A Soft Decode-Compress-Forward Relaying Scheme ...

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A Soft Decode-Compress-Forward Relaying Scheme for Cooperative Wireless Networks Dushantha N. K. Jayakody, Member, IEEE, and Mark F. Flanagan, Senior Member, IEEE

Abstract—This paper proposes a new technique for soft information relaying (SIR) which is based on a soft decode-compressforward (DCF) relay protocol. The proposed system provides a means of using distributed low-density parity-check (LDPC) coding in conjunction with higher order modulation such as pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) which is effective even under poor sourcerelay link conditions. Ordinarily, such schemes suffer from error propagation to the destination caused by incorrect decoding at the relay when the signal-to-noise ratio (SNR) on the sourcerelay link is low; however, our system avoids this problem by generating soft versions of the additional (parity-bearing) PAM symbols for transmission from the relay. The proposed technique of soft compression does not suffer from the problem of parity log-likelihood ratios (LLRs) converging to zero, as do many soft re-encoding techniques for turbo and LDPC codes. In the case of Gray-coded PAM/QAM signaling, we also propose a method of performing exact expectation-based soft modulation with low computational complexity. Furthermore, we propose a new model, which we refer to as the soft scalar model, for the overall source-to-destination channel encountered by the constellation symbols, and this model is used at the destination to compute LLRs for joint decoding of the distributed LDPC code. Simulation results demonstrate that the proposed scheme can provide good coding gain, diversity gain and spectral efficiency under poor source-relay SNR conditions.

I. I NTRODUCTION OOPERATIVE communication for wireless networks promises improved transmit diversity and increased spectral efficiency [1]. The broadcast nature of the wireless network provides unique opportunities for collaborative and distributed signal processing techniques. Intermediate nodes can hear the source at no additional cost and it is very efficient for these nodes to forward information to the destination. Two of the simplest relaying protocols are decodeand-forward (DF) and amplify-and-forward (AF); DF lacks the main advantages of AF and vice versa [2], [3]. In DF, errors at the relay can be corrected and thus error propagation can be avoided under good source-relay channel conditions. But an error-prone relay can destroy the performance of the destination’s decoder as it may forward erroneous information. In contrast, the AF protocol suffers from the problem of noise

C

Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported by the Telecommunications Graduate Initiative (TGI), which is funded by the Irish Higher Education Authority (HEA) under the Programme for Research in Third-Level Institutions (PRTLI) Cycle 5 and co-funded under the European Regional Development Fund (ERDF). The authors are with the School of Electrical, Electronic and Communications Engineering, University College Dublin, Ireland (email: [email protected], [email protected]).

amplification but by reserving any premature decision, AF preserves the soft information content of the received signal. Distributed coding schemes such as distributed LDPC codes [4], [5] and distributed turbo codes (DTC) [6], [7] have been proposed in recent years to improve the efficiency and/or reliability of wireless links. It has been a common feature until recently in most of these distributed coding systems to assume either that the relay can always decode correctly or else that the relay will not forward when it cannot decode correctly. A recently proposed and promising relay protocol is that of soft information relaying (SIR) [8]-[25]; it may mitigate the shortcomings of traditional protocols, such as error propagation to the destination (DF) and noise amplification at the relay (AF). In contrast to DF, in SIR the performance of decoding at the destination is not degraded under a poor channel condition, as the erroneous signal from the relay does not ‘pretend’ to be highly reliable. Usually, soft modulation is performed based on calculating the expected value of the modulated symbol (a method which was shown to be SNR-optimal in the uncoded scenario [9]); this is possibly preceded by iterative (probabilistic) decoding. A notable exception is the proposal of [10], [11] whose soft forwarding technique is based on symbol-wise mutual information (SMI); note however that this work considered only uncoded transmission and binary phase shift keying (BPSK) modulation, thus limiting its practical application. One of the principal ideas behind the coded SIR strategies employed for relay channels is that the relay forwards additional soft parity-check bits to the destination, thus facilitating joint decoding of a stronger channel code at the destination. In [12]-[17], soft re-encoding via a soft-input soft-output (SISO) module was implemented at the relay. However, as was shown in [18] and [19], when iteratively decodable codes are employed the recursive structure of such soft re-encoding procedures means that the reliability of the recursively encoded soft bits depends strongly on the least reliable input bits, causing a decaying LLR profile. Therefore, the resulting soft modulated symbols, being close to the origin in the signal space, are difficult to differentiate from each other in the presence of noise; this problem is even more severe when higher-order modulation is employed. A scheme for soft parity generation was recently proposed in [19] which has the advantage that successively generated soft symbols do not converge to zero in this manner; it was however specific to BPSK signaling. Various models have been proposed in the literature for the overall channel experienced by modulated symbols in their transmission from source to relay to destination, the

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Relay

Soft Demap.

LDPC Dec.

H1

Soft bit map.

