Channel Code Design with Causal Side Information at the Encoder

Abstract

The problem of channel code design for the M -ary input AWGN channel with additive discrete interference where the sequence of i.i.d. interference symbols is known causally at the encoder is considered. The code design criterion at high SNR is derived by defining a new distance measure between the input symbols of the Shannon's associated channel. For the case of binary-input channel, it is shown that it is sufficient to use only two symbols of the associated channel in the encoding as long as the distance spectrum of the code is concerned. This reduces the problem of code design for the binary-input AWGN channel with known interference to the design of binary codes for the AWGN channel with maximum Hamming distance.

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arXiv:0711.1565v1 [cs.IT] 12 Nov 2007
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Channel Code Design with Causal Side
Information at the Encoder
Hamid Farmanbar, Shahab Oveis Gharan, and Amir K. Khandani
Coding and Signal Transmission Laboratory
Department of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario, N2L 3G1
Email: {hamid,shahab,khandani}@cst.uwaterloo.ca
Abstract
The problem of channel code design for the M-ary input AWGN channel with additive
Q-ary interference where the sequence of i.i.d. interference symbols is known causally at the
encoder is considered. The code design criterion at high SNR is derived by defining a new
distance measure between the input symbols of the Shannon’s associated channel. For the case
of binary-input channel, i.e., M = 2, it is shown that it is sufficient to use only two (out of
2Q) input symbols of the associated channel in the encoding as far as the distance spectrum
of code is concerned. This reduces the problem of channel code design for the binary-input
AWGN channel with known interference at the encoder to design of binary codes for the binary
symmetric channel where the Hamming distance among codewords is the major factor in the
performance of the code.
Index Terms
Causal side information, Shannon’s associated channel, channel coding, pairwise error
probability.
This work was presented in part at the IEEE Canadian Workshop on Information Theory, Edmonton, Alberta,
Canada, June 6-8, 2007.

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I. INTRODUCTION
Information transmission over channels with known interference at the transmit-
ter has recently found applications in various communication problems such as digital
watermarking [1] and broadcast schemes [2]. A remarkable result on such channels
was obtained by Costa, who showed that the capacity of the additive white Gaussian
noise (AWGN) channel with additive Gaussian i.i.d. interference where the sequence of
interference symbols is known non-causally at the transmitter is the same as the capacity
of the AWGN channel [3]. Therefore, the Gaussian interference does not incur any loss in
the capacity. This result was extended to arbitrary (random or deterministic) interference
in [4] by using a precoding scheme based on multi-dimensional lattice quantization.
Following Costa’s “Writing on Dirty Paper” famous title [3], coding for the channel
with non-causally known interference at the transmitter is referred to as “dirty paper
coding” (DPC). By analogy, coding for the channel with causally-known interference at
the transmitter is sometimes referred to as “dirty tape coding” (DTC). The result obtained
by Costa does not hold for the case that the sequence of interference symbols is known
causally at the transmitter.
Recently, dirty paper coding has emerged as a building block in multiuser communi-
cation. In particular, there has been considerable research studying the application of dirty
paper coding to broadcast over multiple-input multiple-output (MIMO) channels. In such
systems, for a given user, the signals sent to other users are considered as interference.
Since all signals are known to the transmitter, successive “dirty paper” cancelation can be
used in transmission after some linear preprocessing [2]. It was shown that DPC in fact
achieves the sum capacity of the MIMO broadcast channel [5], [6], [7]. Most recently,
it has been shown that the same is true for the entire capacity region of the MIMO
broadcast channel [8].
These developments motivate finding realizable dirty paper coding techniques. Build-
ing upon [4], Erez and ten Brink [9] proposed a practical code design based on vector

