From Frequency to Quefrency: A History of the Cepstrum

Abstract

The idea of the log spectrum or cepstral averaging has been useful in many applications such as audio processing, speech processing, speech recognition, and echo detection for the estimation and compensation of convolutional distortions. To suggest what prompted the invention of the term cepstrum, this article narrates the historical and mathematical background that led to its discovery. The computations of earlier simple echo representations have shown that the spectrum representation domain results does not belong in the frequency or time domain. Bogert et al. (1963) chose to refer to it as quefrency domain and later termed the spectrum of the log of a time waveform as the cepstrum. The article also recounts the analysis of Al Oppenheim in relation to the cepstrum. It was in his theory for nonlinear signal processing, referred to as homomorphic systems, that the realization of the characteristic system of homomorphic convolution was reminiscent of the cepstrum. To retain both the relationship to the work of Bogart et al. and the distinction, the term power cepstrum was eventually applied to the nonlinear mapping in homomorphic deconvolution . While most of the terms in the glossary have faded into the background, the term cepstrum has survived and has become part of the digital signal processing lexicon.

Full text

dsp history
Alan V. Oppenheim and Ronald W. Schafer
From Frequency to Quefrency: A History of the Cepstrum
IEEE SIGNAL PROCESSING MAGAZINE
n their classic paper enti-
tled The Quefrency Alanysis
of Time Series for Echos:
Cepstrum, Pseudo-Auto-
covariance, Cross-Cepstrum,
and Saphe Cracking [1], Bogert,
Healy and Tukey were perhaps in a
very playful frame of mind when
they coined the term cepstrum,
along with a complete glossary that
included, in addition to those in the
title of their paper, terms such as
rahmonics and liftering. The motiva-
tion for this strange new terminolo-
gy, where familiar words were
paraphrased by interchanging con-
sonants, was succinctly stated as fol-
lows: “In general, we find ourselves
operating on the frequency side in
ways customary on the time side and
vice versa” [1].
To suggest what prompted the
invention of the term cepstrum,
note that a signal with a simple echo
can be represented as
x(t) = s (t ) + αs (t − τ). (1)
The Fourier spectral density (spec-
trum) of such a signal is given by
|X ( f )|
2
=|S ( f )|
2
[1 + α
2
+ 2αcos(2π f τ)].(2)
Thus, we see from (2) that the
spectral density of a signal with an
echo has the form of an envelope
(the spectrum of the original signal)
that modulates a periodic function
of frequency (the spectrum contri-
bution of the echo). By taking the
logarithm of the spectrum, this
product is converted to the sum of
two components; specifically
C ( f ) = log|X ( f )|
2
= log|S ( f )|
2
+ log[1 + α
2
+ 2 α cos(2π f τ)].(3)
Thus,
C ( f )
viewed as a wave-
form has an additive periodic com-
ponent whose “fundamental
frequency” is the echo delay
τ
. In
conventional analysis of time wave-
forms, such periodic components
show up as lines or sharp peaks in
the corresponding Fourier spec-
trum. Therefore, the “spectrum” of
the log spectrum would likewise
show a peak when the original time
waveform contained an echo. This
new “spectral” representation
domain was not the frequency
domain, nor was it really the time
domain. So, looking to forestall
confusion while emphasizing con-
nections to familiar concepts,
Bogert et al. chose to refer to it as
the quefrency domain, and they
termed the spectrum of the log of
the spectrum of a time waveform
the cepstrum. While most of the
terms in the glossary at the end of
the original paper have faded into
the background, the term cepstrum
has survived and become part of the
digital signal processing lexicon.
In the early 1960s, totally unre-
lated to, and independent of, the
work by Bogert et al., Al Oppen-
heim was pursuing his doctoral
research on a class of nonlinear sig-
nal processing techniques inspired
by the concept of homomorphic
(i.e., linear in a generalized sense)
mappings between algebraic groups
and vector spaces. His dissertation,
“Superposition in a Class of
Nonlinear Systems” [2] completed
at MIT in May, 1964, developed a
theory for nonlinear signal process-
ing referred to as homomorphic sys-
tems. The use of such systems for
signal processing was termed homo-
morphic filtering.