Soft Compressor H2

Soft Mod.

ysr

hrd

nsr S-R link

R-D link

hsr Source

u

LDPC Enc. H1

ysd

S-D link

hsd Mod.

nrd yrd

Soft Demap.

nsd Destination LDPC Dec. H

û

Figure 1. The proposed half-duplex soft DCF relaying system.

model in each case being matched to the soft relaying scheme employed. The work of [13] represents the effective (‘soft’) noise introduced by the relay as Gaussian noise; however, it was shown in [14] that the Gaussian assumption made in [13] is not very accurate by demonstrating numerically the LLR distribution at the destination for DTC. In [20] a soft fading model was presented for the overall source-relay-destination channel, and a rate-compatible LDPC code structure was also proposed. These models published so far are specific to binary phase-shift keying (BPSK) signaling, as is most existing work on SIR in the literature [8]-[25]. However, it is of great interest in practice to use more spectrally efficient modulation schemes such as PAM and QAM and, within this context, to limit the computational complexity of computing the soft higherorder modulation symbols at the relay. The work of [22] generalized the idea of expectation-based soft forwarding, first proposed for BPSK modulation in [13] and [14], to higherorder (PAM/QAM) modulation. However, in this scheme the relay simply transmits soft versions of the source symbols (it does not perform soft re-encoding), and maximum-ratio combining (MRC) is correspondingly used for reception at the destination. In this paper, we propose a simple but effective protocol called soft decode-compress-forward (DCF) for generating additional soft (parity) PAM/QAM symbols at the relay. The soft modulation method is based on estimating the expected value of the modulated symbols, a method which has proven to be very effective for uncoded systems [15], [23] and which was applied to coded systems in [22]. Crucially, in contrast to the work of [22], in the proposed scheme the soft symbols transmitted from the relay do not correspond simply to a repetition of the source’s parity symbols; thus additional (distributed) code structure is introduced which is then exploited at the destination. Finally, in the case of a particular form of Graycoded PAM/QAM signaling, a technique is proposed which is mathematically proven to perform exact expectation-based soft modulation with low computational complexity. The scheme does not exhibit the problem of decaying LLR

profiles at the relay, and is thus able to exploit coding gain while still preserving reliability information. We also introduce a model, called the soft scalar model, for the overall channel experienced by these soft PAM/QAM symbols and hence compute the corresponding LLRs for the destination’s joint decoding. Finally, we present extensive simulation results, covering a range of higher-order modulations, which show significant gains for the proposed scheme when the sourcerelay SNR link is in a poor condition. In the simulations, we used LDPC coding; note however that any other coding scheme which uses iterative decoding may be used in order to reap the benefits offered by the proposed cooperative communication system. The remainder of this paper is organized as follows. Section II presents the system model description, and is followed by the proposed SIR scheme in Section III. Simulation results are presented in Section IV, and the paper is concluded in Section V. II. S YSTEM M ODEL In this paper, we consider a three-terminal relay channel configuration. It is assumed that there is a direct transmission from the source to the destination. In the first time slot, the source transmits a message to both relay and destination, and in the second time slot the relay aids the destination by transmitting a processed version of the signals received in the first time slot. We assume that all nodes have only one antenna working in a half-duplex mode. The source, relay and destination processing for the proposed soft DCF scheme is shown in Fig. 1. For ease of presentation, in this section and the next we limit our exposition to PAM signaling; however, note that the proposed technique, including the soft modulation method and its efficient implementation for the case of Gray coding, can very easily be extended to work for QAM, provided separate bits are used to modulate the real and imaginary dimensions. In Section IV, simulation results will be demonstrated for both PAM and QAM signaling.

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In the first time slot, the source bit vector1 u of length K is encoded by an LDPC encoder of rate R1 = K/N1 to give the codeword c1 of length N1 . This codeword c1 is defined by a (N1 −K)×N1 parity-check matrix H1 via H 1 c1 T = 0. The bits c1 are mapped to a vector x1 of symbols from the M -ary PAM constellation with M = 2q , i.e., each consecutive q-tuple p of bits is mapped to a symbol from the constellation Aq / Eq where

where H 2 is an N2 × N1 binary matrix and I is the N2 × N2 identity matrix. The code associated with H has code rate R = K/N where N = N1 + N2 . III. S OFT INFORMATION RELAYING In this section, we explain how SIR is performed for the proposed soft DCF scheme. In particular, we provide details of the relay processing function f (y sr ).

Aq = {+(2q − 1), +(2q − 3), . . . , +3, +1, − 1, −3, . . . , −(2q − 3), −(2q − 1)} , (1)

A. Calculation of soft code bits at the relay

and the multiplier Eq = (22q − 1)/3 normalizes the PAM constellation to have unit energy. Since M = 2q , the length of x1 is N1 /q. The received signals at the relay and destination in the first time slot are, respectively, p (2) y sr = Ps hsr x1 + nsr

The first step required for SIR is the calculation of the a posteriori LLRs λ(c1,i |y sr ). These may be calculated from y sr using a standard soft demapper followed by an LDPC decoder which uses the parity-check matrix H 1 . For soft compression, these LLRs are then represented in the form of “soft bits”, i.e., c˜1,i = tanh(λ(c1,i |y sr )/2).

and

B. Soft compression

y sd =

p

Ps hsd x1 + nsd ,

(3)

