The Capacity of Linear Channels with Additive Gaussian Noise

The standard method of computing the mutual information between two stochastic processes with finite energy replaces the processes with their Fourier coefficients. This procedure is mathematically justified here for random signals w,(ω) square-integrable in the product space t × ω where t ∊ O, T and w is an element of a probability space. A natural notion of the sigma field generated by w, (ω) is presented and it is shown to coincide with the sigma field generated by the random Fourier coefficients of w,(ω) in any complete orthonormal system in L 2 O, T. This justifies the use of Fourier coefficients in mutual information computations. Capacity is calculated for finite and infinite-dimensional channels, where the output signal consists of a filter (general Hilbert-Schmidt operator) operating on the input signal with additive Gaussian noise. The finite-dimensional optimal signal is obtained. In the infinite-dimensional case capacity can be approached arbitrarily closely with finite-dimensional inputs. The question of the existence of an infinite-dimensional signal which achieves capacity is considered. There are channels for which no signal achieves capacity. Some results are obtained when the noise coordinates are independent in the eigensystem of the filter.