ON COMPUTING THE MINIMUM DISTANCE FOR FASTER THAN NYQUIST SIGNALING

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 percent 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 to find the smallest value of δ, 0 < δ < 0.5, for which the best L2 approximation to the constant 1 on the interval (— δ,δ) is 1, when using a linear combination of the functions exp(i2πnθ) to approximate, n ≥ 0, and restricting the coefficients to be 0, ±1. The smallest value of δ is 0.401. © 1990 IEEE