Capacity of the Gaussian Two-way Relay Channel to within 1/2 Bit
aa r X i v : . [ c s . I T ] F e b Capacity of the Gaussian Two-way Relay Channelto within Bit
Wooseok Nam,
Student Member, IEEE,
Sae-Young Chung,
Senior Member, IEEE, andYong H. Lee,
Senior Member, IEEE
School of EECS, KAIST,Daejeon, Republic of KoreaE-mail: [email protected], [email protected], [email protected]
Abstract
In this paper, a Gaussian two-way relay channel, where two source nodes exchange messages with each otherthrough a relay, is considered. We assume that all nodes operate in full-duplex mode and there is no direct channelbetween the source nodes. We propose an achievable scheme composed of nested lattice codes for the uplink andstructured binning for the downlink. We show that the scheme achieves within bit from the cut-set bound for allchannel parameters and becomes asymptotically optimal as the signal to noise ratios increase. Index Terms
Two-way relay channel, wireless networks, network coding, lattice codes
I. I
NTRODUCTION
We consider a two-way relay channel (TRC), as shown in Fig. 1 (a). Nodes 1 and 2 want to exchangemessages with each other, and a relay node facilitates the communication between them. This TRC canbe thought of as a basic building block of general wireless networks, along with the relay channel [1],the two-way channel [2], etc. Recently, there have been a great deal of interest in the capacity of wirelessnetworks. Inspired by network coding [3], TRC has been studied in the context of network coding forwireless networks due to its simple structure. However, the capacity region of the general TRC is stillunknown.In [4], several classical relaying strategies for the one-way relay channel [1], such as amplify-and-forward (AF), decode-and-forward (DF), and compress-and-forward (CF), were extended and applied tothe TRC. AF relaying is a very simple and practical strategy, but due to the noise amplification, it cannotbe optimal in throughput at low signal to noise ratios (SNRs). DF relaying requires the relay to decodeall the source messages and, thus, does not suffer from the noise amplification. In [5], it was shown thatthe achievable rate region of DF relaying can be improved by applying network coding to the decodedmessages at the relay. This scheme is sometimes optimal in its throughput [7], but it is generally subjectto the multiplexing loss [6].In general, in relay networks, the relay nodes need not reconstruct all the messages, but only need topass sufficient information to the destination nodes to do so. CF or partial DF relaying strategies for theTRC, in which the relay does not decode the source messages, were studied in [8], [9]. It was shownthat these strategies achieve the information theoretic cut-set bound [23] within a constant number of bitswhen applied to the Gaussian TRC. In [10], [11], structured schemes that use lattice codes were proposedfor the Gaussian TRC, and it was shown that these schemes can achieve the cut-set bound within bit.In this paper, we focus on the Gaussian TRC with full-duplex nodes and no direct communication linkbetween the source nodes. Such a Gaussian TRC is shown in Fig. 1 (b), and it is essentially the same as This work was supported by the IT R&D program of MKE/IITA. [2008-F-004-01, 5G mobile communication systems based on beamdivision multiple access and relays with group cooperation]
Encoder 1Decoder 1 Relay Encoder 2Decoder 2
Node 1 Node 2Node 1 Node 2Relay (a)(b)
Fig. 1. Gaussian two-way relay channel those considered in [8]-[11]. For the uplink, i.e., the channel from the source nodes to the relay, we proposea scheme based on nested lattice codes [19] formed from a lattice chain. This scheme is borrowed fromthe work on the relay networks with interference in [12], [13]. By using nested lattice codes for the uplink,we can exploit the structural gain of computation coding [15], which corresponds to a kind of combinedchannel and network coding. For the downlink, i.e., the channel from the relay to the destination nodes,we see the channel as a BC with receiver side information [7], [16], [17], since the receiver nodes knowtheir own transmitted messages. In such a channel, the capacity region can be achieved by the randombinning of messages [16]. In our strategy, a structural binning of messages, rather than the random one,is naturally introduced by