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quantization via trellis shaping and using powerful channel codes. Due to the complexity
of implementation, their scheme uses the knowledge of interference up to six future
symbols rather than the whole interference sequence. Bennatan et al. [10] gave another
design based on superposition coding and successive cancelation decoding. Their design
uses a trellis coded quantizer with memory length nine and a low density parity check
(LDPC) code as channel code. Wei Yu et al. [11] gave a design based on convolutional
shaping and channel codes.
The schemes that use the interference sequence up to the current symbol can be
used as low-complexity solutions for the dirty paper problem. For example, in [1], scalar
lattice quantization is proposed for data-hiding even though in that context, the host signal
in clearly known non-causally.
In this paper, we consider the problem of channel code design for the M-ary
input AWGN channel with additive causally-known discrete interference. The discrete
interference model is more appropriate for many practical applications. For example,
in the MIMO broadcast channel where the transmitter uses a finite constellation, the
interference caused by other users is discrete rather than continuous.
Our design does not rely on the suboptimal (in terms of capacity) precoding scheme
based on scalar lattice quantization for the dirty tape channel [4], [12]. Instead, we
consider a new approach based on code design for the Shannon’s associated channel
over all possible input symbols. Another distinction between our work and the related
research in the field is that we consider a finite channel input alphabet rather than a
continuous one.
This paper is organized as follows. In the next section, we summarize Shannon’s
work on channels with causal side information at the transmitter. In section III, we
introduce the channel model. In section IV, we derive the code design criterion for the
AWGN channel with causally-known discrete interference at the encoder. In section V,
we consider channels with binary input for which we show that the design criterion
derived in section IV reduces to maximizing the Hamming distance. In section VI, we

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consider a special case for which the result for the binary channel also holds for the
M-ary channel. In section VII, we consider a more general channel model for which the
main results of this work hold. We conclude this paper in section VIII.
II. CHANNELS WITH SIDE INFORMATION AT THE TRANSMITTER
Channels with known interference at the transmitter are special case of channels
with side information at the transmitter which were considered by Shannon [13] in the
causal knowledge setting and by Gel’fand and Pinsker [14] in the non-causal knowledge
setting.
Shannon considered a discrete memoryless channel (DMC) whose transition matrix
depends on the channel state. A state-dependent discrete memoryless channel (SD-DMC)
is defined by a finite input alphabet X, a finite output alphabet Y, and transition prob-
abilities p(y|x,s), where the state s takes on values in a finite alphabet S. The block
diagram of a state-dependent channel with state information at the encoder is shown in
fig. 1.
In the causal knowledge setting, the encoder maps a message w into Xnas
xi= fi(w,s1,...,si),
1 ≤ i ≤ n.
(1)
Shannon showed that it is sufficient to consider the coding schemes that use only
the current state symbol in the encoding process to achieve the capacity of an SD-DMC
with i.i.d. state sequence known causally at the encoder [13].
The SD-DMC can be used in the way shown in fig. 2 to transmit information. A
precoder is added in front of the SD-DMC. A message w is mapped into Tn, where T
is a new alphabet. The output of the precoder ranges over X and depends on the current
interference symbol. The regular (without state) channel from T to Y is defined by the
transition probabilities
q(y|t) =
?
s∈S
p(s)p(y|x = t(s),s),
(2)

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Encoder
State
Channel
Generator
Decoder
p(y|x,s)
w
ˆ w
s
y ∈ Y
x ∈ X
s ∈ S
Fig. 1.SD-DMC with state information at the encoder.
State
Channel
Generator
DecoderEncoderPrecoder
p(y|x,s)ˆ wwt ∈ T
s ∈ S
s
x ∈ X
y ∈ Y
Fig. 2.The associated regular DMC.
where p(s) is the probability of the state s. The DMC defined in (2) is called the
associated channel. The codes for the associated channel describe the codes for the
SD-DMC that use only the current state symbols in the encoding operation. In order to
describe all coding schemes for the SD-DMC that use only the current state symbol in
the encoding process, T must include all functions from the state alphabet to the input
alphabet of the state-dependent channel. There are a total of |X||S|of such functions,
where |.| denotes the cardinality of a set. Any of the functions can be represented by a
|S|-tuple (x1,x2,...,x|S|) composed of elements of X, implying that the value of the
function at state s is xs,s = 1,2,...,|S|.
III. THE CHANNEL MODEL
We consider data transmission over the channel
Y = X + S + N,
(3)