Protocols and Performance Limits for Half-Duplex Relay Networks
aa r X i v : . [ c s . I T ] F e b SUBMITTED TO IEEE TRANS. COMM. 1
Protocols and Performance Limits forHalf-Duplex Relay Networks
Peter Rost,
Member, IEEE, and Gerhard Fettweis,
Fellow, IEEE
Abstract
In this paper, protocols for the half-duplex relay channel are introduced and performance limits are analyzed.Relay nodes underly an orthogonality constraint, which prohibits simultaneous receiving and transmitting on the sametime-frequency resource. Based upon this practical consideration, different protocols are discussed and evaluated usinga Gaussian system model. For the considered scenarios compress-and-forward based protocols dominate for a widerange of parameters decode-and-forward protocols. In this paper, a protocol with one compress-and-forward and onedecode-and-forward based relay is introduced. Just as the cut-set bound, which operates in a mode where relaystransmit alternately, both relays support each other. Furthermore, it is shown that in practical systems a randomchannel access provides only marginal performance gains if any.
I. I
NTRODUCTION
In [1] Cover and El Gamal introduced two basic coding strategies for the three-terminal relay channel, whichstill serve as basis for most relaying protocols today: decode-and-forward (DF), where the relay decodes the sourcemessage and provides additional, redundant information, and compress-and-forward (CF), where the relay nodequantizes its channel output and forwards the quantization to the destination. Relays using DF and CF are operatingin a digital relaying mode, which affects both physical and medium access layer.
Analog relays, by contrast, workas a repeater and simply amplify and forward the received signal. As the former mode offers more flexibility withrespect to coding and resource assignment strategies, this paper focuses on digital relaying approaches.Practical requirements such as power, cost, and space efficiency imply the necessity for small, low-cost terminals,which implement low-complexity protocols. These restrictions result in an insufficient separation of transmit andreceive path on the same time-frequency resource. As the transmission power is usually of much higher order thanthe received signal power, a severe drop of the signal-to-interference-and-noise ratio (SINR) is caused. Therefore,we assume in the following half-duplex constrained relay nodes. Generally, the half-duplex constraint requires thateach terminal cannot listen and transmit on the same resource simultaneously but only on orthogonal resources.
Manuscript submitted August 20, 2018. Part of this work has been published in the proceedings of 2008 IEEE Global CommunicationsConference and presented at Asilomar Conference on Signals, Systems, and Computers 2008.
August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 2
A. Related Work
One of the first works analyzing the capacity of the half-duplex relay channel is [2], which derives upper andlower bounds on the capacity of wireless half-duplex single-relay networks. At the same time [3] studied an upperand lower bound on the capacity of a wireless relay network, where an arbitrary number of relays support a singlesource-destination pair. Later, [4], [5] derived an upper bound on the achievable rates for general relay networkswith practical constraints, which are modeled by an arbitrary number of possible states at each node. Kramer [6]introduced the idea of exploiting the randomness of channel states to transmit information and hence counteractthe half-duplex loss.This work particularly discusses alternately transmitting relays nodes in a two-relay network. Alternately trans-mitting analog relay nodes were at first discussed in [7] and later in [8], [9], where an interference cancellationemployed at the destination was introduced. In order to overcome the inter-relay interference, [10], [11] analyzedalternately transmitting relay nodes for systems using code-division multiple access (CDMA) under the assumptionof perfect separation on the inter-relay link. In [12] the authors propose a scheme based on superposition coding,which explicitly exploits the inter-relay interference to improve diversity and multiplexing gain.
B. Contribution and Outline of this Work
In [13] a general framework for the full-duplex multiple relay channel has been introduced and combined theideas of partial DF and CF. This paper extends [13] by applying known protocols to a half-duplex multiple relaychannel and introducing new approaches. Among others, we investigate the benefits of a random channel access[6] in a network of half-duplex nodes. Additionally, a new regular encoding CF strategy is introduced, whichovercomes some of the drawbacks of the separation of source and channel coding. In addition, we derive a protocolwith alternately transmitting relay nodes, which also considers a direct link compared to the diamond channel [14].After an introduction of the notation, definitions, and system model in Section II, we discuss the individualapproaches in Section III. Results for these approaches are discussed in Section IV and the paper is concluded inSection V. II. N
OTATIONS , D
EFINITIONS , AND S YSTEM M ODEL
In this paper, we use non-italic uppercase letters X to denote random variables, and italic letters ( N or n ) todenote real or complex-valued scalars. Ordered sets are denoted by X , the cardinality of an ordered set is denotedby kX k and [ b : b + k ] is used to denote the ordered set of numbers b, b + 1 , · · · , b + k . Let X l be a random variableparameterized using l . Then X C denotes the vector of all X l with l ∈ C (this applies similarly to sets of events).Matrices are denoted by boldface uppercase letters K and the element in the i -th row and j -th column of matrix K is denoted by [ K ] i,j . Furthermore, I (X; Y | Z) denotes the mutual information between random variables X and Y given Z and C ( x ) abbreviates log (1 + x ) . All logarithms are taken to base .This paper considers a network of N + 2 nodes: the source node s = 0 , the set of N relays R := [1; N ] , and thedestination node d = N + 1 . With set R we express an arbitrary numbering of all relay nodes, which is subject August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 3 to an optimization (which is not explicitly noted in the following presentation). We focus on a Gaussian channelsetup where d l ′ ,l is the distance between nodes l ′ and l = l ′ and θ is the path loss exponent, such that the gainfactor between both nodes is given by h l ′ ,l = d − θ/ l ′ ,l . The channel input at node l and time instances t ∈ [1; n ] is given by the n -length sequence of complex Gaussian r.v.s { X tl } nt =1 with zero mean and variance P l , denotedby X l n ∼ CN (0 , P l ) . Node states are denoted by M s , M , . . . M N with M l ∈ { L, T } , and L, T representing thelistening and transmitting state. The channel output at node l ∈ [1; N + 1] and time instance t ∈ [1; n ] is given by Y tl = 1 (cid:0) M tl = L (cid:1) · X l ′ ∈ [0; N ] \ l h l ′ ,l X tl ′ + Z tl , (1)where · ) is the indicator function returning if its argument is true and otherwise, and Z l n ∼ CN (0 , N l ) is additive white Gaussian noise. From the orthogonality constraint follows that (M tl = L ) → (X tl = 0) and (M tl = T ) → (Y tl = 0) . This implies that each node l ∈ [0; N ] must fulfill the power constraint (M tl = T ) → E n | X tl | o = P l , which requires that each node can switch from transmit to receive state in arbitrarily short time. Incontrast to bursty relay approaches where power is concentrated on a small portion of the overall block, we definea peak power and do not normalize the overall spent energy.III. P ROTOCOLS FOR THE H ALF -D UPLEX C HANNEL
