LR Rabiner and RW Schafer, June 3

DRAFT: L. R. Rabiner and R. W. Schafer, June 3, 2009
Chapter 8
The Cepstrum and
Homomorphic Speech
Processing
8.1 Introduction
In 1963, Bogert, Healy and Tukey published a chapter with one of the most
unusual titles to be found in the literature of science and engineering, namely
“The Quefrency Alanysis of Time Series for Echoes” [1]. In this paper, the
authors observed that the logarithm of the Fourier spectrum of a signal plus an
echo (delayed and scaled replica) consisted of the logarithm of the signal spectrum
plus a periodic component due to the echo. They suggested that further
Fourier analysis of the log spectrum could highlight the periodic component in
the log spectrum and thus lead to a new indicator of the occurrence of an echo.
Specifically they made the following observation:
“In general, we find ourselves operating on the frequency side in
ways customary on the time side and vice versa.”
As an aid in formalizing this new point of view, where the time- and frequencydomains
are interchanged, Bogert et al. created a number of new terms by
transposing letters in familiar engineering terms. For example, to emphasize
that the log spectrum was to be viewed as a waveform to be subjected to Fourier
analysis, they defined the term “cepstrum” of a signal as the power spectrum of
the logarithm of the power spectrum of a signal. Similarly, the term “quefrency”
was introduced for the independent variable of the cepstrum. In Reference [1],
Bogert et al. proposed a number of other terms with the goal of emphasizing
the parallels that arise when the time and frequency domains are interchanged.
The definitions of a subset of these terms are given in Table 8.1. A few of these
terms have survived the test of time and will be used in the discussions of this
chapter.
Bogert et al. based their definition of the cepstrum on a rather loose interpretation
of the spectrum of an analog signal. In fact, their simulations used
425