Exact Relation Between Continuous and Discrete Linear Canonical Transforms

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Öktem S. F. , Özaktaş M. H.

IEEE SIGNAL PROCESSING LETTERS, vol.16, no.8, pp.727-730, 2009 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 16 Issue: 8
  • Publication Date: 2009
  • Doi Number: 10.1109/lsp.2009.2023940
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.727-730
  • Keywords: Bicanonical width product, fractional Fourier transform, linear canonical series, linear canonical transform, FRACTIONAL FOURIER-TRANSFORM, DIGITAL COMPUTATION, SERIES EXPANSION, SIGNALS, CONVOLUTION, THEOREMS


Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.