Cross-Terms Analysis in the Modified Instantaneous Power Spectrum
This correspondence examines cross terms in the modified version of the instantaneous power spectrum (MIPS), which is a smoothed form of the real part of the Rihaczek distribution. MIPS was reported recently to be less sensitive to the poor resolution of spectro- gram and the cross terms in the pseudo Wigner-Ville distribution (PWVD). However, in this correspondence we will show that like PWVD, the MIPS possesses cross terms too. The only differences be- tween them are the location and the oscillating frequency of the cross terms. We deduce that the cross terms of the MIPS are located in about the same region of the autoterms, while the cross terms of the WVD are located midway between the autoterms. The undesirable cross terms and artifact between different spectral components are much less se- vere and interfere less with each other in MIPS than in PWVD cases. I. INTRODUCTION Page originally defined the instantaneous power specturm (IPS) as the derivative of the running energy spectrum (I), which satis- fied many of the properties of time-frequency distribution. How- ever, its ringing effect blocks its application. Hippenstiel and De Oliveira (2) advocated the use of the Fourier transform of an av- eraged correlation function estimate. This estimate is further mod- ified by applying a sliding window after forming the correlation estimate. The result, a smoothed version of the real part of the Rihaczek distribution (3), is a modified version of IPS. Hippenstiel shows that the MIPS can be less sensitive to the time-frequency resolution tradeoff problem occurring in the spectrogram 141 and the cross-term interference problem occurring in Wigner-Ville dis- tribution (WVD) (5)-(7). Moreover, the modified IPS (MIPS) can avoid the ringing effect which is a major problem in Page's IPS method. However, in this correspondence we will show that. like PWVD, the MIPS possesses cross terms too. Compared with WVD, the only differences between them are the location and the oscil- lating frequency of the cross terms. This correspondence will also show why the modified IPS reduces the ringing effect.