Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Local polynomial Fourier transform <s> The local polynomial Fourier transform (LPFT) and the local polynomial periodogram (LPP) are proposed in order to estimate the rapidly time-varying instantaneous frequency (IF) /spl Omega/(t) of a harmonic signal. The LPFT gives the time-frequency power distribution over the t-(/spl Omega/(t), d/spl Omega/(t)/dt,...,d/sup m-1//spl Omega/(t)/dt/sup m-1/) space, where m is a degree of the LPFT. The LPFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting the local polynomial approximation of the frequency which implements a high-order nonparametric regression. The a priori information about bounds for the frequency and its derivatives can be incorporated to improve the accuracy of the estimation. The asymptotic mean square errors, bias and covariance, of the estimators of d/sup s//spl Omega/(t)/dt/sup s/, s=0, 1,2,...,m/sup -1/, are obtained. The considered estimators are high-order generalization of the short-time Fourier transform. The comparative study of the asymptotic variance and bias of the estimates is presented. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Local polynomial Fourier transform <s> The local polynomial approximation (LPA) of the time-varying phase is used to develop a new form of the Fourier transform and the local polynomial periodogram (LPP) as an estimator of the instantaneous frequency (IF) /spl Omega/(t) of a harmonic complex-valued signal. The LPP is interpreted as a time-frequency energy distribution over the t-(/spl Omega/(t), /spl Omega//sup 1/(t)),...,/spl Omega//sup m-1/(t) space, where m is a degree of the LPA. The variance and bias of the estimate are studied for the short- and long-time asymptotic behavior of the IF estimates. In particular, it is shown that the optimal asymptotic mean squared errors of the estimates of /spl Omega//sup k-1/(t) have orders O(N/sup -(2k+1)/) and O(N/sup -/2(m-k+1)/2m+3), k=1.2,...,m, respectively, for a polynomial /spl Omega/(t) of the degree m-1 and arbitrary smooth /spl Omega/(t) with a bounded mth derivative. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Local polynomial Fourier transform <s> The principal axes of the time-frequency representation of a signal are defined as those mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum. The time-frequency kernels used in the Cohen class of time-frequency representations are then rotated in the time-frequency plane, in order to align the kernels' preferred axes to the signal's principal axes. It is shown that the resulting time-frequency representations show a better reduction of cross-terms without too severely degrading the autoterms than the corresponding original time-frequency representations. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Local polynomial Fourier transform <s> Many useful relations between the Gabor transform (GT) and the fractional Fourier transform (FRFT) have been derived. First, we find that, like the Wigner distribution function (WDF), the FRFT is also equivalent to the rotation operation of the GT. Then, we show that performing the scaled inverse Fourier transform (IFT) along an oblique line of the GT of f(t) can yield its FRFT. Since the GT is closely related to the FRFT, we can use it for analyzing the characteristics of the FRFT. Compared with the WDF, the GT does not have the cross-term problem. This advantage is important for the applications of filter design, sampling, and multiplexing in the FRFT domain. Moreover, we find that if the GT is combined with the WDF, the resultant operation [called the Gabor-Wigner transform (GWT)] also has rotation relation with the FRFT. We also derive the general form of the linear distribution that has rotation relation with the FRFT. <s> BIB004
It is also of our interest to establish the relationship between the FRFT and the local polynomial Fourier transform (LPFT) BIB001 BIB002 . First, we rewrite the FRFT as where x w ðtÞ ¼ xðt þ tÞwðtÞ. If the LPFT is given by BIB001 BIB002 From these equations it can be clearly seen that the LPFT provides a broad generalization of the FRFT. Similar observations have been made in BIB004 . Additionally, we can relate the FRFT to the local polynomial periodogram (LPP), which is defined as the squared magnitude of the LPFT: To understand the relationship between the FRFT and the LPP, let us consider the LPP in more detail. The LPP is a distribution belonging to the Cohen class of distributions BIB001 BIB002 . Its kernel is defined as the AF of w g ðtÞ ¼ wðtÞ which is a generalization of the kernel used for the spectrogram (i.e., the squared magnitude of the short-time Fourier transform), where the kernel function is defined as the AF of w(t). This generalization stems from the fact that the short-time Fourier transform is equal to the LPFT for M =1. Hence, to relate the LPP to BIB003 , let us consider the second order case (i.e., M = 2): 2 Àj2pf tÞ dt where AF w (t,f) is the ambiguity function of the used window. Hence, the LPP rotates AF w (t,f) in the ambiguity plane and can adjust the resulting time-frequency representation according to the signal of interest. Similarly can be stated for any localized form of the FRFT with a proper selection of the a (angle) parameter. Eqs. (13)- (22) demonstrate that various coordinate transformations have been implemented to relate the FRFT to different transformations. This implies that any coordinate transformation can be used, which is demonstrated in the next section, where we relate the FRFT with the linear canonical transform.