Partial decoding for synchronous and asynchronous Gaussian multiple relay channels A. del Coso, C. Ibars
Publication: Vol.: No.: Date:
in Proc. IEEE International Conference in Communications Jun., 2007
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Partial Decoding for Synchronous and Asynchronous Gaussian Multiple Relay Channels Aitor del Coso and Christian Ibars Centre Tecnol`ogic de Telecomunicacions de Catalunya (CTTC) Parc Mediterrani de la Tecnologia – 08860 – Castelldefels – Barcelona (Spain) email:
[email protected],
[email protected]
Abstract— Relaying diversity is a key technology to increase capacity in wireless networks. In this paper, the capacity of a single source-destination wireless channel, aided by a set of N potential relays, is studied. For such a link (analyzed under fullduplex and half-duplex constraint), we derive a capacity upper bound based upon the max-flow-min-cut theorem stated by Cover in [1]. Furthermore, a lower bound is proposed considering the achievable with partial decoding at the relay nodes. To maximize the later, optimum relay selection is carried out. Both bounds are then applied to Gaussian channels with channel knowledge at both transmitter and receiver sides, and two synchronization modes are addressed: the synchronous mode, where relays perform coherent beamforming, and the asynchronous mode. Results show that, surprisingly, partial decoding is more spectrally efficient for the half-duplex scheme, becoming the capacity achieving technique for low number of relays.
I. I NTRODUCTION The wireless multiple relay channel consists of a single source-destination (s, d) pair aided in its communication by a set of N wireless relay nodes (see Fig. 1). Such communication scheme has been shown to increase the robustness of wireless links [2], as well as the capacity of wireless networks [3], when appropriately allocating network resources [4]. The use of relays to increase the spectral efficiency of wireless communications was first proposed by Cover and Gamal in [5]. In that work, the discrete single-relay channel was analyzed in terms of achievable rates and capacity upper bounds. Decode-and-forward, compress-and-forward and partial decoding techniques were firstly proposed therein, and theoretical milestones presented: i) block-Markov coding/ backward decoding to deploy decode-and-forward relaying, ii) application of the max-flow-min-cut theorem to upper bound the capacity, and iii) use of partial decoding (PD) to increase the spectral efficiency. Our contribution focusses on this former technique, applied to communication channels with more than one relay. With PD, the relay nodes just partially decode the source data, which turns out a definite advantage when the amount of decoded data can be adapted to the sourcerelays channel quality. This technique was shown to achieve capacity for single-relay channels with half-duplex constraint and high SNR [4], and also, claimed to be capacity achieving for single-relay channels with orthogonal components [6], [7]. This work was partially funded by Generalitat de Catalunya under grant SGR-2005-00690, and by the European Comission under project IST-4027402 (WIP).
(a) Full-duplex mode Fig. 1.
(b) Half-duplex mode
Multiple relay channel
The extension of the single-relay channel in [5] to the scenario with multiple parallel relays has attracted huge efforts [3], [8]–[10]. For this channel under AWGN, Gastpar showed in [3] that the max-flow-min-cut bound scales as the logarithm of the number of relays. However, few achievable rates in AWGN have been reported so far: for N relays with orthogonal components, decode-and-forward is studied in [9], and for N -relay case with half duplex constraint is analyzed in [11]. The aim of this paper is, therefore, to contribute with a new achievable rate for the multiple relay channel, based upon partial decoding at the relay nodes. Two multiple relay channels are analyzed in this paper: the full-duplex scheme in Fig. 1(a), and the time-division half-duplex scheme in Fig. 1(b). For the later, the communication is arranged into two consecutive, non-identical time slots: the first slot accounting for SIMO1 communication s → d, 1, · · · , N and the second for MISO transmission s, 1, · · · , N → d. Our contribution upper bounds the capacity of both channels by means of the max-flow-min-cut theorem, and derives their achievable rates considering partial decoding at the relay nodes. We also derive the optimum relay selection. Later, the proposed capacity bounds are applied to Gaussian channels, assuming an overall energy constraint. In our analysis, two synchronization scenarios are addressed: i) the synchronous mode, where source and relays have perfect symbol and phase synchronization, and ii) the asynchronous mode, where wireless nodes have symbol but no phase synchronization due to hardware limitations. The remainder of this paper is organized as follows: in Section II we introduce the channel model and notation. In 1 SIMO = Single Input Multiple Output; MISO = Multiple Input Single Output
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Section III we derive capacity theorems for the multiple relay channel based upon max-flow-min-cut and partial decoding. The application of the capacity bounds to full-duplex and halfduplex Gaussian channels is carried out in Sections IV and V, respectively. Finally, Section VI describes numerical results and Section VII summarizes conclusions.
