On the minimum distance problem for faster-than-Nyquist signaling

The authors reconsider the problem of determining the minimum distance between output sequences of an ideal band-limiting channel that are generated by uncoded binary sequences transmitted at a rate exceeding the Nyquist rate. For signaling rates up to about 25% faster than the Nyquist rate, it is shown that the minimum distance does not drop below the value which it would have in the ideal case wherein there is not intersymbol interference. Mathematically, the problem is to decide if the best L² Fourier approximation to the constant 1 on the interval (-σπ, σπ), 0<σ⩽1, using the functions exp(inx), n>0, with coefficients restricted to be =1 or =0, occurs when all coefficients are zero. This is shown to be optimal for 0.802...⩽σ⩽1