Multi-Layer Transmission and Hybrid Relaying for Relay Channels with Multiple Out-of-Band Relays
aa r X i v : . [ c s . I T ] A p r Multi-Layer Transmission and Hybrid Relayingfor Relay Channels withMultiple Out-of-Band Relays
Seok-Hwan Park, Osvaldo Simeone, Onur Sahin and Shlomo Shamai (Shitz)
Abstract
In this work, a relay channel is studied in which a source encoder communicates with a destinationdecoder through a number of out-of-band relays that are connected to the decoder through capacity-constrained digital backhaul links. This model is motivated by the uplink of cloud radio access networks.In this scenario, a novel transmission and relaying strategies are proposed in which multi-layer transmis-sion is used, on the one hand, to adaptively leverage the different decoding capabilities of the relays and,on the other hand, to enable hybrid decode-and-forward (DF) and compress-and-forward (CF) relaying.The hybrid relaying strategy allows each relay to forward part of the decoded messages and a compressedversion of the received signal to the decoder. The problem of optimizing the power allocation across thelayers and the compression test channels is formulated. Albeit non-convex, the derived problem is foundto belong to the class of so called complementary geometric programs (CGPs). Using this observation,an iterative algorithm based on the homotopy method is proposed that achieves a stationary point of theoriginal problem by solving a sequence of geometric programming (GP), and thus convex, problems.Numerical results are provided that show the effectiveness of the proposed multi-layer hybrid schemein achieving performance close to a theoretical (cutset) upper bound.
S.-H. Park and O. Simeone are with the Center for Wireless Communications and Signal Processing Research (CWC-SPR), ECE Department, New Jersey Institute of Technology (NJIT), Newark, NJ 07102, USA (email: {seok-hwan.park,osvaldo.simeone}@njit.edu).O. Sahin is with InterDigital Inc., Melville, New York, 11747, USA (email: [email protected]).S. Shamai (Shitz) is with the Department of Electrical Engineering, Technion, Haifa, 32000, Israel (email:[email protected]).
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Index Terms
Relay channel, multi-layer transmission, hybrid relaying, out-of-band relaying, cloud radio accessnetworks.
I. I
NTRODUCTION
The multiple relay network, in which a source encoder wishes to communicate with a destina-tion through a number of relays, as seen in Fig. 1, has been actively studied due to its wide rangeof applications. Most of the activity, starting from [1], focuses on Gaussian networks in whichthe first hop amounts to a Gaussian broadcast channel from source to relays and the secondhop to a multiple access channel between relays and receivers. The literature on this subject isvast and includes the proposal of various transmission strategies, including decode-and-forward (DF) [1]-[3], compress-and-forward (CF) [1]-[9], amplify-and-forward (AF) [2][3][8] and hybridAF-DF [2][8].In this paper, we are concerned with a variation of the more classical multi-relay channeldiscussed above in which the relays are connected to the destination through digital backhaul linksof finite-capacity. The motivation for this model comes from the application to so called cloudradio cellular networks, in which the base stations (BSs) act as relays connected to the centraldecoder via finite-capacity backhaul links [10][11]. This model was studied in [4]-[7][9][12] (seealso review in [13]). References [4][6][7][9] focus on CF strategies, while [5] considers hybridDF-CF strategies and [12] studies schemes based on compute-and-forward . A. Contributions
In this paper, we propose a novel transmission and relaying strategy in which multi-layertransmission is used, on the one hand, in order to properly leverage the different decodingcapabilities of the relays similar to [2], and, on the other hand, to enable hybrid DF and CFrelaying. In the proposed hybrid relaying strategy, each relay forwards part of the decodedmessages and a compressed version of the received signal. The multi-layer strategy is designedso as to facilitate decoding at the destination based on the information received from the relays.To this end, the proposed design is different from the classical broadcast coding approach of[14] in which each layer encodes an independent message. Instead, in the proposed scheme,
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Figure 1. Illustration of the considered channel with multiple relays connected to the decoder via out-of-band digital backhaullinks with given capacities. each layer encodes an appropriately selected set of independent messages. It is emphasized thatthe hybrid DF-CF approach studied in [5] is based on single-layer transmission.The problem of optimizing the power allocation across the layers and the compression testchannels is formulated. Albeit non-convex, the derived problem is found to belong to the class ofso called complementary geometric programs (CGPs) (see [15, Sec. 3.2] for more detail). Usingthis observation, an iterative algorithm based on the homotopy method is proposed that achievesa stationary point of the original problem by solving a sequence of geometric programming (GP)[16], and thus convex, problems. Numerical results are provided that show the effectiveness ofthe proposed multi-layer hybrid scheme in achieving performance close to a theoretical cutsetupper bound [17, Theorem 1].
