Gaussian Relay Channel Capacity to Within a Fixed Number of Bits

Transcription

1 Gaussian Relay Channel Capacity to Within a Fixed Number of Bits Woohyuk Chang, Sae-Young Chung, and Yong H. Lee arxiv:.565v [cs.it] 23 Nov 2 Abstract In this paper, we show that the capacity of the threenode Gaussian relay channel can be achieved to within and 2 bit/sec/hz using compress-and-forward and amplify-and-forward relaying, respectively. I. INTRODUCTION Although both relay channels and interference channels are fundamental building blocks for constructing multiuser networks, their single-letter capacity characterization has been open for decades. In [], instead of struggling to find the exact capacity utilizing complicated achievable schemes, the authors show that a simple Han Kobayashi scheme can achieve the capacity of the two-user Gaussian interference channel to within bit/sec/hz for all values of the channel parameters. In [2], this approach is further developed and generalized. The authors first consider a deterministic channel model related to a given Gaussian channel and develop a scheme to achieve the capacity of such a deterministic channel. After getting an insight from the achievable scheme for the deterministic channel, they then develop a scheme to achieve the original Gaussian channel capacity to within a constant number of bits for all values of the channel parameters. This approach is called the deterministic approach. As an example, they showed that the decode-and-forward DF relaying scheme originally proposed by Cover and El Gamal in [3] achieves the capacity of the three-node relay channel to within bit/sec/hz and a simple partial DF relaying scheme achieves the capacity of the diamond Gaussian relay channel to within 2 bits/sec/hz. In this paper, we first show that the compress-and-forward CF relaying scheme by Cover and El Gamal in [3] can achieve the three-node Gaussian relay channel capacity to within bit/sec/hz for all values of the channel parameters. We also show a simple amplify-and-forward AF relaying scheme the relay amplifies and forwards the received signal only when the channel from the source to the relay is stronger than the channel from the source to the destination also achieves the Gaussian relay channel capacity to within 2 bits/sec/hz regardless of the channel parameters. The rest of the paper is organized as follows. In Section II, we introduce the Gaussian relay channel. Section III shows that the CF relaying scheme achieves the Gaussian relay channel capacity to within bit/sec/hz regardless of the channel parameters. In Section IV, the AF relaying scheme Woohyuk Chang is with the Center for High-Performance Integrated Systems, KAIST, Daejeon 35-7, Republic of Korea whchang@kaist.ac.kr. Sae-Young Chung and Yong H. Lee are with the Dept. of EE, KAIST, Daejeon 35-7, Republic of Korea {sychung, yohlee}@ee.kaist.ac.kr. is proposed and shown to achieve the capacity to within 2 bits/sec/hz. II. GAUSSIAN RELAY CHANNEL We consider a Gaussian relay channel in Fig.. For simplicity, we assume a full-duplex relay as in [2] [4]. The received signals at the relay and the destination are given by Y 2 = h 2 X +Z 2, Y 3 = h 3 X +h 32 X 2 +Z 3, 2 respectively, h 2, h 3, and h 32 are complex constants, Z 2 CN, and Z 3 CN, are noises at the relay and at the destination, respectively, that are independent of each other, E[ X 2 ] P, and E[ X 2 2 ] P 2. From [3], the upper bound on