One-Bit Sampling In Fractional Fourier Domain

The fractional Fourier transform has found applications in a variety of topics linked with science and engineering. In this context, sampling theory is one of the most well-studied subjects. Since the fractional Fourier transform or the FrFT generalizes the notion of bandlimitedness, extension of Shannon's sampling theorem to the FrFT domain generalizes the classical result for the Fourier domain. These ideas have further been extended to the class of non-bandlimited functions via shift-invariant subspaces and sparse models. In this paper, we discuss a different approach to sampling theory in the FrFT domain. For the first time, we propose sampling and recovery of bandlimited functions in the FrFT domain that is based on one-bit samples. Our work is inspired by the Sigma--Delta quantization scheme. In particular, we capitalize on the idea of noise shaping and develop a one-bit sampling architecture that allows for recovery of bandlimited functions in the FrFT domain by pushing quantization noise to the higher frequencies. Since the FrFT generalizes the Fourier transform, our work results in a generalized Sigma--Delta architecture. We validate our theoretical concepts through computer experiments and provide an approximation theoretic error bound.