36Cepstrum - Cepstrum Analysis1 Terminology The terms have...

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Cepstrum Analysis 1 Terminology : The terms have been derived by paraphrasing the existing similar words. Derived Terms Original Terms Cepstrum Spectrum Quefrency Frequency Rahmonics Harmonics Gamnitude Magnitude Saphe Phase Lifter Filter Short-pass Lifter Low-pass Filter Long-pass Lifter High-pass Filter The word ‘cepstrum’ was to mean the ‘spectrum of a natural logarithmic (amplitude) spectrum’, which was first introduced by B. P. Gogert, M. J. R. Healy and J. W. Tukey in 1963. Methodology : 1. Detection: determination of an echo delay time or sideband spacing. 2. Editing: by removal of certain components in the cepstrum, it is possible to remove information of the effects of echoes from a spectrum using the complex cepstrum. Applications : 1. Processing of signals containing echoes: land-based and marine seismology, aero engine noise, loudspeaker measurements in a reverberant environment, and measurement of the properties of a reflecting surface. 2. Speech analysis: formant and voice pitch tracking, vocoding and speech synthesis. 3. Machine diagnostics: detection of families of harmonics and sidebands, for example, in gearbox and turbine vibration signals. 4. Calculation of the minimum phase spectrum corresponding to a given log amplitude spectrum incorporated with Hilbert transform. Basic Theory : The cepstrum of a real signal x t () is defined as, for < < t , 1. Original power cepstrum: { } 2 ) ( log ) ( f S c xx F = τ 2. New power cepstrum: {} { } { } [ ] ) ( log Re ) ( log ) ( log ) ( 1 1 f G f S f S c xx - xx xx - p F F F = = = 1 R. B. Randall and Jens Hee, Cepstrum Analysis , B&K Technical Review, No.3, 1981
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MAE 591 RANDOM DATA Thus, the formula for the envelope cepstrum, since { } [ ] ) ( log Im 1 f G xx - F is the Hilbert transform of { } [ ] ) ( log Re 1 f G xx - F , becomes { } ) ( log ) ( 1 f G c xx - e F = τ 3. Amplitude cepstrum 2 : Amplitude cepstrum of the one-sided spectrum best reveals the periodic structure of the spectrum { } ) ( ) ( ) ( log ) ( p xx a c c f G c = = = F The peak in power cepstrum will always indicate the true spacing of components in the spectrum, independent of their displacement along the frequency axis. On the other hand, it would not be possible to edit in these amplitude cepstra and return to the power spectra, unlike the complex cepstrum. The word ‘high quefrency’ is used for representing rapid fluctuations in the spectrum (i.e. small
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36Cepstrum - Cepstrum Analysis1 Terminology The terms have...

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