Notation : We use p ( y | x ) to denote conditional probability density function (pdf) of ran-dom variable X given Y . All logarithms are in base two unless specified. Given a sequence X , . . . , X m , we define a set X S = { X j | j ∈ S} for a subset S ⊆ { , . . . , m } ; we set X φ as theempty set. II. S YSTEM M ODEL
We consider a relay channel in which a source encoder wishes to communicate with adestination decoder through a number M of relays as illustrated in Fig. 1. We denote the set ofrelays by M = { , . . . , M } . The relays operate out of band in the sense that each i th relay is February 16, 2018 DRAFT connected to the receiver via an orthogonal finite-capacity link of capacity C i in bits per channeluse (c.u.). The encoder transmits a signal X which is subject to power constraint E [ | X | ] ≤ P .Each relay i receives a signal Y i which is given as Y i = h i X + Z i (1)with a complex channel coefficient h i = √ g i e jθ i and independent additive white Gaussian noise(AWGN) Z i ∼ CN (0 , for i = 1 , . . . , M . We assume that the channel coefficients h , . . . , h M are constant over a transmission block and are perfectly known to all nodes. Without loss ofgenerality, the channel powers g , . . . , g M are assumed to be sorted such that g ≤ . . . ≤ g M . (2)III. M ULTI -L AYER T RANSMISSION WITH H YBRID R ELAYING
In this section, we propose a transmission strategy that is based on multi-layer transmission andhybrid relaying. Hybrid relaying is performed by having each relay forward part of the decodedmessages, which amounts to partial decode-and-forward (DF), along with a compressed versionof the received signal, thus adhering also to the compress-and-forward (CF) paradigm. The multi-layer strategy used at the source is designed so as to facilitate decoding at the destination basedon the information received from the relays, as detailed below.
A. Multi-Layer Transmission
The amount of information decodable at the relays depends on the generally different fadingpowers g , . . . , g M . To leverage the different channel qualities, we enable flexible decodingat the relays by adopting a multi-layer transmission strategy at the encoder. This approachwas also considered in [2] for the case of two relays that communicate to the decoder viamultiple access Gaussian channels. We assume that the transmitter splits its message into M + 1 independent submessages, say W , . . . , W M +1 , with corresponding rates R , . . . , R M +1 in bit/c.u.,respectively. The idea is that message W will be decoded by all relays, message W onlyby relays , . . . , M , and so on. This way, relays with better channel conditions decode moreinformation. Message W M +1 is instead decoded only at the destination.To encode these messages, the encoded signal is given by X = M +1 X k =1 p P k X k , (3) February 16, 2018 DRAFT where the signals X , . . . , X M +1 are independent and distributed as CN (0 , , and the powercoefficients P , . . . , P M +1 are subject to the power constraint P M +1 k =1 P k ≤ P . The signal X encodes message W , signal X encodes both message W and W , and so on, so that signal X k encodes messages W , . . . , W k for k = 1 , . . . , M . Note that, unlike classical multi-layertransmission [14][18], here signal X k does not only encode message W k . The reason for thischoice will be clarified below. Finally, signal X M +1 encodes message W M +1 .Relay 1 decodes message W from X ; relay 2 first decodes message W from X and thenmessage W from X using its knowledge of W ; and so on, so that relay k decodes messages W , . . . , W k for k = 1 , . . . , M . From standard information-theoretic considerations, the followingconditions are sufficient to guarantee that rates R k are decodable by the relays [14] R k ≤ I ( X k ; Y k | X , . . . , X k − ) , (4)for k = 1 , . . . , M . This is because, by (3), condition (4) with k = 1 , namely R ≤ I ( X ; Y ) ensures that not only relay 1 but all relays can decode message W ; and, generalizing, theinequality (4) for a given k guarantees that not only relay k can decode message W k afterhaving decoded W , . . . , W k − , but also all relays k + 1 , . . . , M can. The signal X M +1 , and thusmessage W M +1 is decoded by the destination only as it will be described in the next subsection. B. Hybrid Relaying