The essential idea of homomor-
phic system theory was that many
signal processing operations satisfy
the same algebraic postulates as
addition. Therefore, homomorphic
mappings between signal spaces in
which these other operations play
the role of signal (vector) addition
are, in essence, linear mappings in a
generalized sense. This suggested a
new approach to a variety of prob-
lems in separating signals that had
been nonadditively combined, such
as through convolution or multipli-
cation. Various potential applica-
tions of homomorphic signal
separation were actively considered,
primarily for deconvolution and
demultiplication.
The first step in the development
of homomorphic filtering for
deconvolution was to work out
both the basic theory and ways to
implement homomorphic systems
for convolution. It was shown in Al
Oppenheim’s Ph.D. dissertation
that all homomorphic systems have
a canonic representation consisting
of a cascade of three systems. The
first system is an invertible nonlinear
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characteristic operator (system) that
maps a nonadditive combination
operation such as convolution into
ordinary addition. The second sys-
tem is a linear system obeying addi-
tive superposition, and the third
system is the inverse of the first
nonlinear system. Thus, for signals
combined by convolution, a homo-
morphic deconvolution system
maps convolution into addition,
then addition into addition, and
finally addition into convolution.
For convolution (denoted
∗
)
, the
characteristic operator
D
∗
[][][]
has
the property that
D
∗
[x
1 ∗
x
2
] =
ˆ
x
1
+
ˆ
x
2
,
where
D
∗
[x
1
] =
ˆ
x
1
and
D
∗
[x
2
] =
ˆ
x
2
.
One intuitively
appealing approach to the imple-
mentation of the nonlinear mapping
D
∗
[][][]
is through the complex-val-
ued logarithm of the complex-val-
ued Fourier transform. That is, if
two signals are convolved, their
dsp history
continued
heirs is a historical collaboration. It started in 1965, it con-
tinues to-date, and it is planned to continue for at least
another 40 years. From this collaboration resulted the coau-
thored books Digital Signal Processing (1975), Discrete-Time
Signal Processing (1989, 1999), Computer-Based Exercises
for Signal Processing Using MATLAB 5 (1998), and numerous
research articles, all of which testify to the creative enthusi-
asm and challenging work of their authors. Their numerous
research and teaching awards, including the IEEE Education
Medal received by each clearly acknowledge their great love
and talent for both teaching and research. The quiz is easy,
Dear Reader, and you may have guessed the answer right
away. Our guests today, in a double feature of the “DSP
History” column, are Dr. Al Oppenheim and Dr. Ron Schafer.
Al Oppenheim was born on 11 November 1937, in New
York City. He obtained his S.B. (1961), S.M. (1961), and Sc.D.
(1964) degrees from the Massachusetts Institute of
Technology (MIT), Cambridge. He has been with the
Department of Electrical Engineering and Computer Science at
MIT for his entire academic career, with a two-year leave of
absence at MIT Lincoln Laboratory (1967–1969) and two
years part time as an associate division head at MIT Lincoln
Laboratory (1978–1980). Dr. Oppenheim’s research focuses
on algorithms for signal processing motivated by applications
in speech processing, image processing, acoustics, communi-
cations, and more recently drifting toward biology. In addition
to the books mentioned earlier, he also coauthored:
Signals
Signals
and Systems
and Systems (1982, 1997), authored two widely used video
lecture series, and edited and coedited other books. Dr.
