A Note on Mutual Information in a White Gaussian Channel with Linear Feedback

Let a message m = {m(t)} be a Gaussian process. We consider the transmission of m over a white Gaussian channel with a linear feedback. The channel output in question is given by Y ( t ) = ∫ 0 t ( m ( s ) − f ( s ) ) d s + W ( t ) , where f ( s ) = ∫ s 0 f ( s, u ) dY ( u ) is a causal linear functional of Y and the noise W ( t ) is a Brownian motion. We shall prove the following: Even if such a linear feedback is taken into account, the amount of mutual information I ( m, Y ) between m and Y never increases.