As discussed, relay i decodes messages W , . . . , W i . Then, each i th relay transmits partial information about the decoded messages to the destination via the backhaul links. The rate atwhich this partial information is transmitted to the destination is selected so as to enable thelatter to decode messages W , . . . , W M jointly based on all the signals received from the relays.This step will be detailed below. We denote as C DF i ≤ C i the portion of the backhaul capacitydevoted to the transmission of the messages decoded by relay i .Beside the rate allocated to the transmission of (part of) the decoded messages, relay i utilizes the residual backhaul link to send a compressed version ˆ Y i of the received signal Y i . The compression strategy at relay i is characterized by the test channel p (ˆ y i | y i ) accordingto conventional rate-distortion theory arguments (see, e.g., [19]). Moreover, since the receivedsignals at different relays are correlated with each other, it is beneficial to adopt a distributedsource coding strategy. Here, similar to [7][9][20], we use successive decoding via Wyner-Ziv February 16, 2018 DRAFT compression with a given order ˆ Y π (1) → . . . → ˆ Y π ( M ) , where π ( i ) is a given permutation of therelays’ indices M . Thus, the decoder can successfully retrieve the descriptions ˆ Y , . . . , ˆ Y M ifthe conditions [21] I (cid:16) Y π ( i ) ; ˆ Y π ( i ) | ˆ Y { π (1) ,...,π ( i − } (cid:17) ≤ C CF π ( i ) (5)are satisfied for all i = 1 , . . . , M , where we defined C CF i ≤ C i as the capacity allocated by relay i to communicate the compressed received signal ˆ Y i to the decoder. It is recalled that (5) is therate needed to compress Y π ( i ) as ˆ Y π ( i ) given that the destination has side information given bythe previously decompressed signals ˆ Y π (1) , . . . , ˆ Y π ( i − .Without claim of optimality, we assume Gaussian test channel p (ˆ y i | y i ) , so that the compressedsignal ˆ Y i can be expressed as ˆ Y i = Y i + Q i , (6)where the compression noise Q i ∼ CN (0 , σ i ) is independent of the received signal Y i to becompressed. We observe that assumption of the Gaussian test channels (6) does not involve anyloss of optimality if the relays are allowed to perform only the CF strategy [6][22][23]. Weremark that the compression strategy (6) at relay i is characterized by a single parameter σ i . C. Decoding
The destination decoder is assumed to first recover the descriptions ˆ Y , . . . , ˆ Y M from the signalsreceived by the relays. This step is successful as long as conditions (5) are satisfied. Havingobtained ˆ Y M = { ˆ Y , . . . , ˆ Y M } , the destination decodes jointly the messages W , . . . , W M basedon the partial information about these messages received from the relays and on the compressedreceived signals ˆ Y M . Finally, message W M +1 is decoded. The following lemma describes theset of tuples ( R , . . . , R M +1 ) that is achievable via this strategy. Lemma 1.
A rate tuple ( R , . . . , R M +1 ) is achievable by the proposed multi-layer strategywith hybrid relaying if the following conditions are satisfied for some values of C DF i ∈ [0 , C i ] , February 16, 2018 DRAFT i = 1 , . . . , M : R i ≤ I ( X i ; Y i | X , . . . , X i − ) , i = 1 , . . . , M, (7) C DF π ( i ) + I (cid:16) Y π ( i ) ; ˆ Y π ( i ) | ˆ Y { π (1) ,...,π ( i − } (cid:17) ≤ C π ( i ) , i = 1 , . . . , M, (8) M X j = k R j ≤ M X j = k C DF j + I (cid:16) X { k,...,M } ; ˆ Y M | X { ,...,k − } (cid:17) , k = 1 , . . . , M, (9) and R M +1 ≤ I (cid:16) X M +1 ; ˆ Y M | X M (cid:17) . (10) Proof:
The constraint (7) corresponds to (4) and guarantees correct decoding at the relays.Constraint (8) follows from (5) and the backhaul constraint. The inequalities in (9) ensure that themessages W , . . . , W M are correctly decoded by the destination based on the partial informationreceived from the relays and the compressed signals ˆ Y M . This is a consequence of well-knownresults on the capacity of multiple access channels with transmitters encoding given subsets ofmessages [24] (see also [25]), as recalled in Appendix A. We observe here that the sufficiencyof (9) for correct decoding hinges on the fact that signal X k encodes messages W , . . . , W k for k = 1 , . . . , M , and not merely W k as in the more conventional multi-layer approach [18][14].Finally, constraint (10) ensures the correct decoding of message W M +1 based on all the decodedsignals X M and the compressed received signals ˆ Y M .IV. O PTIMIZATION
