Abbas El Gamal
580 California St., Suite 400
San Francisco, CA, 94104
Capacity of a Class of Relay Channels With Orthogonal Components
2005, IEEE Transactions on Information Theory
https://doi.org/10.1109/TIT.2005.846438visibility
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Abstract
The capacity of a class of discrete-memoryless relay channels with orthogonal channels from the sender to the relay receiver and from the sender and relay to the receiver is shown to be equal to the max-flow min-cut upper bound. The result is extended to additive white Gaussian noise (AWGN) relay channels where the channel from the sender to the relay uses a different frequency band from the channel from the sender and the relay to the receiver. Index Terms-Additive white Gaussian noise (AWGN) relay channel, discrete memoryless relay channel.
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 5, MAY 2005
Capacity of a Class of Relay Channels
With Orthogonal Components
1815
Definition: A discrete-memoryless relay channel is said to have orthogonal components if the sender alphabet X = XD 2 XR and the
channel can be expressed as
j
Abbas El Gamal, Fellow, IEEE, and
Sina Zahedi, Student Member, IEEE
p(y; y1 x; x1 )
= p(y
for all
(xD ; xR ; x1 ; y; y1 )
Abstract—The capacity of a class of discrete-memoryless relay channels
with orthogonal channels from the sender to the relay receiver and from
the sender and relay to the receiver is shown to be equal to the max-flow
min-cut upper bound. The result is extended to additive white Gaussian
noise (AWGN) relay channels where the channel from the sender to the
relay uses a different frequency band from the channel from the sender
and the relay to the receiver.
Index Terms—Additive white Gaussian noise (AWGN) relay channel, discrete memoryless relay channel.
I. INTRODUCTION
The discrete-memoryless relay channel denoted by (X 2
j
Y 2 Y1 ) consists of a sender X 2 X , a receiver
Y 2 Y , a relay sender X1 2 X1 , a relay receiver Y1 2 Y1 , and a family
of conditional probability mass functions p(y; y1 jx; x1 ) on Y 2 Y1 ,
one for each (x; x1 ) 2 X 2 X1 . A (2nR ; n) code for the channel
consists of: i) a set of messages f1; 2; . . . ; 2nR g, ii) an encoding
function that maps each message w into a codeword xn (w) of length
n, iii) relay encoding functions x1i = fi (y11 ; y12 ; . . . ; y1(i01) ), for
1 i n, and iv) a decoding function that maps each received
n
^ (y ). A rate R is achievable if there
sequence y n into an estimate w
(n)
nR
^ 6= W g ! 0,
exists a sequence of (2 ; n) codes with Pe = PfW
as n ! 1. Channel capacity C is defined as the supremum over the
set of achievable rates.
The relay channel was introduced by van der Meulen in [1]. The capacity of degraded and reversely degraded relay channels as well as
upper and lower bounds on the capacity of the general relay channel
were established in [2]. The capacity of the general relay channel, however, is not known. Recent work on relay channel capacity has been
motivated by wireless communication scenarios. Since in a practical
wireless communication system, a node cannot transmit and receive at
the same time or over the same frequency band, there has been much interest in channels with orthogonal components [4], [5]. In [4], a half-duplex additive white Gaussoan noise (AWGN) relay channel model is
introduced where the relay transmits and receives at nonoverlapping
time slots and the effect of multipath fading on the performance for several simple cooperative schemes is studied. In [5], a simple network of
AWGN multiple-access channel with generalized feedback is considered where orthogonal channels carry partial feedback information and
transmitters can partially cooperate through the feedback paths. Upper
and lower bounds on the capacity region of this channel are presented
and performance of several practical coding schemes are studied under
fading conditions.
In this note, we consider the following class of discrete-memoryless
relay channels with orthogonal channel components from the sender to
the relay and from the sender and relay to the receiver.
X
1 ; p(y; y1 x; x1 );
Manuscript received May 7, 2004; revised December 13, 2004. This work
was supported by the Stanford Network Research Center (SNRC).
The authors are with the Department of Electrical Engineering, Stanford
University, Stanford, CA 94305 USA (e-mail: [email protected]; sina@
stanfordalumni.org).
Communicated by R. W. Yeung, Associate Editor for Shannon Theory.
Digital Object Identifier 10.1109/TIT.2005.846438
j
j
x D ; x 1 )p(y 1 x R ; x 1 )
2X 2X 2X 2Y 2Y
D
1
R
1:
Our main result is to establish the capacity of this class of relay
channels.
Theorem: The capacity of the relay channel with orthogonal components is given by
C
= max min
f
I (X D ; X 1 ; Y
); I (X R ; Y 1
j
X1 )
+ I (X D ; Y
j
X1 )
g
where the maximum is taken over all joint probability mass functions
of the form p(xD ; xR ; x1 ) = p(x1 )p(xD jx1 )p(xR jx1 ):
We also extend this result to the class of AWGN relay channels where
the channel from the sender to the relay uses a different frequency band
from the channel from the sender and the relay to the receiver.
