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On the capacity of the multiple-hop relay channel with linear relaying
Accepted Manuscript On the capacity of the multiple-hop relay channel with linear relaying Zhixiang Deng, Bao-Yun Wang PII: DOI: Reference:
S1874-4907(13)00055-4 http://dx.doi.org/10.1016/j.phycom.2013.07.001 PHYCOM 206
To appear in:
Physical Communication
Received date: 12 October 2011 Revised date: 17 March 2013 Accepted date: 20 July 2013 Please cite this article as: Z. Deng, B.-Y. Wang, On the capacity of the multiple-hop relay channel with linear relaying, Physical Communication (2013), http://dx.doi.org/10.1016/j.phycom.2013.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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On the Capacity of the Multiple-hop Relay Channel with Linear Relaying Zhixiang Denga,d, Bao-Yun Wang *a,b,c a
College
of
Telecommunications
and
Information
Engineering,
Nanjing
University
of
Posts
and
Telecommunications, Nanjing, 210003, China b
National Mobile Communications Research Laboratory, Southeast University, Nnanjing, 210096, China
c
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
d
College of Computer and Information Engineering, Hohai University, Changzhou, 213022, China
Corresponding Author: Bao-Yun Wang, Tel: +86-18951896152, Email: [email protected], Address: College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China.
Abstract: The relay nodes with linear relaying transmit linear combination of their past received signals. The capacity of the multiple-hop Gaussian relay channel with linear relaying is derived, assuming that each node in the channel only communicates with its nearest neighbor nodes. The capacity is formulated as an optimization problem over the relaying matrices and the covariance matrix of the signals transmitted from the source. It is proved that the solution to this optimization problem is equivalent to a “single-letter” optimization problem when some certain conditions are satisfied. We also show that the solution to the “single-letter” optimization problem has the same form as the expression of the rate achieved by time-sharing amplify-and-forward (TSAF). In order to solve this equivalent problem, we give an iterative algorithm. Simulation results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively. Keywords: linear relaying; multiple-hop relay channel; time-sharing AF
*
Corresponding author
Email: [email protected] (Z. X. Deng), [email protected] (B.Y. Wang)
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1. Introduction The broadcast nature of the wireless channel brings convenience for cooperation between nodes. The communication efficiency can be increased with cooperation. Meulen first introduced three-terminal communication channels [1]. After that, Cover et al. [2] proposed two fundamental relaying schemes: decode-and-forward (DF) and compress-and-forward (CF), derived the capacity of the degraded and reversely degraded relay channel and gave the max-flow min-cut upper bound on the capacity for general relay channel. Several other schemes such as amplify-and-forward (AF), partial-decode-and-forward (PDF), and linear relaying (LR) are discussed in literatures [3]-[6]. Since the introduction of relays can enlarge the coverage of wireless networks, the achievable rate of multiple-level or multiple-hop relay channels are studied in [7]-[12]. The achievable rate of multiple-level relay channel was derived in [7] with regular encoding/sliding window decoding schemes. In [11], the achievable rate of two-level relay channel with irregular encoding/jointly decoding methods was derived. The achievable secrecy rate of multiple-level relay-eavesdropper channel under the constraint that message must be kept secret from the eavesdropper was obtained in [12]. The traditional wireless communication problem is to design effective encoding and decoding techniques to enable reliable communication at data rates approaching the capacity of a channel. With cooperative communications, not only is there a need to optimally design the encoder and decoder at the source and destination, but also to design the channel itself by optimally choosing the functionality of the intermediate relay nodes. The most desirable schemes are those that approach the limits of cooperative capacity with minimal processing complexity at the relays [13]. In [8], it was shown that DF achieved the capacity when the relay node was close to the source, while CF achieved the capacity when the relay node was close to the destination node. Compared with DF and CF, AF scheme is simple relatively, since decoding is not needed at the relay nodes. However, the relay nodes with AF inject more noise than signal at low signal-to-noise ratio (SNR), thus with poor performance. The authors in [14] proposed to use a “modulo function” or a “triangular function” to implement the mapping at the relay which was so called “sawtooth relaying”, and demonstrated significantly improved achievable rates over the AF relaying. In [15], the “sawtooth relaying” was studied further and a piecewise linear relaying function was proposed to approximate the theoretically optimal relaying
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function for the relay channel with orthogonal components. With this piecewise linear relaying function, the achievable rates can be close to that with CF relaying, but at a much lower complexity. However, the relaying schemes in both [14] and [15] rely on the side information by the direct link between the source and the destination. Therefore, when the direct link is poor, or there is no direct link, the rate performance will be deduced significantly. In [6], the relaying function was restricted to be the linear combination of the past received signals at the relay, namely linear relaying, and it has been proved that the problem for the capacity of the frequency-division additive white Gaussian noise (AWGN) relay channel with LR can be reduced to a “single-letter” nonconvex optimization problem, but the problem for general relay channel can’t be reduced as such. In [16], the optimal covariance matrix of the signals transmitted from the source was derived for fixed relaying matrices and the suboptimal relaying matrices were also designed. In this paper, we study the performance of multiple-hop relay channel with linear relaying in the view of information theory. Each node in the channel is assumed to communicate with its nearest neighbor nodes only. The relaying function we concern about is the same as that in [6], namely LR. With linear relaying, the relays transmit the combination of their past received signals. In this set up, i) the relaying schemes proposed in [14] and [15] cannot be used here, because there is no direct link to provide the side information, and ii) two parameters can be optimized to achieve higher achievable rate: the linear relaying function (relaying matrix) at each relay node and the covariance matrix of the signal transmitted from the source. We prove that the achievable rate for the Gaussian multiple-hop relay channel with LR can be formulated as a “single-letter” optimization problem under some certain conditions. And the solution to this “single-letter” optimization problem has the same form as the expression of the rate achieved by TSAF. With TSAF, the time for a block transmission is split into several slots. During each slot, the source transmits with certain average power while the relay nodes amplify and forward their received signals with corresponding amplifying coefficients under the total average power constraint of each. The amplifying coefficients and the average transmission power at the source in each slot are strictly relevant. We present an iterative algorithm to compute these amplifying coefficients and ratios of the average transmission power of the source in each slot. Thus, the processing complexity of TSAF at each relay node is almost as low as that of AF. The numerical results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively. In the wireless networks such as wireless sensor
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networks, of which the processing ability is poor, the transmission power is limited and channel gain varies significantly among different paths, the TSAF scheme achieves a good trade-off between processing complexity and achievable rates. Notations: Bold uppercase letters denote matrices or vectors and bold lowercase letters denote column vectors; transpose is represented by ()T ; IM is the identity matrix of size M M ; variables are denoted by italic letters; N(μ, N0) denotes circularly symmetric Gaussian distribution with mean μ and variance N0;
( ) denotes expectation; I ( ; ) denotes the mutual information between two
random variables, h( X ) denotes the differential entropy of the continuous random variable X , and
M denotes the determinant of matrix M .
2. Multiple-hop relay channel model and capacity 2.1 Channel model and problem setup The multiple-hop channel model introduced in this paper is shown in Fig. 1. Let the source node be denoted by 0, the destination node by M, and let the other M–1 relay nodes be donated sequentially as 1, 2, …, M-1. Each node communicates with its neighbor nodes, and there is no direct link between the other nodes. Assume each node k∈{0, 1, …, M-1} sends xk(i)∈k and receives ym(i)∈m (m = 1, …, M) at time i, where k and m are the corresponding input and output alphabets for corresponding nodes. We have the following definitions: A (2 nR, n) -code for the M-hop relay channel consist of i) a set of messages
{1, 2,...,2nR } ;
ii) a codebook {x n (1), x n (2),..., x n (2nR )} consisting of codewords of length n; ii) encoding function at the source
:1,2,...,2nR
0
(1)
iii) a sequence of relaying functions at the relay node k,
k ,i : for i = 1, 2, …, n, where k = 1, 2, …, M-1.
i 1 k
k
(2)
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iv) a decoding function at the destination
n :
n M
{1,2,...,2nR }
(3)
The average probability of decoding error is defined as
Pe( n )
1 2nR
2nR
P
n
( yMn )
j | x n ( j )is transmitted
(4)
j 1
We assume the average power constraint of the source node power constraints of the relay nodes
1 n E ( X 0,2 i ) P , and the average n i 1
1 n E ( X k2,i ) P , k=1, 2, …, M-1. n i 1
A rate R is said to be achievable if there exits a sequence of (2nR ,n) -codes satisfy the average power constrains defined above, such that Pe( n )
0 as n
. The capacity is defined as the
supremum of the set of achievable rates.
Fig. 1 The multiple-hop relay channel with linear relaying
During n uses of the channel, node k transmits a sequence of random symbols
Xnk =[X k ,1 , X k ,2 , ..., X k ,n ]T (k = 0, 1, 2, …, M-1) and receives Ykn
[Yk ,1 , Yk ,2 , ..., Yk , n ]T (k = 1, …, M).
As mentioned earlier, the relay nodes transmit the combination of their past received symbols. Thus, at i
time i, the transmitted relay symbol at relay k (k = 1, 2, …, M-1) can be expressed as xk ,i
d i(,kj ) yk , j , j 1
where d i(,kj ) are the coefficients chosen such that the average power constraint for the relay k is satisfied. For a transmission block length n, Xnk
Dk Ykn , where Dk (namely, relaying matrix) is a
lower triangular weight matrix of relay k. 2.2 Capacity for the multiple-hop relay channel with linear relaying Capacity for the multiple-hop relay channel is defined as the supremum over all the achievable rates [6], which is expressed as (5).
CLR
lim
n
1 sup n p ( x0 , x1 , ..., xM
I ( X0n ; YMn )
(5)
1)
The maximization in (5) is achieved when X 0n is Gaussian. Here, we consider AWGN channel,
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where hk denotes the channel coefficient between node k and k+1, and the Gaussian white noise zk ~ N(0, N0). Let P be the average power constraint for the source node while Denote Σs
P for the relay nodes.