In this section, we are interested in optimizing the power allocation P , . . . , P M +1 , the com-pression test channels characterized by the compression noise variances σ , . . . , σ M and thebackhaul capacity allocation between DF and CF relaying, with the aim of maximizing thesum-rate R sum = P M +1 k =1 R k . Based on Lemma 1, this problem is formulated as maximize π, { P k , R k ≥ } M +1 k =1 , { σ i , C DF i ≥ } Mi =1 M +1 X k =1 R k (11) s . t . (7) − (10) , M +1 X k =1 P k ≤ P. February 16, 2018 DRAFT
In (11), the optimization space includes the ordering π used for decompression at the decoder,along with the mentioned power and backhaul allocations and the compression noises. Due tothe inclusion of the ordering π , the problem is combinatorial. Therefore, in this section, wefocus on the optimization of the other variables for fixed ordering π . Optimization of π willthen have to be generally performed using an exhaustive search procedure or using a suitableheuristic method.Under the assumption of the multi-layer transmission (3), the Gaussian test channels (6) andgiven ordering π , the problem (11) can be written as maximize { R i , C DF i ≥ , β i ∈ [0 , } Mi =1 , { P i ≥ } M +1 i =1 M X k =1 R k + log (cid:0) P M +1 ¯ β M (cid:1) (12a) s . t . R i ≤ log (cid:18) g i ¯ P i g i ¯ P i +1 (cid:19) , i = 1 , . . . , M, (12b) C DF i + log (cid:18) P ¯ β π − ( i ) P ¯ β π − ( i ) − (cid:19) − log (1 − β i ) ≤ C i , i = 1 , . . . , M, (12c) M X j = k R j ≤ M X j = k C DF j + log (cid:18) P k ¯ β M P M +1 ¯ β M (cid:19) , k = 1 , . . . , M, (12d) ¯ P ≤ P, (12e)where we have defined variables β i = 1 / (1 + σ i ) ∈ [0 , for i = 1 , . . . , M , the cumulativepowers ¯ P k = P M +1 j = k P j for k = 1 , . . . , M + 1 , the cumulative variables ¯ β i = P ij =1 g π ( j ) β π ( j ) for i = 1 , . . . , M and the function π − ( j ) returns the position of the index j ∈ { , . . . , M } in theordering π . The problem (12) is not easy to solve due to the non-convexity of the constraints(12b)-(12d). In Sec. IV-A, we propose an iterative algorithm to find a stationary point of theproblem (12). A. Proposed Algorithm
Here we propose an iterative algorithm for finding a stationary point of problem (12). We firstsimplify the problem by proving the following lemma.
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Lemma 2.
Imposing equalities on the constraints (12b) and (12c) induces no loss of optimality.Proof:
Suppose that the constraints (12b) or (12c) are not satisfied with equality. Then,we can decrease the transmission powers P , . . . , P M +1 or increase the backhaul usage until theconstraints are tight without decreasing the achievable rate.With Lemma 2 and some algebraic manipulations, the problem (12) can be written as minimize { ¯ P i ≥ } M +1 i =1 , { ¯ β i ,γ i ≥ } Mi =1
11 + P M +1 ¯ β M M Y i =1 g i ¯ P i +1 g i ¯ P i (13a) s . t . P M +1 ¯ β M P Mi = k C i (cid:0) P k ¯ β M (cid:1) M Y i = k ( (cid:0) g i ¯ P i (cid:1) (cid:0) P ¯ β π − ( i ) (cid:1) γ i (cid:0) g i ¯ P i +1 (cid:1) (cid:0) P ¯ β π − ( i ) − (cid:1) ) ≤ , k = 1 , . . . , M, (13b) P ¯ β π − ( i ) C i (cid:0) P ¯ β π − ( i ) − (cid:1) γ i ≤ , i = 1 , . . . , M, (13c) ¯ P P ≤ , (13d) ¯ P i +1 ¯ P i ≤ , ¯ β i − ¯ β i ≤ , i = 1 , . . . , M, (13e) ¯ β i g π ( i ) + ¯ β i − ≤ , g i γ i + ¯ β π − ( i ) g i γ i + ¯ β π − ( i ) − ≤ , i = 1 , . . . , M, (13f)where we characterized the problem over the cumulative variables { ¯ P i } M +1 i =1 and { ¯ β i } Mi =1 , andintroduced auxiliary variables γ i = 1 − ( ¯ β π − ( i ) − ¯ β π − ( i ) − ) /g i for i = 1 , . . . , M .Problem (13) is not a standard GP [16] since the denominators in the left-hand side of (13b),(13c) and (13f) are not monomials. However, the problem is a class of CGP problems [15, Sec.3.2], and thus a stationary point of (13) can be found by applying the homotopy method [15,Sec. 3.2], which solves a sequence of GPs obtained by locally approximating the posynomialdenominators as monomial expressions (see, e.g., [15, Lemma 3.1]). The resulting algorithm issummarized in Table Algorithm 1. V. S PECIAL C ASES
Here we discuss some relevant special cases of the proposed scheme. A GP can be converted into an equivalent convex problem (see [16, Sec. 4.5.3] for more detail).
