OPTIMAL DISTRIBUTED LINEAR TRANSCEIVERS FOR SENDING INDEPENDENTLY CORRUPTED COPIES OF A COLORED SOURCE OVER THE GAUSSIAN MAC Onkar Dabeer School of Technology and Computer Science Tata Institute for Fundamental Research Mumbai, India Email: onkar@tcs.tifr.res.in Aline Roumy, Christine Guillemot * IRISA-INRIA 35042 Rennes Cedex France ABSTRACT We consider distributed linear transceivers for sending a second- order wide-sense stationary process observed by two noisy sensors over a Gaussian multiple-access channel (MAC). We derive the min- imum mean-square error (MSE) distributed linear transceiver. The optimal linear transmitter exploits bandwidth expansion by repeat- ing transmission and the transmitters at the two sensors are the same except for a constant factor. When the source is white, uncoded transmission is the best linear code for any SNR. But for a col- ored source, whitening transmit filter is sub-optimal. In high SNR regime, the magnitude response of the optimal transmission filter is inversely proportional to fourth-root of the power spectrum of the process (while that for the whitening filter is inversely proportional to the square-root of the spectrum). In the special case of a single sensor with Gaussian source, we also quantify the performance loss of linear source-channel codes with respect to the Shannon limit. Index Terms— Colored source, linear codes, joint source-channel coding, multiaccess communications, sensor networks 1. INTRODUCTION Motivated by applications in sensor networks, several researchers are considering the transmission of dependent sources over multiple- access channels. The problem is particularly interesting because the source-channel separation theorem does not hold in general ([1]). In fact, it is known that separation can be exponentially worse than joint source-channel coding ([2]). In this paper, we look at linear (over the real field) joint source-channel codes. Our motivation is two-fold. First, linear processing is simple to implement and by now there is vast experience in efficient hardware implementation of linear pro- cessing. This is important to keep the sensors simple and reduce their cost. Second, it has recently been shown that for transmitting memo- ryless, bivariate Gaussian sources over the Gaussian MAC, uncoded transmission is optimal below a certain SNR threshold ([3]). The class of linear transmitters includes uncoded transmission, and it is insightful to understand the nature of the best linear transmitters. We note that there is extensive literature on linear transceiver optimiza- tion for sending independent, memoryless sources over the MAC (see [4] and references therein). In [4], the problem is formulated as a semi-definite program, and numerical algorithms are proposed for the same. In contrast, in this paper we consider a single colored source being observed by two sensors with independent noises and * This research was partly funded by French National Research Agency (ANR) under the Essor project. derive closed-form solution to the optimal linear transceiver. The problem of distributed source coding of multiple independently cor- rupted copies of a source is commonly referred to as the CEO prob- lem ([5]). Thus in this paper we consider linear joint source-channel codes for the CEO problem over a Gaussian MAC. Our goal is to find the performance limit of linear source-channel codes for a colored source. So we consider non-causal transceivers, which may be viewed as the limit of block transceivers as the block size goes to infinity. Our results show that bandwidth expansion is exploited by the optimal transmitter by repeating the transmission. Moreover the two sensors employ the same transmit filter (except for a scale factor), which in effect reduces the two sensor case to a single sensor case. In the high SNR regime, the optimal transmitter filter has magnitude response inversely proportional to the fourth-root of the power-spectrum (while that for the whitening filter is inversely proportional to the square-root of the spectrum). We also provide an expression for the least MSE in the high SNR regime, which shows that the MSE is proportional to the integral of the square-root of the spectrum. In the case of a single sensor with Gaussian source, we also quantify the loss with respect to the Shannon limit. For exam- ple, in the high SNR regime, for a Gaussian first-order Markov pro- cess with correlation sequence 0.7 |k| , the optimal linear transceiver is about 1.4 dB from Shannon limit but it is 1.5 dB better than the whitening filter. The remainder of this paper is organized as follows: the precise problem definition is given in Section 2, the main results are given in Section 3, and the conclusion in Section 4. Notation: All vectors are column vectors. Superscript T denotes transpose and superscript H denotes conjugate transpose. The mini- mum MSE achievable using linear codes is denoted by MSE*, while that achievable using any code is denoted by MSE**. 2. PROBLEM DEFINITION Consider a stationary second-order stochastic process {st } ∞ t=-∞ with zero mean and covariance function c(t), c(0) = 1. We assume that c(t) is integrable, and this implies that the process has a continu- ous bounded spectral density φ(ω), ω ∈ (−π,π] ([6]). The process is observed by two sensors in the presence of additive noise. The observations at sensor i are xi,t = ai st + vi,t , where ai are the signal amplitudes, the additive noise is i.i.d. N (0,σ 2 o ) and the noise processes are independent across the sensors. These