Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals

Abstract

We examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples. This sampling scheme, which has been studied previously, has an interesting optimality property that uniform sampling lacks: one can sample and reconstruct the class () of multiband signals with spectral support , at rates arbitrarily close to the Landau minimum rate equal to the Lebesgue measure of , even when does not tile under translation. Using the conditions for exact reconstruction, we derive an explicit reconstruction formula. We compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band-limited to the "span of " (the smallest interval containing) which is a bigger class than the valid signals (), band-limited to. We also examine the performance of the reconstruction system when the input contains additive sample noise.

Time-Frequency Localization and Sampling of Multiband Signals

“…When Σ is a finite, pairwise disjoint union of intervals a number of sampling methods have been developed. We will concentrate on two such methods, those due to Venkataramani and Bresler [40] and to Behmard and Faridani [3] (cf. also [4] for more recent work).…”

“…A detailed analysis of the aliasing errors that arise from these techniques is also provided in [40]. .…”

“…The Nyquist sampling rate is the reciprocal of the length of this interval. Several approaches to effective sampling of multiband signals have been proposed in recent years, e.g., [13,40,3]. Among them, works of Venkataramani and Bresler and of Behmard and Faridani establish formulas or algorithms for interpolating multiband signals from samples and their methods will be reviewed in Section 7.…”

Digital filtering of band-limited signals using Periodic Nonuniform Sampling

“…The second realization of the filter is a direct extension of the reconstruction method for PNS [10]. If the implementation is performed along the branches of PNS, that is there is a synthesis filter ( ) Ψ l f after each upsampler, the implementation scheme can be described as:…”

“…Landau [6] proved that the minimum sampling rate for arbitrary sampling and reconstruction of multiband signals asymptotically reaches the signal's total bandwidth B which may be significantly lower than the Nyquist rate. This result was exploited and several methods for sampling and reconstruction of passband and multiband signals were introduced ( [7], [8], [9], [10], and [11]). …”