III. C APACITY T HEOREMS In this section, we derive capacity theorems for full-duplex and half-duplex multiple relay channels. Theorem 1: The capacity of full-duplex multiple relay channels with N potential relays is lower and upper bounded by C ≥ max min {I (Xs , X1:n ; Yd ) , 1≤n≤N
I (U ; Yn |X1:n ) + I (Xs ; Yd |U, X1:n )} (3)
II. M ODEL AND N OTATION Topologies depicted in Fig. 1(a) and Fig. 1(b) are both {N + 2}-Relay networks, with a source terminal s, a destination terminal d, and a set of potential relays N = {1, 2, · · · , N }. Time invariant and memoryless channels among network nodes are considered and, as shown in the Fig. 1, no wireless connectivity among relays. In the full duplex model (See Fig. 1(a)), Xs denotes the complex signal transmitted by the source; Xi , the complex signal transmitted by relay i ∈ N ; and Yd and Yi , the channel output at the destination node and at relay i ∈ N , respectively. √ √ √ We define as,d , as,i and ai,d , as the complex channel coefficients from source to destination, source to relay i ∈ N and relay i to destination, respectively. Furthermore, Zd is the additive noise at the receiver node, and Zi the additive noise at relay i ∈ N . Thus, full-duplex signal model remains: Yi =
√
as,i · Xs + Zi for i ∈ N N √ √ ai,d · Xi + Zd . Yd = as,d · Xs +
(1)
i=1
For the half-duplex model of Fig. 1(b), we assume a timedivision scheme with one slot of duration α and a second slot of duration 1 − α. During the first slot, the source communicates with relays and destination. Next, during the second time slot, source and relays jointly transmit to destination. In this case, Xst denotes the complex signal transmitted by the source during slot t ∈ {1, 2} and Xi2 the complex signal transmitted by relay i ∈ N during the second slot; Ydt is the channel output at destination node during time slot t ∈ {1, 2}, and Yi1 , the channel output at relay i ∈ N during the first slot. Finally, we define Zdt as additive noise at the destination node during slot t ∈ {1, 2}, and Zi1 as the additive noise at relay i ∈ N during the first slot. Thus, the signal model for the half-duplex scheme remains: √ as,i · Xs1 + Zi1 for i ∈ N √ 1 1 1 Yd = as,d · Xs + Zd √ √ Yd2 = as,d · Xs2 + N ai,d · Xi2 + Zd2 . i=1
C ≤ min {I (Xs , X1:N ; Yd ) , I (Xs ; Y1:N , Yd |X1:N )}
(4)
where source-relays path gains have been ordered as: as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N ,
(5)
and both bounds are maximized over all joint probability mass functions p (xs , x1 , · · · , xN , u). Remark 1: n in the Theorem denotes the cardinality of the decoding set (DS), i.e, the number of relays that are able to decode (their intended message) and retransmit. Due to the broadcast nature of the source transmission, a DS with cardinality n is composed of the n relays with the highest source-relay channels. Remark 2: Maximization over n obtains the optimum number of relays to belong to the DS, and so performs the relay selection. Notice that the left hand side of minimization in (3) increases with the cardinality n, while the right hand side decreases. Therefore, the optimum relay selection n∗ is the one who better trade among the two of them, and it is composed of the n∗ relays with the best source-relay channels. Proof: See Appendix I for details. Theorem 2: The capacity of half-duplex multiple relay channels with N potential relays is lower and upper bounded by C ≥ max max min I Xs , X21:n ; Yd , 1≤n≤N α (6) α · I Xs1 ; Yn1 + (1 − α) · I Xs2 ; Yd2 |U, X21:n 2 C ≤ max min I Xs , X1:N ; Yd , α I Xs ; Y11:N , Yd |X21:N (7) where source-relays path gains have been ordered as: as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N ,
(8)
and both bounds are maximized over all joint probability mass functions p x1s , x2s , x21 , · · · , x2N , u . Proof: See Appendix II for details.