where each vector nij contains i.i.d. real Gaussian random 2 variables, each having zero mean and variance σij = N0 /2, i, j ∈ {s, r, d}, Pi is the transmit power constraint from node i, and hij is the Rayleigh fading coefficient between the nodes i and j where i, j ∈ {s, r, d}. We denote by SNRij the SNR on the link connecting node i to node j. In the second time slot, the relay employs a soft  demapper to  p(c =0|ysr,i ) calculate the a priori LLRs λ(c1,i |ysr,i ) = ln p(c1,i 1,i =1|ysr,i ) – here ysr,i denotes the received value containing c1,i . These are then input to the LDPC decoder, which uses H 1 for decoding the codeword c1 . In a “hard” DCF scheme, assuming perfect decoding of c1 , it is possible to generate additional parity bits c2 by compression of this codeword via cT2 = H 2 cT1 . These additional bits may then be modulated to form a vector of M -ary PAM symbols x2 , which are transmitted from relay to destination in order to achieve diversity (this was the basis for the work in [26], where BPSK modulation was assumed for retransmission). However, at low sourcerelay SNR, c1 is often decoded incorrectly, and this hard compression procedure can result in erroneous symbols being propagated to the destination. Therefore, our proposed relay transmits a “soft” version xr = f (y sr ) of the symbols x2 , where the function f (.) will be elucidated in Section III. The destination’s received signal in the second time slot is p y rd = Pr hrd f (y sr ) + nrd (4) Note that the overall codeword at the destination, denoted by c, satisfies   H1 0 T Hc = [c1 c2 ]T = 0 (5) H2 I 1 Unless otherwise stated, in this work s, r, d stand for source, relay, and destination respectively. Throughout the paper all vectors are taken to be row vectors. Also, vectors and matrices are denoted as bold letters and the ith element as an italic letter. We use regular letters to denote scalars and random variables. For a random variable x, we use E[x] to denote the expected value of x. The vector of soft symbols corresponding to symbol vector a is ˜. represented by a

The equation cT2 = H 2 cT1 can be implemented in the “soft domain” (i.e., on the soft bits) as follows [20]:   Y c˜2,j =  c˜1,k  for j = 1, . . . , N2 (6) kAj

where Aj is the set of positions of the ones in the j th row of H 2 . The complexity of the soft compression procedure depends on the density of H 2 . C. Soft M -ary PAM modulation The next step, required for M > 2, is to generate the ˜ 2 which will be transmitted to soft M -ary PAM symbols x the destination. Note that in the case of soft BPSK, the generated soft compressed BPSK symbols are same as the ˜2 = c ˜2 . soft compressed bits, i.e., x First, the bit LLRs can be generated from the soft compressed bits via λ(c2 |y sr ) = 2 tanh−1 (˜ c2 ). Next, the underlying bit probabilities can be computed using p(c2,i = exp(λ(c2,i |y sr )) and p(c2,i = 1|y sr ) = 0|y ) = 1+exp(λ(c  sr  2,i |ysr )) 1 1+exp(λ(c2,i |y sr )) . ˜2 Finally, in general, in order to generate the soft symbols x (for M > 2), we use the expectation of the PAM symbol calculated using the known probabilities of the underlying compressed bits, i.e., X α · p(x2,i = α|y sr ) . (7) x ˜2,i = E(x2,i |y sr ) = √ α∈Aq /

Eq

Note that the soft symbol estimate in (7) minimizes the conditional mean square error (MSE) between the soft symbol x ˜2,i and the actual symbol x2,i , i.e., E{(˜ x2,i − x2,i )2 |y sr }, as was shown in [22]. In the following, in order to simplifypthe mathematical notation, we denote the scaled symbol x2,i Eq by zq (this is in order to emphasize the dependence of this variable on q; also, the subsequent analysis is made easier by the fact that

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a1 0 1 1 0

a2 0 0 1 1

4

z2 +3 +1 -1 -3

Table I G RAY- CODED MODULATOR FOR 4-PAM ( INTEGER CONSTELLATION ).

a1 0 1 1 0 0 1 1 0

a2 0 0 1 1 1 1 0 0

a3 0 0 0 0 1 1 1 1

(7) may be calculated using the values of the underlying soft bits {˜ ai : i = 1, 2, . . . , q} as follows: z˜1 = a ˜1 and for 1 ≤ j ≤ q − 1,
z˜j+1 = a ˜j+1 z˜j + 2j . (10) Proof: The result z˜1 = a ˜1 is trivial to prove. For any j > 1, we have by (7) (for convenience of notation, here conditioning of the bit probabilities on y sr is omitted)

z3 +7 +5 +3 +1 -1 -3 -5 -7

z˜j+1 =

j+1 2X

(2j+1 )

Ll

j+1 Y

(2j+1 )

p(ai = gi,l

)

i=1

l=1

j

= p(aj+1 = 0) ·

2 X

(2j+1 )

Ll

+ p(aj+1 = 1) ·

(2j )

p(ai = gi,l )

i=1

l=1

Table II G RAY- CODED MODULATOR FOR 8-PAM ( INTEGER CONSTELLATION ).

j Y

2j  X

(2j+1 )

−Ll

j Y i=1

l=1

zq lies in the integer-valued PAM constellation Aq ). Also, we denote the q bits underlying the symbol zq by {a1 , a2 , . . . , aq }. In general, the evaluation of (7) requires computation, for all 2q PAM symbol values α, of the product of the a posteriori probabilities of the underlying bits. However, in the following we show that for a particular form of Gray-coded PAM, there is a very efficient technique to implement (7), i.e., to perform soft modulation, using only the q corresponding soft bit values a ˜j = p(aj = 0|y sr ) − p(aj = 1|y sr )

(8)

where j ∈ {1, 2, . . . , q}. Note that (8) may be rewritten as p(a =0|y ) a ˜j = tanh(λ(aj |y sr )/2), where λ(aj |y sr ) = log p(a1j =1|ysr ) . sr We next describe the details of the required Gray-coded PAM mapping, together with the corresponding efficient soft modulator. (M ) For M = 2q -level PAM, a 2q ×q Gray =  code  matrix G 0 (2q ) (2) G is constructed using G = and 1 ! j j G(2 ) 0(2 ) (2j+1 ) G = for 1 ≤ j ≤ q − 1 , (9) j j R(G(2 ) ) 1(2 ) where R(A) denotes the matrix A with the order of its rows reversed, and where 0(l) and 1(l) are column vectors of length l consisting entirely of zeros and of ones, respectively. Also, (2j ) denote by gl,i the (binary) entry in the l-th row and the i-th j

column of G(2 ) . Then, the modulator for M = 2q -level PAM works as follows: if the input bit sequence a = (a1 a2 . . . aq ) matches q row l of G(2 ) , then the modulated symbol zq is equal to q (2 ) level Ll which is the l-th symbol in the constellation Aq as listed in (1). For example, for q = 2 and q = 3 we obtain the modulation mapping shown in Tables I and II respectively. The following theorem shows how to perform soft modulation via (7) efficiently for this case of Gray-coded PAM modulation, using only the soft bits which are given by (8). Theorem 1. For the Gray coded PAM scheme described above, the soft M = 2q -level PAM symbol z˜q = x ˜2,1 of

