Compress-and-Forward Strategy for The Relay Channel With Non ...

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

Compress-and-Forward Strategy for The Relay Channel With Non-Causal State Information Bahareh Akhbari, Mahtab Mirmohseni, Mohammad Reza Aref Information Systems and Security Lab (ISSL) Electrical Engineering Department, Sharif University of Technology, Tehran, Iran. Email: {b akhbari, mirmohseni}@ee.sharif.edu, [email protected] Abstract—In this paper, we consider a discrete memoryless state-dependent relay channel with non-causal Channel State Information (CSI). We investigate three different cases in which perfect channel states can be known non-causally: i) only to the source, ii) only to the relay or iii) both to the source and to the relay node. For these three cases we establish lower bounds on the channel capacity (achievable rates) based on using Gel’fandPinsker coding at the nodes where the CSI is available and using Compress-and-Forward (CF) strategy at the relay. Furthermore, for the general Gaussian relay channel with additive independent and identically distributed (i.i.d) states and noise, we obtain lower bounds on the capacity for the cases in which CSI is available at the source or at the relay. We also compare our derived bounds with the previously obtained results which were based on Decodeand-Forward (DF) strategy, and we show the cases in which our derived lower bounds outperform DF based bounds, and can achieve the rates close to the upper bound.

I. I NTRODUCTION In the last decade, state-dependent channels have attracted considerable attention [1]. In the case of non-causal Channel State Information at Transmitter (CSIT), Gel’fand and Pinsker [2] derived the capacity of state-dependent single-user channel by a coding scheme which is called Gel’fand-Pinsker (GP) coding. Non-causal CSIT can be considered in a context of coding for a computer memory with defective cells, where their locations are known a priori to the encoder [1]. Costa [3] showed that the capacity of power constrained Additive White Gaussian Noise (AWGN) channel with additive interference S, known as CSIT in the non-causal manner, equals to that of the same Gaussian channel without S (Costa’s coding is also called dirty paper coding (DPC)). Due to the use of multiuser models in different applications, analyzing these models with both causal and non-causal Channel State Information (CSI) have recently received high attention [1], [4]-[7]. In state-dependent multiuser models, in addition to broadcast [5], multiple-access [6] and cognitive interference channels with states [7], some results have also been obtained for the relay channels, recently [8]-[11]. In [10] it has been shown that for the degraded Gaussian Relay Channel (RC), similar to the Costa’s result in the single-user AWGN channel, additive Gaussian interference (S) added to both the direct and the relay path, does not decrease the capacity as long as the source and the relay have the same This work was partially supported by Iranian National Science Foundation (INSF) under contract No. 84,5193-2006 and by Iran Telecommunication Research Center (ITRC) under contract No. T500/20958.

978-1-4244-4313-0/09/$25.00 ©2009 IEEE

and full non-causal knowledge of the S. Based on Decode-andForward (DF) strategy ([12, Theorem 1]) for the relay channel, in [11] a lower bound on the capacity of T -node relay network ((T − 2) relays, one transmitter and one receiver) in the case where the source and the relays have non-causal and identical CSI, has been established. But, in many naturally interesting cases, only the source or the relay may be informed about CSI. In [13] and [14] in order to assume this asymmetry, the authors have considered that only the source or the relay has CSI in a non-causal manner. Therefore only the informed node can combine GP coding with DF strategy. In [13] and [14] lower and upper bounds on the capacity have been derived. Up to our best knowledge, in all the previous works on the state-dependent relay channel with non-causal CSI, only DF strategy (or Partial DF (PDF) in [13]) has been considered. Since in conventional DF (or PDF [15]), the relay decodes all (or part) of the message and then cooperate with the transmitter to send the decoded part, if for a scenario, the channel from the sender to the relay is worse than the direct channel, or even if these two links are similar in average, DF strategy cannot be beneficial and another approach is needed. So in a completely different approach (known as CF strategy [12, Theorem 6], [16]) the relay does not decode any part of the transmitted message, instead, it compresses the received message and forwards it to the receiver. In [17] it was shown that for the classic Gaussian relay channel, CF outperforms DF strategy for the mentioned scenario. In this paper we focus on using CF strategy in statedependent RC for two reasons: 1) We expect that in the state-dependent RC, similar to the classic RC, CF encoding scheme outperforms the DF strategy when the link between the source and the relay is worse than the direct link, and CF scheme achieves the rates close to the upper bound; 2) In the conventional DF (or PDF), the source must know the relay input in order to have cooperation with the relay. But when the CSI is asymmetric (e.g., informed relay only) the source cannot exploit the CSI. Hence, it does not know what the relay exactly sends and this makes some loss in the coherence gain that we expect to achieve in DF strategy, and as it has been shown in [14] although they used codeword splitting in their scheme, their DF based lower bound cannot be tight in the degraded Gaussian case. On the other hand, CF encoding is a completely different approach in which independent codebooks are used at the source and the relay.