Oppenheim is the recipient of numerous awards for excel-
lence, including the IEEE Third Millenium Medal (2000), the
IEEE Centennial Medal (1984), the IEEE Education Medal
(1988), and election to the National Academy of Engineering
(1987). In collaborators, he appreciates “creative, out-of-the-
box thinking, willingness to assume that any idea is a poten-
tially good idea, ability to listen and be open-minded, but also
to challenge in a friendly and constructive way.” His hobbies
include a variety of sports pursued intensely. Over the years
these have included skiing, squash, tennis, offshore sailing and
racing, windsurfing, biking, and flying. His current nonprofes-
sional passions are magic (illusions), windsurfing, and flying
(he received his private pilot’s license in 2003). He is a playful
soul, even more so on St. Patrick’s day, when (in true Irish
spirit), his friends and colleagues call him “O’Ppenheim.”
Ron Schafer was born on 17 February 1938 in Tecumseh,
Nebraska. He obtained the B.S.E.E. (1961) and M.S.E.E.
(1962) degrees from the University of Nebraska, Lincoln, and
the Ph.D. degree (1968) from MIT. After a career-defining six
years with Bell Telephone Laboratories (1968–1974), he
joined the Georgia Institute of Technology (Georgia Tech) in
1974 as John and Marilu McCarty Professor. In 2004, he
retired from Georgia Tech as Professor Emeritus and took a
position as distinguished technologist at Hewlett-Packard
Laboratories in Palo Alto, California. Ron’s work has focused
on speech processing, image processing, and applications of
DSP to medicine and biology. In addition to the books men-
tioned earlier, he also coauthored Digital Processing of
Speech Signals (1978), DSP First: A Multimedia Approach
(1998), and Signal Processing First (2003). He is the recipient
of several awards for excellence, including the IEEE Emanuel
R. Piore Award (1980), the IEEE Education Medal (1992), the
IEEE Signal Processing Society Education Award (2000), and
election to the National Academy of Engineering (1992). In
addition to his almost four-decade collaboration with Al, he
has had long, fruitful, and greatly valued collaborations over
many years with Larry Rabiner, Tom Barnwell, Russ
Mersereau, and Jim McClellan. Ron appreciates in collabora-
tors their ability to push and motivate him and their tolerance
of his tendency to procrastinate. To his own children and a
multitude of students, he has often quoted an instruction
from the MIT catalog of 1963: “A student is expected to study
at least three hours outside of class for every hour in the
classroom.” To his chagrin, this suggestion was not generally
met with an enthusiastic response. Dr. Schafer’s hobbies
include photography, fly fishing, reading, walking, and volun-
teering.
We invite you to follow Dr. Al Oppenheim and Dr. Ron
Schafer in their pursuit of reconstructing the captivating his-
tory of the cepstrum. Spell-checkers are not recommended...
—Adriana Dumitras and
George Moschytz
“DSP History” column editors
adrianad@ieee.org,
moschytz@isi.ee.ethz.ch
T

Fourier transforms are multiplied,
and a suitably defined complex log-
arithm will produce the sum of two
log Fourier transforms. The inverse
Fourier transform of a sum is the
sum of the individual inverse trans-
forms, so the cascade of — Fourier
transform
→→→
complex logarithm
→→→
inverse Fourier transform — maps
convolution into a sum of corre-
sponding signals. Interest in this
implementation fortunately coincid-
ed nicely with the publication of the
1965 paper by Cooley and Tukey
[3] on efficient implementation of
the Fourier transform on digital
computers, i.e., the fast Fourier
transform (FFT). The FFT provided
an efficient means of implementing
the Fourier transform computations
needed to implement the nonlinear
mapping
D
∗
[[[]]]
.
It was a fortuitous discussion in
1965 between Jim Flanagan of Bell
Telephone Laboratories and Al
Oppenheim that connected the
work going on at MIT to the devel-
opment of the cepstrum at Bell
Laboratories. After hearing about
homomorphic deconvolution from
Oppenheim, Flanagan noted that
the characteristic system for homo-
morphic convolution was reminis-
cent of the spectrum of the log of
the spectrum (i.e., the cepstrum) as
proposed by Bogert et al.