For the sake of readability and comprehensibility, this section only treats a network with N = 2 relays while theAppendix extends the results to networks with an arbitrary number of relay nodes. A. Decode-and-Forward Protocols
Assume that the states M l of source and relay nodes are chosen randomly. If they are interpreted as a bit pattern,we can be exploit them as an additional information carrier [6]. However, in order to obtain a significant gain, thesystem must provide a high granularity of resources. The first protocol class, which is considered in this paper, is anapplication of DF and the idea of randomized channel access to the half-duplex multiple-relay channel. The sourceintends to communicate a message W s , which is mapped to the message tuple (M s , U s, , U s, , U s, ) consistingof the source’ state M s and three different, superimposed messages with individual rates R s,k . The first relayonly decodes the source state and the first message level U s, , while the second relay additionally decodes thesecond message level U s, , and finally the destination needs to decode the complete tuple in order to correctlyreconstruct the source message. Relay supports the first message level by transmitting additional, redundantinformation represented by the tuple (M , V , ) . If the source has channel knowledge for the complete network, itcan coherently support the transmission of relay . Relay exploits this additional information in order to decode (M s , U s, ) and then also provides additional redundant information for the first two source message levels to thedestination node.Using the previously introduced notation of the considered Gaussian system model, the channel input at the August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 4 source and both relays is the following superposition of signals: X ts = 1 (cid:0) M ts = T (cid:1) p P s X k =1 √ α ( s,s ) ,k U ts,k + 1 (cid:0) M t = T (cid:1) √ α ( s, , V t , + X k =1 (cid:0) M t = T (cid:1) √ α ( s, ,k V t ,k ! (2) X t = 1 (cid:0) M t = T (cid:1) p P (cid:0) √ α (1 , , V t , + 1 (cid:0) M t = T (cid:1) √ α (1 , , V t , (cid:1) (3) X t = 1 (cid:0) M t = T (cid:1) p P (cid:0) √ α (2 , , V t , + √ α (2 , , V t , (cid:1) , (4)where α ( l ′ ,l ) ,k denotes the fraction of power spent by node l ′ for the support of message level k sent by relay l and is assumed to be constant for all transmission phases, which might result in an average transmit power of node l less than P l . An adaptive power fraction results in an enormous parameter space and in case of non-coherenttransmission (which appears to be more practically relevant) no power savings are obtained anyway.The differential entropy for the channel output Y l ′ if the channel states of nodes L are known and L = [0; N ] \{L , l } are unknown is denoted by h k ( l,l ′ ) ( m L ) and defined in detailed in Appendix A. Furthermore, P DF denotesthe set of channel input pdfs, which assign the different power levels α ( l ′ ,l ) ,k such that the power constraints inSection II are satisfied and assign the probabilities to the node states M l (a more detailed definition of P DF is givenin Appendix A). Theorem 1:
The achievable rates for the previously described partial DF protocol are given by R = sup p ∈P DF ( R s, + R s, + R s, ) , (5)with the individual rate constraints R s, ≤ min n Q s, ( L ) , Q s, ( L ) + Q , ( L ) ,Q s,d ) ( L ) + Q ,d ) ( L ) + Q ,d ) ( L d ) o , (6) R s, ≤ min n Q s, ( L ) , Q s,d ) ( L ) + Q ,d ) ( L ) o (7) R s, ≤ Q s,d ) ( L ) . (8)where L l = [ l : 2] is the set of nodes for which the state is known. The mutual information function Q k ( l,l ′ ) ( L ) isgiven by Q l,l ′ ) ( L ) = X m L ∈M L : m l ′ = L p ( m L ) (cid:16) h l,l ′ ) ( m L ) − X m l ∈M l p ( m l | m L ) h l,l ′ ) (cid:16) m {L ,l } (cid:17)(cid:17) , (9) Q k ( l,l ′ ) ( L ) = X m L ∈M L : m l ′ = L p ( m L ) (cid:16) h k − l,l ′ ) ( m L ) − h k ( l,l ′ ) ( m L ) (cid:17) . (10) Proof:
The theorem is an application of the more general Theorem 3 given in Appendix A and describing theachievable rates for an arbitrary number of relay nodes.Eq. (6) is the minimum of the three cuts for the first source message level: from source to relay , from source andrelay to relay , and from source, relay and to the destination. However, we can see that the transmit-diversity August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 5 gain is increasing with the number of nodes, which already decoded the message. The function Q k ( l,l ′ ) ( L ) givesthe mutual information in the half duplex channel between nodes l and l ′ and message level k . In case of a fixedchannel access the channel state of all nodes is known, hence only Q k ( l,l ′ ) ([0; 2]) is used. Nonetheless, in case ofa random channel access we face the difficulty to evaluate an integral of the form ∞ Z X k a k λ k π e − λ k y ! log X k a k λ k π e − λ k y ! d y, (11)which can only be loosely upper and lower bounded (using log-sum inequality and Jensen’s inequality). Therefore,the results presented in Section IV follow from a numerical evaluation of this integral.If we use only a subset of P DF , which includes only those input pdfs with deterministic state probabilities, theprevious theorem gives the achievable rates for a fixed transmission schedule. Such a schedule is preferable as itneeds no additional complexity and hardware to detect the node states (only wireline based networks can supportthis detection at reasonable complexity). Furthermore, consider an orthogonal frequency division multiplex (OFDM)system with groups of F c subcarriers, which are assigned to users. Then the actual advantage through a randomchannel access is reduced by a factor /F c , which makes a fixed transmission schedule an even more preferablechoice. Finally, consider a multihopping approach with reuse factor / k [15]. This implies that one resource is onlyoccupied by / k -th of all nodes, or that one node only uses / k -th of the available resources. Applied to a half-duplexrelay network this implies that all p ∈ P DF must satisfy ∀ l ∈ [0; N ] : Pr m l = T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ [0; N ] m j = T ) > (cid:22) k ( N + 1) (cid:23) = 0 . B. Compress-and-Forward Protocols
In this section, we discuss a CF based approach, where, by contrast to DF, relay nodes need not to decode thesource messages but forward their quantized channel output. Due to the fact that the channel input of each relaycannot be predicted, we assume a fixed transmission schedule known at each node. In comparison to previous work,we introduce a CF approach using joint source-channel coding to overcome the drawbacks of separating both [16].More specifically, both relays l ∈ [1; 2] create for each possible quantization ˆY l a corresponding broadcastmessage X l . Depending on the channel output Y in block b , relay searches for a jointly typical quantizationand then transmits in block b + 2 the corresponding broadcast message. Similarly, relay transmits in block b + 1 the broadcast message corresponding to the quantization in block b . This shift of blocks allows the destinationto use quantizations of relay for the decoding of the broadcast message transmitted by relay . To decode thequantization index for instance of relay for block b , the destination has to create two sets. The first set containsall those indices of broadcast messages which are jointly typical with Y d in block b + 1 and the second set ofthose indices such that the quantization is jointly typical with Y d in block b . Using the intersection of both sets isthen the index of the correct quantization of relay in block b . Similarly, the destination proceeds to decode thequantization of the first relay node where ˆY is exploited to improve the quality of ˆY and X . August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 6