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Thus, using CF approach in asymmetric scenario seems to be reasonable. In order to analyze this approach, we will consider three different situations in the non-causal case: Perfect CSI is known i) to the source only, ii) to the relay only and iii) both to the source and to the relay node. In each case, we derive the corresponding lower bound on the capacity (achievable rate), based on the CF strategy and GP coding at the informed node. Furthermore, we specialize our results to the Gaussian case with non-causal CSI and under power constraint. In order to compare our derived lower bounds with the established DF based bounds for state-dependent Gaussian relay channel, we consider those obtained in [13] and [14], and we show the cases in which CF approach outperforms DF strategy. In the Gaussian case similar to [14] we use Generalized Dirty Paper Coding (GDPC) that allows arbitrary correlation between the available CSI and the codeword at the informed encoder. This paper is organized as follows. In section II, we present the state-dependent relay channel and its related definitions. In section III, we investigate the non-causal case for three different situations and for each situation we establish the corresponding lower bound. Section IV specializes the results to the Gaussian case. The paper is concluded in section V. II. P RELIMINARIES AND DEFINITIONS To specify the state-dependent discrete memoryless relay channel, we define five finite sets: (X , X 1 , Y1 , Y, S). A probability transition matrix p(y, y 1 |x, x1 , s) is also defined for all (x, x1 , y1 , y, s) ∈ X × X1 × Y1 × Y × S. In this model the n-sequences X n and X1n are the source and the relay inputs, respectively. Y n and Y1n are the outputs of the receiver and the relay, respectively. We also assume that channel states S n are i.i.d and drawn from a given probability distribution p(s). The source wants to transmit a message W (in n uses of the channel) which is assumed to be uniformly distributed over the set W = {1, ..., M }. The rate R is defined as lognM bits per transmission. In this paper, random variables are denoted by upper case letters (e.g., X) which takes value on a set X ; Lower case letters indicate the realization of random variables (e.g., x) and pX (x) denotes the probability mass function (p.m.f.) of X on X , where occasionally subscript X is omitted. A n (X, Y ) denotes the set of -strongly, jointly typical n-length sequences based on p(x, y) which may be indicated as A n in the sequel, when it is obvious from the context. III. M AIN R ESULTS In this section, based on CF strategy we establish lower bounds on the capacity (achievable rate) of state-dependent RC in three different cases. The first two cases (Theorems 1 and 2) are related to the asymmetric situations in which noncausal CSI is only at the source or at the relay, respectively. The proofs are mainly based on combining the CF strategy and the GP coding at the informed encoder. Furthermore, in the Theorem 3 we investigate the case where CSI is available at both the source and the relay node. Moreover, as stated in [5], if the CSI is available at the receiver, it can be viewed as part of the channel output which highly simplifies the analysis. Hence, we do not investigate this case here.

A. Non-causal CSI at the source only Theorem 1: The capacity of the discrete memoryless statedependent relay channel with CSI available non-causally only at the source, is lower bounded by: C ≥ Rsource = sup I(U ; Yˆ1 , Y |X1 ) − I(U ; S) (1) in

s.t. I(X1 ; Y ) ≥ I(Yˆ1 ; Y1 |X1 , Y ),

(2)