its channel capacity can be found as C + = max ρ min{ C + ρ,c+ 2 ρ} 3 C + ρ = log 2 + ρ 2 h h 3 2 P, 4 C 2 + ρ = log 2 + h 3 2 P + h 32 2 P 2 +2ρ h 3 2 h 32 2 P P 2, [ ] 5 E X X 2 ρ = E [ X 2] E[ X 2 2], 6 X 2 is the complex conjugate of X 2. Since C + ρ decreases while C 2 + ρ increases as ρ increases, and C+ C 2 +, there exist two possible cases for the optimalρ values depending on whether h 2 2 P h 32 2 P 2 or not as shown in Fig 2. If h 2 2 P h 32 2 P 2, C + ρ C+ 2 ρ for all ρ, and hence ρ = and C + = C +. If h 2 2 P > h 32 2 P 2, ρ is determined such that C + = C + ρ = C 2 + ρ. III. GAUSSIAN RELAY CHANNEL CAPACITY TO WITHIN ONE BIT: CF RELAYING SCHEME In [2], the authors introduce a deterministic relay channel model corresponding to the Gaussian relay channel in Section II as shown in Fig. 3. Each circle at the transmitter of each node represents a signal level and a binary digit can be put on each circle for transmission. n i represents the received signal-to-noise ratio SNR for path i in db scale. More

2 2 Z 2 R Y : X 2 2 R n 2 h 32 S h 2 Z 3 h 3 X Y 3 D S D n 3 n 32 Fig.. Gaussian relay channel. a b Fig. 3. Deterministic relay channel. C + ρ C 2 + ρ.5 ρ C + ρ C 2 + ρ.5 ρ Fig. 2. C + ρ and C+ 2 ρ: a h 2 2 P h 32 2 P 2, b h 2 2 P > h 2 32 P 2. specifically, n 3 = log 2 h 3 P, n 32 = log 2 h 32 P 2 and n 2 = log 2 h 2 P. Then, the transmitted bits at the first n i signal levels are received clearly through the path i at the corresponding receiver while the remaining bits at the other signal levels are not delivered to the receiver through the path i. This is motivated to mimic the AWGN channel since the effect of background Gaussian noise can be simplified such that the first n i bits including the most significant bit MSB are above noise level at the receiver while the remaining bits including the least significant bit LSB are below noise level. The capacity of this deterministic relay channel is then found in [2] as C d = min{max{n 2,n 3 },max{n 32,n 3 }} = n 3 +[min{n 2,n 32 } n 3 ] + 7 [x] + = max{x,}. 7 implies a capacity-achieving scheme such that first n 3 bits are directly delivered to the destination from the source while the remaining [min{n 2,n 3 } n 3 ] + bits are routed from the source to the destination through the relay. This motivates the authors in [2] to propose a DF-based relaying scheme such that it chooses one of two schemes depending on whether h 3 2 > h 2 2 or not as follows. If h 3 2 > h 2 2, the relay is ignored and the achievable rate is equal to log 2 + h 3 2 P. If h 3 2 h 2 2, the block-markov encoded DF scheme in [3] is used and hence its achievable rate is equal to min{log 2 + h 2 2 P,log 2 + h 3 2 P + h 32 2 P 2 }. Hence, the overall achievable rate is given by R DF = max { log 2 + h 3 2 P,min{log 2 + h 2 2 P, log 2 + h 3 2 P + h 32 2 P 2 } }, 8 and C + R DF is shown to be satisfied for all values of the channel parameters. In [4] [6], the achievable rate of the CF relaying scheme for the Gaussian relay channel in Section II is explicitly given by R CF = log 2 + h 3 2 h 2 2 h 32 2 P P 2 P + + h h 2 2 P + h P 2 9 Then, R CF in 9 can be rewritten as R CF = log 2 + h3 2 P +log 2 +δ min{ } h 2 2 P, h 32 2 P 2 δ = + h 3 2 P, max { h 2 2 P, h 32 2 P 2 } + h h 2 2 P + h 32 2 P 2 <, and δ as max { h 2 2 P, h 32 2 P 2 }. Interestingly, we can see that is very similar to 7 such that log 2 + h3 2 P bits are achieved from the direct path between the source and the