Due to the regular encoding , we are able to alleviate the drawbacks of source-channel coding separation. Thedifference of irregular and regular encoding is illustrated in Fig. 1. Consider for instance the multiple-descriptionproblem [17] and assume two receivers and an irregular encoding . In this case, both decoders are forced to decode atfirst the broadcast and then the quantization messages, where the weaker source-to-destination link is the bottleneckfor the achievable broadcast message rates. Now consider a regular encoding scheme. This time, the worse source-to-destination link can be balanced out with stronger side information while the better source-to-destination linkallows for weaker side information.Using the previously introduced Gaussian system model, the channel input at both relays using CF is given by X t = 1 (cid:0) M t = T (cid:1) p P β , W t , (12) X t = 1 (cid:0) M t = T (cid:1) p P β , W t , (13)with the broadcast messages W , , W , n ∼ CN (0 , and their fractional power factors β , , β , . Since we onlyuse one source message level, the channel input at the source node is simply given by X ts = 1 (cid:0) M ts = T (cid:1) q P s α ( s,s ) , U ts, . (14)In addition, we need the following auxiliary variables describing receive power variances: • The received power at node l , which originates from the transmission of nodes L ⊂ [0; N ] is given by Γ ( L ,l ) (cid:16) m [0; N ] (cid:17) . • The covariance of the channel outputs at nodes l and l ′ for the transmission sent by nodes L is given by ˜Γ L , ( l,l ′ ) (cid:16) m [0; N ] (cid:17) . • Finally, let K L , L ′ (cid:16) m [0; N ] (cid:17) be the covariance matrix of all quantizations at nodes l ∈ L ′ and the destination’schannel output when all messages from nodes L c = [0; 2] \ L are known.For the benefit of readability, the arguments of Γ ( L ,l ) (cid:16) m [0; N ] (cid:17) and ˜Γ L ′ , ( l,l ′ ) (cid:16) m [0; N ] (cid:17) are dropped. Theorem 2:
The regular CF approach achieves any rates R ≤ sup p X m [0:2] ∈M [0:2] p (cid:16) m [0:2] (cid:17) log (cid:13)(cid:13)(cid:13) K s, [1:2] (cid:16) m [0:2] (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K ∅ , [1:2] (cid:16) m [0:2] (cid:17)(cid:13)(cid:13)(cid:13) , (15)subject to X { m [0:2] ∈M [0:2] : m − = L } p (cid:16) m [0:2] (cid:17) log (cid:13)(cid:13) K [0:1] , (cid:13)(cid:13) N , (cid:13)(cid:13) K [0:1] , ∅ (cid:13)(cid:13) ! ≤ X { m [0:2] ∈M [0:2] : m = T } p M [0:2] (cid:16) m [0:2] (cid:17) log (cid:13)(cid:13) K [0:2] , ∅ (cid:13)(cid:13)(cid:13)(cid:13) K [0:1] , ∅ (cid:13)(cid:13) ! (16) August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 7 for the quantization at relay and X { m [0:2] ∈M [0:2] : m − = L } p (cid:16) m [0:2] (cid:17) " C Γ (2 , Γ (0 , + N , + N ! + log (cid:13)(cid:13) K s, [1:2] (cid:13)(cid:13) N , k K s, k ! ≤ X { m [0:2] ∈M [0:2] : m = T } p M [0:2] (cid:16) m [0:2] (cid:17) log (cid:13)(cid:13) K [0:1] , (cid:13)(cid:13) k K s, k ! . (17)for the quantization at relay . Proof:
The theorem is an application of the more general Theorem 4 given in Appendix B and describing theachievable rates for an arbitrary number of relay nodes.Eq. (16) and (17) reflect the side condition on the quantization quality. The right hand side of both inequalitiesgives the channel coding constraint, and the left hand side gives the source coding constraint. Both quantizationnoise variances must be determined iteratively in descending order, starting with (16).
C. Alternately Transmitting Relays
This section introduces a protocol for two alternately transmitting relay nodes of which one relay node istransmitting while the other relay is listening. By contrast to the previous two protocols, we apply a mixed approachwhere one relay supports the source using CF and one node employs DF. The major bottleneck in such a network isthe inter-relay interference, which can, however, be exploited if the destination uses the CF transmission to decodenot only the source but also the DF-relay transmission. Nonetheless, we still face the problem that the DF relayis interfered by the CF relay, which we mitigate using the previously introduced regular encoding approach, i. e.,both DF relay and the destination decode the transmission of the CF relay but use different side information.Fig. 2 illustrates the setup: the overall transmission period is divided into two phases with probabilities p (phase in Fig. 2) and p (phase in Fig. 2) such thatPr (M s = T ) = 1 , Pr (M = T | M = L ) = 1 , Pr (M = L | M = T ) = 1 ,p = Pr (M = T ) , p = Pr (M = T ) = 1 − p , with each phase of length n = n · p and n = n · p , respectively. The source message is divided in two parts X s, and X s, with rates R DF and R CF , respectively. Fig. 3 illustrates the coding procedure: relay supports the sourcemessage in the second phase of each block using its quantized channel output ˆY for which in the first phase ofthe next block the corresponding broadcast message is transmitted (again using regular encoding). Relay decodesthis quantization by taking account for the fact that it depends on its own transmission signal in the previous block.Alternatively, if the inter-relay channel is weak, relay simply treats the transmission as noise. After removingthe interference from relay , relay decodes the source message X s, and transmits in the next phase redundantinformation to support it.The decoding process starts with decoding the quantization index of relay for block b + 1 , which containsinformation for the relay transmission supporting the source message of block b . It can then use this quantization August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 8 and its own channel output to decode the source message transmitted in block b . After this message is known,the destination decodes X s, for which it uses again the quantization of relay (after subtracting the previouslydecoded signals) as well as its own channel output.Again, we apply this protocol to the Gaussian setup described in Section II. Let the source message use the twoindividual messages X s, , X s, n ∼ CN (0 , of lengths n = p · n and n = p · n . Then the source channel inputis given by X ts = p P s (cid:2) t ≤ n ) √ α ( s,s ) , X ts, + 1 ( t > n ) (cid:0) √ α ( s,s ) , X ts, + √ α ( s, , V t , (cid:1)(cid:3) . (18)Relay supports the source message X s, with V , n ∼ CN (0 , with rate R DF , such that the channel input isgiven by X t = 1 ( t > n ) q P α (1 , , V t , . (19)Assume that quantization signals at relay are generated according to ˆY , n ∼ CN (cid:0) , σ + N , (cid:1) . Relay usesthe broadcast messages W , n ∼ CN (0 , (both with codebook size n ∆ , ) such that its channel input is given by X t = 1 ( t ≤ n ) p P β , W t , . (20)In the following theorem, we reuse the definitions of Section III-A and III-B, e. g., the covariance matrices K = K { s, } , (cid:0) M [1 , = { T, L } (cid:1) of the quantized channel output at relay and the channel output at the destinationbefore the transmission of relay is decoded, and K = K s, (cid:0) M [1 , = { T, L } (cid:1) after this transmission is decoded.Now we can define the following corollary of Theorems 1 and 2: Corollary 1:
The previously presented combined protocol achieves any rate R = sup p ∈P ( R DF + R CF ) (21)where the rate achieved by the DF phase is constrained by R DF ≤ min (cid:26) p C (cid:18) Γ ( s,d ) N d (cid:19) + p log (cid:18) k K kk K k (cid:19) , p C (cid:18) Γ ( s, N (cid:19)(cid:27) , (22)if node decodes the quantization of node , and R DF ≤ min (cid:26) p C (cid:18) Γ ( s,d ) N d (cid:19) + p log (cid:18) k K kk K k (cid:19) , p C Γ ( s, N + Γ (2 , !(cid:27) , (23)otherwise. The rate achieved by the CF phase is limited by R CF ≤ p C (cid:18) Γ ( s, N + N , + Γ ( s,d ) N d (cid:19) , (24)and subject to a lower bound on the quantization noise: N , ≥ Γ ( { s, } , + N − (cid:16) ˜Γ { s, } , (2 ,d ) (cid:17) Γ ( { s, } ,d ) + N d (cid:16) ˆ R , /p − (cid:17) − (25)with the broadcast message rate ˆ R , = p C Γ (2 ,d ) N d + Γ ( s,d ) ! , (26) August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 9 and if relay decodes ˆY also subject to N , ≥ (cid:16) Γ ( s, + N (cid:17) · (cid:16) ˆ R , /p − (cid:17) − with ˆ R , = p C Γ (2 , N + Γ ( s, ! . (27) Proof:
Eq. (22) and (23) are an application of the DF rates where relay decodes X s, . The latter of bothequations needs to consider the additional interference of relay as its quantization is not decoded. Eq. (24)-(27)follow from an application of the CF rates where relay quantizes the channel output and, in addition to Theorem2, the previously decoded messages of relay is used as additional side information.IV. R ESULTS AND D ISCUSSION
In order to evaluate the performance of the previously introduced protocols, we present in this part results forthe linear network illustrated in Fig. 4, i. e., we consider a system of N = 2 relay nodes, equal transmission power P s = P = P and noise power N = N = N d such that ρ s,d = P s / N d = 10 dB . Both relays are symmetricallyplaced such that d s, = d ,d = | r | and d , = 1 − r , i. e., if r > both relays are placed between source anddestination while r < implies that neither relay is placed between source and destination. Unless otherwise noted,we consider a path loss exponent θ = 4 . Our analysis presents results for the cut-set bound [1] applied to thehalf-duplex relay network, compress-and-forward, Partial Decode-and-Forward (PDF) using all degrees of freedomprovided by the DF introduced in Section III-A, a simpler Decode-and-Forward (DF) with one source message leveland either full resource reuse where all nodes can transmit on all available resources or no resource reuse whereeach node transmits on an orthogonal resource, and finally the introduced approach for two alternately transmittingrelays. A. Achievable Rates for Two Relays
Fig. 5 shows the achievable rates for coherent and non-coherent transmission as well as for fixed and randomtransmission schedules. DF with a random transmission schedule achieves a performance improvement of up to . over DF with a fixed transmission schedule, which is much less than the theoretical maximum of .The superposition coding of PDF provides only for r < gains over the less complex DF, which result from amode where relay is turned off.Coherent transmission for DF does not provide any gains for r & . , which implies that the additionalcomplexity is not beneficial. For a large range of r , the combined strategy provides the maximum performanceclose to the cut-set bound for r ≈ . . Interestingly, at r ≈ . the cut-set bound uses two alternately transmittingrelay nodes. There is a significant performance drop of the combined strategy for < r < . due to the increasedinterference between both relays, which is not strong enough to be decoded and not weak enough to be ignored. B. Full vs. Half-Duplex Relaying
Fig. 6 shows the achievable rates of a single-relay
Gaussian half-duplex and full-duplex relay network, where weonly consider relay and permanently turn off relay . Compared to a full-duplex relay network, DF with a fixed August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 10 transmission schedule achieves rates which are up to about 2.5 bpcu lower. By contrast to full-duplex relaying, inhalf-duplex relaying CF is able to dominate DF for all r although the difference is not significant. In addition, noneof the protocols is able to achieve the upper bound for r > . For r > . a simple multihop protocol withoutany resource reuse achieves the maximum DF performance. On the other hand, for r < . DF with full resourcereuse provides a significant gain, which implies that for those scenarios where the source-to-relay link is of highquality, it is preferable to form virtual transmit-antenna arrays. Fig. 6 does not show the performance of partial DFas it does not provide any performance gain over single-level DF. A comparison of Fig. 5(b) and Fig. 6(b) revealsthat at r = 0 . (mid-way placed relays) two half-duplex DF relays are not able to achieve the same performanceas one full-duplex DF relay. Only the combined strategy with two alternately transmitting relays is able to achievethe same performance as one full-duplex relay.Fig. 7 shows the achievable rate of the individual protocols for an increasing number of relays placed in equaldistances. By contrast to the full-duplex channel [18], DF is unable to achieve the cut-set bound for an increasingnumber of relays. The advantage of a random transmission does not increase with the network size, which makesstatic schedules even more attractive. An open challenge is the design of a protocol, which is able to achieve thesame performance as the cut-set bound or at least the same within a non-increasing interval. DF faces the problemthat it needs to decode the source message, while CF increases the effective noise. Hence, the optimal protocolwould be a DF protocol, which needs not to decode the complete source message but can still provide noise-freeredundant information.Finally, consider Fig. 8 showing the achievable rates depending on the path loss exponent. The performancegain not only increases with the path loss exponent but also the gap between N = 3 and N = 1 is increasing in θ , which underlies that it is highly beneficial to add relay nodes in case of strong shadowing and path loss.V. C ONCLUSIONS
This paper introduced and analyzed different half-duplex protocols using DF and CF and compared their perfor-mance with the cut-set bound. In contrast to full-duplex networks, CF is able to dominate DF. But, the theoreticalperformance of CF is not achieved by practical codes yet [19], [20]. Besides, DF can use standard codes such asturbo-codes, which are able to closely approach channel capacity. Furthermore, DF uses standard encoding anddecoding algorithms, which might be less complex than the decoding algorithms used for Wyner-Ziv coding. Infading channels CF additionally has the problem that the quantization levels must be constantly adjusted in order toachieve a reasonable performance, which further limits its applicability due to the high signaling overhead. Anotherimportant advantage of DF is its higher flexibility regarding the number of antennas and deployment, which letsDF seem to be favorable over CF. Nonetheless, in the case we use mobile relay terminals or cooperation on userterminal level, CF might be an attractive alternative as the offered performance gains are remarkable.R
EFERENCES[1] T. Cover and A. E. Gamal, “Capacity theorems for the relay channel,”
IEEE Transactions on Information Theory , vol. 25, no. 5, pp.572–584, September 1979.