(2j )

p(ai = gi,l )

= [p(aj+1 = 0) − p(aj+1 = 1)] · j

2  X

(2j )

Ll

+ 2j

j Y i=1

l=1

=a ˜j+1 ·

X 2j

+ 2j

(2j )

p(ai = gi,l )

l=1 j Y

j Y (2j )

Ll

(2j )

p(ai = gi,l )

i=1

 [p(ai = 0) + p(ai = 1)]

i=1
=a ˜j+1 · z˜j + 2j .

Here, in the second equality we have used the structure (9) of the Gray code; the third equality uses the fact that the first half of the PAM levels of the 2j+1 -ary constellation may be obtained by shifting the 2j -ary constellation by 2j ; and the second product term in the penultimate equality comes from the fact that all length-j bit patterns are represented in the j Gray code matrix G(2 ) . As two illustrative examples, in the case of 4-PAM (q = 2) whose modulation mapping is given by Table I, the soft PAM symbols are computed via       1 λ(a2 |y sr ) λ(a1 |y sr ) x ˜2,i = √ tanh · tanh +2 , 2 2 5 while in the case of 8-PAM (q = 3) whose modulation mapping is given by Table II, the soft PAM symbols are computed via   λ(a3 |y sr ) 1 · x ˜2,i = √ tanh 2   21       λ(a2 |y sr ) λ(a1 |y sr ) tanh · tanh +2 +4 . 2 2 (11) Finally, the signal transmitted from the relay (prior to power scaling) can be written as ˜2 xr = f (y sr ) = β x

(12)

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where the factor β is chosen to satisfy the transmit power constraint at the relay, i.e., s 1 (13) β= PN2 2 1 ˜2,i i=1 x N2 ˜ 2 carries extra parity information to the destiNote that x nation. As can be seen from (5), a feature of the proposed scheme is that the generated soft parity symbols will not become progressively smaller in magnitude, as happens with some recursive soft parity generation methods in the regime of low source-relay SNR [13]-[20] (the assignation of a time slot to such a frame being a questionable use of resources). Indeed, as we shall see in Section IV, this makes the proposed soft compression mechanism effective in the regime of low source-relay SNR.

The destination receives two different signals via two independent fading routes, i.e., y sd and y rd . The formation of the LLRs at the destination corresponding to the relay is non-trivial when we use SIR; we describe this below for the proposed cooperative communication system. The relationship ˜ 2 is between the correct symbols x2 and the soft symbols x modeled in [13] by x ˜2,i = x2,i (1 − n ¯i) ,

(14)

where n ¯ i is a soft noise variable whose mean and variance 2 can be measured offline as µn¯ = N12 (ΣN ˜2,l − x2,l ) and l=1 x N2 1 2 2 ˜2,l x2,l − µn¯ ) ] respectively. Note that σn¯ = N2 [Σl=1 (1 − x this model was used in [13] for soft BPSK but here we will apply it as a reference model for soft M -ary PAM/QAM also. Assuming this model, the normalization factor βqof (9) may be estimated from the soft noise statistics as β = (1−µn¯1)2 +σ2 . n ¯ Also, the received signal at the destination in the second time slot can be written as p p yrd,i = Pr hrd β x ˜2,i +nrd,i = Pr hrd βx2,i (1−µn¯ )+ n ¯ rd,i (15) √ where n ¯ rd,i = nrd,i − Pr hrd βx2,i (¯ ni −µn¯ ) is the equivalent 2 (zero-mean) noise at the destination with variance σ ¯rd = 2 2 2 2 σrd +Pr hrd β σn¯ – this equivalent noise is modeled as having a Gaussian distribution. The variance σn2¯ of the soft noise variable is computed offline numerically, and is stored at the destination for computation of the destination LLR during realtime transmission (therefore, the relay does not need to provide any extra information to the destination). The soft demapper at the destination is matched to this model; for example, for 4-PAM with modulation mapping given by Table I, λ(a1 |yrd,i ) operates according to p(a1 = 0|yrd,i ) (16) p(a = 1|yrd,i )  1  p(z1 = +3|yrd,i ) + p(z1 = −3|yrd,i ) = log p(z1 = +1|yrd,i ) + p(z1 = −1|yrd,i ) h ¯ i h ¯ i  3Kyrd,i −3Kyrd,i exp + exp 2 ¯ 2 2 4K σ ¯ σ ¯ h ¯ rd i h ¯ rd i  = − 2 + ln  Kyrd,i −Kyrd,i σ ¯rd exp + exp 2 2 log

σ ¯rd

σ ¯rd

(17) √ ¯ = Pr hrd β(1 − µn¯ )/ 5 - here yrd,i denotes the where K received value which contains c2,i (in this context, c2,i is equal to either a1 or a2 ). For soft BPSK, the corresponding LLR λ(c2,i |y rd ) under this model is √ 2 Pr hrd β(1 − µn¯ ) √ 2 yrd,i . (18) λ(c2,i |yrd,i ) = 5¯ σrd √