the lattice codes used in the uplink. Thus, at each destination node, togetherwith the side information on its own message, this binning can be exploited for decoding.In fact, as stated above, our work is not the first to apply lattice codes to the Gaussian TRC. However,we assume more a general TRC model compared to the other works. In [11], it was assumed that thechannel is symmetric, i.e., all source and relay nodes have the same transmit powers and noise variances.In [10], a lattice scheme for the asymmetric Gaussian TRC was proposed. However, the scheme requiresthe existence of a certain class of lattices to achieve a bit gap to the cut-set bound. This paper extendsthose previous works and shows that we can in fact achieve the cut-set bound within bit for any channelparameters, e.g., transmit powers and noise variances. Moreover, the gap vanishes as the uplink SNRsincrease.This paper is organized as follows. In Section II, we present the channel model and define relatedparameters. The cut-set bound on the capacity region is given in Section III. Section IV illustrates ourachievable scheme and computes the achievable rate region. Section V concludes the paper.II. S YSTEM M ODEL
We consider a Gaussian two-way relay channel, as shown in Fig. 1 (b). We assume that the source andrelay nodes operate in full-duplex mode and there is no direct path between the two source nodes. Thevariables of the channel are as follows: To be exact, this implies bit per dimension. For a complex-valued system, as considered in [8], we have 1 bit gap per complexdimension. • W i ∈ (cid:8) , . . . , nR i (cid:9) : message of node i , • X i = h X (1) i , . . . , X ( n ) i i T : channel input of node i , • Y R = h Y (1) R , . . . , Y ( n ) R i T : channel output at the relay, • X R = h X (1) R , . . . , X ( n ) R i T : channel input of the relay, • Y i = h Y (1) i , . . . , Y ( n ) i i T : channel output at node i , • ˆ W i ∈ (cid:8) , . . . , nR i (cid:9) : estimated message of node i ,where i ∈ { , } , n is the number of channel uses, and R i denotes the rate of node i . We assume that themessages W and W are independent of each other. Node i transmits X ( t ) i at time t to the relay throughthe uplink channel specified by Y ( t ) R = X ( t )1 + X ( t )2 + Z ( t ) R ,where Z ( t ) R is an independent identically distributed (i.i.d.) Gaussian random variable with zero mean andvariance σ R . The transmit signal X ( t ) i is determined as a function of message W i and past channel outputs Y t − i = n Y (1) i , . . . , Y ( t − i o , i.e., X ( t ) i = f ( t ) i (cid:0) W i , Y t − i (cid:1) . There are power constraints P i , i ∈ { , } onthe transmitted signals n n X t =1 (cid:16) X ( t ) i (cid:17) ≤ P i , i = 1 , .At the same time, the relay transmits X ( t ) R to nodes 1 and 2 through the downlink channel specified by Y ( t ) i = X ( t ) R + Z ( t ) i , i ∈ { , } ,where Z ( t ) i is an i.i.d. Gaussian random variable with zero mean and variance σ i . The power constraintat the relay is given by n n X t =1 (cid:16) X ( t ) R (cid:17) ≤ P R .Since the relay has no messages of its own, X ( t ) R is formed as a function of past channel outputs Y t − R = { Y (1) R , . . . , Y ( t − R } , i.e., X ( t ) R = f ( t ) R (cid:0) Y t − R (cid:1) . At node , the message estimate ˆ W = g ( W , Y ) is computed from the received signal Y and its message W . The decoding of node is performedsimilarly. Now, the average probability of error is defined as P e = Pr n ˆ W = W or ˆ W = W o .For the aforementioned TRC, we say that a rate pair ( R , R ) is achievable if a sequence of encodingand decoding functions exists such that the error probability vanishes as n tends to infinity. The capacityregion of the TRC is defined as the convex closure of all achievable rate pairs.III. A N UPPER BOUND FOR THE CAPACITY REGION