August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 11 [2] A. H. st Madsen, “On the capacity of wireless relaying,” in
IEEE Vehicular Technology Conference (VTC) , vol. 3, Vancouver (BC), Canada,September 2002, pp. 1333–1337.[3] M. Gastpar and M. Vetterli, “On the capacity of wireless networks: The relay case,” in
The 21st Annual Joint Conference of the IEEEComputer and Communications Societies (INFOCOM) , vol. 3, June 2002, pp. 1577–1586.[4] M. Khojastepour, A. Sabharwal, and B. Aazhang, “Bounds on achievable rates for general multi-terminal networks with practicalconstraints,” in
Information Processing in Sensor Networks , Palo Alto (CA), USA, April 2003.[5] ——, “On the capacity of Gaussian ’cheap’ relay channel,” in
IEEE Global Communications Conference , San Francisco (CA), USA,December 2003, pp. 1776–1780.[6] G. Kramer, “Models and theory for relay channels with receive constraints,” in , Monticello (IL), USA, September 2004.[7] T. Oechtering and A. Sezgin, “A new cooperative transmission scheme using the space-time delay code,” in
ITG Workshop on SmartAntennas , Munich, Germany, March 2004.[8] B. Rankov and A. Wittneben, “Spectral efficient protocols for non-regenerative half-duplex relaying,” in , Monticello (IL), USA, October 2005.[9] ——, “Spectral efficient protocols for half-duplex fading relay channels,”
IEEE Journal on Selected Areas in Communications , vol. 25,no. 2, pp. 379–389, February 2007.[10] A. Ribeiro, X. Cai, and G. B. Giannakis, “Opportunistic multipath for bandwidth-efficient cooperative networking,” in
IEEE InternationalConference on Acoustics, Speech and Signal Processing , Montr`eal, Canada, May 2004.[11] A. Ribeiro, X. Cai, and G. Giannakis, “Opportunistic multipath for bandwidth-efficient cooperative multiple access,”
IEEE Transactionson Wireless Communications , vol. 5, no. 9, pp. 2321– 2327, September 2006.[12] C. Wang, Y. Fan, I. Krikidis, J. Thompson, and H. Poor, “Superposition-coded concurrent decode-and-forward relaying,” in
IEEEInternational Symposium on Information Theory (ISIT) , Toronto, Canada, July 2008.[13] P. Rost and G. Fettweis, “Analysis of a mixed strategy for multiple relay networks,”
IEEE Transactions on Information Theory , vol. 55,no. 1, pp. 174–189, January 2009.[14] F. Xue and S. Sandhu, “Cooperation in a half-duplex Gaussian diamond relay channel,”
IEEE Transactions on Information Theory , vol. 53,no. 10, pp. 3806–3814, October 2007.[15] P. Herhold, E. Zimmermann, and G. Fettweis, “Cooperative multi-hop transmission in wireless networks,”
Journal on Computer Networks ,vol. 49, no. 3, pp. 299–324, October 2005.[16] M. Gastpar, “To code or not to code,” Ph.D. dissertation, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, December2002.[17] A. E. Gamal and T. Cover, “Achievable rates for multiple descriptions,”
IEEE Transactions on Information Theory , vol. 28, no. 6, pp.851–857, November 1982.[18] P. Rost, “Opportunities, benefits, and constraints of relaying in mobile communication systems,” Ph.D. dissertation, Technische Universit¨atDresden (TUD), Dresden, Germany, June 2009.[19] J. Li and R. Hu, “Slepian-Wolf cooperation: A practical and efficient compress-and-forward relay scheme,” in
Allerton Conference onCommunications , St. Louis (MO), USA, November 2005.[20] Z. Liu, V. Stankovic, and Z. Xiong, “Wyner-Ziv coding for the half-duplex relay channel,” in
IEEE Intnl. Conf. on Acoustics, Speech andSignal Processing (ICASSP) , vol. 5, Philadelphia (PA), USA, March 2005, pp. 1113–1116.[21] T. Cover and J. Thomas,
Elements of Information Theory . John Wiley & Sons, Inc., 1991. A PPENDIX
A. Proof of Theorem 1
In order to prove Theorem 1, we will derive the more general achievable rate region for an arbitrary numberof relay nodes N . The proof is done in two steps: at first we derive the rates for the discrete memoryless relaychannel and apply then the derived rates to the Gaussian system model introduced in Section II. The achievable August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 12 rates for the half-duplex discrete memoryless relay channel are an application of [13, Theorem 3], which derivesthe DF rates for the full-duplex relay channel and are described by the following corollary:
Corollary 2:
Using the partial decode-and-forward strategy presented in Section III-A we achieve any rate R = sup p ∈P DF N +1 X m =1 R s,m , (28)which satisfies R s, ≤ min l ∈ [1: N +1] I (cid:18) M s , U s, ; Y l (cid:12)(cid:12) n V [ i : N ] ,i o li =1 , M [1: N ] (cid:19) + l − X j =1 I (cid:18) M j , V j, ; Y l (cid:12)(cid:12) n V i, [1; i ] o li = j +1 , V [ l : N ] , [1: l ] , M [ j +1: N ] (cid:19) (29) R s,m ≤ min l ∈ [ m : N +1] I (cid:18) U s,m ; Y l (cid:12)(cid:12) U s, [1: m − , n V [ i : N ] ,i o li =1 , M [0: N ] (cid:19) + l − X j = m I (cid:18) V j,m ; Y l (cid:12)(cid:12) V j, [1: m − , n V i, [1: i ] o li = j +1 , V [ l : N ] , [1: l ] , M [ j : N ] (cid:19) (30)for m ∈ [2 : N + 1] . The set P DF is the set of all joint pdf of the form p (cid:16) y [1: N +1] , u s, [1: N +1] , v l ∈ [1: N ] , [1: l ] , m [0: N ] (cid:17) = p (cid:16) y [1: N +1] (cid:12)(cid:12) u s, [1: N +1] , v l ∈ [1: N ] , [1: l ] (cid:17) · N Y l =0 p (cid:16) m l (cid:12)(cid:12) m [ l +1: N ] (cid:17) · N Y l =1 l Y k =1 p (cid:16) v l,k (cid:12)(cid:12) v l, [1: k − , v [ l +1: N ] ,k , m [ l : N ] (cid:17) · N +1 Y k =1 p (cid:16) u s,k (cid:12)(cid:12) u s, [1: k − , v l ∈ [ k : N ] ,k , m { s, [ k : N ] } (cid:17) . (31) Proof:
