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Linearity conditions for optimal estimation from multiple noisy measurements

Source estimation from several noisy observations, each of which are corrupted with additive independent noise is arguably among the most fundamental problems in estimation theory. Estimating multiple independent sources, corrupted by identical noise realization is important in various communication applications. It is well-known for both cases that when all sources and noises are Gaussian, linear estimator minimizes the mean square estimation error. This paper analyzes the conditions for linearity of optimal estimation in these two settings, for general source and noise distributions and distortion measures. Specifically, we show that these settings depart from the single source-single channel setting in that Gaussianity of all system components is necessary to render the Lp optimal estimator linear at a given signal-to-noise ratio (SNR). Moreover, we show for both settings that at asymptotically high SNR, for Gaussian sources the optimal estimator converges to linear, irrespective of the distribution of the noises; similarly, at low SNR, it is asymptotically linear for Gaussian noises regardless of the sources.