Yi1 =
IV. F ULL -D UPLEX G AUSSIAN C HANNELS (2) T
Notation: for model, X1:n = [X1 , · · · , Xn ] √the full-duplex √ ∈ {1, · · · , N }. For and h1:n = a1,d , · · · , an,d with n T 1 2 T the half-duplex model, X = X , X , Yd = Yd1 , Yd2 , s s s T √ √ X21:n = X12 , · · · , Xn2 and h1:n = a1,d , · · · , an,d with n ∈ {1, · · · , N }. Moreover, in the paper I (A; B) denotes mutual information between random variables A and B, C (x) = log2 (1 + x), b† denotes the conjugate transpose of vector b and b∗ denotes the conjugate of b.
In the following, we extend result in Theorem 1 to bound the capacity of Gaussian full-duplex multiple relay channels. We consider random variables {Zd , Z1 , · · · , ZN } as independent, identically distributed (i.i.d.) AWGN with power σo2 , and we use the following energy constraint: N E {Xs · Xs∗ } + i=1 E {Xi · Xi∗ } =γ . (9) σo2 As mentioned earlier, this paper addresses two synchronization modes: i) the synchronous mode, which assumes transmit
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CSI at every network node, and perfect phase synchronization among transmitters, and ii) the asynchronous mode, which considers transmit channel knowledge but no phase synchronization due to hardware limitations. Proposition 1 and Proposition 2 analyzes first and second scenario, respectively. Proposition 1: The capacity of full-duplex, synchronous, Gaussian multiple relay channels with N potential relays, full channel state information and overall power constraint, is lower and upper bounded by n C ≥ max max min {C ((as,d + β (1 − η) i=1 ai,d ) γ) , (10) 1≤n≤N (η,β)
γ C as,n βη + C (as,d (1 − β) γ) 1 + as,n (1 − β) γ
On the other hand, at the relay node n, the received signal is √a ∗ √ s,d √ Xbf + Xnew + as,n Xsd + Zd , (15) Yn = as,n |h| U
where U , as defined in Appendix I, is the part of the source message that accounts for the relayed communication. Therefore, we derive: I (U ; Yn |X1:n ) = I (Xnew ; Yn |Xbf )
∗ a E{Xnew Xnew } = C σ2s,n ∗ o +as,n E {Xsd Xsd }
γ = C as,n βη . 1 + as,n (1 − β) γ
(16)
N C ≤ max min C as,d + ρ i=1 ai,d γ , (11) Finally, from (13), we obtain: ρ
N C as,d + i=1 as,i (1 − ρ) γ I (Xs ; Yd |U, X1:n ) = I (Xsd ; Yd |Xnew , Xbf ) = C (as,d (1 − β) γ) . (17) where 0 ≤ η ≤ 1, 0 ≤ β ≤ 1, 0 ≤ ρ ≤ 1 and source-relay path Applying Theorem 1, this shows the achievable rate given gains have been ordered as: the subset of relays Rn . However, noting that we may select (12) the subset from {R1 , · · · , RN }, and we may optimize the as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N . Remark 3: Remarks 1 and 2 also apply here. power allocation on codes, i.e., the values of η and β, then Remark 4: Optimization problem in (10) is not an straight- it concludes the proof for the lower bound. The upper bound forward task and requires a separate analysis. However, due to is straightforward derived applying (4) to Gaussian channels, in [13, Appendix A] and defining space limitations, we postpone this analysis and solution (as using standard arguments ∗ well as for the other optimization problems in the paper) to ρ = | E{Xs Xi } ∗ |2 for all i ∈ [1, · · · , N ] with E{Xs Xs∗ }E {Xi Xi } further contribution. Hereafter, we assume that optimizations E{Xi Xj } | | = 1 for all i, j ∈ [1, · · · , N ]. are solved numerically. E {Xi Xi∗ }E {Xj Xj∗ } Proof: Let us assume, as previously, that only the subset Proposition 2: The capacity of full-duplex, asynchronous, Rn = {1, 2, · · · , n} ⊂ N is active, with relays ordered as (12), Gaussian multiple relay channels with N potential relays, full and let us construct a coding scheme with three independent channel state information and overall power constraint, is lower Gaussian codebooks. The first one, Xsd ∼ CN (0, (1 − β) P ), and upper bounded by accounts for the part of the source message directly transmitted max max (18) to destination without relaying. The other two, Xnew and Xbf , C ≥ 1≤n≤N (η,β)