destination while log 2 +δ min { h2 2 P, h 32 2 P 2 } / + h3 2 P bits are additionally achieved through the relaying path. This makes us conjecture that C + R CF is also satisfied for all values of the channel parameters. In this paper, we show that C + R CF is indeed satisfied for all values of the channel parameters. For simplicity, we define a h 3 2 P, b h 32 2 P 2, and c h 2 2 P. We first consider the case of c b and then the case of b < c. A. The case of c b h 2 2 P h 32 2 P 2 Since +a+b+c increases as b increases, R CF = log 2 +a+ +a+b+c c log 2 +a = log +a+2c 2 +a+2c.2

3 3 Defining C + R CF, we get C log 2 +a+2c + 2 = log 2 + log 2 +a+2c +a+2c = log 2 + log 22 =. 3 Hence, the CF relaying scheme achieves the Gaussian relay channel capacity to within one bit when c b. Moreover, when c = b, we get. On the other hand, when b only,, i.e., the capacity is asymptotically achieved. B. The case of b < c h 32 2 P 2 < h 2 2 P Since ρ in C + is determined to satisfy C + = C + ρ = C 2 + ρ, log 2 + ρ 2 = log 2 +a+b+2ρ ab ρ 2 +2 abρ +b c = a b+cc ab ρ =, 4 ab < a b+cc for b < c. Hence, C + = log 2 +a+b+ 2 aa b+c 2ab. 5 Then, +a+b+c +a+b+ 2 aa b+c 2ab = log 2 +a+a+b+c+ = log 2 + B + C, 6 A A = +a+a+b+c+, 7 B = ab a a 2 b+b 2 c ab 2, 8 C = 4+a+b+c 2 aa b+c. 9 Note that if B+ C A,. Since A B and C, showing A B 2 C is equivalent to showing B+ C A. After some manipulation, we obtain A B 2 C = 2 α +2α c+α 2 c 2 + α 3, 2 α = a 4 +4a 3 b++6a 2 b+ 2 +b ab 3 +b 2 +b+, 2 α = a+ 3 +a 2 +b a+b 2 b 3, 22 α 2 = a b 2 +2a+b+, 23 α 3 = 4ab+2a 2 b+2ab 2 +2a 2 b 2 +ab Since > and α 3, it is sufficient to show fc α +2α c+α 2 c 2 for all c. After some manipulation, we obtain the discriminant D of fc as D = α 2 α α 2 = 4ab { 2a 3 +3b+5a 2 +8b 2 +b+4a +b+ 2 3b+ }, 25 which implies that fc for all c since α 2 >. Hence, when c > b. Finally, we conclude the CF relaying scheme achieves the Gaussian relay channel capacity to within bit/sec/hz for all values of the channel parameters. Fig. 4 shows for various h 2 2 P values whenp = P 2, h 3 2 =. h 2 2 and h 32 2 =.5 h 2 2. This explains the gap is always less than one bit for the case of h 2 2 P h 32 2 P 2. Fig. 5 shows for various h 2 2 P values when P = P 2, h 3 2 =. h 2 2 and h 32 2 =.8 h 2 2. In this case of h 2 2 P > h 32 2 P 2, the gap becomes quite close to one as h 2 2 increases, but still less than one. IV. GAUSSIAN RELAY CHANNEL CAPACITY TO WITHIN TWO BITS: AF RELAYING SCHEME Although the DF and CF relaying schemes work well in the Gaussian relay channel, both of them need a smart relay that can decode or compress the received signal and reencode it. In this section, we propose a very simple AF-based relaying scheme for a dumb relay and show that it can achieve the Gaussian relay channel capacity to within 2 bits/sec/hz regardless of the channel parameters. We first find an explicit expression for the achievable rate of the AF relaying scheme. For simplicity, we use previously defineda,band c with θ a h 3, θ b h 32 and θ c h 2. After the relay amplifies the received signaly 2,i at time i and forwards it to the destination at time i as X 2,i = = P2 c+ Y 2,i P2 c+ c P e jθc X,i +Z 2,i, 26 the destination receives Y 3,i at time i as a b Y 3,i = e jθa X,i + e jθ b X 2,i P P 2 = ae X jθa,i + P c+ ejθ b+θ X c,i P b + c+ ejθ b Z 2,i +Z 3,i, 27 we assume E[ X,i 2 ] = P and E[ X 2,i 2 ] = P 2. For simplicity, we normalize noise power by dividing Y 3,i by b c+ + as Y 3,i Ỹ 3,i = H k X,i k + Z 3,i, 28 b c+ + k=