Furthermore, he suggested looking
at work by Michael Noll [4], [5] in
the Journal of Acoustical Society of
America. Noll credits Manfred
Schroeder (who was aware of the
work of Bogert et. al.) with suggest-
ing to him that it might be interest-
ing to apply cepstrum analysis on a
short-time basis to speech signals.
In the Journal of Acoustical Society
of America papers, Noll applied the
cepstrum as a basis for pitch detec-
tion. The problem of pitch detec-
tion is very similar to detecting echo
times in the sense that the basic
speech model consists of represent-
ing speech as the convolution of the
vocal tract impulse response with
the quasi-periodic train of glottal
pulses. The basic idea of cepstral
pitch detection is illustrated in
Figures 1 and 2.
The success of cepstral pitch
detection suggested to us the strong
potential of applying homomorphic
deconvolution to deconvolve speech,
i.e., to separate the vocal tract
impulse response, the glottal pulse
shape, and the periodic excitation.
Bogert et al. had introduced the con-
cept of liftering, i.e., linear filtering
of the log spectrum, as a way of
emphasizing the periodic component
of the log spectrum so as to enhance
the detectability of echos [1]. The
concept of homomorphic filtering
clearly suggested how liftering could
be used to actually separate the vari-
ous convolutional components, such
as separating the vocal tract filter
response from the periodic excitation
spectrum. By applying a low-pass
lifter to the cepstrum in Figure 2 to
extract the low quefrency compo-
nents below the first rahmonic peak,
the slowly varying curve (in red,
upper graph) results. The low que-
frency components thus correspond
to the resonance structure of the
vocal tract and high frequency falloff
of the speech spectrum due to the
glottal pulse.
This example shows that the gen-
eral concept of homomorphic
deconvolution is fundamentally dif-
ferent from echo or pitch detection
in a number of ways. One, of course,
is that the ultimate objective in
homomorphic deconvolution is sig-
nal separation and recovery rather
than detection; i.e., the nonlinear
mapping is followed by linear filter-
ing and subsequently by the inverse
of the nonlinear mapping. While the
spectral density as used by Bogert et
al. loses phase information, the
objective of homomorphic deconvo-
lution requires that phase informa-
tion be retained or reconstructed.
Consequently, in homomorphic
deconvolution the complex-valued
Fourier transform and complex loga-
rithm must in general be used. To
retain both the relationship to the
work of Bogert at al. and the distinc-
tion, the term complex cepstrum was
eventually applied to the nonlinear
mapping in homomorphic deconvo-
lution, and the term power cepstrum
or real cepstrum used for the cep-
strum as originally defined by
Bogert et al.
Shortly after Al Oppenheim
joined the MIT faculty and became
aware of the work in [1] and [4], he
and Ron Schafer were introduced to
each other by Tom Stockham, a
IEEE SIGNAL PROCESSING MAGAZINE
SEPTEMBER 2004 97
▲ 1. A segment of voiced speech with a Hamming window during a voiced (vowel-like)
time interval of 50 ms. Short-time analysis of speech involves the analysis of a succession
of such segments taken sequentially in time. At this particular time in a vowel sound, the
pitch period is approximately 12.5 ms.
1
0.5
0
–0.5
–1
50
Windowed Speech Segment
12.5
Fundamental Period
051015 20 25 30 35 40 45
Time [ms]

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faculty member at MIT and a mem-
ber of Oppenheim’s dissertation
committee. At that time, Ron
Schafer was a graduate student in
search of an interesting doctoral dis-
sertation topic. Thus began a close
and wonderfully productive collabo-
ration between a young professor
and an enthusiastic graduate stu-
dent, exploring and developing the
theoretical concepts, techniques, and
applications of the complex cep-
strum. The resulting doctoral disser-
tation “Echo Removal by Discrete
Generalized Linear Filtering” [6] by
Ron Schafer at MIT in 1968,
focused on the issues of the dis-
crete-time formulation of the com-
plex cepstrum, phase computation,
recursion relations, and applications
to echo removal from speech. In our
development of the complex cep-
strum, a variety of alternate imple-
mentations of the complex cepstrum
and relationships between the power
cepstrum and complex cepstrum
were developed, both in general and
for minimum-phase, maximum-
phase and all-pass sequences. The
work also led to the interpretation in
the cepstral domain of the Hilbert
transform relationship between
Fourier transform magnitude and
phase for minimum phase signals.
