In this paper, we examine the effects of imperfect channel estimation at the receiver and no channel knowledge at the transmitter on the capacity of the fading Costa's channel with channel state information non-causally known at the transmitter. We derive the optimal Dirty-paper coding (DPC) scheme and its corresponding achievable rates with the assumption of Gaussian inputs. Our results, for uncorrelated Rayleigh fading, provide intuitive insights on the impact of the channel estimate and the channel characteristics (e.g. SNR, fading process, channel training) on the achievable rates. These are useful in practical scenarios of multiuser wireless communications (e.g. Broadcast Channels) and information embedding applications (e.g. robust watermarking). We also studied optimal training design adapted to each application. We provide numerical results for a single-user fading Costa's channel with maximum-likehood (ML) channel estimation. These illustrate an interesting practical trade-off between the amount of training and its impact to the interference cancellation performance using DPC scheme.
Consider the problem of communicating over a Gaussian channel corrupted by an additive Gaussian interfering signal that is non-causally known at the transmitter. This variation of the conventional additive white Gaussian noise (AWGN) channel is commonly known as channel with state information at the transmitter. The state S is a random Gaussian variable with power Q and independent of the Gaussian noise Z. The channel input is the message m ∈ {1, . . . , ⌊2 nR ⌋} and its output is Y = X + S + Z, where R is the rate in bit per transmission. The capacity expression of single-user channels with random parameters has been derived by Gel'fand and Pinsker in [1]. The authors show that the capacity of such a channel {W (y|x, s), x ∈ X, s ∈ S} with state information S non-causally available at the transmitter is C = sup p(u,x|s)
U is an auxiliary random variable chosen so that U (X, S) Y form a Markov Chain and p(u, x|s) = δ x-f (u, s) p(u|s).
In “Writing on Dirty Paper” [2], Costa applied this result to an AWGN channel corrupted by an additive white Gaussian interfering signal S. He showed that choosing U = X + αS, with an appropriate value for α (α * = P /( P + σ 2 Z ), σ 2 Z being the AWGN variance). This coding scheme, referred as Dirty-paper coding (DPC), allows one to achieve the same capacity as if the interfering signal S was not present, i.e. C = 1 2 log 2 1 + P σ 2
. This result has gained considerable attention during the last years, mainly because of its potential use in communication scenarios where interference cancellation at the transmitter is needed. In particular, multiuser interference cancellation for Broadcast Channels (BC) and information embedding (digital watermarking for multimedia security applications) are instances of such scenarios. In the recent years, the Gaussian Multiple-Input-Multiple-Output Broadcast Channel (MIMO-BC) has been extensively studied. In [3], the authors based on DPC have established an achievable rate region, referred to as Dirty-paper coding region. Recently in [4], the DPC region was proved to be equal to the capacity.
Most of the literature focuses on the information-theoretic performances of DPC under the assumption on the availability of perfect channel information at both transmitter and receiver. However, it is well-known that the performances of wireless systems are severely affected if only a noisy estimate that differs from the true channel is available (cf. [5], [6] and [7]). Of particular interest is the issue of the effect of this imperfect channel knowledge if interference cancellation or Dirty-paper coding is used. The problem may even be more serious in the practical situations where no channel information is available at the transmitter, i.e., no feedback information from the receiver back to the transmitter with the channel estimates.
Throughout this paper, we consider a wireless or watermarked channel modeled as Y = H(X + S) + Z, where H is the random channel, which neither the transmitter nor the receiver know. We assume that the receiver estimates H during a phase of independent training, by using maximumlikelihood (ML) channel estimation (Section III). Whereas, the transmitter does not know this estimate. Then, we observe that depending on the targeted application, e.g. Broadcast Channel or robust watermarking, two different training scenarios are relevant. In this work, we determine the tradeoff between the amount of training required for channel estimation and the corresponding achievable rates using DPC (Section IV). We address this problem through the notion of reliable communication based on the average of the error probability over all channel estimation errors. This allows to make an equivalence with the capacity of a composite (more noisy) channel. Our proposed framework is sufficiently general to involve the most important information embedding and multiuser communication scenarios. Finally, Section V uses a Rayleigh-fading Costa’s channel to illustrate average rates over all estimates, for different amount of training.
First consider a general model for communication under channel uncertainty over discrete memoryless channels (DMCs) with input alphabet X , output alphabet Y and channel states S (cf. [1] and [8]). A specific instance of the unknown channel is characterized by a transition probability mass (PM) W (•|x, s, θ) ∈ W Θ with a random state s ∈ S perfect known by the transmitter and a fixed but unknown channel θ ∈ Θ ⊆ C d . Here, W Θ = W (•|x, s, θ) : x ∈ X , s ∈ S , θ ∈ Θ is a family of conditional transition PMs on Y , parameterized by a vector θ ∈ Θ, which follows i.i.d. θ ∼ ψ(θ). It is assumed that the receiver only knows an estimate θ of the channel and a characterization of the estimator performance in terms of the conditional probability density function (pdf) ψ(θ| θ) (this can be obtained using W Θ and the a priori distribution of θ). On another side, the transmitter does not know the estimate θ, it only knows its
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