(X0n X0nT ) as the covariance matrix of X 0n .
From Fig. 1, the received vector at node k can be expressed as:
Ykn where, Xnk
1
hk 1Xkn
1
z kn , k
(6)
1, 2, ..., M
is the transmitted vector at node k-1. With linear relaying, Xnk
is given by (7).
1
Xnk 1 = Dk 1Ykn 1
(7)
By (6) and (7), the received vector at the destination node is M 1M 1
M 1
YMn
hm DM
1~1
X0n +
m 0
where， D p ~ q
Dp Dp
1~ l
z ln
z nM
(8)
l 1 m l
Dq , DTp ~ q
1
hm DM
DTq DTq
1
DTp .
Since the random variables in (8) are jointly Gaussian, with Theorem 8.4.1 of [17] we can compute the achievable rate R as
R
1 I( X0n ; YMn ) n
1 h(YMn ) h(YMn | X0n ) n
1 1 log (2 e) n K YM n 2
1 log (2 e) n K YM | X 0 2 M 1M 1
M 1
1 log 2n
(9)
hm2 DM
1 ~1
Σ s DTM
1 ~1
hm2 DM
N 0 (I
m 0
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
hm2 DM
N 0 (I
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
M 1
hm2 DM
where, K YM
T 1 ~1 Σ s DM
1 ~1
hm2 DM
N 0 (I
m 0
1 ~l
DTM
1 ~l
) is the covariance matrix of
l 1 m l M 1M 1
the Gaussian random vector YMn , and K YM | X 0
hm2 DM
N 0 (I
1 ~l
DTM
1 ~l
) is the covariance
l 1 m l M 1M 1
hm DM
matrix of
1~ l
z ln
z nM .
l 1 m l
By
(5),
CLR ( P, P)
capacity
for
Gaussian
multiple-hop
relay
channel
lim Cn ( P, P) , where, Cn ( P, P) is expressed as (10).
n
with
linear
relaying
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1 n
Cn ( P, P)
I ( X0n ; YMn )
max
Σ s ，{Dk }1M 1
M 1M 1
M 1
1 max log 2n {Dk }1M 1
hm2 DM
1 ~1
Σ s DTM
hm2 DM
N 0 (I
1 ~1
m 0
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
hm2 DM
N 0 (I
Σs
1 ~l
DTM
1 ~l
）
l 1 m l
s.t. Σ s
0, tr ( Σ s )
k 1
hm2 Dk ~1 Σ s DTk ~1
tr{ m 0
k
(10)
nP
k 1
k 1
n P,
m l
l 1
1, 2, ..., M
hm2 Dk ~ l DTk ~ l N 0 Dk DTk }
N0
1
3. Two-hop relay channel with linear relaying In this section, we firstly analyze two-hop relay channel with linear relaying by letting M=2 shown in Fig. 1. By (10), the capacity for two-hop relay channel with linear relaying is 2 CLR ( P, P )
1 max I ( X0n ; Y2n ) n Σs，D1
h12 h02 D1 Σ s D1T 1 max log 2n Σs，D1 N 0 (I s.t. Σ s
0, tr ( Σ s )
tr{h D1 Σ s D 2 0
T 1
N 0 (I
h12 D1D1T )
(11)
h12 D1D1T )
nP
N 0 D1D1T }
n P
For fixed D1 , (11) is a concave function over Σ s , however nonconcave over D1 or ( D1 , Σ s ). We consider the block length n=2 as a special case. The relaying matrix and input covariance matrix are expressed as
d 0 , Σs 0 d
D1
where d
P /(h02 P
(1
2P (1
)
)
(12)
1
N0 ) is selected to satisfy the power constraint at the relay,
( 1
1)
is the correlation coefficient between X1 and X2 which are the signals of the two transmissions at each block respectively,
P and (1
)P are the average power of X1 and X2 respectively.
2 ( P, P) is a concave function over Σ s , and the According to [16], for fixed relaying matrix D1 , CLR
optimal covariance matrix Σ sopt
P 0 . 0 P
In the case above, the achievable rate with linear relaying at optimal point is equivalent to that
h12 h02 d 2 P 1 ) . In [6], it was pointed out that as the with AF, and the achievable rate is R2 = log（1 + 2 (1 h12 d 2 ) N 0 power P is decreased, linear relaying injects more noise than signal, thus becomes less helpful. The
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same problem occurs in the two-hop relay channel with linear relaying mentioned in this paper. When the signal power is lower than that of the noise, noise is amplified more than signal by linear relaying, which causes decrease in achievable rate. The reason is that at optimal point the amplifying coefficients at the relay and the average transmission power at the source remain equal during each use of channel. This can be seen from (12) and the expression of Σ sopt . The amplifying coefficients at the relay nodes are chosen to satisfy the average power constraint regardless of the SNR of the received signals at the relay node. Therefore, the power is not utilized fully. Suppose that if the power at the source is changeable of each slot under its total power constraint, the relay nodes can amplify the received signals accordingly to the SNR under their power constraints. The received signals with higher SNR will be amplified more at the relay. In such a way, the power efficiency at both the source and the relay nodes will be improved and a higher achievable rate can be derived. Specifically speaking, the time for a block transmission is split into several slots. During each slot, the source transmits with some certain average power while the relay nodes amplify and forward their received signals with corresponding amplifying coefficients under the total average power constraint. The amplifying coefficients and the average transmission power at the source in each slot are strictly relevant. By optimization methods, the optimal amplifying coefficients, the optimal ratio of the average power at the source in each slot and the optimal number of slots can be derived. We call this relaying scheme time-sharing AF (TSAF), which is explained vividly as Fig. 2.
(a) Time-sharing transmission with different power level at the source
(b) The corresponding amplifying coefficients of the relay in each slot
Fig. 2 Time-sharing transmission among slots
In Fig. 2, θi and αi are ratio of the transmission power ratio and the number of channel uses of slot i at the source node respectively, where θ1+θ2+θ3=1 and α1+α2+α3=1; ηi denotes the corresponding
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2 i
amplifying coefficient under the average power constraint
(h02 i P
i
N0 )
P . Optimizing
i
over the parameters θ, α and η, we can compute the lower bound of the capacity of the two-hop relay channel with linear relaying, which is shown in (13). 2 CLR ( P, P )
s.t.
i i
max α ,θ , η
1,
1 2
i i
P h12 h02 h12 i N0 1 i
i log(1 i 2 i
1,
2 0 i
(h
P
i
N)
2 i 2 i
) (13)
P
i
To verify the effectiveness of TSAF, assuming the block length n=2, we compare the achievable rate with TSAF with the maximal rate which is computed by search over the relaying matrix D1 and the input covariance matrix Σ s exhaustively, where
d1,1 d 2,1
D1
Σs
(1
2P (1
1
2
and
1
0 . d 2,2
For the optimization problem in (13), parameters are defined as α = [
η=[
)
)
1
2
]T , θ = [
1
2
]T and
]T . It is a nonconvex problem. However, the optimization problem in (13) is convex in each
set of α, θ, η if the other two are fixed. A Mote Carlo algorithm similar to that described in [6] can be used to derive a local maximum. Fig. 3 compares the achievable rate of TSAF with the maximal rate, where h0
h1
1 . From
the figure, it can be found that the two rates are almost equal to each other for any SNR, which demonstrates the effectiveness of TSAF scheme.
Fig. 3 Comparison between the achievable rate with TSAF and maximal rate by exhaustive search
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4. Multiple-hop Gaussian relay channel with LR 4.1 Optimization problem on Gaussian multiple-hop relay channel with LR In [6], it has been proved that the optimization problem for LR in the case of orthogonal relay channel can be reduced to a “single-letter” optimization problem. In this section, we will prove that the optimization for multiple-hop relay channel with linear relaying is equivalent to a “single-letter” optimization problem too, as shown in Theorem 1, in which each block is divided into several time slots. During each slot, the source node sends symbols at a given average power while the relay nodes work at AF mode with corresponding amplifying coefficients. Theorem 1: For the M-hop Gaussian relay channel with linear relaying, the SVD (singular value
Fk L k QTk , (k
decomposition) of the relaying matrix at relay k is Dk 3, …, M-1}, if FmT 1Qm
QTm Fm
1, 2, ..., M
1) , for any m∈{2,
I , the expression shown in (10) is equivalent to the following
1
optimization problem: M 1 N
C
L LR
maxM -1
α , θ ,{ ηk }k =1
P ( N i 0
h02
2 m m 1 M 1M 1
i
i i 1
h
m ,i
) 2 m
1
h
m ,i
l 1 m l
s.t.
i , i ,
k ,i
N
0,
N i
N
1,
i
i 1
i 1 k 1 k 1
k 1 k ,i
(h02 i P
hm2
m ,i
i
m 1
i 1
hm2
N0 (
m ,i
where, N is the number of time slots, (x)=1/2log(1+x), α
ηk
[
k ,1
,
k ,2
, ...,
k,N
1))
P, k
1, 2,..., M
[ 1,
2
, ...,
DM 1 ~l ）
1 ~1
N
]T , θ
]T (k=1, 2, …, M-1).