By the cut-set bound [23], if the rate pair ( R , R ) is achievable for a general TRC, a joint probabilitydistribution p ( x , x , x R ) exists such that R ≤ min { I ( X ; Y R , Y | X R , X ) , I ( X , X R ; Y | X ) } , (1a) R ≤ min { I ( X ; Y R , Y | X R , X ) , I ( X , X R ; Y | X ) } . (1b) In particular, for the Gaussian TRC, we can use the fact that there is no direct path between nodes 1and 2, i.e., ( X , X , Y R ) → X R → ( Y , Y ) , and that X R → ( X , X ) → Y R . This induces another upperbound from (1), given by R ≤ min { I ( X ; Y R | X ) , I ( X R ; Y ) } , (2a) R ≤ min { I ( X ; Y R | X ) , I ( X R ; Y ) } , (2b)for some p ( x , x , x R ) . It can be easily seen that, for the Gaussian TRC with transmit power constraints,all terms under the minimizations in (2) are maximized by the product distribution p ( x , x , x R ) = p ( x ) p ( x ) p ( x R ) , where p ( x ) , p ( x ) , and p ( x R ) are Gaussian probability density functions with zeromeans and variances P , P , and P R , respectively. The resulting upper bound on the capacity region isgiven by R ≤ min (cid:26)
12 log (cid:18) P σ R (cid:19) ,
12 log (cid:18) P R σ (cid:19)(cid:27) , (3a) R ≤ min (cid:26)
12 log (cid:18) P σ R (cid:19) ,
12 log (cid:18) P R σ (cid:19)(cid:27) . (3b)IV. A N ACHIEVABLE RATE REGION FOR THE G AUSSIAN
TRCIn this section, we compute an achievable rate region for the Gaussian TRC. For the uplink, we considerusing nested lattice codes, which are formed from a lattice chain. For the downlink, we use a structuredbinning of messages at the relay, which is naturally introduced by the nested lattice codes. The destinationnodes decode each other’s message using this binning and the side information on their own transmittedmessages.The main result of this section is as follows:
Theorem 1:
For a Gaussian TRC, as shown in Fig. 1 (b), we can achieve the following region: R ≤ min ((cid:20)
12 log (cid:18) P P + P + P σ R (cid:19)(cid:21) + ,
12 log (cid:18) P R σ (cid:19)) , (4a) R ≤ min ((cid:20)
12 log (cid:18) P P + P + P σ R (cid:19)(cid:21) + ,
12 log (cid:18) P R σ (cid:19)) , (4b)where [ x ] + , max { x, } .Note that the achievable rate region in Theorem 1 is within bit of the upper bound (3), regardlessof channel parameters such as the transmit powers and noise variances. Moreover, as the uplink SNRs P σ R and P σ R increase, the gap vanishes and our achievable region asymptotically approaches the capacityregion of the Gaussian TRC.We prove Theorem 1 in the following subsections. A. Lattice scheme for the uplink
For the scheme for the uplink, we consider a lattice coding scheme. We will not cover the full detailsof lattices and lattice codes due to page limitations. For a comprehensive review, we refer the reader to[19]-[21] and the references therein.A nested lattice code is defined in terms of two n -dimensional lattices Λ nC and Λ n , which form a latticepartition Λ nC / Λ n , i.e., Λ n ⊆ Λ nC . The nested lattice code is a lattice code which uses Λ nC as codewordsand the Voronoi region of Λ n as a shaping region. For Λ nC / Λ n , we define the set of coset leaders as C = { Λ nC mod Λ n } , { Λ nC ∩ R} , O Fig. 2. Example of a lattice chain and sets of coset leaders. C ⊆ C ⊆ Λ C . where R is the Voronoi region of Λ . Then the coding rate of the nested lattice code is given by R = 1 n log |C| = 1 n log Vol(Λ n )Vol(Λ nC ) ,where Vol( · ) denotes the volume of the Voronoi region of a lattice. For the TRC, we should design twonested lattice codes, one for each source node. This subsection will show how the nested lattice codesare formed. In the following argument, we assume that P ≥ P without loss of generality. Now, let usfirst consider a theorem that is a key for our code construction. Theorem 2:
For any P ≥ P ≥ and γ ≥ , a sequence of n -dimensional lattice chains Λ n ⊆ Λ n ⊆ Λ nC exists that satisfies the following properties.a) Λ n and Λ n are simultaneously Rogers-good and Poltyrev-good while Λ nC is Poltyrev-good (for thenotion of goodness of lattices, see [20]).b) For any ǫ > , P i − ǫ ≤ σ (Λ ni ) ≤ P i , i ∈ { , } , for sufficiently large n , where σ ( · ) denotes thesecond moment per dimension associated with the Voronoi region of the lattice.c) The coding rate of the nested lattice code associated with the lattice partition Λ nC / Λ n can approach anyvalue as n tends to infinity, i.e., R = 1 n log |C | = 1 n log (cid:18) Vol (Λ n )Vol (Λ nC ) (cid:19) = γ + o n (1) , (5)where C = { Λ nC mod Λ n } and o n (1) → as n → ∞ . Furthermore, the coding rate of the nested latticecode associated with Λ nC / Λ n is given by R = 1 n log |C | = 1 n log (cid:18) Vol (Λ n )Vol (Λ nC ) (cid:19) = R + 12 log (cid:18) P P (cid:19) + o n (1) ,where C = { Λ nC mod Λ n } . Proof:
See the proof of Theorem 2 in [13].For instance, a lattice chain and the corresponding sets of coset leaders are visualized in Fig. 2 for thetwo-dimensional case.