account for the relayed communication, and define a block+ Markov coding scheme such as that presented in [5, Theorem min C as,d (1 − β + βη) + a[s,1:n],d β (1 − η) γ ,
1]: Xnew ∼ CN (0, βηP ) is sent from source to relays (and γ C as,n βη + C (as,d (1 − β) γ) destination) to contribute with new information, while Xbf ∼ 1 + as,n (1 − β) γ CN (0, β (1 − η) P ) is jointly sent by source and relays to
destination to relay data. We define P = γσo2 as the total transmin C as,d (1 − ρ) + a+ · ρ γ , (19) mitted power, and we notice that the overall transmitted power C ≤ max [s,1:n],d ρ
satisfies (9). Considering transmit channel knowledge at every N C as,d + i=1 as,i (1 − ρ) γ network node, and perfect phase synchronization, the optimum way to jointly transmit Xbf is by implementing √ an optimum as,d , h1:n , the beamforming [12]. Therefore, defining h = received signal at the destination node is: Yd = h
h† √ √ Xbf + as,d Xnew + as,d Xsd + Zd . |h|
(13)
Therefore, the first term of (3) is rewritten as: I (Xs , X1:n ; Yd ) = I (Xsd , Xnew , Xbf ; Yd )
∗ ∗ ∗ a E{Xsd Xsd }+as,d E{Xnew Xnew }+|h|2 E{Xbf Xbf } = C s,d 2 σo n = C ((as,d + β (1 − η) i=1 ai,d ) γ) . (14)
where 0 ≤ η ≤ 1, 0 ≤ β ≤ 1, 0 ≤ ρ ≤ 1, a+ [s,1:n],d = maxt={s,1,···,n} at,d and source-relay path gains have been ordered as: as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N . (20) Proof: The sketch of the proof for the lower bounds is equivalent to that in Proposition 1. However, one difference arises: since no phase synchronization is possible among network nodes, the transmission of Xbf can not be based upon the optimum beamforming. In fact, the optimum strategy for transmitting Xbf , considering transmit CSI and no phasesynchronization, is to force the node (source or relay belonging
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to Rn ) with better channel to destination to individually transmit the codeword. We skip the proof due to space limitations. Therefore, adapting (13), the received signal at the destination node is: √ √ as,d Xnew + as,d Xsd + Zd . (21) Yd = a+ [s,1:n],d Xbf + with a+ [s,1:n],d = maxt={s,1,···,n} at,d . Considering the received output, we can rewrite (14) to prove the lower bound in Proposition 2. The same consideration has to be done for the upper bound.
U
V. H ALF -D UPLEX G AUSSIAN C HANNELS Bounds on the capacity of Gaussian half-duplex multiple relay channels are derived here using results in Theorem we consider all random variables 1 2.2 As1 previously, 1 as i.i.d. AWGN with power σo2 . MoreZd , Zd , Z1 , · · · , ZN ∗ ∗ E {Xs1 (Xs1 ) } E {Xs2 (Xs2 ) } and γ2 = + over, we define γ1 = 2 2 σ σ o o N 2 2 ∗ E {Xi (Xi ) } i=1 as the transmitted powers (normalized with σo2 respect to the noise level) over the first and second time slots, respectively, and we use the following energy constraint: α · γ1 + (1 − α) · γ2 = γ .
two and accounting for the part of the source message directly transmitted to destination without relaying. We have defined Pi = γi σo2 , i ∈ {1, 2}. As in Proposition 1, the optimum solution to transmit √ Xbf is the optimum beamforming. Thus, as,d , h1:n , the received signal at relay and defining h = destination nodes during both slots reads: √ Yn1 = as,n Xnew + Zn1 √ Yd1 = as,d Xnew + Zd1 h† √ Xbf + as,d Xsd + Zd2 . Yd2 = h (26) |h| Being U as defined in Appendix II. Therefore, we may rewrite: I Xs , X21:n ; Yd = αI Xnew ; Yd1 + (1 − α) I Xbf , Xsd ; Yd2 = α · C (as,d γ1 ) n + (1 − α) · C ((as,d + β i=1 ai,d ) γ2 ) 1 1 = I Xnew ; Yn1 I Xs ; Yn I
Xs2 ; Yd2 |U, X21:n
(22)
= C (as,n γ1 ) = I Xsd ; Yd2 |Xbf = C (as,d (1 − β) γ2 ) .
