ISIT 2009, Seoul, Korea, June 28 - July 3, 2009
Simultaneous Partial And Backward Decoding Approach for Two-Level Relay Networks Leila Ghabeli, Mohammad R. Aref Information Systems and Security Lab Department of Electrical Engineering Sharif University of Technology, Tehran, Iran Email:
[email protected], aref @sharif.edu
it achieves the capacity of new forms of degraded multiple relay networks. In [12], mixed strategy consisting of partial decoding scheme and compress-and-forward is developed for multiple relay networks. In [13] and [14], partial decode-andforward scheme was extended to multiple relay network and the capacity of multilevel semi-deterministic and orthogonal relay network were established. In [15], a comprehensive partial decoding scheme based on regular encoding/sliding window decoding analysis is proposed, despite [12]-[14], all possible partial decoding states are considered between the different parts of the messages of the source and the relays in a two-level relay network. In [16] and [17], an achievable rate was proposed for two-relay networks based on symmetric relaying, in which the relays are arranged in the same position and no priorities are considered for any of them over the others in receiving the message of the source or the message of the other relays. In this work, we propose a new achievable rate for twolevel relay networks based on using simultaneous partial and backward decoding strategy at the relays and the receiver. In the proposed method, each relay in the network and the receiver can partially decode the message parts transmitted by the previous relay and the transmitter and also these different message parts are simultaneously decoded at the relays and the receiver. Similar to [15] and in contrast with [12], we consider a comprehensive scenario where all possible partial decoding states are considered between different parts of the messages of the source and the relays in a two-level relay network. Simultaneous partial and backward decoding strategy is shown to yield better rates than the previously proposed methods based on decode-and-forward [7], partial decode-andforward based on regular encoding/sliding window decoding [15] and also the rate proposed in [12] with the omission of compress-and-forward part from it.
1Abstract- In this paper, we propose a new achievable rate for two-level relay networks by introducing a new strategy named simultaneous partial and backward decoding. In the proposed method, benefitting from regular encoding/backward decoding strategy, different message parts transmitted in the network are simultaneously decoded at the relays and the receiver. The proposed strategy is shown to achieve better rates than the previously proposed methods based on decode-and-forward and partial decode-and-forward.
I. INTRODUCTION
Decode-and-forward protocol, first introduced by Cover and El Gamal in 1979 for relay channel [1], is a relaying protocol in which the relay can decode the transmitted message by the sender. Partial decode-and-forward was defined in [2] as a special case of the proposed coding scheme by Cover and El Gamal [1, Theorem 7]. In this scheme, the relay can partially decode the transmitted message by the sender. In [3], generalized backward decoding strategies were proposed for the relay channel based on sequential backward decoding and simultaneous backward decoding. The achievable rate of the second strategy was shown to include that of [1, Theorem 7]. There have been a lot of works which apply the proposed decode-and-forward schemes in [1] and [2], to multiple relay networks [4]-[17]. In [5], Gupta and Kumar applied irregular encoding/successive decoding to multirelay networks in a manner similar to [4]. In [6] and [7], Xie and Kumar developed regular encoding/sliding-window decoding for multiple relays, and showed that their scheme achieves better rates than those of [4] and [5]. Regular encoding/backward decoding first proposed in [8], was generalized to the relay networks in [9]. The achievable rates of the two regular encoding strategies tum out to be the same, however, the delay of sliding-window decoding is much less than that of backward decoding. In [10] and [11], parity-forwarding protocol is introduced and a structured generalization of decode-and-forward strategy for multiple relay networks with feed-forward structure based on such protocol is proposed. In their method, each relay chooses a selective set of previous nodes in the network and decodes all messages of those nodes. Parity forwarding was shown to improve previous decode-and-forward strategies, and
II. THE PROPOSED ACHIEVABLE RATE FOR Two-LEVEL RELAY NETWORKS
In this section, we propose a partial decode-and-forward scheme based on simultaneous regular encoding/backward decoding. In the proposed method, each relay in the network can partially decode the message transmitted by the sender and the previous relay and also different message parts are simultaneously decoded at the relays and the receiver. Individual
1 This work was supported by Iranian National Science Foundation (INSF) under contract no. 84,5193-2006 and Iranian Telecommunication Research Center (ITRC) under contract no. 500/1892.