Encoding
Let us think of a lattice chain (more precisely, a sequence of lattice chains) and sets of coset leadersas described in Theorem 2. We use C and C for nodes 1 and 2 respectively. For node i , the message set (cid:8) , . . . , nR i (cid:9) is one-to-one mapped to C i . Thus, to transmit a message, node i chooses W i ∈ C i associated with the message and sends X i = ( W i + U i ) mod Λ i ,where U i is a random dither vector with U i ∼ Unif( R i ) and R i denotes the Voronoi region of Λ i (wesuppressed the superscript ‘ n ’ for simplicity). The dither vectors U i , i ∈ { , } , are independent of eachother and also independent of the messages and the noise. We assume that each U i is known to the sourcenodes and the relay. Note that, due to the crypto-lemma [21], X i is uniformly distributed over R i andindependent of W i . Thus, the average transmit power of node i is equal to σ (Λ i ) , which approaches P i as n tends to infinity, and the power constraint is met. Decoding
The received vector at the relay is given by Y R = X + X + Z R ,where Z R = h Z (1) R , . . . , Z ( n ) R i T . Upon receiving Y R , the relay computes ˜ Y R = α Y R − X j =1 U j ! mod Λ = " X j =1 ( W j + U j ) mod Λ j − X j =1 X j + α X j =1 X j + α Z R − X j =1 U j mod Λ = (cid:16) T + ˜ Z R (cid:17) mod Λ ,where T = " X j =1 ( W j − Q j ( W j + U j )) mod Λ = [ W + W − Q ( W + U )] mod Λ , (6) ˜ Z R = − (1 − α )( X + X ) + α Z R , (7) α ∈ [0 , is a scaling factor, and Q j ( · ) denotes the nearest neighbor lattice quantizer associated with Λ j .If we let α be the minimum mean-square error (MMSE) coefficient α = P + P P + P + σ R ,the variance of the effective noise (7) satisfies n E (cid:26)(cid:13)(cid:13)(cid:13) ˜ Z R (cid:13)(cid:13)(cid:13) (cid:27) ≤ ( P + P ) σ R P + P + σ R .From the chain relation of lattices in Theorem 2, it follows that T ∈ C . Moreover, using the crypto-lemma,it is obvious that T is uniformly distributed over C and independent of ˜ Z R [13, Lemma 2].The relay attempts to recover T from ˜ Y R instead of recovering W and W separately. Thus, thelattice scheme inherits the idea of computation coding [15] and physical-layer network coding [18]. Also,by not requiring the relay to decode both messages, W and W , we can avoid the multiplexing loss[5] at the relay. The method of decoding we consider is Euclidean lattice decoding [19]-[22], which finds the closest point to ˜ Y R in Λ C . Thus, the estimate of T is given by ˆ T = Q C (cid:16) ˜ Y (cid:17) , where Q C ( · ) denotes the nearest neighbor lattice quantizer associated with Λ C . Then, from the lattice symmetry andthe independence between T and ˜ Z R , the probability of decoding error is given by p e = Pr n ˆ T = T o = Pr n ˜ Z R mod Λ / ∈ R C o , (8)where R C denotes the Voronoi region of Λ C . We then have the following theorem. Theorem 3:
Let R ∗ = (cid:20)
12 log (cid:18) P P + P + P σ R (cid:19)(cid:21) + .For any ¯ R < R ∗ and a lattice chain as described in Theorem 2 with R approaching ¯ R , i.e., R =¯ R + o n (1) , the error probability under Euclidean lattice decoding (8) is bounded by p e ≤ e − n “ E P “ R ∗ − ¯ R ” − o n (1) ” ,where E P ( · ) is the Poltyrev exponent [22]. Proof:
