**Definitions of CEPSTRUM ( a re-attangement / anagram of
the letters of spectrum / spelled as septrum earlier )**

**return to Breakthru
in Measuring INTERNAL COHERENCE.Teaching Biological Oscillators
SUSTAINABILITY -**

(note -- it appears particularly that the lower frequency bandwidth within which the significant power densitites of the EKG exist - allow the cepstrum technique - to qunatify internal coherence - specifically quantizing the onset of phase synchrony between internal harmonic constituents.)

**Cepstrum** Definition:

The Fourier transform of the logarithm of the autospectrum.
Useful for determining periodicities in the autospectrum. Additions
in the cepstrum domain correspond to multiplication in the frequency
domain and convolution in the time domain.

**Units:**

V or dB vs. s

**from- http://www.modeshape.com/cvocabulary.htm**

**Cepstrum**

**The cepstrum is the forward Fourier transform of a spectrum.
It is thus the spectrum of a spectrum, and has certain properties
that make it useful in many types of signal analysis. One of its
more powerful attributes is the fact that any periodicities, or
repeated patterns, in a spectrum will be sensed as one or two
specific components in the cepstrum. **If a spectrum contains
several sets of sidebands or harmonic series, they can be confusing
because of overlap. But in the cepstrum, they will be separated
in a way similar to the way the spectrum separates repetitive
time patterns in the waveform. Gearboxes and rolling element bearing
vibrations lend themselves especially well to cepstrum analysis.
The cepstrum is closely related to the auto correlation function.

**from - about
Cepstrum**

1. The term *cepstrum* was introduced by Bogert et al.
and has come to be accepted terminology for the (note insert from
dan winter - this is wrong! it is not inverse) inverse Fourier
transform of the logarithm of the power spectrum of a signal.
(L.R.Rabiner and R.W.Schafer, Digital Processing of Speech Signals,
Prentice Hall, Englewood Cliffs, NJ, 1978)

2. In 1963, Bogert, Healy and Tukey published a paper with
the unusual title "The Quefrency Analysis of Time Series
for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and
Saphe Cracking."

They observed that the logarithm of the power spectrum of a signal
containing an echo has an additive periodic component due to the
echo, and thus the Fourier transform of the logarithm of the power
spectrum should exhibit a peak at the echo delay. They called
this function the *cepstrum*, interchanging letters in the
word *spectrum* because "in general, we find ourselves
operating on the frequency side in ways customary on the time
side and vice versa."

Bogert et al. went on to define an extensive vocabulary to describe
this new signal processing technique; however, only the term *cepstrum*
has been widely used. (A.V.Oppenheim and R.W.Schafer, Discrete-Time
Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1989)

3. The transformation of a signal into its *cepstrum*
is a *homomorphic* transformation, and the concept of the
cepstrum is a fundamental part of the theory of *homomorphic*
systems for processing signals that have been combined by convolution.
(A.V.Oppenheim and R.W.Schafer, Discrete-Time Signal Processing,
Prentice Hall, Englewood Cliffs, NJ, 1989)

4. Books including the subject about *cepstrum* and *homomorphic*
systems

* Discrete-Time Signal Processing, A.V.Oppenheim and R.W.Schafer,
Prentice Hall

* Digital Processing of Speech Signals, L.R.Rabiner and R.W.Schafer,
Prentice Hall

* Discrete-Time Processing of Speech Signals, J.R.Deller,Jr.,
J.G.Proakis and J.H.L.Hansen, Prentice Hall

* Seismic Applications of Homomorphic Signal Processing, J.M.Tribolet,
Prentice Hall

* Onsei Singou Syori (in Japansese, Speech Signal Processing), S.Imai, Morikita Publishing

This book describes modern *Cepstrum* analysis technology
and its applications in speech processing.

For example,

log magnitude approximation (MLA)

6. Example of the Hypersigal worksheet of *power cepstrum*-source
of Hypersignal worksheet

pwrcep1.lst

7. Cepstrum Co., Ltd. is a developer of digital signal processing
system/software in Japan.

Also Cepstrum is the dealer and 3rd party developer of Hypersignal
in Japan.