Proof: Define M 1
hm He
m 0
N0
1
l 1 m l
M 1M 1
hm2 DM
(I l 1 m l
The problem shown in (10) can be written as
T 1 ~ l DM
1 2
VΛVT
[ 1,
2
, ...,
N
]T and
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 log I 2n 0, tr ( Σ s ) nP
Cn
H e Σ s HTe
max
DM 1 ,...,D1 , Σ s
s.t. Σ s
k 1
k 1
hm2 Dk ~1 Σ s DTk ~1
tr{ m 0
k
k 1
N0
n P,
m l
l 1
1, 2, ..., M
(14)
hm2 Dk ~ l DTk ~ l N 0 Dk DTk }
1
(14) is a nonconvex optimization problem. However, for fixed DM 1 , …, D1 , Cn is a concave
1 , the Lagrangian for Cn is expressed as 2n
function on Σ s . Neglecting the factor
L（Σ s，Ω， ， 1,
2
,..., k ）=
H e Σ s HTe
log I
M 1
tr (ΩΣ s ) k 1
k 1 k
hm2 Dk ~1 Σ s DTk ~1
(tr{
nP)
hm2 Dk ~ l DTk ~ l N 0 Dk DTk } n P)
N0
m 0
k 1
(tr ( Σ s ) k 1
m l
l 1
KKT conditions are M 1
k 1
i) I
hm2 DTk ~1Dk ~1
k
ii) tr ( ΩΣs )
0
iii) (tr (Σs )
nP)
0 k 1
k 1 k
hm2 Dk ~1 Σs DTk ~1
(tr{ m 0
where， , 1 ,
2
k 1
N0 l 1
,...,
hm2 Dk ~ l DTk ~ l N0 Dk DTk } n P) = 0,k
1
0.
M 1
QTm Fm
1,...,M
m l
Let the SVD of Dk ,He are Dk = Fk Lk QTk (k assume FmT 1Qm
Ω
m 0
k 1
iv)
HTe (I + H e Σs HTe ) 1 He
1
1, ...,M -1), He
UΛVT respectively, and
I for m = 2, 3, …, M-1, then M 1
hm2 T e
H He where， L2M
1~ l
M 1M 1
Q1L2M
m 0
N0
1~1
(
hm2 L 2M
1~ l
+ I) 1 Q1T
VΛ 2 VT
(15)
l 1 m l
L2M 1...L2l , L1, L2, …, LM-1 and Λ are all diagonal matrices. Thus, the columns of
Q1 and V are engenvectors of HTe H e and VT Q1
Q1T V
I . The first KKT condition can be
simplified to M 1
i) I
k 1 k
k 1
hm2 Λ k ~1
Λ(I + ΛVT Σs VΛ) 1 Λ VT ΩV
m 0
where， Λk = L2k , Λk ~1 = Λk Λk 1...Λ1 . Now, we show that for Σs
VΨVT , if Φ = VΘVT , then tr (Σs Φ)
tr (ΨΘ)
0 , the KKT
conditions hold. The matrices Ψ and Θ are both diagonal matrices. Therefore, (14) is reduced to
M 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Cn
max
Ψ ,Λ1 ,Λ 2 ,..., Λ M 1
s.t. Ψ
0, tr{Ψ}
M 1M 1
ΨΛ M
m 0
N0
1~1
hm2 Λ M
(I
1 ~l
） 1
l 1 m l
(16)
nP, k 1
k 1
k 1
hm2 ΨΛ k ~1
tr{
hm2 Λ k ~ l N 0 Λ k }
N0
m 0
k
hm2
1 log I 2n
1, 2, ..., M
n P
m l
l 1
1
Since all the matrices in (16) are diagonal, the following “single-letter” optimization problem is equivalent to (16). M 1
Cn
1 2n
max 1
,... n
M 1
h02
n
log 1 i 1
0,
i
k ,i
h
i
hm2
N0 1
k ,1 ,..., k ,n k 1
s.t.
2 m m ,i m 1 M 1M 1
m ,i
l 1 m l
(17)
n
0,
nP
i i 1
n
h02
k ,i
hm2
m ,i
+
i
1, 2, ..., M
k 1
hm2
N0
m 1
i 1
k
k 1
k 1
m ,i
N0 n P
m l
l 1
1
Obviously, the rates will approach maximum when the source nodes and the relays transmit at
maximal power, i.e. at optimal point
n
k 1
k 1
h02
k ,i
hm2
m ,i
i
+
m 1
i 1
k 1
hm2
N0 l 1
m ,i
N0
n P ,
m l
n
nP for k
i
1, 2, ..., M
1 . So the corresponding inequality constraints in (17) can be
i 1
substituted by the above equalities respectively. And the Lagrangian is M 1
L(
n 1
,
n 1
, ...,
n M 1
, ,
1 ,...,
h02
n M
1)
log 1 i 1
2 m m ,i m 1 M 1M 1
h
i
n
( 2 m m ,i
N0 1
h
i
nP)
i 1
l 1 m l
M 1
n k
k 1
k 1
k 1 k ,i
h02
hm2
m,i
m 1
i 1
i
+
k 1
hm2
N0 l 1
k 1, i
...
l ,i
N0
n P
m l
Thus, at optimal point the following equalities hold.
L
0
i
L
0 (18)
1, i
L
0
M 1, i
Solve the set of equations shown in (18). For any hM 1 ..., h1 , h0 , assume there’re at most N
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positive roots of the set of equations, which are denoted subsequently as ( 1 ,
( 2,
1,2
,...,
M 1,2
) , …, (
N
,
1, N
,...,
M 1, N
1,1
,
2,1
,...,
M 1,1
),
) . So, for any n≥N, there are at most N distinct roots which
are optimal. Without loss of generality, assume that at optimum, the roots of the first n1 indices are related to ( 1 ,
1,1
,
2,1
,...,
M 1,1
) , while ( 2 ,
,...,
1,2
M 1,2
) related to the roots of the next n2 indices,
analogically, the roots of the last nN indices are related to (
N,
N
1, N ,...,
ni
1, N ) , where
M
n . Let
i 1
i
nP be the total power allocated to the ni (i = 1, 2, …, N) indices, so
nP and ni
N
i i
i
1.
i 1
Therefore, the problem in (17) is the same as (19). M 1
Cn
max
n ,θ , M 1 ηk k 1
s.t.
0,
i
n
1 2n
i nP ni
ni log 1 i 1
N0 1
hm2
m ,i m 1 M 1M 1 2 m l 1 m l
h
m ,i
N
0,
k ,i
h02
i
(19)
1
i 1 n
k 1
k 1
h02
k ,i
m,i i
nP + ni
m 1
i 1
k
hm2
hm2
m ,i
ni N 0 n P
m l
l 1
1, 2, ..., M
1
[ 1 ,..., N ]T , η j
where, n=[n1, n2, …, nN]T, θ
k 1
N0
[
j ,1
As n→∞, taking the limit of (19) and defining
,...,
i
j,N
lim
n
]T . ni , we derive n M 1
n
C
L LR
lim Cn
2 m m 1 M 1M 1
i
max
α ,θ , M 1 i 1 ηk k 1
n
P i N0
hm2
i
h
m ,i
hm2
1
m ,i
l 1 m l
s.t.
i
, i ,
N k ,i
0,
N i
i
i 1 N
( h02 i P
k 1 k 1
hm2 m 1
i 1
k
i 1
k 1 k ,i
1, 2, ..., M
(20)
1
m ,i
i
hm2
N0 (
m ,i
1))
P
l 1 m l
1
This completes the proof. 4.2 Algorithm design Though (20) is “single-letter” characterized, it is still a nonconvex problem. It requires exhaustive search which is computationally expensive. However, the optimization problem in (20) is convex in each set of α, θ, η1 , η2 ,...,ηM
1
if the other M sets are fixed. Furthermore, it is hard to show the exact
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value of N. We use the following numerical method to derive a sub-optimal solution to this problem.
Initialization: Fix the number of time slots N=Nini, randomly choose initial values for the M+1
α0 , θ
sets of variables: α
θ0 , η1
η2,0 , …, ηM
η1,0 , η2
1
ηM
1,0
2, opt
;
Step 1: -1.1 Iteration over variable sets: (1)
Fix α , θ , η1 , …, ηM
2
, optimize over ηM 1 , update ηM 1 =ηM
(2)
Fix α , θ , η1 , …, ηM
3
, ηM 1 , optimize over ηM
(3)
Fix the sets α, θ, η1 , η2 ,...,ηM
1
2
1, opt
;
, update ηM 2 =ηM
excluding ηk , optimize over ηk and update
ηk =ηk ,opt subsequently for k 1,2,...,M
3;
(4)
Fix α , η1 , …, ηM 1 , optimize over θ , update θ
(5)
Fix θ , η1 , …, ηM 1 , optimize over α , update α
θopt ; α opt .
-1.2 Termination: Repeat Step 1.1 until the rate converges to a local maximum.
Step 2: -2.1 Randomly choose initial values for the M+1 sets of variables: α
η2
η2,0 , …, ηM
1
ηM
1,0
α0 , θ
θ0 , η1
η1,0 ,
, repeat Step 1.
-2.2 Termination: Repeat Step 2.1 until the repeating times exceed K (a given number), select the maximal value of Step 2 as the local maximum corresponding to the current number of time slots.
Step 3: -3.1 Increase the number of time slots N = N + 1, repeat Step 2. -3.2 Termination: Repeat Step 3.1 until the number of slots N exceeds Nf
(a given number),
choose the maximal value as the final result. 4.3 Numerical Results Taking a three-hop Gaussian relay channel with linear relaying as example, we present numerical results to analyze the performance of TSAF. In order to run the algorithm in 4.2, we choose the following parameters. The initial and final value of the number of time slots are chosen respectively as
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Nini = 3 and Nf = 10. We further choose the repeating times of Step 2.1 K = 40. Step 1 is terminated if the change in the resulting converged rate of this step is less than 10-4. The received signals at the two relay nodes and the destination are given by
respectively, where,
1 n
n
E ( X 0,2 i )
P,
i 1
Y1n
h0 X0n
Z1n
(21)
Y2n
h1X1n
Z2n
(22)
Y3n
h2 X2n
Z3n
(23)
1 n
n
E ( X 1,2i ) i 1
1 n
n
E ( X 2,2 i )
P and Z1 , Z 2 , Z 3 are
i 1
i.i.d. Gaussian noises with variance N 0 . As known, the capacity of the general 3-hop relay channel is achieved with DF relaying. The capacity is shown as follows
C
min{C (
h02 P h2 P h2 P ), C ( 1 ), C ( 2 )} N0 N0 N0
From (24), it can be found that the minimum value of h0 , h1
and h2
(24) dominates the capacity.
