Trans. Inform. Teory, vol. 53, pp , Oct [7] R. Liu, P. Spasojević, and E. Soljanin, Reliable cannel regions for good binary codes transmitted over parallel cannels, IEEE Trans. on Inform. Teory, vol. 52, pp , April [8] J. P. K. Cu, R. S. Adve, and A. W. Ecford, Using te Battacaryya parameter for design and analysis of cooperative wireless systems. Submitted to IEEE Trans. Wireless Comm., under review. [9] R. Liu, P. Spasojević, and E. Soljanin, Incremental redundancy cooperative coding for wireless networs: Cooperative diversity, coding, and transmission energy gains, IEEE Trans. on Inform. Teory, vol. 54, pp , Mar [10] A. W. Ecford, J. P. K. Cu, and R. Adve, Low-complexity and fractional coded cooperation for wireless networs, IEEE Trans. on Wireless Comm., vol. 7, pp , May [11] F. R. Ksciscang, B. J. Frey, and H.-A. Loeliger, Factor graps and te sum-product algoritm, IEEE Trans. Inform. Teory, vol. 47, pp , Feb [12] S. Lin and D. J. Costello, Error Control Coding: Fundamentals and Applications. Prentice Hall, second ed., [13] H. Jin and R. J. McEliece, RA codes acieve AWGN cannel capacity, in Proc. 13t Int. Symp. on Applied Algebra, Algebraic Algoritms, and Error-Correcting Codes, pp , Nov [14] S. Samai (Sitz) and I. Sason, Variations on te gallager bounds, connections, and applications, IEEE Trans. on Comm., vol. 48, pp , December [15] I. Sason and I. Goldenberg, Coding for parallel cannels: Gallager bounds and applications to turbo-lie codes, IEEE Trans. on Inform. Teory, vol. 53, pp , July [16] D. Divsalar, H. Jin, and R. J. McEliece, Coding teorems for turbolie codes, in Proc. Allerton Conf. on Comm., Control, and Computing, pp , Sept [17] A. Abbasfar, D. Divsalar, and K. Yao, Maximum lieliood decoding analysis of accumulate-repeat-accumulate codes, in Proc. IEEE Globecom, Dec VI. CONCLUSION In addition to single-in single-out cannels, UB-BP can also be used for analysis in relay cannels. In tis paper, we ave presented te BP associated wit AF and DemF repetition