Band-Limited Signal Reconstruction from Irregular Samples with Variable Apertures∗

Sampling plays a critical role in remote sensing and signal analysis. In conventional sampling theory, the signal is sampled at a uniform rate at a minimum of twice the signal bandwidth. Sampling with an aperture function requires a fixed aperture function, which can be removed by deconvolution after signal reconstruction. However, in some cases the signal samples are available only at irregular positions and different samples use different aperture functions. Here we explore the theory of finite-length signal reconstruction with irregular sam-ples and variable apertures in one and two dimensions is considered. We concentrate on discrete sampling and reconstruction. In the one-dimensional case, a band-limited discrete signal can be exactly recon-structed from a finite number of arbitrarily spaced samples with few restrictions on the aperture functions, which can vary between samples. Exact reconstruction in the two-dimensional case requires the sampling location matrix be invertable, and is not always possible. Variable aperture functions, while complicating the process, can enable reconstruction for a broader range of sample loca-tions. Note that two-dimensional reconstruction from general cubic lattice samplings, which include uniform sampling, are always fully invertable in the ideal aperture case. We consider a number of practical issues and provide some numerical examples based on simulated special sensor microwave imager (SSM/I) radiometer measurements which have variable apertures. We note that variable aperture reconstruction has application in a variety of remote sensing and other signal processing problems.