The achievable rate with linear relaying is compared with the capacity in (24) under two assumptions: i) the channel gain of one certain hop is smaller relatively, in this case
h0 =0.5， h1 h0 =h1
=h2
=h2
we choose
=2 ; and ii) the gain of the three hops are comparable, in this case we choose
=2 .
As mentioned in section 3, TSAF is introduced to overcome the low power efficiency of regular AF relaying when the SNR at the relay node is low. We also compare the achievable rate with TSAF
h1 with the achievable rate with regular AF when h0 =0.2，
=1,h2
=2 .
Fig. 4 compares the achievable rate with TSAF with the capacity of the three-hop relay channel. Some optimized parameters α , θ , η1 and η2 for different SNRs are shown in Table 1 and Table 2 respectively. The capacity of the 3-hop relay channel is achieved by DF relaying with which each relay node must decode the message from its upstream node and forwards the decoded message to its downstream node. The decoding and encoding at the relay nodes increase the complexity of these nodes. From Fig. 4, we find that when the gain of one certain hop is smaller than that of the other two hops relatively, the achievable rate with TSAF relaying may be close to the capacity. However, the complexity of the relay nodes with TSAF relaying or linear relaying is low because decoding or
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encoding is not needed at the relay nodes. With TSAF relaying or linear relaying, the relay nodes only have to amplify and forward their received signals with corresponding amplifying coefficients of each slot or transmit the combination of their past received signals to the downstream nodes. Therefore, the complexity of the TSAF relaying is almost as same as that of the simple AF relaying. We conclude that TSAF achieves a good trade-off between the complexity and the achievable rate.
Fig. 4 Comparison between the capacity and achievable rate with linear relaying
Table 1 α * , θ* , η1* , η*2 for different SNR ( h0 =h1
=h2
=2 )
Optimized parameters
SNR = 2
SNR = 4
SNR = 6
SNR = 8
SNR = 10
α
0.2056 0.1694 0.2127 0.4123
0.2346 0.1750 0.5904
0.0804 0.2475 0.6720
0.2994 0.2515 0.4491
0.2994 0.2515 0.4491
θ*
0.2007 0.2032 0.2048 0.3913
0.1945 0.1792 0.6264
0.1171 0.2249 0.6580
0.2717 0.2233 0.5050
0.2717 0.2233 0.5050
η1*
0.2254 0.2007 0.2272 0.2289
0.2653 0.2321 0.2268
0.1845 0.2566 0.2441
0.2598 0.2637 0.2235
0.2617 0.2656 0.2246
η*2
0.2222 0.2233 0.2221 0.2218
0.2348 0.2354 0.2355
0.2406 0.2398 0.2400
0.2422 0.2422 0.2427
0.2437 0.2437 0.2441
*
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Table 2 α * , θ* , η1* , η*2 for different SNR ( h0 =0.5,h1
=h2
=2 )
Optimized parameters
SNR = 2
SNR = 4
SNR = 6
SNR = 8
SNR = 10
α
0.0415 0.0510 0.0720 0.0741 0.0797 0.1154 0.5663
0.0529 0.1050 0.0491 0.0319 0.0731 0.0251 0.6630
0.0691 0.0295 0.1223 0.0237 0.1024 0.6529
0.0876 0.0234 0.0028 0.1547 0.1143 0.6172
0.0077 0.0996 0.1631 0.0827 0.6470
0.0512 0.0585 0.0759 0.0832 0.0882 0.0000 0.6429
0.0531 0.1113 0.0494 0.0360 0.0732 0.0258 0.6512
0.0679 0.0278 0.1204 0.0239 0.0971 0.6629
0.0899 0.0224 0.0025 0.1555 0.1234 0.6063
0.0121 0.1000 0.1637 0.0824 0.6418
η
1.4194 1.4862 1.5169 1.4148 1.4617 0.0000 1.4351
2.0199 1.9236 1.9778 1.8093 2.0036 1.8131 2.0293
2.5053 2.7833 2.4920 2.1988 2.7245 2.3140
2.7785 2.5010 1.5559 2.6891 3.0034 2.5891
2.2163 3.0712 3.0050 2.9055 2.7909
η
0.2389 0.2263 0.2150 0.2239 0.2213 0.0000 0.2260
0.2325 0.2442 0.2381 0.2597 0.2348 0.2624 0.2319
0.2310 0.2107 0.2321 0.2588 0.2144 0.2480
0.0899 0.0224 0.0025 0.1555 0.1234 0.6063
0.2441 0.2282 0.2326 0.2406 0.2501
θ
*
*
* 1
* 2
Fig.
5
h0 =0.2,h1
compares =1,h2
the
achievable
rate
between TSAF and
regular AF
when
=2 . It can be found from Fig. 5 that TSAF outperforms AF when the SNR is
low, e.g. the achievable rate with TSAF is almost twice as high as that of the regular AF when P / N0
1 and triple as high as that of the regular AF when P / N0
0.5 . The lower the SNR at
the relay node is, the greater the rate improvement is. This observation indicates the intention of designing the TSAF relaying scheme, i.e. to improve the performance of AF at low SNR.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Fig. 5 Comparison between the achievable rate of AF and TSAF: h0 =0.2,h1
=1,h2
=2
5. Conclusions We study the performance of linear relaying scheme for multiple-hop relay channel. The capacity expression for the multiple-hop relay channel with linear relaying is derived. It is shown that under some certain conditions, the solution to the capacity expression can be reduced to a “single-letter” optimization problem. However, this optimization problem is a nonconvex optimization problem which requires computationally expensive exhaustive search. We design an iterative algorithm to find a sub-optimal solution to this optimization problem. Numerical results show that if channel gain of one certain hop is smaller than that of all the other hops relatively, the achievable rate with TSAF is close to the capacity which is achieved with DF. The complexity of TSAF, however, is much simpler than DF since block-wise encoding or decoding is not needed at the relay nodes. The simulation results also show that TSAF outperforms the simple AF in achievable rate when the SNR at the relay is low. In the wireless networks such as wireless sensor networks, of which the processing ability is poor, the transmission power is limited, nodes usually communicate through multiple-hop routes and channel gain of each hop varies significantly among different paths, the TSAF scheme can achieve a good trade-off between processing complexity and achievable rates.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant No. 60972045, 61271232, 61071089), the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2012D05), the Natural Science Foundation of Jiangsu Province,
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China (Grant No. BK2010077), the University Postgraduate Research and Innovation Project in Jiangsu Province, China (No. CXZZ11_0395), the Fundamental Research Funds for the Central Universities (No. 2009B32114).
References [1] E C. MEULEN, Three-terminal communication channels, Advances in Applied Probability, 3 (1971)
120-154. [2] T. COVER, A. E.GAMAL, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, 25(5)
(1979) 572-584. [3] A. Host-Madsen, J. Zhang, Capacity bounds and power allocation for wireless relay channels, IEEE Trans.
Inform. Theory, 51(6) (2005) 2020-2040. [4] R. Nabar, H. Bolcskei, F. Kneubohler, Fading relay channels: performance limits and space-time signal
design, IEEE J. Select. Areas Commun., 22(6) (2004) 1099-1109. [5] A. del Coso, C. Ibars, Partial decoding for synchronous and asynchronous Gaussian multiple relay channels,
International Conference on Communications (ICC) Proceeding, Glasgow UK, Jun. 2007. [6] A. El Gamal, M. Mohseni, S. Zahedi, Bounds on capacity and minimum energy-per-bit for AWGN relay
channels, IEEE Trans. Inform. Theory, 52(4) (2006) 1545-1561. [7] L. L. Xie, P. R. Kumar, An achievable rate for the multiple-level relay channel, IEEE Trans. Inf. Theory,
51(4) (2005) 1348-1358. [8] G. Kramer, M. Gastpar, P. Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE
Trans. Inf. Theory, 51(9) (2005) 3037-3063. [9] A. Reznik, S. R. Kulkarni, S. Verdú, Degraded Gaussian multirelay channel: Capacity and optimal power
allocation, IEEE Trans. Inf. Theory, 50(12) (2004) 3037-3046. [10] M. Gastpar, M. Vetterli, On the capacity of large Gaussian relay networks, IEEE Trans. Inf. Theory, 51(3)
(2005) 765-779. [11] P. Razaghi, W. Yu, Parity forwarding for multiple-relay networks, IEEE Trans. on Inf. Theory, 55(1) (2009)
158-173. [12] Y. Chen, B. Y. Wang, Achievable secrecy rate of multiple-level relay-eavesdropper channel, Journal of
Applied Sciences, 29 (2011) 243-250. [13] K. S. Gomadam, S. Ali Jafar, Optimal relay functionality for SNR maximizationin memoryless relay
networks, IEEE J. Selected Areas Commun., 25(2) (2007) 390-400. [14] S. Yao, M. N. Khormuji, M. Skoglund, Sawtooth Relaying, IEEE Communications Letters, 12(9) (2008)
612-614. [15] M. N. Khormuji, M. Skoglund, On Instantaneous Relaying, IEEE Trans. Inf. Theory, 56(7) 2010 3378-3394. [16] A. del Coso, C. Ibars, Achievable rate for Gaussian multiple relay channels with linear relaying functions,
IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2007. [17] T. M. Cover, J. A. Thomas, Elements of Information Theory, 2nd ed. New York: Liley, 2006.
Zhixiang Deng received his B.S. degree in Communication Engineering from Hohai University, China, in 2003, and his M.S. degree in Communication and Information System from Southeast University, China, in 2006. He is now
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
working towards his Ph.D. degree in Information and Signal Processing with Nanjing University of Posts and Telecommunications, China. His main research interests include multiuser information theory and information theoretic security.