Here we also propose an alternative model, where the ˜ 2 are related by correct symbols x2 and the soft symbols x x ˜2,i = ηx2,i + n ˆi

D. Calculation of LLR at the destination

=

and a similar development shows that i h ¯   ¯2 −2Kyrd,i −4K + 1 exp ¯ 2 2Kyrd,i σ ¯  h ¯ rd ¯ 2 i + ln  λ(a2 |yrd,i ) = 2 2Kyrd,i −4K σ ¯rd +1 exp 2

σ ¯rd

(19)

where we refer to the constant η as a soft scalar (this may be viewed as an equivalent fading coefficient) and to n ˆ i as the soft error. Taking the expectation of both sides of (19) (and omitting the subscript i for convenience) yields E(˜ x2 ) = ηE(x2 ) + E(ˆ n) .

(20)

Assuming all codewords are equiprobable for transmission, and by symmetry of the PAM constellation, we have E(x2 ) = 0. Also, using symmetry of the additive white Gaussian noise (AWGN) channel, LDPC decoder and soft modulation process, it may be shown that E(˜ x2 ) = 0. It follows that the mean of n ˆ may be assumed to be zero for any value of η. We choose the value of η which minimizes the mean-square value of the soft error, i.e., ∂ E{(˜ x2 − ηx2 )2 } = 2E(x2 x ˜2 ) − 2ηE[x22 ] = 0 ∂η

(21)

and so η = E[x2 x ˜2 ], as E[x22 ] = 1. In P practice, the soft N2 scalar may be computed offline as η = N12 i=1 [x2,i x ˜2,i ] for any desired source-relay SNR. Also, the variance of the soft error, σn2ˆ , is computed numerically offline for each relevant source-relay SNR. These statistics are then stored at the destination for computation of the destination LLR during realtime transmission. The (minimized) soft error variance may also be calculated via σn2ˆ = [σx2˜2 − η 2 ], where σx2˜2 is the variance of x ˜2 . Assuming this model, the normalization factor β of (9) may be estimated from the soft error statistics as s 1 β= . (22) 2 η + σn2ˆ Also, the destination’s received signal from the relay can be modeled as √ ˆ rd,i (23) yrd,i = P r hrd βηx2,i + n √ where n ˆ rd,i = nrd,i + Pr hrd β n ˆ i is the equivalent zero2 2 mean noise at the destination having variance σ ˆrd = σrd + 2 2 2 Pr hrd β σnˆ – we model this equivalent noise as having a Gaussian distribution. Next, the soft demapper used to

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Finally, the destination then concatenates the relevant LLR vectors λ(c1 |y sd ) and λ(c2 |y rd ), and decodes based on the overall parity-check matrix H. E. Performance Evaluation at the Relay

25

Soft scalar model; BPSK Soft noise model; BPSK

Effective SNR

20

15

10

5

0 −3

−2

−1

5

4

0

1

2

3

SNRsr (dB)

6

Effective SNR

compute the LLRs for the destination’s LDPC decoder is matched to the soft scalar model; for example, for soft-4-PAM modulation this is given by (12) √ where we replace √ and (13), 2 2 ¯ by K ˆ = Pr hrd βη/ 5. For the case of σ ¯rd by σ ˆrd and K soft BPSK, the LLR λ(c2,i |yrd,i ) corresponding to the second time slot transmission is given by √ 2 P r hrd ηˆβ √ 2 yrd,i . (24) λrd (c2,i |yrd,i ) = 5ˆ σrd

Soft scalar model; 4−PAM Soft noise model; 4−PAM Soft scalar model; 8−PAM Soft noise model; 8−PAM

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2

For accurate joint decoding at the destination, the LDPC decoder at the destination requires use of a statistical model for the relay transmission; this model depends on the sourcerelay channel as well as on the (soft) relaying function used. Regardless of whether the system uses the soft noise model of (14) or the soft scalar model (19) described in the previous subsection, the destination will treat the ‘soft noise’ or ‘soft error’ (respectively) as noise (i.e., independent of the signal) even though it contains some signal contribution. This motivates us to define and measure an effective SNR, i.e., the SNR which follows from adopting a particular relaying scheme as well as adopting a corresponding model for the overall channel experienced by the modulated PAM symbol (this is similar to the concept of ‘generalized SNR’ introduced in [9] for the case of uncoded soft relaying). The reason for calculating the effective SNR in each case is to provide a performance indication at the destination for different soft symbol modelling techniques. This allows us to discriminate between the models that can be used at the destination for LLR computation, independently of the destination’s design. It provides a qualitative indicator of the relative performance of different models and modulation schemes. Note that the effective SNR measure is not used by the decoder at the relay or the destination, nor does this parameter fully capture the quality of either model from the destination’s perspective (e.g., it does not indicate the accuracy of the Gaussian approximation underlying LLR formation in either case). The effective SNR for our proposed soft scalar model (19) is γ nˆ = |η|2 /Pnˆ (25) P N2 where Pnˆ = N12 i=1 |ˆ ni |2 . In the case of the soft noise model of (14), the effective SNR may be defined as γ n¯ = 1/Pn¯ PN2

(26)

where Pn¯ = N12 i=1 |¯ ni |2 . Fig. 2 shows the variation of this effective SNR with the SNR of the source-relay channel. It may be seen that the soft scalar model achieves a consistently higher effective SNR than the soft noise model in all cases, thus validating its adoption (qualitatively similar results were observed for 16QAM; these are omitted). In fact, as will be shown in Section

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Figure 2. Comparison of the effective SNR obtained using the soft scalar model (19) with that of the soft noise model (14) for the proposed soft DCF scheme.