See the proof of Theorem 3 in [13].According to Theorem 3, the error probability vanishes as n → ∞ if ¯ R < R ∗ since E p ( x ) > for x > . This implies that the nested lattice code can have any rate below R ∗ for the reliable decoding of T . Thus, by c) of Theorem 2 and Theorem 3, the error probability at the relay vanishes as n → ∞ if R i < (cid:20)
12 log (cid:18) P i P + P + P i σ R (cid:19)(cid:21) + , i = 1 , . (9) B. Downlink phase
We first generate nR n -sequences with each element i.i.d. according to N (0 , P R ) . These sequencesform a codebook C R . We assume one-to-one correspondence between each t ∈ C and a codeword X R ∈ C R . To make this correspondence explicit, we use the notation X R ( t ) . After the relay decodes ˆ T , ittransmits X R ( ˆ T ) at the next block to nodes 1 and 2. We now assume that there is no error in the uplink,i.e., ˆ T = T . Under this condition, ˆ T is uniform over C , and, thus, X R ( ˆ T ) is also uniformly chosen from C R .Upon receiving Y = X R + Z , where Z = h Z (1)1 , . . . , Z ( n )1 i T , node 1 estimates the relay message ˆ T as ˆ T = t if a unique codeword exists in C R, such that ( X R ( t ) , Y ) are jointly typical, where C R, = { X R ( t ) : t = [ W + w − Q ( w + U )] mod Λ , w ∈ C } .Then, from the knowledge of W and ˆ T , node 1 estimates the message of node 2 as ˆ W = (cid:16) ˆ T − W (cid:17) mod Λ . (10)Given ˆ T = T , we have ˆ W = W if and only if ˆ T = ˆ T . Note that |C R, | = 2 nR . Thus, from theargument of random coding and jointly typical decoding [23], we have Pr n ˆ T = ˆ T | ˆ T = T o → (11)as n → ∞ if R <
12 log (cid:18) P R σ (cid:19) . (12) Similarly, at node 2, the relay message is estimated to be ˆ T = t by finding a unique codeword in C R, such that ( X R ( t ) , Y ) are jointly typical, where C R, = { X R ( t ) : t = [ w + W − Q ( W + U )] mod Λ , w ∈ C } .Then the message of node 1 is estimated as ˆ W = h ˆ T − W + Q ( W + U ) i mod Λ . (13)Since |C R, | = 2 nR , we have Pr n ˆ T = ˆ T | ˆ T = T o → (14)as n → ∞ if R <
12 log (cid:18) P R σ (cid:19) . (15)Note that, in the downlink, although the channel setting is broadcast, nodes 1 and 2 achieve theirpoint-to-point channel capacities (12) and (15) without being affected by each other. This is because ofthe side information on the transmitted message at each node and the binning of message. In our scheme,the relation in (6) represents how the message pair ( W , W ) is binned to T . C. Achievable rate region
Clearly, the message estimates (10) and (13) are exact if and only if ˆ T = ˆ T = T . Thus, the errorprobability is given by P e = Pr n ˆ T = T or ˆ T = T o ≤ Pr n ˆ T = ˆ T or ˆ T = ˆ T or ˆ T = T o ≤ Pr n ˆ T = T o + Pr n ˆ T = ˆ T | ˆ T = T o + Pr n ˆ T = ˆ T | ˆ T = T o (16)By Theorem 3, the first term of (16) vanishes as n → ∞ if R i < R ∗ i , i ∈ { , } . Also, by (11) and (14),the second and third terms also vanish as n → ∞ if (12) and (15) hold. Thus, the achievable rate region(4) follows from (9), (12), and (15). V. C ONCLUSION
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