Bao-Yun Wang received Ph.D. degree in electrical engineering from Southeast University, Nanjing, in 1997. In 1997-1999, he was on the faculty of Nanjing University of Posts and Telecommunications. In 1999-2000, he was a research associate at Pohang University of Science and Technology, Pohang, Korea. In 2000-2002, he was with City University of Hong Kong as a research associate. In 2004-2005, he was a visiting research fellow at The University of Western Sydney, Sydney, Australia. Since 2005, he has been on the faculty of Nanjing University of Posts and Telecommunications, where he is the Dean of School of Automation Engineering. His research interests are in the area of information theory, statistical signal processing, and their applications in wireless communications.
* * α * , θ* , η1 , η2 for different SNR ( h0 =h1
=h2
Table 2 α * , θ* , η1* , η*2 for different SNR ( h0 =0.5,h1
=2 )
=h2
=2 )
Fig. 1 The multiple-hop relay channel with linear relaying Fig. 2 Time-sharing transmission among slots: (a) Time-sharing transmission with different power level at the source, (b) The corresponding amplifying coefficients of the relay at each slot Fig. 3 Comparison between the capacity with TSAF and maximal rate by exhaustive search Fig. 4 Comparison between the capacity and achievable rate with linear relaying Fig. 5 Comparison between the achievable rate of AF and TSAF: h0 =0.2,h1
=1,h2
=2
S1874-4907(13)00055-4 http://dx.doi.org/10.1016/j.phycom.2013.07.001 PHYCOM 206
To appear in:
Physical Communication
Received date: 12 October 2011 Revised date: 17 March 2013 Accepted date: 20 July 2013 Please cite this article as: Z. Deng, B.-Y. Wang, On the capacity of the multiple-hop relay channel with linear relaying, Physical Communication (2013), http://dx.doi.org/10.1016/j.phycom.2013.07.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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On the Capacity of the Multiple-hop Relay Channel with Linear Relaying Zhixiang Denga,d, Bao-Yun Wang *a,b,c a
College
of
Telecommunications
and
Information
Engineering,
Nanjing
University
of
Posts
and
Telecommunications, Nanjing, 210003, China b
National Mobile Communications Research Laboratory, Southeast University, Nnanjing, 210096, China
c
College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
d
College of Computer and Information Engineering, Hohai University, Changzhou, 213022, China
Corresponding Author: Bao-Yun Wang, Tel: +86-18951896152, Email: [email protected], Address: College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China.
Abstract: The relay nodes with linear relaying transmit linear combination of their past received signals. The capacity of the multiple-hop Gaussian relay channel with linear relaying is derived, assuming that each node in the channel only communicates with its nearest neighbor nodes. The capacity is formulated as an optimization problem over the relaying matrices and the covariance matrix of the signals transmitted from the source. It is proved that the solution to this optimization problem is equivalent to a “single-letter” optimization problem when some certain conditions are satisfied. We also show that the solution to the “single-letter” optimization problem has the same form as the expression of the rate achieved by time-sharing amplify-and-forward (TSAF). In order to solve this equivalent problem, we give an iterative algorithm. Simulation results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively. Keywords: linear relaying; multiple-hop relay channel; time-sharing AF
*
Corresponding author
Email: [email protected] (Z. X. Deng), [email protected] (B.Y. Wang)
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1. Introduction The broadcast nature of the wireless channel brings convenience for cooperation between nodes. The communication efficiency can be increased with cooperation. Meulen first introduced three-terminal communication channels [1]. After that, Cover et al. [2] proposed two fundamental relaying schemes: decode-and-forward (DF) and compress-and-forward (CF), derived the capacity of the degraded and reversely degraded relay channel and gave the max-flow min-cut upper bound on the capacity for general relay channel. Several other schemes such as amplify-and-forward (AF), partial-decode-and-forward (PDF), and linear relaying (LR) are discussed in literatures [3]-[6]. Since the introduction of relays can enlarge the coverage of wireless networks, the achievable rate of multiple-level or multiple-hop relay channels are studied in [7]-[12]. The achievable rate of multiple-level relay channel was derived in [7] with regular encoding/sliding window decoding schemes. In [11], the achievable rate of two-level relay channel with irregular encoding/jointly decoding methods was derived. The achievable secrecy rate of multiple-level relay-eavesdropper channel under the constraint that message must be kept secret from the eavesdropper was obtained in [12]. The traditional wireless communication problem is to design effective encoding and decoding techniques to enable reliable communication at data rates approaching the capacity of a channel. With cooperative communications, not only is there a need to optimally design the encoder and decoder at the source and destination, but also to design the channel itself by optimally choosing the functionality of the intermediate relay nodes. The most desirable schemes are those that approach the limits of cooperative capacity with minimal processing complexity at the relays [13]. In [8], it was shown that DF achieved the capacity when the relay node was close to the source, while CF achieved the capacity when the relay node was close to the destination node. Compared with DF and CF, AF scheme is simple relatively, since decoding is not needed at the relay nodes. However, the relay nodes with AF inject more noise than signal at low signal-to-noise ratio (SNR), thus with poor performance. The authors in [14] proposed to use a “modulo function” or a “triangular function” to implement the mapping at the relay which was so called “sawtooth relaying”, and demonstrated significantly improved achievable rates over the AF relaying. In [15], the “sawtooth relaying” was studied further and a piecewise linear relaying function was proposed to approximate the theoretically optimal relaying
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function for the relay channel with orthogonal components. With this piecewise linear relaying function, the achievable rates can be close to that with CF relaying, but at a much lower complexity. However, the relaying schemes in both [14] and [15] rely on the side information by the direct link between the source and the destination. Therefore, when the direct link is poor, or there is no direct link, the rate performance will be deduced significantly. In [6], the relaying function was restricted to be the linear combination of the past received signals at the relay, namely linear relaying, and it has been proved that the problem for the capacity of the frequency-division additive white Gaussian noise (AWGN) relay channel with LR can be reduced to a “single-letter” nonconvex optimization problem, but the problem for general relay channel can’t be reduced as such. In [16], the optimal covariance matrix of the signals transmitted from the source was derived for fixed relaying matrices and the suboptimal relaying matrices were also designed. In this paper, we study the performance of multiple-hop relay channel with linear relaying in the view of information theory. Each node in the channel is assumed to communicate with its nearest neighbor nodes only. The relaying function we concern about is the same as that in [6], namely LR. With linear relaying, the relays transmit the combination of their past received signals. In this set up, i) the relaying schemes proposed in [14] and [15] cannot be used here, because there is no direct link to provide the side information, and ii) two parameters can be optimized to achieve higher achievable rate: the linear relaying function (relaying matrix) at each relay node and the covariance matrix of the signal transmitted from the source. We prove that the achievable rate for the Gaussian multiple-hop relay channel with LR can be formulated as a “single-letter” optimization problem under some certain conditions. And the solution to this “single-letter” optimization problem has the same form as the expression of the rate achieved by TSAF. With TSAF, the time for a block transmission is split into several slots. During each slot, the source transmits with certain average power while the relay nodes amplify and forward their received signals with corresponding amplifying coefficients under the total average power constraint of each. The amplifying coefficients and the average transmission power at the source in each slot are strictly relevant. We present an iterative algorithm to compute these amplifying coefficients and ratios of the average transmission power of the source in each slot. Thus, the processing complexity of TSAF at each relay node is almost as low as that of AF. The numerical results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively. In the wireless networks such as wireless sensor
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networks, of which the processing ability is poor, the transmission power is limited and channel gain varies significantly among different paths, the TSAF scheme achieves a good trade-off between processing complexity and achievable rates. Notations: Bold uppercase letters denote matrices or vectors and bold lowercase letters denote column vectors; transpose is represented by ()T ; IM is the identity matrix of size M M ; variables are denoted by italic letters; N(μ, N0) denotes circularly symmetric Gaussian distribution with mean μ and variance N0;
( ) denotes expectation; I ( ; ) denotes the mutual information between two
random variables, h( X ) denotes the differential entropy of the continuous random variable X , and
M denotes the determinant of matrix M .
2. Multiple-hop relay channel model and capacity 2.1 Channel model and problem setup The multiple-hop channel model introduced in this paper is shown in Fig. 1. Let the source node be denoted by 0, the destination node by M, and let the other M–1 relay nodes be donated sequentially as 1, 2, …, M-1. Each node communicates with its neighbor nodes, and there is no direct link between the other nodes. Assume each node k∈{0, 1, …, M-1} sends xk(i)∈k and receives ym(i)∈m (m = 1, …, M) at time i, where k and m are the corresponding input and output alphabets for corresponding nodes. We have the following definitions: A (2 nR, n) -code for the M-hop relay channel consist of i) a set of messages
{1, 2,...,2nR } ;
ii) a codebook {x n (1), x n (2),..., x n (2nR )} consisting of codewords of length n; ii) encoding function at the source
:1,2,...,2nR
0
(1)
iii) a sequence of relaying functions at the relay node k,
k ,i : for i = 1, 2, …, n, where k = 1, 2, …, M-1.
i 1 k
k
(2)
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iv) a decoding function at the destination
n :
n M
{1,2,...,2nR }
(3)
The average probability of decoding error is defined as
Pe( n )
1 2nR
2nR
P
n
( yMn )
j | x n ( j )is transmitted
(4)
j 1
We assume the average power constraint of the source node power constraints of the relay nodes
1 n E ( X 0,2 i ) P , and the average n i 1
1 n E ( X k2,i ) P , k=1, 2, …, M-1. n i 1
A rate R is said to be achievable if there exits a sequence of (2nR ,n) -codes satisfy the average power constrains defined above, such that Pe( n )
0 as n
. The capacity is defined as the
supremum of the set of achievable rates.