IV, the BER advantage is even greater than might be predicted on the basis of these curves; this appears to be due to the tendency of the soft error variable to be closer to Gaussiandistributed than the soft noise variable. At high SNR, both models perform similarly, exhibiting very high effective SNR as the soft modulated symbols are approximately equal to their hard counterparts. IV. S IMULATION RESULTS AND D ISCUSSION In this section, we evaluate the performance of the proposed soft DCF scheme. The information word length is K = 408, and the relevant dimensions of the parity-check matrix are N1 = 816 and N2 = 408, so that the overall code has a block length of N1 + N2 = 1224; therefore, the code rates are R1 = 1/2 at the relay and R = 1/3 at the destination. The LDPC codes used at the source and the relay are regular Gallager codes2 having blocklength 816 and code rate 1/2. In the simulations we consider BPSK, 4-PAM, 8-PAM and 16-QAM signaling, and we assume Pr = Ps = 1. Note that bit error rate (BER) is used as the performance metric for BPSK modulation while symbol error rate (SER) is used for higher order modulation. We simulate two channel scenarios; (i) where all links experience only AWGN, and (ii) where all links experience Rayleigh fading as well as AWGN. All simulations assume SNRrd = SNRsd (this assumption represents a particularly challenging environment for cooperation). Four systems are evaluated and compared: 1) Hard DCF reference system: here the relay makes a binary hard decision on the decoding result at the relay, ˆ1 of c1 . This hard decision vector producing the estimate c 2 The parity-check matrix H1 was that corresponding to the code labeled 816.3.174 in the listing at http://www.inference.phy.cam.ac.uk/mackay/codes/data. html, and the parity-check matrix H 2 was that of the code labeled 816.3.184 in the same listing.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2015.2442459, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY

3 Note that in our simulation setup we choose N − K = N , so that our 1 2 proposed scheme and this competing scheme both transmit the same number of parity symbols from the relay.

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is then compressed to form additional parity symbols via ˆT2 = H 2 c ˆT1 . Ordinary (hard) PAM/QAM modulated symbols c are then relayed. Results for this system are included in order to demonstrate the advantages of soft versus hard DCF. 2) The proposed soft DCF system, using the soft scalar model of (19) to compute the destination’s LLRs. 3) The proposed soft DCF scheme, but using the soft noise model of (14) to compute the destination’s LLRs. Results for this system are included in order to demonstrate the performance advantage of the soft scalar model versus the soft noise model, when using the proposed soft DCF scheme. 4) We also compare the performance of the proposed scheme with that of the soft information relaying scheme of [22] (simulation curves are labeled “Weitkemper et al.”). In this scheme, the relay forwards soft estimates of the same BPSK/PAM symbols as were originally sent by the source. Specifically, the relay performs LDPC decoding using the parity-check matrix H 1 , and the resulting N1 − K a posteriori LLRs for the parity bits (i.e., corresponding directly to the source parity bits) are used as input to the conditional expectation based soft forwarding mechanism3 . Since in this scheme two versions of the same parity packet are transmitted from the relay and the source, the destination adds the corresponding bit LLRs before LDPC decoding. Note finally that the destination’s decoder for the scheme of [22] is simply the decoder for parity-check matrix H 1 , while for our proposed scheme it is the decoder for the stronger code represented by the full matrix H. Fig. 3 compares the performance of the proposed soft DCF scheme with that of hard DCF, where BPSK modulation is used over AWGN links. We fixed the SNR of the source-relay link at different values in order to represent both poor and good channel conditions. It may be seen that the use of soft DCF parity generation process yields considerable gains in the regime of low source-relay SNR (here, erroneous decoding in DCF results in the communication of incorrect symbols to the destination). However, as expected hard DCF shows better BER performance in the higher SNR regime (where relay decoding is reliable). For this system, soft DCF was found to outperform hard DCF for source-relay SNRs below 4dB, for the case of BPSK modulation. Note that if knowledge of the source-relay SNR was available to the relay, it would be possible to switch between soft or hard relaying depending on the SNR. Also, if such information is not available, the proposed soft DCF system is naturally adaptive for channels which exhibit strong variations in SNR, offering an alternative to selective relaying. Fig. 4 presents simulation results for the soft DCF scheme where the soft scalar model of (19) is compared with the soft noise model of (14) for formation of the LLRs at the destination (here the modulation was soft BPSK, and the source-relay SNR was set to 2dB and to 2.5dB). Simulation results are also presented in the figure for the SIR protocol of [22] (curves are labeled “Weitkemper et al.”).

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SNRsd (dB) Figure 3. A comparison of the BER performance of the proposed soft DCF scheme with that of hard DCF, using BPSK modulation. The simulations took place over AWGN links, for different values of source-relay SNR. At the destination node, the soft noise model of (14) was used to obtain bit LLR values.