(27)
Applying Theorem 2, it demonstrates the achievable rate for a given Rn and α. Since we may arbitrary choose both, and optimally allocate power on time slots, it concludes the proof. The upper bound is straightforward obtained from the multiple access cut and thebroadcast for Gaussian channels, defining ρ = |E {Xs2 (Xi2 )∗ }/ E{Xs2 (Xs2 )∗ }E {Xi2 (Xi2 )∗ }|2 for all i. Proposition 4: The capacity of half-duplex, asynchronous, Gaussian multiple relay channels with N potential relays, full (23)channel state information and overall power constraint, is lower and upper bounded by
As for the full-duplex channels, we analyze independently the capacity of synchronous and asynchronous transmission modes. Proposition 3: The capacity of half-duplex, synchronous, Gaussian multiple relay channels with N potential relays, full channel state information and overall power constraint, is lower and upper bounded by C ≥ max
max
1≤n≤N (α,γ1 ,γ2 ,β)
n min {α · C (as,d γ1 ) + (1 − α) · C ((as,d + β i=1 ai,d ) γ2 ) , α · C (as,n γ1 ) + (1 − α) · C (as,d (1 − β) γ2 )} s.t. α · γ1 + (1 − α) · γ2 = γ.
C ≥ max
max
1≤n≤N (α,γ1 ,γ2 ,β)
min {α · C (as,d γ1 ) +
β γ2 , (1 − α) · C as,d (1 − β) + a+ (24) [s,1:n],d
(28)
C ≤ max (α,γ1 ,γ2 ,ρ)
α · C (as,n γ1 ) + (1 − α) · C ((1 − β) as,d γ2 )} N min α · C (as,d γ1 ) + (1 − α) · C as,d + ρ i=1 ai,d γ2 , s.t. α · γ1 + (1 − α) · γ2 = γ
N α · C as,d + i=1 as,i γ1 + (1 − α) · C (as,d (1 − ρ) γ2 ) (29) C ≤ max (α,γ1 ,γ2 ,ρ) s.t. α · γ1 + (1 − α) · γ2 = γ. min {α · C (as,d γ1 ) +
where 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, 0 ≤ ρ ≤ 1, and source-relay (1 − α) · C as,d (1 − ρ) + a+ [s,1:n],d ρ γ2 , path gains have been ordered as:
N α · C as,d + i=1 as,i γ1 + (1 − α) · C (as,d (1 − ρ) γ2 ) as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N . (25) s.t. α · γ1 + (1 − α) · γ2 = γ Proof: let us consider that only the subset Rn = {1, 2, · · · , n} ⊂ N is active, with nodes ordered as in (25), and where 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, 0 ≤ ρ ≤ 1, a+ [s,1:n],d = that the first slot has a time fraction α. To achieve the lower maxt={s,1,···,n} at,d and source-relay path gains have been bound we construct a coding scheme with three independent ordered as: Gaussian codebooks: Xnew ∼ CN (0, P1 ), which is sent from source to relays (and destination) during the first slot to as,1 ≥ · · · ≥ as,n ≥ · · · ≥ as,N . (30) contribute with new information; Xbf ∼ CN (0, βP2 ), which Proof: The proof is equivalent to that in Proposition 3. is jointly sent by source and relays to destination during the However, as in Proposition 2, Xbf is individually transmitted second time slot to relay the message previously transmitted by by the node (source or relay belonging to Rn ) with better Xnew ; and Xsd ∼ CN (0, (1 − β) P2 ), transmitted during slot channel to destination (i.e., no beamforming is performed).
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VI. N UMERICAL R ESULTS
A PPENDIX I P ROOF T HEOREM 1
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UB: HD & Synchronous LB: HD & Synchronous UB: HD & Asynchronous LB: HD & Asynchronous
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2 0
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relays
Fig. 2. Upper (UB) and lower (LB) bounds on the capacity of Gaussian multiple relay channels versus the number of potential relays. SNR = 5 dB. Full−duplex
Half−duplex
4
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UB: FD & Synchronous LB: FD & Synchronous UB: FD & Asynchronous LB: FD & Asynchronous
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3.5
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SNR (dB)
Fig. 3. Upper (UB) and lower (LB) bounds on the capacity of Gaussian multiple relay channel versus transmit SNR. N = 2 potential relays. Full−duplex
Half−duplex
100
90
100
Synchronous Asynchronous
90
80
80
70
70
% of Active Relays
VII. C ONCLUSIONS In this paper the capacity of multiple relay channels was upper and lower bounded by means of the max-flow-mincut theorem and partial decoding, respectively. We derived bounds for Gaussian channels with channel knowledge at both transmitter and receiver sides, considering an overall power constraint. For both full-duplex and half-duplex operation, synchronous and asynchronous cases were analyzed. Results showed that: i) full-duplex capacity upper bounds significantly improve those for half-duplex, ii) partial decoding performs better for half-duplex channels, and iii) the number of active relay nodes is still low in all cases.