Note that if p(y1 jxR ; x1 ) = p(y1 jxR ) for all (y1 ; xR ; x1 ) 2 Y1 2
XR 2 X1 , then our theorem reduces to a special case of the capacity
region of the multiple-access channel with partially cooperating transmitters established in [6]. By setting the capacity of the link from one
of the transmitters to the other to zero, the problem studied in [6] reduces to a special case of the relay channel with orthogonal components
studied here. Therefore, the main contribution of our correspondence
can be viewed as generalizing a special case of [6] and extending it
to the AWGN case. It is also worth noting that using our AWGN extension, the capacity region for AWGN multiple-access channel with
partially cooperating transmitters can be readily established.
II. PROOF OF THEOREM
To prove the theorem, we use two bounds on the capacity of the
general discrete-memoryless relay channel from [2]. The first bound is
the “max-flow min-cut” upper bound
C
max min
p(x;x
)
f
I (X; X1 ; Y
); I (X ; Y ; Y 1
j
g
X1 ) :
(1)
The second bound is a special case of Theorem 7 in [2], which yields
the lower bound
C
max
p(u;x;x
)
min
f
I (X; X1 ; Y
); I (U ; Y1
j
X1 )
+ I (X ; Y
j
X1 ; U )
g
:
(2)
This lower bound is achieved using a generalized block-Markov coding
scheme, where in each block, the relay decodes part of the new message
(represented by U ) and cooperatively sends enough information with
the sender (represented by X ) to help the receiver decode the previous
message (U then X ). Note that in [7], this lower bound was shown to
be optimal and equal to the max-flow min-cut upper bound for the class
of semi-deterministic relay channels.
Achievability: We show that any R < C is achievable using
the generalized block-Markov encoding scheme. Substituting
X = (XD ; XR ) and U = XR in (2) and assuming joint probability mass function of the form p(x1 )p(xR jx1 )p(xD jx1 ), we obtain
I (X; X1 ; Y
) = I (X R ; X D ; X 1 ; Y )
= I (X D ; X 1 ; Y ) + I (X R ; Y
j
XD ; X1 )
= I (X D ; X 1 ; Y )
0018-9448/$20.00 © 2005 IEEE
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1816
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 5, MAY 2005
Fig. 1. Frequency-division AWGN relay channel.
and
I (U ; Y1 jX1 ) + I (X ; Y jX1 ; U )
= I (XR ; Y1 jX1 ) + I (XD ; XR ; Y jX1 ; XR )
(a)
= I ( X R ; Y 1 jX 1 ) + I ( X D ; Y jX 1 ) ;
where (a) follows by the fact that XR ! X1 ! Y form a Markov
chain.
Converse: We show that C is equal to the max-flow min-cut upper
bound. Clearly
and fZi g are independent each with power N , and the constants a; b >
0 represent the gain of the signal over the path from the sender to the
relay and from the relay to the receiver, respectively, relative to the gain
of the direct channel (which is assumed to be equal to one).
Assuming average power constraints P on X = (XD ; XR ) and P ,
on X1 , for 0, the capacity of this channel is given by1
I (X ; Y; Y1 jX1 )
= I (XD ; XR ; Y; Y1 jX1 )
= I ( X D ; X R ; Y 1 jX 1 ) + I ( X D ; X R ; Y jX 1 ; Y 1 )
= I ( X R ; Y 1 jX 1 ) + I ( X D ; Y 1 jX 1 ; X R )
+ I ( X D ; X R ; Y jX 1 ; Y 1 )
(b)
= I ( X R ; Y 1 jX 1 ) + I ( X D ; X R ; Y jX 1 ; Y 1 )
= I ( X R ; Y 1 jX 1 ) + H ( Y jX 1 ; Y 1 )
0 H ( Y jX 1 ; X D ; X R ; Y 1 )
(c)
= I ( X R ; Y 1 jX 1 ) + H ( Y jX 1 ; Y 1 ) 0 H ( Y jX 1 ; X D )
I ( X R ; Y 1 jX 1 ) + H ( Y jX 1 ) 0 H ( Y jX 1 ; X D )
= I ( X R ; Y 1 jX 1 ) + I ( X D ; Y jX 1 ) ;
where (b) and (c) follow from the fact that XD ! (X1 ; XR ) ! Y1
and (XR ; Y1 ) ! (X1 ; XD ) ! Y each form a Markov chain, respectively. Thus, we have shown that
C max min fI (XD ; X1 ; Y ); I (XR ; Y1 jX1 ) + I (XD ; Y jX1 )g
This completes the proof of the theorem.