It can be seen that the soft noise model consistently outperforms the soft scalar model; this result also correlates with the results presented in Fig. 2 for the effective SNR of these two models as measured at the relay: at the source-relay SNRs of 2dB and 2.5dB, the proposed soft scalar model shows higher effective SNR performance than the soft noise model. Note that at a source-relay SNR of 2.5dB, the effective SNR for these two models are very close (though the soft scalar model shows slightly better performance), but the soft scalar model exhibits a markedly better error rate performance than the soft scalar model at this source-relay SNR. This may be attributed to the fact that in the lower SNR regime, the soft noise variable of the soft noise model fits less accurately to a Gaussian distribution than does the soft error variable in the soft scalar model (results demonstrating this phenomenon are omitted due to space considerations). The proposed scheme also achieves a gain with respect to the scheme of Weitkemper et al. [22] for both of the considered source-relay SNRs; at a sourcerelay SNR of 2.5dB and a destination BER of 10−4 , this gain is 0.44dB. This extra coding gain is due to the fact that the additional relay parity results in a stronger code matrix at the destination for the proposed scheme. Fig. 5 compares the symbol error rate (SER) performance for the proposed soft DCF system using soft 4-PAM modulation over Rayleigh fading links with the corresponding performance for the scheme of [22] as well as for hard DCF. The source-relay link SNR was set to 4dB and 5dB; at these values, the proposed soft DCF scheme is seen to significantly outperform hard DCF (the gain is 5.8dB at a source-relay SNR of 5dB and a BER of 10−4 ). Furthermore, the proposed scheme outperforms the scheme of [22] by 1.7dB at a sourcerelay SNR of 4dB and a BER of 10−3 , and by 1.15dB at a source-relay SNR of 5dB and a BER of 10−4 . This indicates

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2015.2442459, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY

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modulation scheme increases, the issue of error propagation in the hard DCF system becomes more severe, as the likelihood of frame errors in the decoded c1 is increased; however, for the soft DCF scheme the bit LLRs are more difficult to extract from the noisy QAM symbols. Therefore, both systems face difficulties in the QAM environment; however, the overall result is that the soft DCF system achieves better BER performance than the hard DCF at SNRsr = 5dB, the gain being approximately 3.1dB at a BER of 10−3 .

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Figure 6. SER performance results for the hard and soft DCF schemes with 8PAM and 16-QAM modulation over Rayleigh fading links. For soft DCF, the soft scalar model was used to form LLRs at the destination. The source-relay link SNR was kept at 5dB in all cases.

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the superiority of soft DCF in the regime of poor sourcerelay SNR. Also, the proposed soft scalar model was found in this case also to demonstrate a consistently superior SER performance compared to the soft noise model; corroborating the results of Figure 2 (results are omitted). Fig. 6 shows the SER performance for the proposed system with soft 8-PAM and soft 16-QAM over the Rayleigh fading channel, together with the corresponding results for hard DCF. Note that as the number of bits per dimension of the

A new technique has been presented for soft parity symbol relaying for use with LDPC coded half-duplex relaying systems. The proposed scheme, basd on soft decoding, compression and modulation, works for PAM and QAM modulations, and the soft modulation operation has been proven to possess a particularly low-complexity implementation in the case of a Gray-coded constellation. We have also proposed a new “soft scalar” model to facilitate computation of the corresponding bit LLRs for decoding at the destination. Simulation results have demonstrated that the proposed scheme achieves substantially better error rate performance compared to hard decodecompress-forward in the regime of low source-relay SNR, and yields a promising solution in the scenario where the sourcerelay channel condition may be unknown or time-varying. R EFERENCES [1] J. N. Laneman, D. Tse and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Info. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [2] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Info. Theory, vol. 25, no. 5, pp. 572–584, Sept. 1979. [3] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorem for relay networks,” IEEE Trans. Info. Theory, vol. 51, no. 9, pp. 572–584, Sept. 2005.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2015.2442459, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY