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Algorithm 1
Homotopy method for problem (13)1. Initialize the variables { ¯ P (1) i ≥ } M +1 i =1 , { ¯ β (1) i ≥ } Mi =1 to an arbitrary feasible point and set n = 1 .2. Update the variables { ¯ P ( n +1) i ≥ } M +1 i =1 , { ¯ β ( n +1) i ≥ } Mi =1 as a solution of the following GPproblem: minimize { ¯ P ( n +1) i ≥ } M +1 i =1 , { ¯ β ( n +1) i ,γ i ≥ } Mi =1 f (cid:16) P ( n +1) M +1 ¯ β ( n +1) M , P ( n ) M +1 ¯ β ( n ) M (cid:17) M Y i =1 g i ¯ P ( n +1) i +1 f (cid:16) g i ¯ P ( n +1) i , g i ¯ P ( n ) i (cid:17) (14) s . t . M Y i = k (cid:16) g i ¯ P ( n +1) i (cid:17) (cid:16) P ( n +1)1 ¯ β ( n +1) π − ( i ) (cid:17) γ i f (cid:16) g i ¯ P ( n +1) i +1 , g i ¯ P ( n ) i +1 (cid:17) f (cid:16) ¯ P ( n +1)1 ¯ β ( n +1) π − ( i ) − , ¯ P ( n )1 ¯ β ( n ) π − ( i ) − (cid:17) × P ( n +1) M +1 ¯ β ( n +1) M P Mi = k C i f (cid:16) ¯ P ( n +1) k ¯ β ( n +1) M , ¯ P ( n ) k ¯ β ( n ) M (cid:17) ≤ , k = 1 , . . . , M, P ( n +1)1 ¯ β ( n +1) π − ( i ) C i γ i f (cid:16) ¯ P ( n +1)1 ¯ β ( n +1) π − ( i ) − , ¯ P ( n )1 ¯ β ( n ) π − ( i ) − (cid:17) ≤ , i = 1 , . . . , M, ¯ P ( n +1)1 P ≤ , ¯ P ( n +1) i +1 ¯ P ( n +1) i ≤ , ¯ β ( n +1) i − ¯ β ( n +1) i ≤ , i = 1 , . . . , M, ¯ β ( n +1) i g π ( i ) f (cid:16) ¯ β ( n +1) i − /g π ( i ) , ¯ β ( n ) i − /g π ( i ) (cid:17) ≤ , i = 1 , . . . , M,g i γ i + ¯ β ( n +1) π − ( i ) g i f (cid:16) ¯ β ( n +1) π − ( i ) − /g i , ¯ β ( n ) π − ( i ) − /g i (cid:17) ≤ , i = 1 , . . . , M, where the function f ( s, ˆ s ) is a monomial function of s defined as [15, Lemma 3.1] f ( s, ˆ s ) = c (ˆ s ) s a (ˆ s ) (15)with a (ˆ s ) = ˆ s (1 + ˆ s ) − and c (ˆ s ) = ˆ s − a (1 + ˆ s ) .3. Stop if some convergence criterion is satisfied. Otherwise, set n ← n + 1 and go to Step 2. February 16, 2018 DRAFT1
A. Compress-and-Forward
If we impose that the encoder uses only the highest layer X M +1 , i.e., X = √ P X M +1 inlieu of the more general (3), the proposed hybrid scheme reduces to a pure CF scheme withsuccessive decoding as studied in [7][9]. Optimization of the test channels β , . . . , β M under thisassumption and given ordering π can be simplified to maximize β ,...,β M ≥ log P M X j =1 g j β j ! (16) s . t . log (cid:18) P ¯ β i P ¯ β i − (cid:19) − log (cid:0) − β π ( i ) (cid:1) ≤ C π ( i ) , i = 1 , . . . , M, whose solutions β opt1 , . . . , β opt M are directly given, using Lemma 2, as β opt π ( i ) = (cid:0) C π ( i ) − (cid:1) (cid:0) P ¯ β i − (cid:1) C i (cid:0) P ¯ β i − (cid:1) + P g π ( i ) , i = 1 , . . . , M. (17) B. Decode-and-Forward
The DF strategy is a special case of the proposed hybrid relaying scheme obtained by fixing β = . . . = β M = 0 and P M +1 = 0 . A similar approach was studied in [2, Sec. V-B] for M = 2 assuming Gaussian channels for relay-to-destination links. A stationary point of the problem canbe obtained by adopting the homotopy method in Algorithm 1 with minor modifications. Asan interesting special case, we consider DF with single-layer transmission in which multi-layertransmission is not leveraged.Using single-layer transmission, the following rate is achievable by optimizing the selectionof the transmitted layer: max i ∈M min ( log (1 + g i P ) , M X j = i C j ) . (18)We remark that in (18) we have used the fact, as in the more general result of Lemma 1, that allrelays i, . . . , M are able to decode message W i and thus the message can be distributed acrossthe backhaul links in order to be delivered to the destination.VI. N UMERICAL R ESULTS
