Local polynomial Fourier transform: A review on recent developments and applications

The local polynomial Fourier transform (LPFT), as a high-order generalization of the short-time Fourier transform (STFT), has been developed and used for many different applications in recent years. This paper attempts to review previous research work on the following issues of the LPFT. Firstly, the definition, the properties of the LPFT and its relationships with other transforms are reviewed. The LPFT for multicomponent signal is then presented. The polynomial time frequency transform (PTFT), which is the maximum likelihood estimator to estimate the parameters in the LPFT, as well as its properties and fast algorithms are discussed. By comparing with the Fourier transform (FT), the STFT and the Wigner–Ville distribution (WVD), the LPFT has its superiority in obtaining improved SNRs, which can be supported by theoretical analysis and computer simulations. Furthermore, the reassignment method is combined with the LPFT and the robust LPFT to improve the concentration of the signal representation in the time–frequency domain. Performances obtained by using various LPP-related methods are compared for signals in different noise environments, such as the additive white Gaussian noise (AGWN), impulsive noise, and the mixture of AGWN and impulsive noise.