4 4 [H H ] [ ac+ b+c+ ejθa ] b+c+ ejθ b+θ c,29 X,i X,i, 3 P b Z 3,i b c+ ejθ b Z 2,i +Z 3,i c+ + CN,. 3 Hence, the AF relying scheme turns the channel from the source to the destination into a unit-memory intersymbol interference channel [4], [6]. Following [7], [8], the achievable rate for the AF relaying scheme is then written by 2π max Σw Σwdw 2π log 2 +Σw Hw 2 dw 32 Hw is the Fourier transform of H k given by ac+ Hw = + b+c+ ejθa b+c+ ejθ b+θ c w,33 Σwdw is the normalized power constraint. and 2π Although the optimal power allocation Σ w is the wellknown water-filling [6] [8], we here assume uniform power allocation as Σw = for all w. Using lnµ+ν cosx+ydx = 2πln µ+ µ 2 ν 2,34 2 from π lnµ+ν cosxdx = πln µ+ µ 2 ν 2 2 for µ ν > in [9, p.526], we obtain R AF = log 2π 2 + ac+ b+c+ + b+c+ + 2 ac+ b+c+ cosw+θ a θ b θ c dw K + L = log 2, 35 +b+c K = +a+b+c+a+, 36 L = +c { +a+b 2 + a b 2 +2a+b+ c }. 37 Then, R AF in 35 can be rewritten as R AF = +log 2 + h3 2 P +log 2 + h 32 2 P 2 h2 2 h 3 2 P + L + h 3 2 P + h 32 2 P 2 + h 2 2. P 38 Interestingly, we can see that as long as h 2 2 > h 3 2, 38 is similar to C d in 7 and R CF in such that log 2 + h 3 2 P bits are directly delivered from the source to the destination while the remaining bits are additionally delivered through the relaying path. This makes us to conjecture that C + R AF 2 is satisfied for h 2 2 > h 3 2 we use two bits instead of one bit since there is a penalty of bit in 38. For h 2 2 h 3 2, the capacity can be achieved within one bit by simply ignoring the relay as in [2]. In this case, if the relay is active, then the signal-to-noise ratio SNR at the destination becomes worse than that for the inactive relay since too much noise is amplified at the relay and forwarded to the destination. Finally, we can expect the capacity is achieved within 2 bits/sec/hz regardless of the channel parameters. Getting an insight from the above argument, we propose an AF-based relaying scheme as follows. If h 3 2 > h 2 2, the relay is ignored and the achievable rate is equal to log 2 + h 3 2 P. In this case, it is easily shown that C + log 2 +a as in [2] by C + C + = log 2+ log 2 +a+.39 If h 3 2 h 2 2, the relay amplifiesy 2,i and forwards it to the destination as P 2 X 2,i = h 2 2 P + Y 2,i, 4 we assume E[ X,i 2 ] = P and E[ X 2,i 2 ] = P 2. From now on, we show that C + R AF 2 in the case of a c. Especially, we first consider the case of a c b, and then the case of a c and b < c. A. The case of a c b h 3 2 P h 2 2 P h 32 2 P 2 Defining 2 C + R AF, we get 2 = +log 2 M K + L, 4 M = ++b+c. 42 M K+ L Since showing log 2 is equivalent to showing 2K + L M, we first consider 2 K + L M 2K +cb a M = a+3cb+ c+, 43 and let gb a + 3cb + c + a + c. From a+3c, it is notable that if gc, then gb for all b c. Since gc = ++2cc a >, for all a c b. It is also notable when a = b = c, we get 2 2. B. The case of a c and b < c h 3 2 P h 2 2 P and h 32 2 P 2 < h 2 2 P From 5 and 35 37, we get P + Q 2 = +log 2 K +, 45 L

