On faster than Nyquist signaling: Computing the minimum distance

AbstractIn this paper we are concerned with the problem of computing the minimum of the L2 norm taken over the interval (−δ, δ), 0 < δ ⩽ 12, and over all non-trivial linear combinations of the functions exp(2πinθ), n = 0, 1, …; the coefficients in the linear combination being restricted to 0, ±1. Denoting the minimal L2 norm over the interval (−δ, δ) by I(δ), it is trivial by the orthogonality of the exponentials that I(12) is 1. The main result of the paper is to show that there is a neighborhood of half for which I(δ) is 1, despite the non-orthogonality of the exponentials in the interval (−δ, δ). The origin of this problem, arises from certain basic problems in data communications, concerned with studying the behavior of the minimum L2 distance between signals, when data is sent faster than the Nyquist rate over an ideal bandlimited channel. The above mentioned result shows that there is no degradation in the minimum distance for rates somewhat faster than the Nyquist rate