Half−duplex 6
UB: FD & Synchronous LB: FD & Synchronous UB: FD & Asynchronous LB: FD & Asynchronous
5.5
% of Active Relays
In this section, the proposed capacity bounds are computed for a Gaussian multiple relay channel under Rayleigh flat fading. All channel coefficients are defined as i.i.d. zeromean, unitary-power, complex Gaussian random variables, and assumed invariant for the entire frame duration. For all plots, ergodic results are shown, obtained by averaging over the channel distribution. Finally, optimization problems in Proposition 1, 2, 3 and 4 have been solved using exhaustive search. Fig. 2 depicts capacity bounds versus the number of potential relays N , for transmit signal-to-noise ratio (SNR) equal to 5 dB2 . Surprisingly, it is shown that the achievable rates for half-duplex (HD) relaying are greater than those for the full-duplex (FD) scheme. This is explained by noting that codeword Xsd (i.e., the information transmitted by the source and not relayed) becomes interference for relays at the FD scheme but not at the HD communication (This explanation is consistent with time-sharing arguments in [13]). Moreover, we notice that partial decoding is the capacity achieving technique for HD channels with low number of relays. Fig. 3 shows the capacity bounds versus the mean SNR, for N = 2 potential relays. First, we notice that upper and lower bounds are closer for the asynchronous case than for the synchronous mode. Indeed, for the HD asynchronous mode, both converge as SNR grows, describing the real capacity of the link. Moreover, it is shown that there is a constant rate gain between synchronous and asynchronous transmissions of 0.1 bps/Hz in the FD mode and 0.3 bps/Hz in the HD scheme. Finally, Fig. 4 shows the number of active relays (i.e., the subset Rn that maximizes the lower bound) versus the total number of potential relays. It is shown that for the FD mode the number of relays that decodes and retransmits is lower that that for HD mode. This explains the fact that HD scheme outperform FD, and it is due to Xsd as interference in the full-duplex architecture.
Full−duplex 6
60
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A. Lower Bound Achievability We consider the N potential relays in Fig. 1(a) ordered as in (5), and we assume that only the subset of relays Rn = 2 The energy constraints in (9) and (22) are defined as a transmit signal-tonoise ratii.
0 0 10
1
10
Nrelays
2
10
0 0 10
1
10
2
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Nrelays
Fig. 4. Number of active relays to achieve the lower bounds versus the total number of potential relays. SNR = 5 dB.
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{1, 2, · · · , n} ⊂ N is active. We split the message to transmit ω into two independent parts: ωd , which is directly send by the source to the destination, and ωr which is relayed through Rn using a block-Markov coding scheme. We define the signal transmitted by the source as Xs = U +V , with U the codeword that accounts for transmitting ωr and V the one that transmits ωd . For the proposed transmission scheme, all relays belonging to Rn decode codeword ωr if its transmission rate Rr is Rr ≤ min {I (U ; Yi |X1:n )} . i∈Rn
(31)
Nevertheless, considering that relays in Rn are ordered as (5), and that additive noises Zi are i.i.d., then Rr ≤ I (U ; Yn |X1:n ) .
(32)
On the other hand, the destination is able to decode ωr if (assuming block-Markov encoding [5, Theorem 1] and parallel channel decoding as in [14, Section 3.2]): Rr ≤ I (U, X1:n ; Yd ) .
Rr ≤ α · I Xs1 , Yd1 + (1 − α) · I U, X21:n ; Yd2 .
(36)
Once decoded ωr , the destination node is able to decode message ωd if its transmission rate Rd Rd ≤ (1 − α) · I Xs2 ; Yd2 |U, X21:n
(37)
Finally, the transmission rate for the proposed scheme equals R = Rr + Rd . Thus, adding (36) and (37) we obtain the first part of minimization in (6), while by adding (35) and (37) we derive the second part. This demonstrates the achievable rate for a given Rn and α. However, noting that we may select the time fraction α ∈ [0, 1], and subset Rn from the selection {R1 , · · · , RN }, then it concludes the proof.
