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SOFT DECODE AND FORWARD IMPROVES COOPERATIVE COMMUNICATIONS Harold H. Sneessens and Luc Vandendorpe Communications and Remote Sensing Laboratory, Université catholique de Louvain, Louvain-la-Neuve, Belgium. {sneessens, vandendorpe}@tele.ucl.ac.be ABSTRACT Mobile receivers can benefit from an increased spatial diversity through cooperation. Two relaying techniques are typically used : either Amplify and Forward (AF) or Decode and Forward (DF). This contribution presents a new technique which combines the main advantages of both AF and DF. This technique amounts to a DF scheme where all operations are performed in a soft-input soft-output fashion. In particular, an encoding technique which uses soft values to incorporate reliability information is proposed. The resulting Soft-DF relay can be seen as an analog signal-to-noise ratio enhancer. Simulations show that this Soft-DF scheme outperforms both AF and DF. 1. INTRODUCTION Cooperation between mobiles provides spatial diversity and thus improves communications on fading wireless channels. This cooperative diversity is particularly suited to cases where the mobiles cannot support multiple antennas. User cooperation was first investigated in [1, 2], where the authors show the interest of such techniques through an information-theoretic analysis and propose an implementation. Two relaying techniques are presented and compared in [3, 4]. In the first technique, called Amplify and Forward (AF), the relay simply amplifies its partner’s signal and forwards it to the destination. The second technique is called Decode and Forward (DF) and consists in decoding, reencoding and forwarding the partner’s signal to the destination. These two techniques were the object of many analyses and improvements in the literature, for instance in [5–7]. While these works demonstrate the advantages of user cooperation, DF lacks the main advantages of AF and vice versa : DF regenerates the signal while AF does not lose soft information. This contribution presents a new relaying technique which enables both to regenerate the signal and The work reported in this paper has partly been funded by the Belgian interuniversity projects PAI MOTION and by the european Network of Excellence NEWCOM . Harold H. Sneessens would like to thank the Belgian FNRS for its financial support.

to keep soft information. This new technique can be seen as a soft flavor of DF, and will be referred to as Soft-DF in the sequel. Simulations show that this technique outperforms AF and DF. Section 2 introduces the system model and section 3 presents the previously known schemes, AF and DF. The new cooperative technique, Soft-DF, is developed in section 4. Finally, section 5 compares the performance of the new scheme with the two previous ones. 2. SYSTEM MODEL We consider the particular case of two users transmitting to the same destination. The inter-users channels as well as the uplink channels are mutually independent Rayleigh block fading channels. They are frequency flat and orthogonal. After the transmission by each source user in a first time slot, each partner relays the signal in a second time slot. The ratio of the second time slot duration to the total duration is defined as the cooperation ratio. The channels are assumed to be constant during both time slots, and are perfectly known by the corresponding receivers so that coherent detection is used. The destination uses both the direct and relayed signals to make a joint decision. Although both users cooperate in a symmetric fashion, we will focus on one source signal in order to simplify notations. We consider thus one user as the source S and the other as its relay R, transmitting to a destination D. For BPSK modulation, the baseband-equivalent discretetime signal of user i ∈ {S, R} received by user j ∈ {R, D} (i = j) at time n is
(1) rij (n) = hij Es,i · bi (n) + nj (n), which describes the transmission of BPSK symbols b i (n) with an energy per symbol E s,i , experiencing fading of magnitude hij and additive noise at the reception n j (n). The model assumes that the fading magnitudes h ij are independent Rayleigh-distributed samples and that the noise terms nj (n) are samples of a zero-mean white gaussian noise process with two-sided power spectral density N j /2.

3. KNOWN COOPERATIVE SCHEMES : AF AND DF Relays using the AF technique simply amplify and re-transmit their partner coded data. The relay chooses a constant amplification factor such that the average energy per symbol over the consecutive frames equals its average symbol energy Es,R . The destination receives the relayed signal and combines it optimally with the direct path before decoding. With the DF technique, the relay decodes its partner’s signal, makes hard decisions and checks if errors occured thanks to a parity check code. Erroneous frames are not relayed since it would break down the performance at destination, but successfully decoded frames are re-encoded, possibly with another code, and forwarded to the destination. An adequate choice of the codes makes possible to construct a cooperative turbo coded scheme, for which an appropriate iterative decoding algorithm is used at the destination. This happens if both the source user and the relay use convolutional codes, provided that the relay interleaves the data before re-encoding. This leads to the interpretation of DF as the partition of a user’s codeword, whose the first part is transmitted by the source user and the second by the relay. The main advantage of AF over DF is that no hard decisions are made, but on the contrary AF does not regenerate the signal. 4. NEW COOPERATIVE TECHNIQUE : SOFT-DF The new Soft-DF technique presented here keeps the advantages of both previous techniques, i.e. it regenerates the signal while keeping soft information. A relay using this technique first decodes its partner’s signal, keeping soft information, interleaves and re-encodes it with a soft-input soft-output (SISO) encoder. Finally the relay deducts the signal to transmit thanks to an equivalent channel assumption on the outputs of the encoder. This equivalent channel assumption results in interpreting the relay as an analog channel signal-to-noise ratio (SNR) enhancer. In this sense, the soft nature of the output signal enables to convey information about its reliability. We successively describe below the new SISO encoder, the equivalent channel modeling its output and finally the receiver at destination. 4.1. Soft-Input Soft-Output Encoder The proposed SISO encoding algorithm has some similarities with a SISO decoding algorithm. Given a sequence of noisy coded symbols r SR , a DF relay would compute estimates of the K information bits u = [u(1), . . . , u(K)]T and from these produce a sequence of N coded bits x = [x(1), . . . , x(N )] T . In contrast, the