The early work on the applica-
tion of homomorphic filtering to
deconvolution and demultiplication
and consequently on the extension
of the power cepstrum to the com-
plex cepstrum, was described in the
Proceedings of the IEEE [7] and also
developed and described more
extensively in Chapter 10 of our
first jointly authored textbook [8]
published in 1975. As an interesting
side note, throughout the various
stages of proofreading of this book,
we constantly had to maintain vigi-
lance to be certain that this
“strange” term cepstrum wasn’t
inadvertently “corrected” to what
seemed to be more appropriate.
(Our text editing program also con-
tinues to complain as we write this
article). We breathed a sigh of relief
when the last page proofs were
returned to the publisher. When the
first printing of the book appeared,
it was clear that a particularly dili-
gent proofreader at the publisher
had caught the “error” at the last
instant and cepstrum had been
reversed to spectrum throughout.
This unusual term has caused lin-
guistic difficulties in other ways.
Since it was meant as a paraphrase
of the term spectrum, its natural
pronunciation would be kepstrum.
On the other hand, based on its
spelling the natural pronunciation
would be sepstrum, and there have
been lively debates on which should
be used. The adopted pronunciation
among those aware of the origin is
kepstrum, and in fact some authors
have chosen to spell it as kepstrum
to make the pronunciation clear.
Applications of the cepstrum,
complex cepstrum, and homomor-
phic deconvolution have been
explored in a variety of areas includ-
ing audio processing, speech pro-
cessing, geophysics, radar, medical
imaging, and others. Early in the
development of the concept of
homomorphic filtering, Tom
Stockham had shown particular
dsp history continued
▲ 2. Log magnitude (in blue, upper graph), lowpass liftered (linearly filtered) log magni-
tude (in red, upper graph), and cepstrum (in blue, bottom graph) of the segment of
voiced speech illustrated in Figure 1. The rapidly varying curve in the upper graph is the
log magnitude of the discrete-time Fourier transform of the segment of speech in Figure 1.
The lower plot is the cepstrum, i.e., the inverse discrete Fourier transform of the log mag-
nitude in the upper plot. Note the peaks at “rahmonics” of 1/80 = 12.5 ms, the fundamen-
tal quefrency of the quasi-periodic ripples in the upper graph. As can be seen by
comparing the speech segment in Figure 1 to this cepstrum, this fundamental quefrency is
also the period (pitch period) of the time waveform. The search for such peaks in the cep-
strum was the basis for Noll's pitch detection algorithm.
4
2
0
–2
–4
–6
0
0.5
0
–0.5
–1
50
Log Magnitude of Fourier Transform
Fundamental "Period" = 80 Hz
0
0.5 1 1.5 2 2.5 3
3.5 4
Frequency [kHz]
Cepstrum of Segment of Voiced Speech
First Rahmonic Peak at 12.5
051015 20 25 30 35 40 45
Frequency [ms]

interest in exploring potential appli-
cations to both demultiplication and
deconvolution. The work on homo-
morphic demultiplication led to
novel methods for dynamic range
compression, as disclosed in a patent
by Stockham and Oppenheim [9],
and signal enhancement for both
audio and images, the latter of
which was discussed by Stockham in
[10]. An early and very novel appli-
cation of the cepstrum to deconvo-
lution was the work by Stockham et
al. [11] directed at blind deconvolu-
tion such as the restoration of old
phonograph recordings. They
focused in particular on restoration
of recordings by Enrico Caruso.