where the supremum is taken over all joint p.m.f of the form: p(s)p(u|s)p(x|u, s)p(x1 )p(y, y1 |x, x1 , s)p(ˆ y1 |y1 , x1 ). (3) Remark 1: Note that, since the relay does not know the CSI, yˆ1 and s, given (y 1 ,x1 ) are independent. Comparing (1) and (2) with the rate of classic RC in [12, Theorem 6], we see that only the rate (1) has changed, while condition (2) is the same as the condition in that theorem. This happens because GP is used only at the source and not at the relay. Outline of the Proof : Our proof is based on random coding scheme which combines GP coding at the source where CSI is available, and CF strategy at the relay. Consider a block Markov encoding scheme where a sequence of B−1 messages are transmitted in B n-length blocks. Note that as B → ∞, for fixed n, the rate R(B − 1)/B is arbitrarily close to R. Random Coding: For any joint p.m.f defined in (3), gener ate 2n(R+R ) i.i.d un (w, m) sequences at the source, where  w ∈ [1, 2nR ] and m ∈ [1, 2nR ]. Generate2nR1 i.i.d xn1 (t) n sequences at the relay each with probability j=1 p(x1j ). Also note that in the CF strategy the codewords at the relay and the ˆ source are independent. For eachx n1 (t), generate 2 nR1 i.i.d n n y1j |x1j ). Randomly yˆ1 (z|t) sequences according to j=1 p(ˆ ˆ1 nR nR1 partition the set {1, ..., 2 } into 2 bins, defined as B(t) where t ∈ [1, 2nR1 ]. Encoding (at the beginning of block i): We assume that the channel state (S n ) in each block is non-causally known to the source node. 1) Let wi be the new message to be sent in block i. The source looks for the smallest m such that (u n (wi , m), sn (i)) ∈ An (U, S). Denote this m with mi . Based on the GP coding [2], there exists such an index m i , if n is large enough and R ≥ I(U ; S).

(4)

n

Then the source transmits i.i.d x (wi ) sequence, drawn with marginal p(x|u, s), given (u n (wi , mi ), sn (i)) in block i. 2) At the relay, assume that (ˆ y 1n (zi−1 |ti−1 ), y1n (i − n n ˆ 1), x1 (ti−1 )) ∈ A (Y1 , Y1 , X1 ) and assume that z i−1 ∈ B(ti ). So the relay transmits xn1 (ti ) in block i. Decoding (at the end of block i): The receiver at the end of block i will decode w i−1 . 1) The relay finds the unique index z such that (ˆ y1n (z|ti ), y1n (i), xn1 (ti )) ∈ An . There exists such an index z with arbitrarily high probability, if n is sufficiently large and ˆ 1 > I(Yˆ1 ; Y1 |X1 ). R (5) 2) The receiver finds the unique tˆi such that (xn1 (tˆi ), y n (i)) ∈ An . This step can be done with arbitrarily small probability of error (i.e., tˆi = ti ), if n is sufficiently large and

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R1 < I(X1 ; Y ).

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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

3) Having known t i−1 (from the previous block), the receiver calculates a set of indices z, denoted by list L(y n (i − 1)), such that (ˆ y1n (z|ti−1 ), xn1 (ti−1 ), y n (i − 1)) ∈ An . Then the receiver declares that zˆi−1 has been sent in block i − 1, if zˆi−1 ∈ B(ti ) ∩ L(y n (i − 1)). With arbitrarily high probability zˆi−1 = zi−1 , if n is sufficiently large and ˆ 1 < I(Y ; Yˆ1 |X1 ) + R1 . R yˆ1n (zi−1 |ti−1 )

(7) n

and y (i − 1) 4) Finally, the receiver uses both and declares that w ˆ i−1 and m ˆ i−1 have been sent in block i − 1 ˆ i−1 ), y n (i − if it is the unique message such that (u n (wˆi−1 , m n n n 1), yˆ1 (zi−1 |ti−1 ), x1 (ti−1 )) ∈ A . With arbitrarily high probˆ i−1 = mi−1 , if n is sufficiently ability w ˆi−1 = wi−1 and m large and R + R < I(U ; Y, Yˆ1 |X1 ). (8) Now (5), (6) and (7) result in (2). Combining (8) and (4) also yeilds (1). Thus the rate in (1) s.t. (2) is achievable. B. Non-causal CSI at the relay only Now, we consider the case where CSI is available only at the relay. Note that in conventional DF (or PDF) strategies, the source must know the relay input in order to cooperate with it. This allows the source and the relay to utilize a joint codebook to transmit cooperative information. However, when only the relay is informed of CSI, the states are unknown to the source, so the source and relay cannot cooperate. But, in CF relaying, codewords at the relay and the source are independent and the relay estimates some information from its received sequence and re-encodes it to send to the receiver. So, it seems that when CSI is only at relay, using CF strategy is a reasonable choice and can be helpful. The next theorem, states corresponding achievable rate, using GP coding and CF strategy. Theorem 2: The capacity of the discrete memoryless statedependent relay channel with CSI available non-causally only at the relay, is lower bounded by: relay C ≥ Rin = sup I(X; Yˆ1 , Y |U1 ) s.t. I(U1 ; Y ) − I(U1 ; S) ≥ I(Yˆ1 ; Y1 , S|U1 , Y ),