soft-DF relay keeps the soft information onthe information  bits, i.e. the a posteriori probabilities P u(k)|rSR and produces  a sequence of a posteriori probabili from these ties P x(n)|rSR on the coded bits x(n). For the sake of notational simplicity in this section, we assume that the encoder produces only one parity bit per information bit (i.e. N = K). Generalization to other rates is straightforward. For a convolutional code, the representation of this soft encoding operation within the framework of factor graphs and the application of the sum-product algorithm lead to an encoding algorithm quite similar to that of the well-known MAP decoding algorithm. More precisely, the soft encoder aims to compute    P x(k)|rSR = P (x|rSR )

k = 1, . . . , K, (2)

∼x(k)

 where ∼x(k) denotes the summation on all variables but x(k). A convenient factorization of P (x|r SR ) must be found to enable an efficient computation of the marginals P (x(k)|r SR ) thanks to the sum-product algorithm. Since x does not depend on rSR once u is assumed, rSR → u → x forms a Markov chain and we have :    P x(k)|rSR = P (x|u) · P (u|rSR ), (3) ∼x(k)

where P (x|u) equals 1 if the codeword x corresponds to the information bits sequence u and 0 otherwise. For a convolutional code, this factor expands in the same way as for a SISO decoder. In order to obtain the second factor of (3) and to factorize it further we need to make the assumption that the u(k) are independent given r SR : P (u|rSR ) =

K 

  P u(k)|rSR ,

(4)

k=1

where the P (u(k)|rSR ) are computed by the SISO decoder. Simulations suggest that this approximation does not have any noticeable impact. For a convolutional code, the application of the sumproduct algorithm to the factor graph described by (3)-(4) gives slightly modified versions of the standard forward recursion α(·) and reverse recursion β(·) of the MAP decoding algorithm. For an encoder of state s(k) at time k, we can write the following recursions where each term in the summations has to correspond to a combination of (s(k − 1), u(k), x(k), s(k)) satisfying the code constraints:   α s(k) = 

 β s(k − 1) =



    α s(k − 1) · P u(k)|rSR ,

∼s(k)



∼s(k−1)

    β s(k) · P u(k)|rSR .

We finally compute          α s(k − 1) β s(k) P u(k)|rSR . P x(k)|rSR = ∼x(k)

4.2. Equivalent Channel Model We need the signal lying behind the probabil  to re-transmit ities P x(k)|rSR . Since we lack any analytical model for the output of the SISO encoding/decoding process, we deduce this signal from the log likelihood ratios (LLRs) under the assumption that they have the same form as at the output of an equivalent AWGN channel with BPSK modulation :     P (x(n) = 0|rSR ) = Leq bR (n)+neq (n) . LLR x(n)
log P (x(n) = 1|rSR ) (5) The amplitude factor L eq reflects the signal reliability, b R (n) = 2x(n) − 1 is the symbol corresponding to the bit x(n) and neq (n) is a zero-mean white gaussian noise sample of vari2 2 . Both amplitude L eq and variance σ eq of the noise ance σeq depend on the signal-to-noise ratio inside a block on the inter-user channel and therefore they change at each frame. Since no analytical characterization exists their relation with the noise variance on the inter-user channel has to be computed empirically. From this interpretation of the LLRs, the relay constructs and forwards to the destination the signal β R (bR (n)+neq (n)) which contains the payload plus an additive gaussian noise. The factor βR is constant over consecutive frames and is computed in order to normalize the transmitted energy averaged over the realizations of the inter-user channel :  Es,R , (6) βR = 2 ] E[σb2 + σeq

Normalization at the relay

hRD

rRD (n) = hRD (n)βR (bR (n) + neq (n)) + nD (n),

(7)

which is simply under the previous assumptions the output of a cascade composition of two channels corrupted by gaussian noise. Of course the receiver needs to estimate the parameters of the compound channel only and not the parameters of both channels separately, thus no channel state information for the individual channels is required. The decoding process of this signal jointly with that from the direct path is achieved by a standard turbo decoding operation.