Using [13, Theorem 3] we apply the substitutions U s, (U s, , M s ) and V l, (V l, , M l ) andremove the CF part, yielding the joint pdf in (31). Eq. (29) can be slightly simplified by modifying (31) such thatthe Markov condition M s ↔ U s, ↔ U s, [2: N +1] is satisfied (and similarly for all relay messages) which yields theresults given in [6].While the rates for the discrete memoryless channel are easily formulated using a simple modification of thefull-duplex channel, the derivation for the Gaussian setup is more intricate as the random channel access must beappropriately modeled. Consider the following, more general formulation of Theorem 1: Theorem 3:
The achievable rate R = sup p ∈P DF N +1 X k =1 R s,k (32)in the Gaussian half-duplex relay network using partial decode-and-forward, a random transmission schedule, anda specific power assignment must satisfy R s, ≤ min l ∈ [1: N +1] Q s,l ) ( L ) + l − X j =1 Q j,l ) ( L j +1 ) (33) R s,k ≤ min l ∈ [ k : N +1] Q k ( s,l ) ( L ) + l − X j = k Q k ( j,l ) ( L j ) (34)where L l = [ l : N ] is the set of nodes for which the state is known. The supremum in (32) must be applied overall those joint pdf satisfying the individual power constraints and state probabilities as described in Section III-A. August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 13
Proof:
Let Y n ∼ CN (0 , σ ) with Y = A + j B such that A = ℜ (Y) and B = ℑ (Y) . We can give the pdf of Y as follows p ( y ) = 1 πσ exp (cid:18) − a + b σ (cid:19) . (35)The entropy of Y is given by h (Y) = − Z Y p ( y ) log ( p ( y )) d y. (36)Now let a = r cos ϕ , b = r sin ϕ , and r = √ a + b . Using these substitutions we have h (Y) = − ∞ Z π Z p ( y ) log ( p ( y )) d ϕ d r (37) = − π ∞ Z p ( r ) log ( p ( r )) d r (38)with p ( r ) = 1 πσ exp (cid:18) − r σ (cid:19) . (39)following from the fact the complex Gaussian distribution is circularly symmetric. Now let r ′ = r we have h (Y) = − π ∞ Z p ( r ′ ) log ( p ( r ′ )) d r ′ (40)with p ( r ) = 1 πσ exp (cid:18) − r ′ σ (cid:19) . (41)Now let Γ k ( l,l ′ ) (cid:16) m [0; N ] (cid:17) denote the overall received power at node l ′ for message level k sent by node l as afunction of the actual realization of the individual channel states m [0; N ] . Let further σ k ( l,l ′ ) (cid:16) m [0;2] (cid:17) denote thevariance of Y l ′ when all V l, [1; k ] and V j, [1; l ′ ] , for j ∈ [ l + 1; N ] , are known and were subtracted from Y l ′ : σ k ( s,l ) (cid:16) m [0: N ] (cid:17) = Γ [ k +1: N +1]( s,l ) (cid:16) m [0: N ] (cid:17) + Γ [ l +1: j ]( j ∈ [ l +1: N ] ,l ′ ) (cid:16) m [0: N ] (cid:17) + N l , (42) σ k ( j,l ) (cid:16) m [0: N ] (cid:17) = Γ [1: N +1]( s,l ) (cid:16) m [0: N ] (cid:17) + Γ [1: j ′ ]( j ′ ∈ [1: j − ,l ) (cid:16) m [0: N ] (cid:17) +Γ [ k +1: j ]( j,l ) (cid:16) m [0: N ] (cid:17) + Γ [ l +1: j ′ ]( j ′ ∈ [ l +1: N ] ,l ) (cid:16) m [0: N ] (cid:17) + N l , (43) Γ k ( j,l ) (cid:16) m [0: N ] (cid:17) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ′ ∈{ s, [ k : j ] } m j ′ = T ) · (cid:16) h j ′ ,l q α ( j ′ ,j ) ,k P t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (44)Then, the differential entropy for the channel output Y l ′ if the channel states of nodes L are known and L =[0; N ] \ {L , l } are unknown is given by h k ( l,l ′ ) ( m L ) = − π ∞ Z p k ( l,l ′ ) ( y, m L ) log (cid:16) p k ( l,l ′ ) ( y, m L ) (cid:17) d y (45)with p k ( l,l ′ ) ( y ′ , m L ) = X m L ∈M L p ( m L | m L ) 1 πσ k ( l,l ′ ) (cid:16) m [0; N ] (cid:17) exp − y ′ σ k ( l,l ′ ) (cid:16) m [0; N ] (cid:17) . (46) August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 14
Since I (cid:0) X; Y (cid:12)(cid:12) Z (cid:1) = h (cid:0) Y (cid:12)(cid:12) Z (cid:1) − h (cid:0) Y (cid:12)(cid:12) X , Z (cid:1) we can state I (cid:18) U s ; Y l (cid:12)(cid:12) n V i [ i : N ] o li =1 , M [1: N ] (cid:19) = Q s,l ( L ) (47) I (cid:18) V j ; Y l (cid:12)(cid:12) n V [1: i ] i o li = j +1 , V [1: l ][ l : N ] , M [ j +1: N ] (cid:19) = Q j,l ( L j +1 ) (48) I (cid:18) U ks ; Y l (cid:12)(cid:12) U [1: k − s , n V i [ i : N ] o li =1 , M { s, [1: N ] } (cid:19) = Q ks,l ( L ) (49) I (cid:18) V kj ; Y l (cid:12)(cid:12) V [1: k − j , n V [1: i ] i o li = j +1 , V [1; l ][ l : N ] , M [ j : N ] (cid:19) = Q kj,l ( L j ) , (50)which is sufficient to apply the results of the discrete memoryless half-duplex relay channel in (29) and (30) to theGaussian half-duplex relay channel. B. Proof of Theorem 2
In the same way as we proved the previous theorem, we derive again at first the achievable data rates for thediscrete memoryless channel using the same regular CF approach as explained in Section III-B. Afterwards, weuse the derived rates and apply them to the Gaussian system model.