Fig. 1 The multiple-hop relay channel with linear relaying
During n uses of the channel, node k transmits a sequence of random symbols
Xnk =[X k ,1 , X k ,2 , ..., X k ,n ]T (k = 0, 1, 2, …, M-1) and receives Ykn
[Yk ,1 , Yk ,2 , ..., Yk , n ]T (k = 1, …, M).
As mentioned earlier, the relay nodes transmit the combination of their past received symbols. Thus, at i
time i, the transmitted relay symbol at relay k (k = 1, 2, …, M-1) can be expressed as xk ,i
d i(,kj ) yk , j , j 1
where d i(,kj ) are the coefficients chosen such that the average power constraint for the relay k is satisfied. For a transmission block length n, Xnk
Dk Ykn , where Dk (namely, relaying matrix) is a
lower triangular weight matrix of relay k. 2.2 Capacity for the multiple-hop relay channel with linear relaying Capacity for the multiple-hop relay channel is defined as the supremum over all the achievable rates [6], which is expressed as (5).
CLR
lim
n
1 sup n p ( x0 , x1 , ..., xM
I ( X0n ; YMn )
(5)
1)
The maximization in (5) is achieved when X 0n is Gaussian. Here, we consider AWGN channel,
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where hk denotes the channel coefficient between node k and k+1, and the Gaussian white noise zk ~ N(0, N0). Let P be the average power constraint for the source node while Denote Σs
P for the relay nodes.
(X0n X0nT ) as the covariance matrix of X 0n .
From Fig. 1, the received vector at node k can be expressed as:
Ykn where, Xnk
1
hk 1Xkn
1
z kn , k
(6)
1, 2, ..., M
is the transmitted vector at node k-1. With linear relaying, Xnk
is given by (7).
1
Xnk 1 = Dk 1Ykn 1
(7)
By (6) and (7), the received vector at the destination node is M 1M 1
M 1
YMn
hm DM
1~1
X0n +
m 0
where， D p ~ q
Dp Dp
1~ l
z ln
z nM
(8)
l 1 m l
Dq , DTp ~ q
1
hm DM
DTq DTq
1
DTp .
Since the random variables in (8) are jointly Gaussian, with Theorem 8.4.1 of [17] we can compute the achievable rate R as
R
1 I( X0n ; YMn ) n
1 h(YMn ) h(YMn | X0n ) n
1 1 log (2 e) n K YM n 2
1 log (2 e) n K YM | X 0 2 M 1M 1
M 1
1 log 2n
(9)
hm2 DM
1 ~1
Σ s DTM
1 ~1
hm2 DM
N 0 (I
m 0
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
hm2 DM
N 0 (I
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
M 1
hm2 DM
where, K YM
T 1 ~1 Σ s DM
1 ~1
hm2 DM
N 0 (I
m 0
1 ~l
DTM
1 ~l
) is the covariance matrix of
l 1 m l M 1M 1
the Gaussian random vector YMn , and K YM | X 0
hm2 DM
N 0 (I
1 ~l
DTM
1 ~l
) is the covariance
l 1 m l M 1M 1
hm DM
matrix of
1~ l
z ln
z nM .
l 1 m l
By
(5),
CLR ( P, P)
capacity
for
Gaussian
multiple-hop
relay
channel
lim Cn ( P, P) , where, Cn ( P, P) is expressed as (10).
n
with
linear
relaying
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1 n
Cn ( P, P)
I ( X0n ; YMn )
max
Σ s ，{Dk }1M 1
M 1M 1
M 1
1 max log 2n {Dk }1M 1
hm2 DM
1 ~1
Σ s DTM
hm2 DM
N 0 (I
1 ~1
m 0
1 ~l
DTM
1 ~l
)
l 1 m l M 1M 1
hm2 DM
N 0 (I
Σs
1 ~l
DTM
1 ~l
）
l 1 m l
s.t. Σ s
0, tr ( Σ s )
k 1
hm2 Dk ~1 Σ s DTk ~1
tr{ m 0
k
(10)
nP
k 1
k 1
n P,
m l
l 1
1, 2, ..., M
hm2 Dk ~ l DTk ~ l N 0 Dk DTk }
N0
1
3. Two-hop relay channel with linear relaying In this section, we firstly analyze two-hop relay channel with linear relaying by letting M=2 shown in Fig. 1. By (10), the capacity for two-hop relay channel with linear relaying is 2 CLR ( P, P )
1 max I ( X0n ; Y2n ) n Σs，D1
h12 h02 D1 Σ s D1T 1 max log 2n Σs，D1 N 0 (I s.t. Σ s
0, tr ( Σ s )
tr{h D1 Σ s D 2 0
T 1
N 0 (I
h12 D1D1T )
(11)
h12 D1D1T )
nP
N 0 D1D1T }
n P
For fixed D1 , (11) is a concave function over Σ s , however nonconcave over D1 or ( D1 , Σ s ). We consider the block length n=2 as a special case. The relaying matrix and input covariance matrix are expressed as
d 0 , Σs 0 d
D1
where d
P /(h02 P
(1
2P (1
)
)
(12)
1
N0 ) is selected to satisfy the power constraint at the relay,
( 1
1)
is the correlation coefficient between X1 and X2 which are the signals of the two transmissions at each block respectively,
P and (1
)P are the average power of X1 and X2 respectively.
2 ( P, P) is a concave function over Σ s , and the According to [16], for fixed relaying matrix D1 , CLR
optimal covariance matrix Σ sopt
P 0 . 0 P
In the case above, the achievable rate with linear relaying at optimal point is equivalent to that
h12 h02 d 2 P 1 ) . In [6], it was pointed out that as the with AF, and the achievable rate is R2 = log（1 + 2 (1 h12 d 2 ) N 0 power P is decreased, linear relaying injects more noise than signal, thus becomes less helpful. The
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same problem occurs in the two-hop relay channel with linear relaying mentioned in this paper. When the signal power is lower than that of the noise, noise is amplified more than signal by linear relaying, which causes decrease in achievable rate. The reason is that at optimal point the amplifying coefficients at the relay and the average transmission power at the source remain equal during each use of channel. This can be seen from (12) and the expression of Σ sopt . The amplifying coefficients at the relay nodes are chosen to satisfy the average power constraint regardless of the SNR of the received signals at the relay node. Therefore, the power is not utilized fully. Suppose that if the power at the source is changeable of each slot under its total power constraint, the relay nodes can amplify the received signals accordingly to the SNR under their power constraints. The received signals with higher SNR will be amplified more at the relay. In such a way, the power efficiency at both the source and the relay nodes will be improved and a higher achievable rate can be derived. Specifically speaking, the time for a block transmission is split into several slots. During each slot, the source transmits with some certain average power while the relay nodes amplify and forward their received signals with corresponding amplifying coefficients under the total average power constraint. The amplifying coefficients and the average transmission power at the source in each slot are strictly relevant. By optimization methods, the optimal amplifying coefficients, the optimal ratio of the average power at the source in each slot and the optimal number of slots can be derived. We call this relaying scheme time-sharing AF (TSAF), which is explained vividly as Fig. 2.
(a) Time-sharing transmission with different power level at the source
(b) The corresponding amplifying coefficients of the relay in each slot
Fig. 2 Time-sharing transmission among slots
In Fig. 2, θi and αi are ratio of the transmission power ratio and the number of channel uses of slot i at the source node respectively, where θ1+θ2+θ3=1 and α1+α2+α3=1; ηi denotes the corresponding
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2 i
amplifying coefficient under the average power constraint
(h02 i P
i
N0 )
P . Optimizing
i
over the parameters θ, α and η, we can compute the lower bound of the capacity of the two-hop relay channel with linear relaying, which is shown in (13). 2 CLR ( P, P )
s.t.
i i
max α ,θ , η
1,
1 2
i i
P h12 h02 h12 i N0 1 i
i log(1 i 2 i
1,
2 0 i
(h
P
i
N)
2 i 2 i
) (13)
P
i
To verify the effectiveness of TSAF, assuming the block length n=2, we compare the achievable rate with TSAF with the maximal rate which is computed by search over the relaying matrix D1 and the input covariance matrix Σ s exhaustively, where
d1,1 d 2,1
D1
Σs
(1
2P (1
1
2
and
1
0 . d 2,2
For the optimization problem in (13), parameters are defined as α = [
η=[
)
)
1
2
]T , θ = [
1
2
]T and
]T . It is a nonconvex problem. However, the optimization problem in (13) is convex in each
set of α, θ, η if the other two are fixed. A Mote Carlo algorithm similar to that described in [6] can be used to derive a local maximum. Fig. 3 compares the achievable rate of TSAF with the maximal rate, where h0
h1
1 . From
the figure, it can be found that the two rates are almost equal to each other for any SNR, which demonstrates the effectiveness of TSAF scheme.
Fig. 3 Comparison between the achievable rate with TSAF and maximal rate by exhaustive search
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4. Multiple-hop Gaussian relay channel with LR 4.1 Optimization problem on Gaussian multiple-hop relay channel with LR In [6], it has been proved that the optimization problem for LR in the case of orthogonal relay channel can be reduced to a “single-letter” optimization problem. In this section, we will prove that the optimization for multiple-hop relay channel with linear relaying is equivalent to a “single-letter” optimization problem too, as shown in Theorem 1, in which each block is divided into several time slots. During each slot, the source node sends symbols at a given average power while the relay nodes work at AF mode with corresponding amplifying coefficients. Theorem 1: For the M-hop Gaussian relay channel with linear relaying, the SVD (singular value
Fk L k QTk , (k
decomposition) of the relaying matrix at relay k is Dk 3, …, M-1}, if FmT 1Qm
QTm Fm
1, 2, ..., M
1) , for any m∈{2,
I , the expression shown in (10) is equivalent to the following
1
optimization problem: M 1 N
C
L LR
maxM -1
α , θ ,{ ηk }k =1
P ( N i 0
h02
2 m m 1 M 1M 1
i
i i 1
h
m ,i
) 2 m
1
h
m ,i
l 1 m l
s.t.
i , i ,
k ,i
N
0,
N i
N
1,
i
i 1
i 1 k 1 k 1
k 1 k ,i
(h02 i P
hm2
m ,i
i
m 1
i 1
hm2
N0 (
m ,i
where, N is the number of time slots, (x)=1/2log(1+x), α
ηk
[
k ,1
,
k ,2
, ...,
k,N
1))
P, k
1, 2,..., M
[ 1,
2
, ...,
DM 1 ~l ）
1 ~1
N
]T , θ
]T (k=1, 2, …, M-1).