(33)
Moreover, once decoded ωr , the destination node is able to decode message ωd if the transmission rate for that message Rd is Rd ≤ I (Xs ; Yd |U, X1:n )
where second equality follows from relay order in (8). Furthermore, considering time-division channels, the destination is able to decode ωr if:
B. Upper Bound Derivation The proof is equivalent to that in Appendix I-B.
(34)
Finally, the transmission rate equals R = Rr + Rd . Thus, adding (33) and (34) we obtain the first part of minimization in (3), while by adding (32) and (34) we derive the second part. This demonstrates the achievable rate for a given Rn . However, notice that we may choose the decoding set Rn from the selection {R1 , · · · , RN }, then it concludes the proof. B. Upper Bound Derivation The upper bound follows directly from the max-flow-mincut theorem in [1, Theorem 14.10.1], considering the multiple access cut and the broadcast cut, respectively. A PPENDIX II P ROOF T HEOREM 2 A. Lower Bound Achievability The proof is similar to Appendix I-A. Assume that the N potential relays are ordered as (8) and let α to be the fraction of time for the first slot. Additionally, assume that only the subset of relays Rn = {1, 2, · · · , n} ⊂ N is active. As in Appendix I, we split the message to transmit, ω, into two independent parts: ωd , which is directly send by the source to the destination during the second time slot, and ωr which is relayed through Rn . ωr is sent by the source during both slots, and by the relays during the second. We define the signal transmitted by the source during the second slot as Xs2 = U + V , with U the codeword that accounts for transmitting ωr and V the one that transmits ωd . During the first time slot, all relays belonging to Rn decode codeword ωr if its transmission rate Rr is Rr ≤ α · min I Xs1 ; Yi1 i∈Rn = α · I Xs1 ; Yn1 , (35)
R EFERENCES [1] T. Cover and J. Thomas, Elements of Information Theory, Wiley Series in Telecommunications, 1991. [2] J.N. Laneman and G. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. on Information Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [3] M. Gastpar and M. Vetterli, “On the capacity of large Gaussian relay networks,” IEEE Trans. on Information Theory, vol. 51, no. 3, pp. 765– 779, Mar. 2005. [4] A. Host-Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Trans. on Information Theory, vol. 51, no. 6, pp. 2020–2040, Jun. 2005. [5] T. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. on Information Theory, vol. 25, no. 5, pp. 572–584, Sep. 1979. [6] A. El Gamal and S. Zahedi, “Capacity of a class of relay channels with orthogonal components,” IEEE Trans. on Information Theory, vol. 51, no. 5, pp. 1815–1817, May 2005. [7] Y. Liang and V.V. Veeravalli, “Gaussian orthogonal relay channels: optimal resource allocation and capacity,” IEEE Trans. on Information Theory, vol. 51, no. 9, pp. 3284–3289, Sep. 2005. [8] B. Schein and R. Gallager, “The Gaussian parallel relay network,” in Proc. International Symposium on Information Theory, Sorrento, Italy, Jun. 2000, p. 22. [9] I. Maric and R. Yates, “Forwarding strategies for Gaussian parallel-relay networks,” in Proc. International Symposium On Information Theory, Chicago, IL, USA, Jun. 2004. [10] A.F. Dana, R. Gowaikar, B. Hassibi, M. Effros, M. Medard, and R. Koetter, “Should we break a wireless network into sub-networks?,” in Proc. 41st Allerton Conference on Communication, Control and Computing, Monticello, IL, USA, Oct. 2003. [11] A. del Coso and C. Ibars, “Distributed antenna channels with regenerative relaying: relay selection and asymptotic capacity,” EURASIP Journal on Wireless Communications, 2007, submitted. [12] I. Telatar, “Capacity of multi-antenna Gaussian channel,” European Trans. on Telecommunications, vol. 10, no. 6, pp. 585–595, Nov. 1999. [13] A. El Gamal, M. Mohseni, and S. Zahedi, “Bounds on capacity and minimum energy-per-bit for AWGN relay channels,” IEEE Trans. on Information Theory, vol. 52, no. 4, pp. 1545–1561, Apr. 2006. [14] A. Host-Madsen, “Capacity bounds for cooperative diversity,” IEEE Trans. on Information Theory, vol. 52, no. 4, pp. 1522–1544, Apr. 2006.
