Abstract
We consider the reconstruction of signals from nonuniformly spaced samples using a projection onto convex sets (POCSs) implemented with the evolutionary time-frequency transform. Signals of practical interest have finite time support and are nearly band-limited, and as such can be better represented by Slepian functions than by sinc functions. The evolutionary spectral theory provides a time-frequency representation of nonstationary signals, and for deterministic signals the kernel of the evolutionary representation can be derived from a Slepian projection of the signal. The representation of low pass and band pass signals is thus efficiently done by means of the Slepian functions. Assuming the given nonuniformly spaced samples are from a signal satisfying the finite time support and the essential band-limitedness conditions with a known center frequency, imposing time and frequency limitations in the evolutionary transformation permit us to reconstruct the signal iteratively. Restricting the signal to a known finite time and frequency support, a closed convex set, the projection generated by the time-frequency transformation converges into a close approximation to the original signal. Simulation results illustrate the evolutionary Slepian-based transform in the representation and reconstruction of signals from irregularly-spaced and contiguous lost samples.
Highlights
The problem of signal reconstruction from Nonuniformly spaced samples is central in many practical problems in image and signal processing [1,2,3,4,5,6,7,8,9,10,11]
Assuming that the signal of interest is square summable and that the discrete evolutionary transform (DET) projects a signal into another square summable signal, under joint time-frequency constraints we develop an iterative projection onto convex sets (POCSs) algorithm to recover the signal from partial information of it
For a signal with bandpass characteristics, the signal can be represented by a small number of bandpass discrete prolate spheroidal sequences (DPSSs) coefficients and restored by small number of projection iteration compared to baseband
Summary
The problem of signal reconstruction from Nonuniformly spaced samples is central in many practical problems in image and signal processing [1,2,3,4,5,6,7,8,9,10,11]. Constraining the solution to satisfy time and frequency conditions iteratively, a close approximation to the signal, with the given samples, is obtained This is the basic idea of the projection onto convex sets (POCSs) method. We show why the PSWF basis is more appropriate than the sinc basis for the reconstruction from Nonuniform samples when the signal is of finite time support and essentially band-limited. Assuming that the baseband components of a bandpass signal has finite support in time and frequency, a DET based in PSWF or Slepian functions is possible.
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