Proof: Define M 1
hm He
m 0
N0
1
l 1 m l
M 1M 1
hm2 DM
(I l 1 m l
The problem shown in (10) can be written as
T 1 ~ l DM
1 2
VΛVT
[ 1,
2
, ...,
N
]T and
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1 log I 2n 0, tr ( Σ s ) nP
Cn
H e Σ s HTe
max
DM 1 ,...,D1 , Σ s
s.t. Σ s
k 1
k 1
hm2 Dk ~1 Σ s DTk ~1
tr{ m 0
k
k 1
N0
n P,
m l
l 1
1, 2, ..., M
(14)
hm2 Dk ~ l DTk ~ l N 0 Dk DTk }
1
(14) is a nonconvex optimization problem. However, for fixed DM 1 , …, D1 , Cn is a concave
1 , the Lagrangian for Cn is expressed as 2n
function on Σ s . Neglecting the factor
L（Σ s，Ω， ， 1,
2
,..., k ）=
H e Σ s HTe
log I
M 1
tr (ΩΣ s ) k 1
k 1 k
hm2 Dk ~1 Σ s DTk ~1
(tr{
nP)
hm2 Dk ~ l DTk ~ l N 0 Dk DTk } n P)
N0
m 0
k 1
(tr ( Σ s ) k 1
m l
l 1
KKT conditions are M 1
k 1
i) I
hm2 DTk ~1Dk ~1
k
ii) tr ( ΩΣs )
0
iii) (tr (Σs )
nP)
0 k 1
k 1 k
hm2 Dk ~1 Σs DTk ~1
(tr{ m 0
where， , 1 ,
2
k 1
N0 l 1
,...,
hm2 Dk ~ l DTk ~ l N0 Dk DTk } n P) = 0,k
1
0.
M 1
QTm Fm
1,...,M
m l
Let the SVD of Dk ,He are Dk = Fk Lk QTk (k assume FmT 1Qm
Ω
m 0
k 1
iv)
HTe (I + H e Σs HTe ) 1 He
1
1, ...,M -1), He
UΛVT respectively, and
I for m = 2, 3, …, M-1, then M 1
hm2 T e
H He where， L2M
1~ l
M 1M 1
Q1L2M
m 0
N0
1~1
(
hm2 L 2M
1~ l
+ I) 1 Q1T
VΛ 2 VT
(15)
l 1 m l
L2M 1...L2l , L1, L2, …, LM-1 and Λ are all diagonal matrices. Thus, the columns of
Q1 and V are engenvectors of HTe H e and VT Q1
Q1T V
I . The first KKT condition can be
simplified to M 1
i) I
k 1 k
k 1
hm2 Λ k ~1
Λ(I + ΛVT Σs VΛ) 1 Λ VT ΩV
m 0
where， Λk = L2k , Λk ~1 = Λk Λk 1...Λ1 . Now, we show that for Σs
VΨVT , if Φ = VΘVT , then tr (Σs Φ)
tr (ΨΘ)
0 , the KKT
conditions hold. The matrices Ψ and Θ are both diagonal matrices. Therefore, (14) is reduced to
M 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
Cn
max
Ψ ,Λ1 ,Λ 2 ,..., Λ M 1
s.t. Ψ
0, tr{Ψ}
M 1M 1
ΨΛ M
m 0
N0
1~1
hm2 Λ M
(I
1 ~l
） 1
l 1 m l
(16)
nP, k 1
k 1
k 1
hm2 ΨΛ k ~1
tr{
hm2 Λ k ~ l N 0 Λ k }
N0
m 0
k
hm2
1 log I 2n
1, 2, ..., M
n P
m l
l 1
1
Since all the matrices in (16) are diagonal, the following “single-letter” optimization problem is equivalent to (16). M 1
Cn
1 2n
max 1
,... n
M 1
h02
n
log 1 i 1
0,
i
k ,i
h
i
hm2
N0 1
k ,1 ,..., k ,n k 1
s.t.
2 m m ,i m 1 M 1M 1
m ,i
l 1 m l
(17)
n
0,
nP
i i 1
n
h02
k ,i
hm2
m ,i
+
i
1, 2, ..., M
k 1
hm2
N0
m 1
i 1
k
k 1
k 1
m ,i
N0 n P
m l
l 1
1
Obviously, the rates will approach maximum when the source nodes and the relays transmit at
maximal power, i.e. at optimal point
n
k 1
k 1
h02
k ,i
hm2
m ,i
i
+
m 1
i 1
k 1
hm2
N0 l 1
m ,i
N0
n P ,
m l
n
nP for k
i
1, 2, ..., M
1 . So the corresponding inequality constraints in (17) can be
i 1
substituted by the above equalities respectively. And the Lagrangian is M 1
L(
n 1
,
n 1
, ...,
n M 1
, ,
1 ,...,
h02
n M
1)
log 1 i 1
2 m m ,i m 1 M 1M 1
h
i
n
( 2 m m ,i
N0 1
h
i
nP)
i 1
l 1 m l
M 1
n k
k 1
k 1
k 1 k ,i
h02
hm2
m,i
m 1
i 1
i
+
k 1
hm2
N0 l 1
k 1, i
...
l ,i
N0
n P
m l
Thus, at optimal point the following equalities hold.
L
0
i
L
0 (18)
1, i
L
0
M 1, i
Solve the set of equations shown in (18). For any hM 1 ..., h1 , h0 , assume there’re at most N
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positive roots of the set of equations, which are denoted subsequently as ( 1 ,
( 2,
1,2
,...,
M 1,2
) , …, (
N
,
1, N
,...,
M 1, N
1,1
,
2,1
,...,
M 1,1
),
) . So, for any n≥N, there are at most N distinct roots which
are optimal. Without loss of generality, assume that at optimum, the roots of the first n1 indices are related to ( 1 ,
1,1
,
2,1
,...,
M 1,1
) , while ( 2 ,
,...,
1,2
M 1,2
) related to the roots of the next n2 indices,
analogically, the roots of the last nN indices are related to (
N,
N
1, N ,...,
ni
1, N ) , where
M
n . Let
i 1
i
nP be the total power allocated to the ni (i = 1, 2, …, N) indices, so
nP and ni
N
i i
i
1.
i 1
Therefore, the problem in (17) is the same as (19). M 1
Cn
max
n ,θ , M 1 ηk k 1
s.t.
0,
i
n
1 2n
i nP ni
ni log 1 i 1
N0 1
hm2
m ,i m 1 M 1M 1 2 m l 1 m l
h
m ,i
N
0,
k ,i
h02
i
(19)
1
i 1 n
k 1
k 1
h02
k ,i
m,i i
nP + ni
m 1
i 1
k
hm2
hm2
m ,i
ni N 0 n P
m l
l 1
1, 2, ..., M
1
[ 1 ,..., N ]T , η j
where, n=[n1, n2, …, nN]T, θ
k 1
N0
[
j ,1
As n→∞, taking the limit of (19) and defining
,...,
i
j,N
lim
n
]T . ni , we derive n M 1
n
C
L LR
lim Cn
2 m m 1 M 1M 1
i
max
α ,θ , M 1 i 1 ηk k 1
n
P i N0
hm2
i
h
m ,i
hm2
1
m ,i
l 1 m l
s.t.
i
, i ,
N k ,i
0,
N i
i
i 1 N
( h02 i P
k 1 k 1
hm2 m 1
i 1
k
i 1
k 1 k ,i
1, 2, ..., M
(20)
1
m ,i
i
hm2
N0 (
m ,i
1))
P
l 1 m l
1
This completes the proof. 4.2 Algorithm design Though (20) is “single-letter” characterized, it is still a nonconvex problem. It requires exhaustive search which is computationally expensive. However, the optimization problem in (20) is convex in each set of α, θ, η1 , η2 ,...,ηM
1
if the other M sets are fixed. Furthermore, it is hard to show the exact
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value of N. We use the following numerical method to derive a sub-optimal solution to this problem.
Initialization: Fix the number of time slots N=Nini, randomly choose initial values for the M+1
α0 , θ
sets of variables: α
θ0 , η1
η2,0 , …, ηM
η1,0 , η2
1
ηM
1,0
2, opt
;
Step 1: -1.1 Iteration over variable sets: (1)
Fix α , θ , η1 , …, ηM
2
, optimize over ηM 1 , update ηM 1 =ηM
(2)
Fix α , θ , η1 , …, ηM
3
, ηM 1 , optimize over ηM
(3)
Fix the sets α, θ, η1 , η2 ,...,ηM
1
2
1, opt
;
, update ηM 2 =ηM
excluding ηk , optimize over ηk and update
ηk =ηk ,opt subsequently for k 1,2,...,M
3;
(4)
Fix α , η1 , …, ηM 1 , optimize over θ , update θ
(5)
Fix θ , η1 , …, ηM 1 , optimize over α , update α
θopt ; α opt .
-1.2 Termination: Repeat Step 1.1 until the rate converges to a local maximum.
Step 2: -2.1 Randomly choose initial values for the M+1 sets of variables: α
η2
η2,0 , …, ηM
1
ηM
1,0
α0 , θ
θ0 , η1
η1,0 ,
, repeat Step 1.