5 5 Since P = +b+c+a+b 2ab, 46 Q = 4+b+c 2 aa b+c. 47 2K P = a 2 b+c+a+b+c 2 P+ Q +c+c bb+c, 48 showing log 2 K+ is equivalent to showing L 2 Q 2K+ L P. To do it, we first consider the case of a b < c and then the case of b < a c. The case of a b < c: After some manipulation, we obtain 2K + L P 2 Q 2K +cb a P 2 Q = β +β c+β 2 c 2 +β 3 c 3, 49 β = a a 2 5 2b+b 2 ++b 2 5+2b+b 2 +2a5+5b b 2 b 3, 5 β = 5+8b+2b 2 +b 4 +2a 3 3+b+a 2 2+2b b 2 +4a5+8b+5b 2 +2b 3, 5 β 2 = +a 3 +a 2 2 8b+22b+6b 2 6b 3 +a2+4b+3b 2, 52 β 3 = 5+6a+a b 2 +4b+8bb a. 53 Since sc β +β c+β 2 c 2 +β 3 c 3 is a cubic function with β 3 >, it is notable that if sb, s b and s b, then sc for all c b. From b a, we get sb = a 3 5+4b+a 2 +3b+2b 2 +b5+8b+29b 2 +2b 3 +a5+32b+63b 2 +44b 3 +4ab 2 a b 2 +4b 3 b 2 +2ab 3a 2, 54 s b = 5+6a 3 +28b+6b 2 +54b 3 +a b +a2+74b+66b 2 +4aba b 2 +6b 2 b 2 a 2 >, 55 s b = 2+24a 2 +74b+96b 2 +a42+64b +2aa b 2 +6b7b 2 6ab a 2 >. 56 Hence, we have 2 2 for all c > b a. 2 The case of b < a c: Similarly, we obtain 2K + L P 2 Q 2K +ca b P 2 Q = γ +γ c+γ 2 c 2 +γ 3 c 3, 57 γ = a a 2 5 2b+b 2 ++b 2 5+2b+b 2 +2a5+5b b 2 b 3, 58 γ = 5+a 3 4 6b+8b+2b 2 +b 4 +4a5+6b+b 2 +a 2 29+b+5b 2, 59 γ 2 = 28a 2 +9a 3 +a29 2b b b+3b 2 +b 3, 6 γ 3 = 5+4a+a b 2 +6b+8aa b. 6 From a b, tc γ +γ c+γ 2 c 2 +γ 3 c 3 is a cubic function with γ 3 >. Since ta = 2a5+28a 3 +b+6b 2 +2b 3 +a b +4a5+6b+b 2 +2aa 2 b a 4 a b, 62 t a = 5+2a 3 +8b+2b 2 +6a 2 7+4b +4a+3b+4b 2 +2b 2 a b 2 +37a 4 36a 3 b b 4 >, 63 t a = a 2 +7b+3b 2 +a22+8b +4ba b 2 +8a9a 2 8ab b 2 >, 64 tc for all c a. Hence, 2 2 for all c a > b. Finally, we conclude the proposed AF relaying scheme achieves the Gaussian relay channel capacity to within 2 bits/sec/hz regardless of the channel parameters. Moreover, when a = b and c, we have 2 2. Fig. 6 shows 2 for various h 2 2 P values whenp = P 2, h 3 2 =. h 2 2 and h 32 2 =.5 h 2 2. This explains the gap 2 is always less than two bits for the case of h 3 2 P h 2 2 P h 32 2 P 2. Fig. 7 shows 2 for various h 2 2 P values when P = P 2, h 3 2 =.48 h 2 2 and h 32 2 =.5 h 2 2. In this case of h 3 2 P h 2 2 P and h 32 2 P 2 < h 2 2 P, since h 3 2 P is quite close to h 32 P 2, the gap 2 also becomes quite close to two as h 2 2 increases, but still less than two. REFERENCES [] R. H. Etkin, D. N. Tse, and H. Wang, Gaussian interference channel capacity to within one bit: the general case, in Proc. IEEE Int. Symp. Inf. Theory ISIT, Nice, France, June 27. [2] A. S. Avestimehr, S. N. Diggavi, and D. N. C. Tse, A deterministic approach to wireless relay networks, in Proc. Allerton Conf. Commun., Contr., Comput., Monticello, USA, Sept. 27. [3] T. M. Cover and A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp , Sept [4] G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Trans. Inf. Theory, vol. 5, no. 9, pp , Sept. 25. [5] A. Host-Madsen and J. Zhang, Capacity bounds and power allocation for wireless relay channels, IEEE Trans. Inf. Theory, vol. 5, no. 6, pp , June 25. [6] G. Kramer, I. Marić, and R. D. Yates, Cooperative Communications. Hanover, MA, USA: now Publishers Inc., 27. [7] W. Hirt and J. L. Massey, Capacity of the discrete-time Gaussian channel with intersymbol interference, IEEE Trans. Inf. Theory, vol. 34, no. 3, pp , May 988. [8] R. S. Cheng and S. Verdú, Gaussian multiaccess channels with ISI: Capacity region and multiuser water-filling, IEEE Trans. Inf. Theory, vol. 39, no. 3, pp , May 993. [9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA, USA: Academic Press, 2.

6 h 2 P [db] 2 Fig. 4. for various h 2 2 P values with P = P 2, h 3 2 =. h 2 2 and h 32 2 =.5 h h 2 2 P [db] h 2 2 P [db] Fig. 5. for various h 2 2 P values with P = P 2, h 3 2 =. h 2 2 and h 32 2 =.8 h 2 2. Fig for various h 2 2 P values with P = P 2, h 3 2 =.48 h 2 2 and h 32 2 =.5 h h 2 2 P [db] Fig for various h 2 2 P values with P = P 2, h 3 2 =. h 2 2 and h 32 2 =.5 h 2 2.