Energy Optimization of Band-Limited Nyquist Signals in the Time Domain

This paper presents a theoretical consideration of the optimal design of band-limited Nyquist-type signal shapes for data transmission, which maximizes its energy in a given time interval and which generates no intersymbol interference at the periodic sampling instants. A method based on a completely analytical approach is given for design of such signals. The optimal signal is a solution of an inhomogeneous linear integral equation of Fredholm type. A technique for solving this equation is given. The computation is straightforward and involves the determination of eigenvalues and eigenfunctions of a positive definite and symmetric kernel in terms of prolate spheroidal wave functions. The constraint for intersymbol interference is shown to be easily included into the problem. Finally, some numerical examples are given and the performance of the optimal signal shapes is compared to that resulting from the use of the "raised-cosine" type of signals. It is also concluded that especially for small values of rolloff factor, the optimal signals, thus obtained, are almost maximally immune to small timing offsets at the sampling instants.