III. EXTENSION TO AWGN RELAY CHANNEL
The result of the theorem can be extended to establish the capacity of
the AWGN relay channel in Fig. 1, where the channel from the sender
to the relay uses a different frequency band from the channel from the
sender and relay sender to the receiver. The AWGN processes fZ1i g
+b
2
+ 2b
p
N
2
+
)P
C
(1
;
0 )P
2
N
where C (x) = 12 log2 (1 + x). Again achievability is established using
the generalized block-Markov scheme, where we let XD N (0; P )
and XR N (0; (1 0 )P ) be independent, and X1 N (0; P ) be
independent of XR but jointly Gaussian with XD with E (X1 XD ) =
p P . New information is sent to the relay through XR and to the
receiver through part of XD (with power (1 0 2 )P ), and the relay
and the rest of XD cooperatively send information to remove the uncertainty of the receiver about the previous message.
To prove the converse, first note that from the theorem, given any
(n)
nR
(2 ; n) sequence of codes with Pe
!0
C minfI (XD ; X1 ; Y ); I (XR ; Y1 jX1 ) + I (XD ; Y jX1 )g
(3)
for some joint probability distribution
p ( x D ; x R ; x 1 ) = p ( x 1 ) p ( x D jx 1 ) p ( x R jx 1 ) :
Since the channel structure in this case has the special form
p(y; y1 jx; x1 ) = p(yjxD ; x1 )p(y1 jxR )
it follows that
I ( X R ; Y 1 jX 1 ) = H ( Y 1 jX 1 ) 0 H ( Y 1 jX 1 ; X R )
= H ( Y 1 jX 1 ) 0 H ( Y 1 jX R )
H ( Y 1 ) 0 H ( Y 1 jX R ) = I ( X R ; Y 1 ) :
where the maximization is over the choice of joint probability mass
function p(xD ; xR ; x1 ). Without loss of generality, we can restrict the
joint probability mass functions to be of the form
p ( x D ; x R ; x 1 ) = p ( x 1 ) p ( x D jx 1 ) p ( x R jx 1 ) :
(
C a (1 N0 )P
I (X; X1 ; Y ) = I (XD ; X1 ; Y ):
Next, we consider the second term under the min in (1)
C
C (P; P ) = 0max
min
;1
Equation (3) then reduces to
C minfI (XD ; X1 ; Y ); I (XR ; Y1 ) + I (XD ; Y jX1 )g
(4)
for some joint distribution p(xD ; x1 )p(xR ). Note that this result can
also be viewed as a special case of the capacity problem studied in [6].
2
2
) + E (XR ) P
Now the power cosntraints require that E (XD
2
2
and E (X1 ) P . Thus, for some 0 1, E (XD ) P and
1This
result was reported in [8].
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 5, MAY 2005
E (XR2 ) (1 0 )P . Define to be the correlation coefficient between
XD and X1 . It is now straightforward to show that
I (X D ; X 1 ; Y ) C
(
+b
2
p
+ 2b
N
)P
:
The second term under the minimum in (4) can be similarly upperbounded to yield
2
I (XR ; Y1 )+ I (XD ; Y jX1 ) C a (1 0 )P
N
+
C
(1
0 2 )P :
N
This completes the proof of converse.
IV. CONCLUSION
This correspondence establishes the capacity of the class of relay
channels with orthogonal channels from the sender to the relay and
from the sender and relay to the receiver. As for all other classes of
relay channels with known capacity, e.g., [2], [7], the capacity for this
class is equal to the max-flow min-cut upper bound.
Our result points to the main difficulty in establishing the capacity
of the general relay channel, which is finding the optimal broadcasting
strategy from the sender to the relay Y1 and the receiver Y . For the
class of relay channels discussed here, the optimal strategy is to split
the message into two parts, one is decoded by the relay and sent cooperatively with the sender to the receiver and the other is sent directly to
the receiver. This strategy, however, is not optimal in general. In [9], it
was shown that for AWGN relay channels, even when the channel from
the sender to both the relay and receiver is orthogonal to the channel
from the relay to the receiver, other strategies such as linear relaying
and side information coding can achieve higher rates.
1817
On Channels With Partial Channel State
Information at the Transmitter
Aviv Rosenzweig, Yossef Steinberg, Member, IEEE, and
Shlomo Shamai (Shitz), Fellow, IEEE
Abstract—We consider an independent and identically distributed (i.i.d.)
state-dependent channel with partial (rate-limited) channel state information (CSI) at the transmitter (CSIT) and full CSI at the receiver (CSIR).