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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009
is directly decoded by the receiver. The message of first relay is also divided into two parts. The first part is decoded by second relay and the receiver can only make an estimate of it, while the second part is directly decoded by the receiver. The sender and the relays cooperate in next transmission blocks to remove the receiver's uncertainty about the previous parts of the message. Fig. 1. Two-relay network with the individual parts of the transmitted message by the source and the relays shown in it, as expressed in Theorem 1.
parts of the messages transmitted by the sender and the sender are shown in Fig. 1. In this figure, U6l denotes common part of the source message that is decoded by both relays. U6l denotes private part of the source message that is decoded by the first relay. U02 denotes private part of the source message that is decoded by the second relay. U 12 denotes part of the transmitted message by the first relay that is decoded by the second relay. The role of U12 is to transmit U6l to the second relay. The role of U20 is to transmit U6l to the receiver. The rates of U6l' U12 are shown by R6l' The rates of U6l and U02 are shown by R5l and R 02 respectively. The rate of the part of the message source that is directly decoded by the receiver is shown by Roo. As it is shown next we don't need to define any auxiliary random variable for this part of the source message. In the proposed method, U6l and U6l are decoded simultaneously at the first relay. U6l and U02 are decoded simultaneously at the second relay. U6l' U6l and U02 are decoded simultaneously at the receiver. The proposed rate is shown by the next theorem. Theorem 1: For any relay networks (Xo x Xl x X2,p(YO,Yl,Y2Ixo,Xl,X2),YO X Yl x Y2), the capacity C is lower bounded by
C 2::
sup P(U61 ,U02,U61 ,U12,U20,XO,Xl,X2)
min{I(XOXlX2; Yo),
I(U6l U6l;Yl1XlU12U20) + I(U02; Y 2 U6lU12X2U20)+ I(Xo; YoIU6l U02U6lXlU12X2U20), I(U6l; YlIU6l XlU12U20) + I(U02U6l U12; Y 2IX2U20)+ I(X o; YoIU6l U02U6l XlU12X2U20), I(U02U6l U12; Y 21X2U20) + I(XOX l; YOIX2U02U6l U12U20), I(U6l U6l; Yl1XlU12U20) + I(XoX2; Y OIXl U6lU6lU12U20)} 1
(1) where the supremum is over all joint probability mass functions
We make use of regular block Markov superposition encoding and for decoding we make use of backward decoding scheme [7]. We consider B blocks of transmission, each of n symbols. A sequence of B-2 messages, WOO,i x W6l,i X 2 W 01 i X W 02,1.". E [1 2 n R oo] x [1 , 2 n R 61] x [1 , 2 n R 61] x [1 "2 n R o2] i ~ 1,2, ... , B-2, will be sent over the channel in nB transmissions. W6l,i is decoded by two relays, W5l,i is decoded by the first relay. W02,i is decoded by the second relay. In each n-block b == 1,2, ... , B, we shall use the same set of codewords. The random codewords to be used in each block are generated as follows:
Random Coding: For any joint probability mass function satisfying (2) and (3), Choose 2n R 61 i.i.d. u'2o each with probability
m20 E [1,
n P(U20,i). i=l n
p(u'2o)
2 n R61 ].
Label
these
as
For each u'2o (m20), generate
x'2 each with probability p(x'2lu'2o) == Label these as x'2(m02I m20), m02 For each u'20(m20), generate 2n R 61 n
u'20(m20), 2 n R o2
i.i.d.
n P(X2,il u20,i). n
i=l E
[1,2 n R o2].
i.i.d
u l 2 with
ll p( U12,i IU20,i). Label these i=l as uI2(m12Im20), m12 E [1, 2n R 1Jl ] . For every u'20(m20) and u l 2(m121 m20), generate 2nR~1 i.i.d Xl each with probability
probability p(u I2Iu'20)
n
Il P(Xlil u12,i, U20,i).