[4] A. Chakrabarti, A. de Baynast, A. Sabharwal, and B. Aazhang, “LDPC code-design for the relay channel,” IEEE Journal of Select. Areas Commun.- Special Issue on Cooperative Communications and Networking, vol. 25, no. 2, pp. 280–291, Feb. 2007. [5] H. Jun and T. Duman, “Low density parity check codes over wireless relay channels,” IEEE Trans. on Wireless Commun., vol. 6, no.9, pp. 3384–3394, Sept. 2007. [6] B. Zhao and M. C. Valenti, “Distributed turbo coded diversity for relay channel,” IEE Electron. Lett., vol. 39, no. 10, pp. 786–787, May 2003. [7] M. Janani, A. Hedayat, T. Hunter, A. Nosratinia, “Coded cooperation in wireless communications: space-time transmission and iterative decoding,” IEEE Trans. on Sig. Proce., vol. 52, pp. 362–371, Feb. 2004. [8] I. Abou-Faycal and M. Medard, “Optimal uncoded regeneration for binary antipodal signaling”, Proc. IEEE International Conference on Communications (ICC 2004), pp. 742–746, 2004. [9] Gomodam and S. Jafar, “Optimal relay functionality for SNR maximization in memoryless relay networks,” IEEE Journal on Sel. Areas in Comm., vol. 25, no. 2, Feb. 2007. [10] M. A. Karim, T. Yang, J. Yuan, Z. Chen and I. Land, “A novel soft forwarding technique for memoryless relay channels based on symbolwise mutual information,” IEEE Commun. Lett., vol. 14, no. 10, pp. 927–929, Oct. 2010. [11] J. Li, M. A. Karim, J. Yuan, Z. Chen, Z. Lin, B. Vucetic, “Novel Soft Information Forwarding Protocols in Two-Way Relay Channels,” IEEE Trans. on Veh. Tech., vol. 62, no. 5, pp. 2374–2381, 2013. [12] H. H. Sneessens and L. Vandendorpe, “Soft decode and forward improves cooperative communications,” 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, pp. 157–160, Dec. 2005. [13] Y. Li, B. Vucetic, T. F. Wong and M. Dohler, “Distributed turbo coding with soft information relaying in multihop relay networks,” IEEE Journal on Sel. Areas in Comm., vol. 24, no. 11, pp. 2040–2050, Nov. 2006. [14] P. Weitkemper, D. Wübben, V. Kühn and K.-D. Kammeyer, “Soft information relaying for wireless networks with error-prone source-relay link,” in Proc. 7th International ITG Conference on Source and Channel Coding, Jan. 2008. [15] P. Weitkemper, D. Wübben and K.-D. Kammeyer, “Minimum MSE relaying in coded networks,” International ITG Workshop on Smart Antennas (WSA), Darmstadt, Germany, 26–27 Feb. 2008. [16] D. N. K. Jayakody, J. Li, B. Chen and M. F. Flanagan, “A Multilevel Soft Quantize-and-Forward Scheme for Multiple Access Relay Systems”, 25th IEEE Annual Symposium on Personal, Indoor and Mobile Radio Communications Conference, (PIMRC 2014), Washington DC, USA, 25 Sept 2014. [17] T. Zhou, F. Wang, J. Xu and J. Lilleberg, “Soft symbol estimation and forward scheme for cooperative relaying”, IEEE 20th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2009), Tokyo, Japan, pp. 3069–3073, Sept. 2009. [18] M. M. Molu and N. Görtz, “A comparison of soft-coded and hard-coded relaying,” European Transactions on Emerging Telecommunications Technologies. DOI: 10.1002/ett.2562. [19] D. N. K. Jayakody and M. F. Flanagan, “LDPC coding with soft information relaying in cooperative wireless networks,” IEEE Wireless Communications and Networking Conference (WCNC), Shanghai, China, April 2013. [20] M. H. Azmi, J. Li. J.Yuan and R. Malaney, “Soft decode-and-forward using LDPC coding in half-duplex relay channels”, in Proc. IEEE International Symposium on Information Theory (ISIT), St. Petersburg, Russia, pp. 1479–1483, Aug. 2011. [21] D. N. K. Jayakody, M. F. Flanagan, “A Soft Decode-Compress-Forward Relaying Scheme for Cooperative Wireless Networks”, 24th IEEE Annual Symposium on Personal, Indoor and Mobile Radio Communications Conference, (WDN-PIMRC 2013), London, UK, pp. 205 – 209, Sept 2013. [22] P. Weitkemper, D. Wübben, K.-D. Kammeyer, “Minimum MSE Relaying for Arbitrary Signal Constellations in Coded Relay Networks,” in IEEE 69th Vehicular Technology Conference 2009 (VTC2009-Spring), Barcelona, Spain, Apr. 2009. [23] C. I. Serediuc, J. Lilleberg and B. Aazhang, “MAP detection with soft information in an estimate and forward relay network”, in Proc. 44th Asilomar Conference on Signals, Systems and Computers (ASILOMAR), California, USA, pp. 121–125, Nov. 2010. [24] F. Hu and J. Lilleberg, “Novel soft symbol estimate and forward scheme in cooperative relaying networks,” in Proc. IEEE 20th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC ’09), pp. 2691–2694, Sept. 2009.

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[25] E. A. Obiedat and L. Cao, “Soft information relaying for distributed turbo product codes,” IEEE Signal Process. Lett., vol. 17, no. 4, pp. 363–366, Apr. 2010. [26] M. Wu, P. Weitkemper, D. Wübben and K.-D. Kammeyer, “Comparison of distributed LDPC coding schemes for decode-and-forward relay channels,” in 2010 International ITG Workshop on Smart Antennas (WSA), Bremen, Germany, Feb. 2010. [27] M. Fisz, “Probability Theory and Mathematical Statistics”. John Wiley & Sons, Inc., 1963. Dushantha N. K. Jayakody received the B.Eng (2009) from Pakistan and ranked as the best foreign student in Pakistan in the year 2009. He received the MSc (2010) in Electronics and Communications Engineering from the Department of Electrical and Electronics Engineering, Eastern Mediterranean University, Cyprus in 2010 with high honors. He received the Ph.D. (2014) in Electronics and Communications Engineering from the University College Dublin (UCD), Ireland. This work is done when he was a Ph.D. student. Since Oct 2014, he holds a Postdoc position at the Coding & Information Transmission group, University of Tartu, Estonia. In the Spring 2015, he was a Visiting Scientist at the Centre of Excellence in Telecommunications, University of Sydney, Australia. Dr. Jayakody served on the Technical Program Committee at several IEEE International Conferences. He is a Member of the IEEE (Communications and Information Theory Societies). His current research interests include wireless communications, channel coding and signal processing.

Mark F. Flanagan (M’03 -SM’10) received the B.E. (1998) and Ph.D. (2005) degrees in electronic engineering from University College Dublin, Ireland. He worked as a project engineer with Parthus ˘ S1999. Technologies Ltd. in the period 1998âA ¸ Between 2006 and 2008 he held postdoctoral research fellowships at the University of Zurich, the University of Bologna, and the University of Edinburgh. In 2008 he was appointed as SFI Stokes Lecturer in Electronic Engineering at University College Dublin, where he is now a Senior Lecturer. His research interests are in the fields of information theory, wireless communications, and signal processing. In summer of 2014 he was a Visiting Senior Scientist at the Institute of Communications and Navigation of the German Aerospace Center under a DLR-DAAD fellowship. Dr. Flanagan is currently serving as an Editor for the IEEE COMMUNICATIONS LETTERS. He has served on the Technical Program Committee at several IEEE International Conferences. He is a Senior Member of the IEEE (Communications and Information Theory Societies).