Sampling the Flow of a Bandlimited Function

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states $$f_t=\phi _t*f$$ f t = ϕ t ∗ f resulting from the convolution with a kernel $$\phi _t$$ ϕ t . It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $$\widehat{f}$$ f ^ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

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 WT→∞, it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of 2WT degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of WT. 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 2WT. The bounds are asymptotically tight and numerical computations show the gap between them decreases as 2WT increases. Additionally, the possible decrease from 2WT in the available degrees of freedom is upper-bounded by a logarithmic function of 2WT.