p(xlluI2, u'2o)
Label
these
i=l xI(mlolm12, m20), mlO E [1, 2nR~1]. For every u'20(m20) and uI2(m12Im20), generate 2n R 61 i.i.d uBI each with probability
p(uBlluI2, u'2o) ==
n P(UBl,ilu12,i, U20,i). i=l n
Label these as
UBI(W6llm12,m20), W6l E [1,2 n R 61]. For every u'20(m20), ul2(m12I m20), xI(mlo Im12, m20) and uBI( W6ll m12, m20), generate 2nR~1 i.i.d u51 each with probability
p( u511 uBI, Xl, u12, u'2o) ==
n
Il p( U5l,i IUBl,i' Xli, U12,i, U20,i).
i=l
these as u51(W5ll w6l' mlO, m12, m20), W5l E m20), [1, 2nR~1]. For every u'20(m20), x'2(m02I uI2(m12Im20), p(XO, U5l, U02, UBI' Xl, U12, X2, U20) == n R o2 P(XoIU5l' U02, UBI' Xl, U12, X2, U20)P( U5ll uBl' Xl, U12, X2, U20) xI(mlolm12,m20) and UBI(W6llm12,m20), generate 2 p(u02luBl' Xl, U12, X2, U20)p( UBllu12, X2, U20)p(Xllu12, U20) i.i.d U02 each with probability p(u021 UBI, Xl , U12' X'2 , u'20) == n
p(U12I u20)P(X2I u20)p(U20)
on U6l x U02 X U6l
X
U12 X U20 X X o X Xl
Label
X
lI p(U02,il uBl,i' Xli, U12,i, X2,i, U20,i).
i=l
(2) X 2 such that
Label
these as U02(W02I w6l' mlO, m12, m02, m20), W02 E [1,2 n R o2]. For every u'20(m20), x'2(m02Im20), uI2(m12Im20), XI(mlolm12, m20), UBI(W6ll m12,m20), U51(w5ll w6l,mlO,m12,m20) and U02(W02I w6l,mlo,m12,m02,m20), generate 2 n R oo i.i.d Xo each with probability P(xolu51' U02' UBI, Xl, u 12, x'2, u'2o) == n 2 1 ) I Labe i= 1 p(xoiluOl,i' U02,i' UOl,i' Xli, U12,i, X2i, U20,i . these as Xo(wooIW5l,W02,W6l,mlo,m12,m02,m20),
(U5l, U02, U6l, U12, U20) ~ (Xo, Xl, X 2) ~ (Yo, Yl , Y2) (3) form a Markov chain. Proof' In this encoding scheme, the message of the transmitter is divided into four parts. The first part is decoded only fi re I ay, th e by two relays, the second part is deco d ed by therst third part is decoded by the second relay and the fourth part
n
518
Authorized licensed use limited to: Sharif University of Technology. Downloaded on October 24, 2009 at 10:18 from IEEE Xplore. Restrictions apply.
ISIT 2009, Seoul, Korea, June 28 - July 3, 2009
Woo
E
[1,2 n R oo ].
backward manner at the second relay by starting from the block i == B to i == 1. Assume that W61,i' W61,i-1 and W02,i, have been decoded accurately. The second relay determines W61 ,i-2 and W02,i-1 were sent such that
In the defined codewords, the index m20 represents the index m12 of the previous block. The index m12 represents the index W61 of the previous block. The index m10 represents the index W51 of the previous block. The index m02 represents the index W02 of the previous block. Based on this encoding scheme, the total rate transmitted by the sender is expressed as,
R == Roo + R61 + R 02 + R61
n (~ ~1 ~2 ~1 ~1 ) W01,i-1' u Q2 WO;',i Iw01,i'A w01,~-2 , w01,i;-1' A1 A1 Al n A1 Al U01 (wo1,~lw01,i-1A' W01,i-2)' U1~(W01,i-1Iw01,i-2)' A IWA I ) ,Un ( W I ) ,Y2 n ( ~") Xn2 ( W02,i-1 01 i-2 20 A 01 i-2 In
{
,
(4)