Outline of the Proof : A block Markov encoding scheme, similar to the proof of Theorem 1, is used. Random Coding: For any joint p.m.f defined in (11): GenernR i.i.d xn (w) sequences at the source, with probability ate n 2 n(R1 +R1 ) i.i.d un1 (t, m) sequences j=1 p(xj ). Generate 2  nR1 at the relay where t ∈ [1, 2 ] and m ∈ [1, 2nR1 ]. For ˆ 2 nR1 i.i.d yˆ1n (z|t) sequences each each un1 (t, m), generate n with probability j=1 p(ˆ y1j |u1j ). Randomly partition the set ˆ {1, ..., 2nR1 } into 2nR1 bins defined as B(t). Encoding (at the beginning of block i): Let w i be the new message to be sent from the source node in block i. We assume that channel state (S n ) in each block is non-causally known to the relay node. 1) The source simply sends x n (wi ). 2) At the relay, assume that (ˆ y 1n (zi−1 |ti−1 ), y1n (i − n n n 1), u1 (ti−1 , mi−1 ), s (i − 1)) ∈ A , and assume that z i−1 ∈ B(ti ). Knowing ti and sn (i), the relay searches for the smallest m such that (un1 (ti , m), sn (i)) ∈ An . We call this m as mi . Based on the GP coding, for sufficiently large n, there exists such an index m i , if R1 ≥ I(U1 ; S).

(12)

x n1 (ti )

sequence, drawn accordThen the relay transmits i.i.d ing to the marginal p(x 1 |u1 , s), given (un1 (ti , mi ), sn (i)) in block i. Decoding (at the end of block i): The receiver at the end of block i will decode w i−1 . 1) The relay finds a unique index z such that (ˆ y1n (z|ti ), y1n (i), un1 (ti , mi ), sn (i)) ∈ An . There exists such an index z with arbitrarily high probability, if n is sufficiently large and ˆ 1 > I(Yˆ1 ; Y1 , S|U1 ). R

(13)

(9)

2) At first, the receiver finds unique ( tˆi , m ˆ i ) such that ˆ i ), y n (i)) ∈ An . For sufficiently large n, ( tˆi , m ˆ i) = (un1 (tˆi , m (ti , mi ) with arbitrarily small probability of error if

(10)

R1 + R1 < I(U1 ; Y ).

where the supremum is taken over all joint p.m.f of the form: p(x)p(s)p(u1|s)p(x1 |u1 , s)p(y, y1 |x, x1 , s)p(ˆ y1 |y1 , u1 , s). (11)

Remark 2: Note that when only the relay node (middle terminal) knows the CSI, the relay tries to use its knowledge for two different goals: 1) to cancel the effect of channel’s state on its received signal (y 1 ), through acting as a decoder of the source-relay link; 2) to compress the CSI (along with its received signal, y 1 ) and send the result to the receiver and let it to utilize partial CSI, where needed. As a simple example, if y1 = ∅, the relay can compress only the CSI and send it to the receiver. So the receiver can utilize this partial CSI. For this case (Y1 = ∅), Theorem 2 confirms the conjecture in [18], by substituting Yˆ1 = V, U1 = ∅, S = T in (9), (10) and considering that the LHS of (10) is equal to R 0 , which is the rate of the fixed rate relay-receiver link. Hence, in general, in order to achieve the two above mentioned goals, we consider that in (11) yˆ1 is conditioned on s, besides u 1 ,y1 .

(14)

3) Having known t i−1 and mi−1 (from the previous block), the receiver calculates a set of indices z, denoted by list L(y n (i − 1)), such that (ˆ y1n (z|ti−1 ), y n (i − 1), un1 (ti−1 , mi−1 )) ∈ An . Then the receiver declares that zˆi−1 has been sent in block i − 1, if zˆi−1 ∈ B(ti ) ∩ L(y n (i − 1)). With arbitrarily high probability zˆi−1 = zi−1 , if n is sufficiently large and ˆ 1 < R1 + I(Yˆ1 ; Y |U1 ). R

(15)

4) Finally the receiver uses both yˆ1n (zi−1 |ti−1 ) and y n (i − 1), and declares that w ˆ i−1 has been sent in block i − 1, if it is the unique message such that (x n (wˆi−1 ), y n (i − 1), yˆ1n (zi−1 |ti−1 ), un1 (ti−1 , mi−1 )) ∈ An . Thus w ˆi−1 = wi−1 with arbitrarily high probability, if n is sufficiently large and R < I(X; Yˆ1 , Y |U1 ).

