4.14 The periodic behavior of a time series induced by echoes can also be observed in the spectrum of the series; this fact can be seen from the results stated in Problem 4.6(a). Using the notation of that problem, suppose we observe xt = st + Ast−D + nt, which implies the spectra satisfy fx(ω) = [1+A2 + 2Acos(2πωD)]fs(ω) + fn(ω). If the noise is negligible (fn(ω) ≈ 0) then log fx(ω) is approximately the sum of a periodic component, log[1+A2+2Acos(2πωD)], and log fs(ω). Bogart et al. (1962) proposed treating the detrended log spectrum as a pseudo time series and calculating its spectrum, or cepstrum, which should show a peak at a quefrency corresponding to 1/D. The cepstrum can be plotted as a function of quefrency, from which the delaty D can be estimated.

For the speech series presented in Example 1.3, estimate the pitch period using cepstral analysis as follows. The data are in the file speech.dat.

(a) Calculate and display the log-periodogram of the data. Is the periodogram periodic, as predicted?

(b) Perform a cepstral (spectral) analysis on the detrended logged periodogram, and use the results to estimate the delay D. How does your answer compare with the analysis of Example 1.24, which was based on the ACF?