Time-Limited Codewords over Band-Limited Channels: Data Rates and the Dimension of the W-T Space

We consider a communication system whereby T-seconds time-limited codewords are transmitted over a W-Hz band-limited additive white Gaussian noise channel. In the asymptotic regime as [Formula: see text] , it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of [Formula: see text] degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of [Formula: see text]. We use prolate spheroidal wave functions to obtain an information lossless equivalent discrete formulation and then we apply Polyanskiy’s results on coding in the finite block-length regime. We derive upper and lower bounds on the achievable rates and the corresponding degrees of freedom and we numerically evaluate them for sample values of [Formula: see text]. The bounds are asymptotically tight and numerical computations show the gap between them decreases as [Formula: see text] increases. Additionally, the possible decrease from [Formula: see text] in the available degrees of freedom is upper-bounded by a logarithmic function of [Formula: see text].