Sampling and Ergodic Theorems for Weakly Almost Periodic Signals

The theory of abstract harmonic analysis on commutative groups is used to prove sampling and ergodic theorems concerning a particular class of finite-power signals, which are known as weakly almost periodic. The analysis brings to light some noteworthy differences between finite-energy and finite-power signal sampling. It is shown that the bandwidth of the Fourier transform of a weakly almost periodic signal is generally larger than the bandwidth of the power spectrum of the signal. Consequently, the signal power spectrum by itself does not generally provide enough information to determine the value of the time-domain Nyquist rate, that is, the minimum sampling rate necessary for exact signal reconstruction in the time-domain. On the other hand, it is also shown that the minimum sampling rate needed to obtain alias-free spectral estimates is determined by the bandwidth of the power spectrum and, consequently, may be lower than the time-domain Nyquist rate. Finally, the sampling and ergodic theorems established in this paper are used in an analysis of averaged periodogram estimates of the power spectrum of a weakly almost periodic signal. It is shown that the value of the time shift between consecutive windows may contribute to the asymptotic bias of the estimates.