The objective in the restoration of
these old recordings was to com-
pensate for the undesireable fre-
quency response of the “morning
glory” recording horn used in that
era. Since the recording horn was
retuned daily and since its frequency
response was typically subject to
temperature and humidity varia-
tions, the compensation could not
simply be based on modern day
measurements of an archived
recording horn, i.e., the spectral
characteristics would have to be
estimated from the recording itself.
Stockham’s essential idea in the
Caruso restoration was to use log
spectral averaging (essentially cep-
stral averaging) and subtraction of
the log spectral average of a modern
recording of the same arias. The
result was then used as an estimate
of the log spectrum of the recording
horn which was to be compensated
for. This idea of log spectrum, or
equivalently, cepstrum averaging has
been useful in many other applica-
tions for estimating and compensat-
ing for a convolutional distortion.
Also in the context of audio pro-
cessing, speech processing devel-
oped as a major application area for
the cepstrum, both in the form of
the power cepstrum and the com-
plex cepstrum. For example, the
cepstral pitch detector utilizing the
power cepstrum became widely
used. Homomorphic deconvolution
was also successfully applied by us
and others in the late 1960s to sepa-
rate the glottal excitation and the
vocal tract impulse response for
speech modeling in general and for
speech compression in particular.
The cepstrum also plays a signifi-
cant role in many speech recogni-
tion systems. Specifically, the
cepstral coefficients have been
found empirically to be a more
robust, reliable feature set for
speech recognition and speaker
identification than linear predictive
coding (LPC) coefficients or other
equivalent parameter sets. Cepstral
parameter extraction in speech rec-
ognizers is, in some instances, based
on converting LPC parameters to
cepstral coefficients by utilizing the
recursion relationship [7], [8] that
we developed in our early research
to obtain the complex cepstrum of
minimum-phase signals without the
need for explicit computation of the
Fourier transform or phase unwrap-
ping. An alternative use of the cep-
strum in speech recognition is the
mel-frequency cepstrum [12]. The
mel-frequency cepstrum is based on
calculating the cepstrum from the
logarithm of the spectrum obtained
from a filter bank with center fre-
quencies and bandwidths deter-
mined by a constant mel-frequency
interval. Cepstral averaging and
time-differencing of cepstra are now
standard methods of removing the
effects of linear filtering prior to
recognition.
While speech processing has been
one of the more successful applica-
tions of cepstral analysis and homo-
morphic deconvolution, applications
in other areas have been explored
with considerable success. As men-
tioned previously, detection of
echoes in seismic signals was the
motivating application in the original
work of Bogert et. al. Following the
publication of [7], there was consid-
erable interest in exploring homo-
morphic deconvolution in the
context of seismology and explo-
ration geophysics. For example, the
complex cepstrum was used in the
early 1970s in determining seismic
wavelets and in deconvolving seismic
traces [13], [14].
There are many other innova-
tions and applications that can be
traced to the cepstrum and its sub-
sequent development, but space
does not permit an exhaustive dis-
cussion or bibliography. This is not
surprising in light of the ubiquity of
convolution as a model for what
goes on in our physical world. It is
almost certain that we have not seen
the end of new ideas and new appli-
cations of cepstrum analysis and
homomorphic filtering. For exam-
ple, our original theoretical develop-
ment showed that the complex
cepstrum of a signal with a rational
z
-transform could be obtained in
terms of the roots of the numerator
and denominator [8]. This avoids
the issue of phase computation
completely. Steiglitz and Dickinson
demonstrated this in [15] for poly-
nomial
z
-transforms of only moder-
ate length. However, a recent article
[16] by Sitton, Burrus, Fox, and
Treitel in IEEE Signal Processing
Magazine has shown that polynomi-
als of order up to 1 million can be
accurately rooted using methods
based on the FFT. At the time the
complex cepstrum was introduced
and developed, factoring such large
polynomials was not even consid-
ered. With such powerful new tools
it may be time to take another look
at the complex cepstrum in the con-
text of new applications.
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dsp history continued from page 99
References
life sciences continued