(16)

Thus (9) is resulted. Now combining (12)-(15) yields (10). We see that the rate in (9) s.t. (10) is achievable.

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C. Non-causal CSI at the source and the relay Theorem 3: The capacity of the discrete memoryless statedependent relay channel with CSI available non-causally at both the source and the relay is lower bounded by: source-relay C ≥ Rin = sup I(U ; Yˆ1 , Y |U1 ) − I(U ; S) (17) s.t. I(U1 ; Y ) − I(U1 ; S) ≥ I(Yˆ1 ; Y1 , S|U1 , Y ), (18)

where −1 ≤ ρ0s ≤ 1. Yˆ1 = βY1 +Zˆ where β is a constant, and quantization noise Zˆ is a zero-mean Gaussian random variable ˆ which is independent of S, X, X 1 , Z, Z1 . S with variance N is also independent of Z and Z 1 . Evaluation of (1) and (2) with the above specified parameters yields (21) and (22). Theorem 5: For the state-dependent general Gaussian relay channel with non-causal CSI at the relay, the rate 1 ρ1s ,α,β,γ 2

max

where the supremum is taken over all p.m.f of the form:

log2 (1 + P

2 1 ˆ [β 2 (a2 N+N1 )+N]A+P 1 QC(D−aβ( α −b)) ), ˆ (β 2 N1 +N)(G−P A)+D 2 P1 QCN

(23)

p(s)p(x, u|s)p(x1 , u1 |s)p(y, y1 |x, x1 , s)p(ˆ y1 |y1 , u1 , s). (19) Proof: The proof is similar to the proof of the previous theorems and is resulted by combining them. However, it is lengthy and is omitted here for brevity. IV. G AUSSIAN CASE In this section, we consider a general full-duplex Gaussian relay channel with i.i.d and additive states and noise. We consider the two following cases: Gaussian interference is known to 1) the source only, 2) the relay only. We use the results obtained in section III to provide lower bounds on the capacity in each case. The received signals at the relay and the receiver at time j = 1...n are given by: Y1j = aXj + Z1j + Sj and Yj = Xj + bX1j + Zj + Sj ,

(20)

where Xj and X1j are transmitted signals by the sender and the relay with individual average power constraints P and P 1 , respectively. Positive constants a and b are known and denote the channel gains. Z 1j and Zj are i.i.d and independent white Gaussian noise components at the relay and the receiver with powers N1 and N , respectively. The components of S are also i.i.d and zero-mean Gaussian with E[S j2 ] = Q. Theorem 4: For the state-dependent general Gaussian relay channel with non-causal CSI at the source, the rate max

ρ0s ,α,β

ˆ (b2 P1 + A) N 1 , log2 ˆ )[(α − 1)2 Q + BN ] + (aαβ − 1)2 N Q 2 (β 2 N1 + N C (21) β2 2 ˆ s.t. N = 2 [N1 A + N D + (a − 1) CQ], (22) b P1

is achievable, where A  P + Q + 2σ 0s + N , B  P + α2 Q + 2ασ0s , C  P (1 − ρ20s ), D  a2 P + Q + 2aσ0s , −1 ≤ ρ0s , α ≤ 1 and 0 ≤ β ≤ 1. Proof: Our proof is based on the evaluation of the achievable rate and its condition in (1) and (2) with an appropriate choice of input distribution. Note that for the CF encoding similar to the classic Gaussian RC [17], the best choice of probability distribution is not known. So we evaluate the related terms for a Gaussian distribution. Subsequently, X and X1 are assumed to be zero-mean Gaussian with variances P and P1 and E[XX1 ] = 0. Moreover, E[S 2 ] = Q, E[X1 S] = 0 (only the source is informed). Similar to Costa’s initial DPC [3], the auxiliary random variable U is defined as U = X+αS, but similar to [13], [14], arbitrary correlation is assumed between X and S which is called Generalized DPC (GDPC) √ to partially cancel the state. Hence, E[XS] = σ 0s = ρ0s P Q

s.t.

2

ˆ = β N1 G + E , N F −G

(24)

is achievable, where A  P 1 + α2 Q + 2ασ1s , B  (αb − 1)2 (1 −

ρ21s ), C  α2 (1−ρ21s ), D  γ+β , E  P1 QC[P (D−aβ( α1 −b))2 + α 2 2 2 D N ] + a β P N A, F  P1 (1 − ρ21s )(P + N + b2 P1 + Q + 2bσ1s ), G  P1 QB + (P + N )A and −1 ≤ ρ1s , α, γ ≤ 1.

