The degradation suffered when pulses satisfying the Nyquist criterion are used to transmit binary data at a rate faster than the Nyquist rate over the ideal band-limited (brick-wall) channel is studied. The minimum distance between received signals is used as a performance criterion. It is well-known that when Nyquist pulses (i.e. pulses satisfying the Nyquist criterion) are sent at the Nyquist rate, the minimum distance between signal points is the same as the pulse energy. The main result is to show that the minimum distance between received signals is the same as the pulse energy for rates of transmission about 25% beyond the Nyquist rate, which is the best possible result. In fact, one can even identify the precise error event and signaling rate that causes the minimum distance to be no longer equal to the pulse energy. The mathematical formulation of the problem is analyzed.>

Upper and lower bounds on the epsilon -rate of a linear, time-invariant multiple input multiple output channel are derived by using the same volume argument previously used by W.L. Root (1968) for single input single output channels. Because these bounds are not very tight, an approximation to the epsilon -rate is presented which lies between the upper and lower bounds, and can be used to compare epsilon -rates for different channels. The extension considered uses a result due to Lerer (1978) on the eigenvalue distribution of a convolution operator with a matrix kernel (impulse response). The present results are used to assess the increase in data rate attainable by designing input signals which exploit the multidimensional nature of the channel, relative to treating each constituent channel in isolation. Numerical results based upon a simple model for two coupled twisted-pair wires are presented.>