In this section, we present numerical results to investigate the advantage of the proposedmulti-layer transmission scheme with hybrid relaying studied in Sec. III-IV as compared to the
February 16, 2018 DRAFT2 =C [bit/c.u.] a c h i e v ab l e r a t e [ b i t/ c . u .] CFDFHybridcutset upperbound
Figure 2. Achievable rates versus the backhaul capacity C = C in a symmetric network with M = 2 , P = 0 dB and g = g = 10 dB . more conventional schemes reviewed in Sec. V. For reference, we also compare the achievablerates with the cutset upper bound [17, Theorem 1] R cutset = min S⊆{ ,...,M } (X j ∈S C j + log P X j ∈S c g j !) . (19)For ease of interpretation, we focus on the case with two relays, i.e., M = 2 . We mark single-layer schemes with the label ’SL’ and multi-layer schemes with ’ML’. For CF related schemes,the optimal ordering π opt in problem (11) was found via exhaustive search and was observed tobe π = (1 , for all the simulated cases.In Fig. 2, we examine the performance in a symmetric setting by plotting the rate versusthe backhaul capacities C = C when P = 0 dB and g = g = 10 dB . It is seen thatin this symmetric set-up, the optimized hybrid scheme ends up reducing to either the DFor the CF strategy at small and large backhaul capacity, respectively. Note that we have notdistinguished between the single-layer and multi-layer strategies in the figure since they showedthe same performance when the relays experience the same fading power, i.e., g = g . Thisis expected since multi-layer strategies are relevant only when the two relays have differentdecoding capabilities. February 16, 2018 DRAFT3 =C [bit/c.u.] a c h i e v ab l e r a t e [ b i t/ c . u .] CFDF−SLDF−MLHybrid−SLHybrid−MLcutset upperboundHybrid DF CF
Figure 3. Achievable rates versus the backhaul capacity C = C per relay with M = 2 , P = 0 dB and [ g , g ] = [0 ,
10] dB . In Fig. 3, we observe the performance versus the backhaul capacity C = C with P = 0 dB and asymmetric channel powers [ g , g ] = [0 ,
10] dB . Unlike the symmetric setting in Fig. 2,the multi-layer strategy is beneficial compared to the single-layer (SL) transmission for bothDF and Hybrid schemes . Moreover, unlike the setting of Fig. 2, the hybrid relaying strategyshows a performance advantage with respect to all other schemes. This is specifically the casefor intermediate values of the backhaul capacities C = C . It should also be mentioned that,as C = C increases, the performance of DF schemes is limited by the capacity of the betterdecoder, namely log (1 + 10) = 3 . bit/c.u., while CF, and thus also the hybrid strategy, areable, for C = C large enough, to achieve the cutset bound.Finally, in Fig. 4, we plot the achievable rates versus the channel power g of the better relaywhen P = 0 dB , g = 0 dB and C = C = 2 bit/c.u.. As expected, the performance gain ofmulti-layer transmission over the single-layer schemes is more pronounced as g increases, sincea better channel to relay 2 allows to support larger rates for both rates of both DF layers. Infact, single-layer transmission uses only the DF layer decoded exclusively by relay 2 accordingto (18). For the same reason, the rate of single-layer DF is limited by the backhaul capacity Not being based on relay decoding, CF operates only with one layer.
