On dithered A/D conversion of multidimensional smooth non-bandlimited signals

Conventional method for sampling smooth multidimensional non-bandlimited signal requires the use of anti-aliasing lowpass prefilter. In spatial sampling applications, such as in geographical surveillance, sampling operation takes place before filtering. Aliasing of sampled signal is inevitable in such cases. Recently, sampling and quantization of one-dimensional smooth non-bandlimited signals in the absence of a lowpass prefilter has been studied in the literature [1]. Two-dimensional extensions of the one-dimensional non-bandlimited signal sampling results are derived in this work. Smoothness of signal can be translated to a decaying Fourier spectrum. The maximum error between signal and its reconstruction is distortion. The distortion when signal is sampled on a uniform grid with high-precision quantizers, imitating the Nyquist-style sampling, is analyzed and bounded. It is shown that, with the help of a known dither, the signal of interest can be reconstructed with similar reconstruction distortion while using only single-bit quantizers. An example illustrates that the distortion is limited by the slowest spectral decay law among all directions in the Fourier transform domain.