The CSIT comprises two parts, both subject to a rate constraint, and communicated over noiseless way-side channels. The first part is coded CSI,
provided by a third party (a genie), and the second part is the output of
a deterministic scalar quantizer of the state. A single-letter expression for
the capacity of the channel is given. In case the CSIT comprises only the
coded part, we show that the capacity previously suggested in the literature is too optimistic, and suggest a correction as part of our expression
for the information capacity. For the general setting, an optimal coding
scheme based upon multiplexing of several codebooks is presented. It is
proved that the capacity of the channel is the same whether the quantizer’s
output is observed causally or noncausally by the encoder. In the rest of
the correspondence, we focus on the special case where the system employs
a stationary quantizer. Using a rate distortion approach we bound the alphabet’s size of the auxiliary random variable (RV) of the information capacity. Next, we turn to the additive white Gaussian noise (AWGN) channel
with fading, and show that the determination of the capacity region reduces
to finding the optimal genie strategies and the optimal power allocation
distribution along the product alphabet of the auxiliary RV and the quantizer’s output alphabets. The suggested model can be applied, for example,
to an orthogonal frequency-division multiplexing (OFDM) communication
system. Here the fading across frequencies comprises the channel state sequence. Coded fading information is provided to the channel encoder via a
way-side, rate-limited channel. In addition, since coding operation is expensive, a simpler scheme provides the channel encoder with quantized fading
information, e.g., whether each coefficient is above/below a threshold.
Index Terms—Channel with state information, fading channels, power
allocation, rate distortion with side information, rate-limited channel state
information, scalar quantization.
ACKNOWLEDGMENT
I. INTRODUCTION
The authors wish to thank the reviewers for several helpful
comments.
The rapid development of wireless communications systems over
the last decade creates an increasing demand for a definition of
corresponding realistic analytic models and for the exploration of the
fundamental limits on reliable information transmission over these
systems. A common model in the literature assumes a memoryless
channel, whose conditional probability distribution is controlled by a
time-varying state. Many studies have been devoted over the years to
various scenarios related to a state-dependent channel model, where
full or partial channel state information (CSI) is available at the transmitter (CSIT) and/or at the receiver (CSIR). In this correspondence,
we suggest a model of a channel controlled by an independent and
identically distributed (i.i.d.) state process, where the CSIT is subject
to a rate constraint. Our channel model provides a unified framework
for the models considered in [13, Theorem 2d], [4, Proposition 1], and
in [15].
REFERENCES
[1] E. C. van der Meulen, “Three-terminal communication channels,” Adv.
Appl. Probab., vol. 3, pp. 120–154, 1971.
[2] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,”
IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572–584, Sep. 1979.
[3] G. Kramer, M. Gastpar, and P. Gupta, “Capacity theorems for wireless relay channels,” in Proc. 41st Annu. Allerton Conf. Communication,
Control and Computation, Allerton, IL, Oct. 2003, pp. 1074–1083.
[4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity
in wireless networks: Efficient protocols and outage behavior,” IEEE
Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.
[5] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversitypart I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp.
1927–1938, Nov. 2003.
[6] F. M. J. Willems, “The discrete memoryless multiple access channel with
partially cooperating encoders,” IEEE Trans. Inf. Theory, vol. IT-29, no.
3, pp. 441–445, May 1983.
[7] A. El Gamal and M. Aref, “The capacity of semi-deterministic relay
channel,” IEEE Trans. Inf. Theory, vol. IT-28, no. 3, pp. 536–536, May
1982.
[8] A. El Gamal and S. Zahedi, “Minimum energy communication over a
relay channel,” in Proc. IEEE Int. Symp. Information Theory, Yokohama,
Japan, Jun./Jul. 2003, p. 344.
[9] S. Zahedi, M. Mohseni, and A. El Gamal, “On the capacity of AWGN
relay channels with linear relaying functions,” in Proc.IEEE Int. Sym.
Information Theory, Chicago, IL, Jun./Jul. 2004, p. 399.
Manuscript received October 24, 2003; revised November 25, 2004. This
work (No. 61/03) was supported by the Israeli Science Foundation. The material
in this correspondence was presented in part at the 41st Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, Oct. 2003.
The authors are with the Department of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: aviv@tx.
technion.ac.il; [email protected]; [email protected]).
Communicated by A. Lapidoth, Associate Editor for Shannon Theory.
Digital Object Identifier 10.1109/TIT.2005.846422
0018-9448/$20.00 © 2005 IEEE
Authorized licensed use limited to: Stanford University. Downloaded on March 02,2010 at 17:04:56 EST from IEEE Xplore. Restrictions apply.
References (9)
- E. C. van der Meulen, "Three-terminal communication channels," Adv. Appl. Probab., vol. 3, pp. 120-154, 1971.
- T. M. Cover and A. El Gamal, "Capacity theorems for the relay channel," IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572-584, Sep. 1979.
- G. Kramer, M. Gastpar, and P. Gupta, "Capacity theorems for wire- less relay channels," in Proc. 41st Annu. Allerton Conf. Communication, Control and Computation, Allerton, IL, Oct. 2003, pp. 1074-1083.
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