2016/01/20 by Ayush Bhandari, Bhandari, Ayush, Ahmed I. Zayed +1 · 2 citations
Computer Science · Mathematics · #Digital Filter Design and Implementation #FOS: Computer and information sciences #Information Theory (cs.IT) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.1601.05793
openalex publication_date 2016/01/20 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
The Special Affine Fourier Transformation or the SAFT generalizes a number of\nwell known unitary transformations as well as signal processing and optics\nrelated mathematical operations. Shift-invariant spaces also play an important\nrole in sampling theory, multiresolution analysis, and many other areas of\nsignal and image processing. Shannon's sampling theorem, which is at the heart\nof modern digital communications, is a special case of sampling in\nshift-invariant spaces. Furthermore, it is well known that the Poisson\nsummation formula is equivalent to the sampling theorem and that the Zak\ntransform is closely connected to the sampling theorem and the Poisson\nsummation formula. These results have been known to hold in the Fourier\ntransform domain for decades and were recently shown to hold in the Fractional\nFourier transform domain by A. Bhandari and A. Zayed.\n The main goal of this article is to show that these results also hold true in\nthe SAFT domain. We provide a short, self-contained proof of Shannon's theorem\nfor functions bandlimited in the SAFT domain and then show that sampling in the\nSAFT domain is equivalent to orthogonal projection of functions onto a subspace\nof bandlimited basis associated with the SAFT domain. This interpretation of\nsampling leads to least-squares optimal sampling theorem. Furthermore, we show\nthat this approximation procedure is linked with convolution and semi-discrete\nconvolution operators that are associated with the SAFT domain. We conclude the\narticle with an application of fractional delay filtering of SAFT bandlimited\nfunctions.\n