Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Coordinate transformation of other higher-order timefrequency representations <s> A dual form of the Wigner higher order spectra is introduced. Its analysis in the case of multicomponent signals is performed. An efficient distribution for time-frequency signal analysis (L-Wigner distribution) is derived from that analysis. The theory is illustrated on a numerical example. > <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Coordinate transformation of other higher-order timefrequency representations <s> The polynomial Wigner-Ville distribution (PWVD) is a time-frequency signal analysis tool for representing polynomial frequency modulated (FM) signals. We present an optimization procedure to design PWVDs and discuss the need for higher order PWVDs. <s> BIB002
The FRFT and LCT framework can be generalized for some higher order time-frequency representations. Here, we discuss the case of the L-Wigner distribution (L-WD) introduced in BIB001 for achieving high concentration of the time-frequency representations by eliminating the socalled inner interferences. The L-WD can be defined as Signal x L ðtÞ, producing the coordinate transform of the L-WD LWD(at+ bf,ct+ df), can be defined for ba0 as The L-FRFT form can be evaluated using the FRFT as For L= 2, this relationship can be written as In general the L-FRFT of the k +m-th order can always be expressed using the L-FRFT of k-th and m-th orders: Similar coordinate transform can be determined for all time-frequency representations depending on powers of the signal local-autocorrelation xðt þ t=2Þx Ã ðtÀt=2Þ. However, as far as we know there are no reported results for other higher-order time-frequency representations such as the polynomial WD (e.g., BIB002 ) with the usage of several auto-correlations.