Sampling Theory, Phase Space and a Swiss Army Knife
Last year marked the Claude Shannon centenary. One of his many elegant results is linked with the topic of Sampling Theory. Seen from an abstract point of view, if a given signal/function is smooth, then, the sampling theory deals with conditions under which signal reconstruction/approximation is perfect. The constraint that a given signal is bandlimited (or compactly supported in Fourier domain) is a mathematical construct that somehow measures the smoothness of a function. For bandlimited functions, this topic is well understood and goes in the name of Nyquist--Shannon Sampling theorem. In the past four decades---thanks to the wavelet revolution---considerable advancements have been made in this area which now incorporates an alternative viewpoint: sampling theory as approximation of functions and covers the case of non-bandlimited, as well as sparse signals.
The idea that Fourier transform of a function forms a cyclic group---four, consecutive Fourier transforms of a function, produces the same function again---attracted the attention of several mathematicians including Norbert Wiener. This resulted in the formalization of the fractional Fourier transform or the FrFT domain (parametrized by an additional parameter) and later, the Special Affine Fourier Transform.
In this talk, we give a first hand account of new results linked with the extension of Shannon’s Sampling theory for the Special Affine Fourier Transform or the SAFT. We will discuss a version of the convolution operator that results in multiplication of the SAFT spectrums of the signals involved. On way, will discuss approximation theory for finite energy functions that live in shift-invariant subspaces. Lastly, we will show how to can one sample and recover sparse signals in the SAFT domain. The fact that the SAFT parametrically generalizes a number of well known unitary transformations leads to a unifying framework for sampling and approximation theory. All of our results are backward compatible with the Fourier transform.
This is joint work with Ahmed I. Zayed and Yonina Eldar.