Remark 3: Note that when there is no channel states (Q = 0), Theorems 4 and 5 reduce to the achievable rates derived based on the CF in [17] for the general Gaussian relay channel without states. Proof: The proof is similar to the proof of Theorem 4, except that the relay is informed in this case, so√E[XS] = 0, U1 = X1 + αS (DPC), E[X1 S] = σ1s = ρ1s P1 Q where −1 ≤ ρ1s ≤ 1 (GDPC) and Yˆ1 = βY1 +γS + Zˆ (CSI is known to the relay, and it can compress its knowledge and send it to receiver to partially cancel the CSI). β is an arbitrary number and we also let γ to be negative in order to consider the role of relay as a channel state canceler of the source-relay link. Therefore, we should maximize the rate in (23) subject to the condition (24), over the parameters −1 ≤ α,ρ 1s ,γ ≤ 1. Now we plot the derived achievable rates in (21) and (23), for some examples to compare with corresponding DF based lowerbounds and upper bounds (both trivial and non-trivial) for sate-dependent general Gaussian case derived in [13],[14]. In Fig. 1, the lower bound in (21), and for comparison the lower and upper bounds in [13] are plotted for a = 2, b = 1, P = Q = N = N1 = 10dB versus (P1 /N ). This figure shows that, although the source-relay link has a good condition (a = 2, b = 1), at high (P 1 /N ), CF-based lower bound outperforms DF. If we set a = b = 1 in the mentioned example, DF based lower bound performs worse than what has been shown in Fig. 1. Now in Fig. 2, we consider P a = b = 1, Q = N = N1 = 10dB and ( PN1 = N =SNR) varies. We can see that in this case, CF outperforms DF. However, if we increase b (improving multiple-access side of the RC) the CF based lower bound becomes very close to the upper bound at high SNR (e.g., for b = 4 the gap between the CF bound and upper bound is 0.0413 bit at 35dB, and for b = 8, gap=0.0111 bit at 30dB), and can achieve almost tight rates for the general Gaussian case. As illustrated in [13], at high SNR, PDF based lower bounds achieve the same rates as DF based bounds. So at high SNR our derived bounds for these cases outperform PDF based bounds, too. We also observe that our CF based lower bound is a decreasing function of Q, and for high Q, the CF based rate becomes very close to the upper bound. This

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CF lower bound DF lower bound [14] upper bound [14] trivial upper bound [14]

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Fig. 1. State-dependent general Gaussian relay channel with CSI only at the source, a = 2, b = 1, P = Q = N = N1 = 10dB.

Fig. 3. State-dependent general Gaussian relay channel with CSI only at the relay, a = b = 1, P = P1 = 20, Q = N1 = 10dB.

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ACKNOWLEDGMENT CF lower bound DF lower bound [13] upper bound [13] trivial upperbound [13]

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The authors wish to thank M. H. Yassaee for his comments. R EFERENCES

Note: In this case upperbound is very close to the trivial upper bound.

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Fig. 2. State-dependent general Gaussian relay channel with CSI only at the source, a = b = 1, Q = N = N1 = 10dB.

figure is not depicted due to space limitations. In Fig. 3 the lower bound in (23), and also lower and upper bounds in [14] are plotted for a = b = 1, P = P 1 = 20dB, Q = N1 = 10dB, as functions of SNR (P/N ). It is observed that for SNRs higher than 10dB, CF outperforms DF lower bound and can achieve the rates nearly close to the upper bounds. In this case γ is approximately (−β), which shows that the relay tries to achieve near optimality, through canceling the known state ˆ and and then compresses the results ( Yˆ1 = β(Y1 − S) + Z), re-encodes it to send to the receiver. V. C ONCLUSION We proposed a coding scheme for the state-dependent relay channel with non-casual CSI, based on combination of the CF strategy with GP coding at the node informed of CSI. We analyzed three different setups in which the perfect CSI is available non-causally only at the source, only at the relay or both at the source and the relay. In each case, we provided lower bounds by using this coding scheme. We further, illustrated our results via Gaussian examples and showed some cases for the general Gaussian relay channel with states, in which our derived lower bounds outperform the DF based lower bounds derived in [13] and [14], and can achieve the rates nearly close to the derived upper bounds in [13] and [14].

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