Lemma 1:
The regular encoding CF achieves any rate R ≤ sup p ∈P CF I (cid:16) X s ; ˆY [1: N ] , Y d (cid:12)(cid:12) X [1: N ] , M [0: N ] (cid:17) , (51)subject to ∀ l ∈ [0 : N −
1] : I (cid:16) ˆY N − l ; Y N − l (cid:12)(cid:12) M [0: N ] (cid:17) ≤ I (cid:16) ˆY N − l , X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [0: N ] (cid:17) , (52)and with the supremum over the set P CF of all joint pdf of the form p (cid:16) y [1: N +1] , x [0: N ] , ˆ y [1: N ] , m [0: N ] (cid:17) = p (cid:16) y [1: N +1] (cid:12)(cid:12) x [0: N ] , m [0: N ] (cid:17) · N Y l =1 p (cid:16) ˆ y l (cid:12)(cid:12) y l , m [0: N ] (cid:17) · p (cid:16) x l (cid:12)(cid:12) m [0: N ] (cid:17) . (53) Proof:
Let ∆ l denote the rate of the quantization at node l , then we know from rate distortion theory [21,Ch. 13] that it is lower bounded by ∆ l ≥ I (cid:16) ˆY l ; Y l (cid:12)(cid:12) M [0: N ] (cid:17) . (54)To decode the quantization index of node N − l corresponding to the destination channel output in block b − l − ,the destination searches for a quantization that is jointly typical with its channel output, the quantizations of theprevious nodes, and the broadcast message transmitted by node N − l . More formally, it searches for an index ∃ ˆ q b − lN − l : ˆ q b − lN − l = (cid:26) ˜ q b − lN − l : (cid:16) ˆ y N − l (cid:0) ˜ q b − lN − l (cid:1) , (cid:8) ˆ y N − l ′ (cid:0) q b − lN − l ′ (cid:1)(cid:9) l − l ′ =0 , (cid:8) x N − l ′ (cid:0) q b − l − N − l ′ (cid:1)(cid:9) ll ′ =0 , y d ( b − l − (cid:17) ∈ A ∗ ( n ) ǫ (cid:27) ∩ (cid:26) ˜ q b − lN − l : (cid:16) x N − l (cid:0) ˜ q b − lN − l (cid:1) , (cid:8) ˆ y N − l ′ (cid:0) q b − l +1 N − l ′ (cid:1)(cid:9) l − l ′ =0 , (cid:8) x N − l ′ (cid:0) q b − lN − l ′ (cid:1)(cid:9) l − l ′ =0 , y d ( b − l ) (cid:17) ∈ A ∗ ( n ) ǫ (cid:27) , where A ∗ ( n ) ǫ is the ǫ -strongly typical set. The requirement of strong typicality arises from the necessity to applythe Markov lemma [21, Lemma 14.8.1] to prove joint typicality. The previous equation can only be fulfilled iff August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 15 (54) holds and the quantization rate is upper bounded by ∆ N − l ≤ I (cid:16) ˆY N − ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) + I (cid:16) X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [0: N ] (cid:17) ≤ I (cid:16) ˆY N − l , X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [0: N ] (cid:17) . Similarly the destination decodes in block b the source message transmitted in block b − N iff (51) holds. Usingstandard methods extensively discussed in literature [21] and in the previous section, (52) and the proof forachievability follow.Using the previous lemma, we can now derive the achievable rates for the Gaussian system model. Theorem 4:
The regular CF approach achieves in a Gaussian system model any rate R ≤ sup p ∈P CF X m [0: N ] ∈M [0: N ] p (cid:16) m [0: N ] (cid:17) log (cid:13)(cid:13)(cid:13) K s, [1: N ] (cid:16) m [0: N ] (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K ∅ , [1: N ] (cid:16) m [0: N ] (cid:17)(cid:13)(cid:13)(cid:13) , (55)subject to X m [0: N ] ∈M [0: N ] : m N − l = L p (cid:16) m [0: N ] (cid:17) " C Γ ([ N − l : N ] ,N − l ) Γ ([0: N − l − ,N − l ) + N N − l, + N N − l ! +log (cid:13)(cid:13) K [0: N − l − , [ N − l : N ] (cid:13)(cid:13) N N − l, (cid:13)(cid:13) K [0: N − l − , [ N − l +1: N ] (cid:13)(cid:13) ! ≤ X m [0: N ] ∈M [0: N ] : m N − l = T p (cid:16) m [0: N ] (cid:17) log (cid:13)(cid:13) K [0: N − l ] , [ N − l +1: N ] (cid:13)(cid:13)(cid:13)(cid:13) K [0: N − l − , [ N − l +1: N ] (cid:13)(cid:13) ! (56)with the symmetric Matrix K defined as follows: (cid:2) K L , L ′ (cid:0) m [0: N ] (cid:1)(cid:3) , = Γ ( L ,d ) + N d (57) (cid:2) K L , L ′ (cid:0) m [0: N ] (cid:1)(cid:3) j +1 ,j +1 = Γ ( L ,j ) + N j, + N j (58) (cid:2) K L , L ′ (cid:0) m [0: N ] (cid:1)(cid:3) ,j +1 = Γ L , ( d,j ) (59) (cid:2) K L , L ′ (cid:0) m [0: N ] (cid:1)(cid:3) j ′ +1 ,j +1 = Γ L , ( j ′ ,j ) (60)where j ∈ L ′ \ { j ′ ∈ L : m j ′ = T } . The supremum in (55) is over all p ∈ P CF satisfying (53) as well as the powerconstraints given in Section III-B. Proof:
Eq. (51) can be rewritten as R ≤ I (cid:16) X s ; ˆY [1: N ] , Y d (cid:12)(cid:12) X [1: N ] , M [0: N ] (cid:17) (61) = H (cid:16) ˆY [1: N ] , Y d (cid:12)(cid:12) X [1: N ] , M [0: N ] (cid:17) − H (cid:16) ˆY [1: N ] , Y d (cid:12)(cid:12) X [0: N ] , M [0: N ] (cid:17) . (62)The variance of the r.v. in the former term is expressed by K s, [1: N ] (cid:16) m [0: N ] (cid:17) and of the second term by K ∅ , [1: N ] (cid:16) m [0: N ] (cid:17) .Hence, if we use both terms in the previous equation and sum over all possible joint node states we obtain (55)defining the maximum achievable rate. Consider now the side constraint in (52), which can be reformulated as August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 16 follows: I (cid:16) ˆY N − l ; Y N − l (cid:12)(cid:12) M [ s : N ] (cid:17) ≤ I (cid:16) ˆY N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [ s : N ] (cid:17) + I (cid:16) X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [ s : N ] (cid:17) (63) I (cid:16) X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [ s : N ] (cid:17) ≥ I (cid:16) ˆY N − l ; Y N − l (cid:12)(cid:12) M [ s : N ] (cid:17) − I (cid:16) ˆY N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [ s : N ] (cid:17) (64)Let us pay particular attention to the l.h.s of (64), which can be reformulated to I (cid:16) X N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [0: N ] (cid:17) = X m [0: N ] ∈M [0: N ] : m N − l = T p (cid:16) m [0: N ] (cid:17) · (cid:18) H (cid:16) ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l +1: N ] , M [0: N ] = m [0: N ] (cid:17) − H (cid:16) ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] = m [0: N ] (cid:17)(cid:19) . (65)If we again apply the definitions in (57)-(60), we obtain the l.h.s of (56). Now, consider the r.h.s. of (64), whichcan be reformulated tor.h.s of (64) = I (cid:16) ˆY N − l ; Y N − l (cid:12)(cid:12) M [0: N ] (cid:17) − I (cid:16) ˆY N − l ; ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) = H (cid:16) ˆY N − l (cid:12)(cid:12) M [0: N ] (cid:17) − H (cid:16) ˆY N − l (cid:12)(cid:12) Y N − l , M [0: N ] (cid:17) − H (cid:16) ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) + H (cid:16) ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , ˆY N − l , M [0: N ] (cid:17) = H (cid:16) ˆY N − l (cid:12)(cid:12) M [0: N ] (cid:17) − H (cid:16) ˆY N − l (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) − H (cid:16) ˆY [ N − l +1: N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) + H (cid:16) ˆY [ N − l : N ] , Y d (cid:12)(cid:12) X [ N − l : N ] , M [0: N ] (cid:17) − H (cid:16) ˆY N − l (cid:12)(cid:12) Y N − l , M [0: N ] (cid:17) Again, if we apply the definitions of (57)-(60), we obtain the r.h.s of (56).
August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 17 Y E n c od e r p ( y , y | w ) W g Y g ′ ˆW Y ˆˆY g Y g ′ ˆW Y ˆˆY (a) Irregular encoding Y E n c od e r p ( y , y | w ) W g Y Y ˆˆY g Y Y ˆˆY (b) Regular encodingFig. 1. Two different strategies for CF with multiple receivers. Phase 1, [1; n ] :( x s, with rate R DF ) s dx s, x x x s, Phase 2, [ n + 1; n ] :( x s, with rate R CF ) s dx x s, Fig. 2. Example for a half-duplex channel with two alternately transmitting relay nodes. The solid lines indicate actual information exchangewhile the dashed line indicates the interfering transmission from node to . Relay : x ( q b ) n = n · p (Phase 1) y ( b ) ˆy ( q b +12 ) x ( q b +12 ) n = n · p (Phase 2)Relay : y ( b ) x ( q b ) x ( q b ) Source: x s, ( q bs, ) x s, ( q bs, ) Fig. 3. Coding structure for the combined strategy with N = 2 alternately transmitting relays. s d r r Fig. 4. Setup for our analysis.
August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 18 − . − . − . . . . distance r R i nb it s p e r c h a nn e l u s e ( bp c u ) bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcrs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rsld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ρ s,d = 10 dB ρ s,d = 16 dB Full reuseNo reuseUpper boundCombined strategy rs Compress-and-Forward ld Partial DF ut Decode-and-Forward bc Single Hop (a) Coherent transmission − . − . − . . . . distance r R i nb it s p e r c h a nn e l u s e bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcrs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rsld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ρ s,d = 10 dB ρ s,d = 16 dB Full reuseNo reuseUpper boundCombined strategy rs Compress-and-Forward ld Partial DF ut Decode-and-Forward bc Single Hop (b) Non-coherent transmissionFig. 5. Achievable rates for the Gaussian half-duplex two-relay channel. Solid curves indicate a fixed transmission strategy and dashed linesindicate a random transmission schedule. ρ s,d = 16 dB again indicates the power-normalized case. August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 19 − − . − . . . . distance r R i nb it s p e r c h a nn e l u s e ( bp c u ) bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bcld ld ld ld ld ld ld ld ld ld ld ld ρ s,d = 10 dB ρ s,d = 13 dB Full reuseNo reuseUpper boundCompress-and-Forward ld Decode-and-Forward bc Single Hop (a) Half-duplex Network − − . − . . . . distance r R i nb it s p e r c h a nn e l u s e ( bp c u ) bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ld ldrs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs ρ s,d = 10 dB ρ s,d = 13 dB Upper boundCompress-and-Forward ld Decode-and-Forward bc Single Hop (b) Full-duplex NetworkFig. 6. Achievable rates for the Gaussian single-relay channel with non-coherent transmission. Solid curves indicate fixed transmission strategyand dashed lines indicate a random transmission schedule. ρ s,d = 13 dB again indicates the power-normalized case. August 20, 2018 DRAFTUBMITTED TO IEEE TRANS. COMM. 20 number of relays N R i nb it s p e r c h a nn e l u s e ( bp c u ) bc bc bc bc bc bc bcbc bc bc bc bc bc bcld ld ld ld ld ld ld Upper boundCompress-and-Forward ld Decode-and-Forward bc Single-Hop
Fig. 7. Achievable rates depending on the network size. Solid curves indicate fixed transmission strategy a-priori known to all nodes, anddashed lines indicate a random transmission schedule which is chosen randomly at each node. Results for single-hop are power-normalized suchthat the power introduced by additional relay nodes is also used in case of single-hop. path loss exponent θ R i nb it s p e r c h a nn e l u s e ( bp c u ) bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bcld ld ld ld ld ld ld ld ldbc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bcld ld ld ld ld ld ld ld ld N = 3 N = 1 Upper boundCompress-and-Forward ld Decode-and-Forward bc Single-Hop
Fig. 8. Influence of path loss on the achievable rates for N = 1 and N = 3 relay nodes which are distributed in equal distances betweensource and destination. Dashed curves indicate a random transmission schedule and solid lines a fixed schedule.relay nodes which are distributed in equal distances betweensource and destination. Dashed curves indicate a random transmission schedule and solid lines a fixed schedule.