Time–frequency analysis method based on affine Fourier transform and Gabor transform

Abstract

The affine Fourier transform (AFT) plays an important role in many fields of optics and signal processing. The Gabor transform (GT) is a kind of linear time–frequency representation (TFR). Compared with many bilinear TFRs, the GT does not have the cross‐term problem. In this study, the authors propose a time–frequency analysis method based on the AFT and GT. First, they obtain an affine relation between the AFT and the modified GT (MGT). Since the MGT is closely related to the AFT, they can use it as an assistant tool for signal processing in the AFT domain. Moreover, many useful relations between the AFT and the MGT are derived, such as recovery relation, projection relation and power integration relation. Then, they demonstrate that the AFT also has the affine relation with other TFRs, such as the Gabor–Wigner transform and the general class of quadratic distribution. Last, using the new time–frequency analysis method associated with the AFT and MGT, they present the filter design for multiple component chirp signal separation. Moreover, the simulation results illustrate the effectiveness of the proposed method.

Algebraic representation for fractional Fourier transform on one‐dimensional discrete signal models

“…The future work will be focused on the following aspects: (i) extend the algebraic representation for the FT to the short-time FT [42] domain and (ii) extend the algebraic representation for the FRFT to the linear canonical transform [11,38,39,[42][43][44] domain.…”

Multichannel sampling expansions in the linear canonical transform domain associated with explicit system functions and finite samples

“…By making the FT on both sides of (4), due to the generalised form of Poisson's sum formula in the FT domain [19], we obtain (see (5)) where I σ = ( − σ, σ), F(ω), and H k (ω), k = 1, 2, …, M, respectively, denotes the FT of f (t) and of h k (t), k = 1, 2, …, M, and the system function Y k F (ω) denotes the FT of the response function y k…”

“…The linear canonical transform (LCT) is a three free parameter class of linear integral transformation [1,2], and its theory and application attract much attention because it is a powerful tool for the process of non-stationary signals. The LCT has found a large number of applications in the fields of optics [3], image watermarking [4], time-frequency analysis [5], filter design [6], and many others. In the meantime, its essential theories are currently derived [2] including the multichannel sampling expansions (MSEs) [7][8][9] that are of importance and usefulness in a variety of applications in which data is sampled with the LCTband-limited signal through the multichannel data acquisition [10].…”