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. In this paper, the theory of finite-length signal reconstruction with irregular samples and variable apertures in one and two dimensions is considered. In the 1-D case, a band-limited discrete signal can be exactly reconstructed from a finite number of arbitrarily spaced samples with few restrictions on the aperture functions. Exact reconstruction in the 2-D case requires the sampling matrix be invertable, and is not always possible. Variable aperture functions, while complicating the process, can enable reconstruction for a broader range of sample locations. Practical issues are discussed, and numerical examples are provided. Variable aperture reconstruction has application in a variety of remote sensing problems. In this paper, reconstruction from 2-D irregular sampling with variable apertures is illustrated using Special Sensor Microwave/Imager radiometer observations.