February 16, 2018 DRAFT4 [dB] (g =0 dB) a c h i e v ab l e r a t e [ b i t/ c . u .] CFDF−SLDF−MLHybrid−SLHybrid−MLcutset upperboundHybrid DF CF
Figure 4. Achievable rates versus the channel power g with M = 2 , P = 0 dB , g = 0 dB and C = C = 2 bit/c.u.. C of relay 2. Moreover, hybrid relaying is advantageous over all conventional schemes forintermediate values of g . VII. C ONCLUSIONS
We have studied transmission and relaying techniques for the relay channels with multipleout-of-band relays, which are connected to the destination via orthogonal finite-capacity backhaullinks. We proposed a novel transmission and relaying strategies whereby multi-layer transmissionis used at the encoder and hybrid DF-CF relaying is adopted at the relays. The multi-layertransmission is designed so as to adaptively leverage the different decoding capabilities of therelays and to enable the hybrid relaying strategy. As a result, the proposed multi-layer strategyis different from the classical broadcast coding approach of [14], which aims at coping withuncertain fading conditions at the transmitter (see also [8] for an application to a multi-relaysetting).We aimed at maximizing the achievable rate, which is formulated as a non-convex problem.However, based on the observation that the problem falls in the class of so called ComplementaryGeometric Programs (CGPs), we have proposed an iterative algorithm based on the homotopymethod which attains a stationary point of the problem. From numerical results, it was shown
February 16, 2018 DRAFT5 that the proposed multi-layer transmission with the hybrid relaying strategy outperforms moreconventional decode-and-forward, compress-and-forward and single-layer strategies, especiallyin the regime of moderate backhaul capacities and asymmetric channel gains from the source tothe relays. A
PPENDIX AP ROOF OF L EMMA W , . . . , W M at the decoder. To see this, we observe that the destination, when decoding messages W , . . . , W M ,can be regarded as the decoder of a multiple access channel with M sources. Specifically, source k has messages W , . . . , W k for k = 1 , . . . , M and has two inputs to the channel to the destination,namely the signal X k and the information sent at rate C DF k on the noiseless backhaul link. Wedenote the latter as T k , where T k ∈ { , . . . , C DF k } so that the overall channel input of the source k is given by ˜ X k = ( X k , T k ) . The destination observes ˆ Y M and T , . . . , T M . We emphasize thatboth X k and T k in ˜ X k depend on all messages W , . . . , W k .As a result, we have an equivalent multiple access channel in which each source has a specificsubset of all the messages and a hierarchy exists among the sources so that source k has allthe messages also available to sources , . . . , k − . Therefore, using the results in [24][25], thefollowing conditions guarantee correct decoding of messages W , . . . , W MM X j = k R j ≤ I (cid:16) ˜ X { k,...,M } ; ˆ Y M , T { ,...,M } | ˜ X { ,...,k − } (cid:17) , (20)for k = 1 , . . . , M . The achievability of rates (20) is ensured for any joint distribution of theinputs { ˜ X k } Mk =1 [24][25]. To proceed, we take ˜ X k to be independent according to the discussionaround (3), and also take X k to be independent of T k for all k = 1 , . . . , M . It is not hard to seethat this choice maximizes the mutual informations in (20). Under these assumptions, we canwrite the right-hand side of (20) as I (cid:16) X { k,...,M } , T { k,...,M } ; ˆ Y M , T { ,...,M } | X { ,...,k − } , T { ,...,k − } (cid:17) (21) = I (cid:16) X { k,...,M } ; ˆ Y M | X { ,...,k − } (cid:17) + H (cid:0) T { k,...,M } (cid:1) = I (cid:16) X { k,...,M } ; ˆ Y M | X { ,...,k − } (cid:17) + M X j = k C DF j , February 16, 2018 DRAFT6 by the chain rule for mutual informations [17, Theorem 2.5.2]. This proves that inequalities (20)reduce to (9) with the given choices. R
EFERENCES [1] B. Schein, "Distributed coordination in network information theory," Ph.D., MIT, Cambridge, MA, 2001.[2] F. Xue and S. Sandhu, "Cooperation in a half-duplex Gaussian diamond relay channel,"
