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8. Cepstral Analysis

(most slides taken from MIT course by Glass and Zue)

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Homomorphic filtering

Homomorphic filtering/transformation is a nonlineartransformation usually applied to image and speech

processing used to convert a signal obtained from a convolutionof two original signals into the sum of two signals.

In speech processing it can be applied to separate the filter fromthe excitation in the source-filter model

The cepstrum is one such homomorphic transformation that allowsus to perform such separation.

It is an alternativeoption to linear prediction analysis seen before

x[n]= e[n]!h[n]" x[n]= e[n]+ h[n]

x[n]=D(x[n])

s[n]= u[n]!h[n]

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Cepstral analysis is based on the observation that

by taking the log of X(z)

If the complex log is unique and the z transform is valid then, by applying Z-1

the two convolved signals are now additive.

Basics of cepstral analysis

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Basics of cepstral analysis (II)

Consider now that we restrict our signal x[n] to have poles and zerosonly in the unit circle, i.e.:

then if

This is the complex logarithm of X(w)

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Definition of Cepstrum

The real cepstrum is defined as:

Its magnitude is real and non-negative

And the complex cepstrum:

Where arg() represents the phase. We call it complex becauseit uses the complex logarithm, not due to the sequence, whichcan also ne real. In fact, the complex cepstrum of a realsequence is also real

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[ ] ( ) !"!

"

"

!

#$

= log2

1

!"=

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#

$$

$#

)](log[2

1

][

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+=

#

#

$$$

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))](arg()(log[

2

1

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Definition of cesptrum(II)

It can be shown that the the real cepstrum is the even part of thecomplex cepstrum:

In speech processing we generally use the real cepstrum, which isobtained by applying an inverse Fourier Transform of the log-spectrum of the signal.

In fact, the name cepstrum comes from inverting the firstsyllable of the word spectrum. Similarly, the variable n incx[n] is called quefrency, which is the inversion offrequency

2

][][][

*

"+

=

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Properties of cepstrums

From this we can derive the following generalproperties:

1) the complex cepstrum decays at least as

fast as 1/|n|2) it has infinite duration, even if x[n] has

finite duration

3) it is real if x[n] is real (poles and zeros arein complex conjugate pairs)

NOTE: from 2) and 3) we see why usually a finite number of cepstrums is used in speech

processing (12-20 is sufficient), as very high order cepstrums have very small values.

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An example

(Using Taylor seriesexpansion and several tricks)

Given the r in the denominator,it is an infinite train of deltas thatconverges to 0

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(...)

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Computational considerations: using DFT

In digital signals we replace the Fourier Transform by the Discrete Fourier Transform

Aliasing by repetition of thecepstrums with period N

When N> the number of used cepstrums

we do not have a problem (which isusually the case)

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Cepstral analysis of speech

As pointed out at the beginning, we would like to separate theexcitation from the vocal tract filter h(n) by using a

homomorphic transformation.

We can do so easily as the filter parameters usually reside in thelower quefrencies, while the excitation parameters havehigher quefrencies

Consider the problem of recovering a filter's response from a periodic signal (such as a voiced excitation):

The filter response can be recovered if we can separate the output of the homomorphic transformation using a simple filter:

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Cepstral analysis of speech

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13

Cepstrum of a generic voiced signal

magnitude spectrum

cepstrum

Contributions to the cepstrum due to periodic excitation will occur at integermultiples of the fundamental period. NOTE that for children and high-pitchwomen we might have a problem

Contributions due to parameters usually modeled by the filter will concentrate inthe low quefrency region and will decay quickly with n

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Cepstral analysis of speech (voicedsignals)

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Cepstral analysis of speech (unvoicedsignals)

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Cepstral analysis of vowel (rectangularwindow)

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Cepstral analysis of vowel (taperingwindow)

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Cepstral analysis of fricative (rectangularwindow)

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Cepstral analysis of fricative (taperingwindow)

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Use in Speech Recognition

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Statistical properties of cepstral coefficients

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Mel-frequency cepstral representation

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MFCC computation diagram

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Mel-filter bank processing