+

Relay to destination channel

Fig. 1. Equivalent user→relay→destination channel for soft decode and forward. This model suggests to interpret the transmission of coded data on the inter-user channel followed by the soft decoding/reencoding process as the transmission of coded data on a global super-channel with better SNR. This interpretation is depicted in figure 1. The SNR enhancement is quantified in figure 2. This figure presents in its upper part the empirical input-output

Eb/Neq (output SNR) [dB]

Equivalent inter-user channel

nD βR

The destination receives the following signal from the relay :

15 10 5 0 −5 −5

Probability density

+

4.3. Reception and decoding

20

where σb2 is the variance of the symbols b(n). neq

relation of the SISO decoder/re-encoder in terms of SNR, for the simulation parameters of section 5 and under the previous assumption of a gaussian i.i.d. noise corrupting the output. For a coded signal received with a SNR of 5 dB for example, the relay produces an output signal with a SNR of about 12 dB. The limitations of the i.i.d. gaussian noise assumption appear distinctly below 0 dB and above 15 dB where the output SNR is lower than the input SNR. However for common channel parameters, input SNRs lie mainly in the range of interest, as shown in the lower part of figure 2 by the probability density function of the input SNR for a Rayleigh fading channel with mean SNR of 8 dB.

0

5

10

15

20

15

20

0.2 0.1 0 −5

0

5 10 E /N (input SNR) [dB] b

R

Fig. 2. Input-output relation of the decoding/re-encoding process, in terms of instantaneous SNRs (i.e. for a particular realization of the fading).

0

perfect information at the relay, and the performance of no cooperation (turbo-code of rate 1/3 with the same constituent codes as for DF, and an equal total energy expense). These results reflect the advantages of the principles underlying soft-DF : in comparison to DF, soft-DF gains from an increased cooperation level and compared to AF, it wastes less power to transmit noise.

10

−1

BER

10

−2

10

6. CONCLUSION −3

10

−4

10

0

No cooperation AF DF Soft−DF Perfect cooperation 4

8 12 Uplink E /N [dB] b

16

20

D

Fig. 3. BER curves for AF, DF and Soft-DF.

5. PERFORMANCE COMPARISON Simulation results confirm the advantages of the Soft-DF relay technique through the following analysis of bit error rate (BER) curves. Figure 3 compares the performance of AF, DF and Soft-DF with the following parameters. Both users encode their own data blocks of length K = 128 bits with a recursive systematic convolutional (RSC) encoder of rate 1/2, constraint length 4 and generator polynomials (138 , 178 ). The partner relays these data with a cooperation rate 1/3. In order to achieve such a cooperation rate, the AF relay forwards the parity bits only, the DF relay reencodes the decoded bits with the same RSC code and forwards the computed parity bits, and the Soft-DF relay forwards the K parity bits computed using the non-recursive convolutional soft-encoder of constraint length 2 and generator polynomial 3 8 . To decide whether to cooperate or not, the DF relay checks if a frame is error-free thanks to a 16-bit CRC code with generator polynomial given by the coefficients 0x1021 (hexadecimal notation). The bandwidth overhead required to transmit the 16 CRC bits is neglected. The figure compares the different schemes for equal overall energy consumption per information bit. The BER is computed for equal SNRs on the uplinks (abscissa in figure 3) and a SNR fixed to 10 dB on the inter-user channel. Denoting Eb,S the energy spent by the source per information bit, the relay spends an additional average energy E b,S /2 to cooperate. Each point is obtained after the observation of 1000 frame errors. The turbo-decoders of DF and Soft-DF perform three iterations, which is enough to achieve convergence. To achieve a BER of 10 −3 Soft-DF requires about 2.5 dB less power than AF and 2 dB less than DF. For the sake of comparison, the figure shows the performance of DF with

The Soft-DF cooperation technique presented in this paper achieves performance reflecting its advantages over AF and DF. The use of exclusively soft information at the relay as well for decoding as for re-encoding is a key point for this result. This soft process is enabled thanks to the design of a SISO encoder. Dedicated strategies and protocols can be designed to further improve its performances, in the same way as for other cooperative techniques. These strategies can take advantage of the soft nature of this new cooperation technique. 7. REFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity–Part I: System Description,” IEEE Trans. Commun., vol. 51, no. 11, Nov. 2003. [2] ——, “User Cooperation Diversity–Part II: Implementation Aspects and Performance Analysis,” IEEE Trans. Commun., vol. 51, no. 11, Nov. 2003. [3] J. N. Laneman and G. W. Wornell, “Energy-efficient antenna sharing and relaying for wireless networks,” in Proc. IEEE WCNC, Chicago, IL, 2000, pp. 7 – 12. [4] J. N. Laneman, G. W. Wornell, and D. N. C. Tse, “An efficient protocol for realizing cooperative diversity in wireless networks,” in Proc. IEEE ISIT, Washington, DC, 2001, p. 294. [5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inform. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [6] T. Hunter and A. Nosratinia, “Diversity through coded cooperation,” IEEE Trans. Wireless Commun., submitted for publication. [7] M. Janani, A. Hedayat, T. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: Space-time transmission and iterative decoding,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 362 – 371, Feb. 2004.