AN ASYMPTOTIC EXPRESSION FOR THE INFORMATION AND CAPACITY OF A MULTIDIMENSIONAL CHANNEL WITH WEAK INPUT SIGNALS

An asymptotic expression is derived for the Shannon mutual information between the input and output signals for a relatively large class of continuous alphabet memoryless channels in the case of weak input sign als, when the input space is multidimensional. We extend hereby a resu lt of Ibragimov and Khas'minskii from the one-dimensional to the N-dim ensional case. The asymptotic expression obtained relates the Shannon mutual information function and the Fisher information matrix. This ex pression is used to derive an asymptotic expression for the capacity o f continuous alphabet memoryless channels with vector-valued weak inpu t signals. This asymptotic capacity involves the largest eigenvalue of the Fisher information matrix evaluated at the zero input signal.