-2.2 Termination: Repeat Step 2.1 until the repeating times exceed K (a given number), select the maximal value of Step 2 as the local maximum corresponding to the current number of time slots.
Step 3: -3.1 Increase the number of time slots N = N + 1, repeat Step 2. -3.2 Termination: Repeat Step 3.1 until the number of slots N exceeds Nf
(a given number),
choose the maximal value as the final result. 4.3 Numerical Results Taking a three-hop Gaussian relay channel with linear relaying as example, we present numerical results to analyze the performance of TSAF. In order to run the algorithm in 4.2, we choose the following parameters. The initial and final value of the number of time slots are chosen respectively as
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Nini = 3 and Nf = 10. We further choose the repeating times of Step 2.1 K = 40. Step 1 is terminated if the change in the resulting converged rate of this step is less than 10-4. The received signals at the two relay nodes and the destination are given by
respectively, where,
1 n
n
E ( X 0,2 i )
P,
i 1
Y1n
h0 X0n
Z1n
(21)
Y2n
h1X1n
Z2n
(22)
Y3n
h2 X2n
Z3n
(23)
1 n
n
E ( X 1,2i ) i 1
1 n
n
E ( X 2,2 i )
P and Z1 , Z 2 , Z 3 are
i 1
i.i.d. Gaussian noises with variance N 0 . As known, the capacity of the general 3-hop relay channel is achieved with DF relaying. The capacity is shown as follows
C
min{C (
h02 P h2 P h2 P ), C ( 1 ), C ( 2 )} N0 N0 N0
From (24), it can be found that the minimum value of h0 , h1
and h2
(24) dominates the capacity.
The achievable rate with linear relaying is compared with the capacity in (24) under two assumptions: i) the channel gain of one certain hop is smaller relatively, in this case
h0 =0.5， h1 h0 =h1
=h2
=h2
we choose
=2 ; and ii) the gain of the three hops are comparable, in this case we choose
=2 .
As mentioned in section 3, TSAF is introduced to overcome the low power efficiency of regular AF relaying when the SNR at the relay node is low. We also compare the achievable rate with TSAF
h1 with the achievable rate with regular AF when h0 =0.2，
=1,h2
=2 .
Fig. 4 compares the achievable rate with TSAF with the capacity of the three-hop relay channel. Some optimized parameters α , θ , η1 and η2 for different SNRs are shown in Table 1 and Table 2 respectively. The capacity of the 3-hop relay channel is achieved by DF relaying with which each relay node must decode the message from its upstream node and forwards the decoded message to its downstream node. The decoding and encoding at the relay nodes increase the complexity of these nodes. From Fig. 4, we find that when the gain of one certain hop is smaller than that of the other two hops relatively, the achievable rate with TSAF relaying may be close to the capacity. However, the complexity of the relay nodes with TSAF relaying or linear relaying is low because decoding or
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encoding is not needed at the relay nodes. With TSAF relaying or linear relaying, the relay nodes only have to amplify and forward their received signals with corresponding amplifying coefficients of each slot or transmit the combination of their past received signals to the downstream nodes. Therefore, the complexity of the TSAF relaying is almost as same as that of the simple AF relaying. We conclude that TSAF achieves a good trade-off between the complexity and the achievable rate.
Fig. 4 Comparison between the capacity and achievable rate with linear relaying
Table 1 α * , θ* , η1* , η*2 for different SNR ( h0 =h1
=h2
=2 )
Optimized parameters
SNR = 2
SNR = 4
SNR = 6
SNR = 8
SNR = 10
α
0.2056 0.1694 0.2127 0.4123
0.2346 0.1750 0.5904
0.0804 0.2475 0.6720
0.2994 0.2515 0.4491
0.2994 0.2515 0.4491
θ*
0.2007 0.2032 0.2048 0.3913
0.1945 0.1792 0.6264
0.1171 0.2249 0.6580
0.2717 0.2233 0.5050
0.2717 0.2233 0.5050
η1*
0.2254 0.2007 0.2272 0.2289
0.2653 0.2321 0.2268
0.1845 0.2566 0.2441
0.2598 0.2637 0.2235
0.2617 0.2656 0.2246
η*2
0.2222 0.2233 0.2221 0.2218
0.2348 0.2354 0.2355
0.2406 0.2398 0.2400
0.2422 0.2422 0.2427
0.2437 0.2437 0.2441
*
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Table 2 α * , θ* , η1* , η*2 for different SNR ( h0 =0.5,h1
=h2
=2 )
Optimized parameters
SNR = 2
SNR = 4
SNR = 6
SNR = 8
SNR = 10
α
0.0415 0.0510 0.0720 0.0741 0.0797 0.1154 0.5663
0.0529 0.1050 0.0491 0.0319 0.0731 0.0251 0.6630
0.0691 0.0295 0.1223 0.0237 0.1024 0.6529
0.0876 0.0234 0.0028 0.1547 0.1143 0.6172
0.0077 0.0996 0.1631 0.0827 0.6470
0.0512 0.0585 0.0759 0.0832 0.0882 0.0000 0.6429
0.0531 0.1113 0.0494 0.0360 0.0732 0.0258 0.6512
0.0679 0.0278 0.1204 0.0239 0.0971 0.6629
0.0899 0.0224 0.0025 0.1555 0.1234 0.6063
0.0121 0.1000 0.1637 0.0824 0.6418
η
1.4194 1.4862 1.5169 1.4148 1.4617 0.0000 1.4351
2.0199 1.9236 1.9778 1.8093 2.0036 1.8131 2.0293
2.5053 2.7833 2.4920 2.1988 2.7245 2.3140
2.7785 2.5010 1.5559 2.6891 3.0034 2.5891
2.2163 3.0712 3.0050 2.9055 2.7909
η
0.2389 0.2263 0.2150 0.2239 0.2213 0.0000 0.2260
0.2325 0.2442 0.2381 0.2597 0.2348 0.2624 0.2319
0.2310 0.2107 0.2321 0.2588 0.2144 0.2480
0.0899 0.0224 0.0025 0.1555 0.1234 0.6063
0.2441 0.2282 0.2326 0.2406 0.2501
θ
*
*
* 1
* 2
Fig.
5
h0 =0.2,h1
compares =1,h2
the
achievable
rate
between TSAF and
regular AF
when
=2 . It can be found from Fig. 5 that TSAF outperforms AF when the SNR is
low, e.g. the achievable rate with TSAF is almost twice as high as that of the regular AF when P / N0
1 and triple as high as that of the regular AF when P / N0
0.5 . The lower the SNR at
the relay node is, the greater the rate improvement is. This observation indicates the intention of designing the TSAF relaying scheme, i.e. to improve the performance of AF at low SNR.
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Fig. 5 Comparison between the achievable rate of AF and TSAF: h0 =0.2,h1
=1,h2
=2
5. Conclusions We study the performance of linear relaying scheme for multiple-hop relay channel. The capacity expression for the multiple-hop relay channel with linear relaying is derived. It is shown that under some certain conditions, the solution to the capacity expression can be reduced to a “single-letter” optimization problem. However, this optimization problem is a nonconvex optimization problem which requires computationally expensive exhaustive search. We design an iterative algorithm to find a sub-optimal solution to this optimization problem. Numerical results show that if channel gain of one certain hop is smaller than that of all the other hops relatively, the achievable rate with TSAF is close to the capacity which is achieved with DF. The complexity of TSAF, however, is much simpler than DF since block-wise encoding or decoding is not needed at the relay nodes. The simulation results also show that TSAF outperforms the simple AF in achievable rate when the SNR at the relay is low. In the wireless networks such as wireless sensor networks, of which the processing ability is poor, the transmission power is limited, nodes usually communicate through multiple-hop routes and channel gain of each hop varies significantly among different paths, the TSAF scheme can achieve a good trade-off between processing complexity and achievable rates.
Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant No. 60972045, 61271232, 61071089), the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2012D05), the Natural Science Foundation of Jiangsu Province,
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China (Grant No. BK2010077), the University Postgraduate Research and Innovation Project in Jiangsu Province, China (No. CXZZ11_0395), the Fundamental Research Funds for the Central Universities (No. 2009B32114).
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Zhixiang Deng received his B.S. degree in Communication Engineering from Hohai University, China, in 2003, and his M.S. degree in Communication and Information System from Southeast University, China, in 2006. He is now
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working towards his Ph.D. degree in Information and Signal Processing with Nanjing University of Posts and Telecommunications, China. His main research interests include multiuser information theory and information theoretic security.
Bao-Yun Wang received Ph.D. degree in electrical engineering from Southeast University, Nanjing, in 1997. In 1997-1999, he was on the faculty of Nanjing University of Posts and Telecommunications. In 1999-2000, he was a research associate at Pohang University of Science and Technology, Pohang, Korea. In 2000-2002, he was with City University of Hong Kong as a research associate. In 2004-2005, he was a visiting research fellow at The University of Western Sydney, Sydney, Australia. Since 2005, he has been on the faculty of Nanjing University of Posts and Telecommunications, where he is the Dean of School of Automation Engineering. His research interests are in the area of information theory, statistical signal processing, and their applications in wireless communications.
* * α * , θ* , η1 , η2 for different SNR ( h0 =h1
=h2
Table 2 α * , θ* , η1* , η*2 for different SNR ( h0 =0.5,h1
=2 )
=h2
=2 )
Fig. 1 The multiple-hop relay channel with linear relaying Fig. 2 Time-sharing transmission among slots: (a) Time-sharing transmission with different power level at the source, (b) The corresponding amplifying coefficients of the relay at each slot Fig. 3 Comparison between the capacity with TSAF and maximal rate by exhaustive search Fig. 4 Comparison between the capacity and achievable rate with linear relaying Fig. 5 Comparison between the achievable rate of AF and TSAF: h0 =0.2,h1
=1,h2
=2