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Upper and lower bounds on the minimum mean square error for additive noise channels are derived when the input distribution is constrained to be close to a Gaussian reference distribution in terms of the Kullback-Leibler divergence. The upper bound is tight and is attained by a Gaussian distribution whose mean is identical to that of the reference distribution and whose covariance matrix is define...

MMSE Bounds for Additive Noise Channels Under Kullback–Leibler Divergence Constraints on the Input Distribution