IEEE Trans. Inf. Theory , vol. 53,no. 10, pp. 3806-3814, Oct. 2007.[3] A. del Coso and C. Ibars, "Achievable rates for the AWGN channel with multiple parallel relays,"
IEEE Trans. WirelessComm. , vol. 8, no. 5, pp. 2524-2534, May 2009.[4] A. Sanderovich, S. Shamai (Shitz), Y. Steinberg and G. Kramer, "Communication via decentralized processing,"
IEEETrans. Inf. Theory , vol. 54, no. 7, pp. 3008-3023, Jul. 2008.[5] A. Sanderovich, O. Somekh, H. V. Poor and S. Shamai (Shitz), "Uplink macro diversity of limited backhaul cellularnetwork,"
IEEE Trans. Inf. Theory , vol. 55, no. 8, pp. 3457-3478, Aug. 2009.[6] A. del Coso and S. Simoens, "Distributed compression for MIMO coordinated networks with a backhaul constraint,"
IEEETrans. Wireless Comm. , vol. 8, no. 9, pp. 4698-4709, Sep. 2009.[7] S.-H. Park, O. Simeone, O. Sahin and S. Shamai (Shitz), "Robust and efficient distributed compression for cloud radioaccess networks,"
IEEE Trans. Veh. Tech. , vol. 62, no. 2, pp. 692-703, Feb. 2013.[8] M. Zamanin and A. K. Khandani, "Broadcast approaches to the diamond channel," arXiv:1206.3719.[9] L. Zhou and W. Yu, "Uplink multicell processing with limited backhaul via successive interference cancellation," in
Proc.IEEE Glob. Comm. Conf. (Globecom 2012) , Anaheim, CA, Dec. 2012.[10] P. Marsch, B. Raaf, A. Szufarska, P. Mogensen, H. Guan, M. Farber, S. Redana, K. Pedersen and T. Kolding, "Futuremobile communication networks: challenges in the design and operation,"
IEEE Veh. Tech. Mag. , vol. 7, no. 1, pp. 16-23,Mar. 2012.[11] J. Segel and M. Weldon, "Lightradio portfolio-technical overview," Technology White Paper 1, Alcatel-Lucent.[12] B. Nazer, A. Sanderovich, M. Gastpar and S. Shamai (Shitz), "Structured superposition for backhaul constrained cellularuplink," in
Proc. IEEE Intern. Sym. Inf. Theory (ISIT 2009),
Seoul, Korea, Jun. 2009.[13] O. Simeone, N. Levy, A. Sanderovich, O. Somekh, B. M. Zaidel, H. V. Poor and S. Shamai (Shitz), "Cooperative wirelesscellular systems: an information-theoretic view,"
Foundations and Trends in Comm. Inf. Theory, vol. 8, no. 1-2, pp. 1-177.[14] S. Shamai (Shitz) and A. Steiner, "A broadcast approach for a single-user slowly fading MIMO channel,"
IEEE Trans.Inf. Theory , vol. 49, no. 10, pp. 2617-2635, Oct. 2003.[15] P. C. Weeraddana, M. Codreanu, M. Latva-Aho, A. Ephremides and C. Fischione, "Weighted sum-rate maximization inwireless networks: a review,"
Foundations and Trends in Networking, vol. 6, no. 1-2, pp. 1-163, 2012.[16] S. Boyd and L. Vandenberghe,
Convex optimization , Cambridge University Press, 2004.[17] T. Cover and J. Thomas,
Elements of information theory , Wiley Series in Telecomm., 1st ed., 1991.[18] T. M. Cover, "Comments on broadcast channels,"
IEEE Trans. Inf. Theory , vol. 44, no. 6, pp. 2524-2530, Oct. 1998.[19] A. E. Gamal and Y.-H. Kim,
Network information theory , Cambridge University Press, 2011.[20] X. Zhang, J. Chen, S. B. Wicker and T. Berger, "Successive decoding in multiuser information theory,"
IEEE Trans. Inf.Theory , vol. 53, no. 6, pp. 2246-2254, Jun. 2007.[21] A. D. Wyner and J. Ziv, "The rate-distortion function for source coding with side information at the decoder,"
IEEE Trans.Inf. Theory , vol. 22, no. 1, pp. 1-10, Jan. 1976.
February 16, 2018 DRAFT7 [22] G. Chechik, A. Globerson, N. Tishby and Y. Weiss, "Information bottleneck for Gaussian variables,"
Jour. Machine Learn. ,Res. 6, pp. 165-188, 2005.[23] C. Tian and J. Chen, "Remote vector Gaussian source coding with decoder side information,"
IEEE Trans. Inf. Theory ,vol. 55, no. 10, pp. 4676-4680, Oct. 2009.[24] K. de Bruyn, V. V. Prelov and E. van der Meulen, "Reliable transmission of two correlated sources over an asymmetricmultiple-access channel,"
IEEE Trans. Inf. Theory, vol. 33, no. 5, pp. 716-718, Sep. 1987.[25] D. Gunduz and O. Simeone, "On the capacity region of a multiple access channel with common messages," in
Proc. IEEEIntern. Sym. Inf. Theory (ISIT 2010),