Capacity of the Gaussian Channel With Memory: The Multivariate Case

A formula is derived for the capacity of a multi-input, multi-output linear channel with memory, and with additive Gaussian noise. The formula is justified by a coding theorem and converse. The channel model under consideration can represent multipair telephone cable including the effect of far-end crosstalk. For such cable under large signal-to-noise conditions, we show that channel capacity and cable length are linearly related; for small signal-to-noise ratio, capacity and length are logarithmically related. Crosstalk tends to reduce the dependence of capacity on cable length. Moreover, for any channel to which our capacity formula applies, and for large signal-to-noise ratio, there is an asymptotic linear relation between capacity and signal-to-noise ratio with slope independent of the channel transfer function. For small signal-to-noise ratio, capacity and signal-to-noise ratio are logarithmically related. Also provided is a numerical evaluation of the channel capacity formula, using measured parameters obtained from an experimental cable.