cepstrum analysis - mario bonusing the inverse z-transform integral: = ∫ − & q; ] ] g] m [ q...

Guido Geßl 21.01.2004

Cepstrum Analysis


B.P. Bogert, M.J. R. Healy and J.W. Tukey, "The Quefrency Alanysis of Time Series forEchoes: Cepstrum, Psuedo-Autocovariance, Cross-cepstrum and Saphe Cracking", Proceedings of the Symposium on Time Series Analysis, M. Rosenblat, Ed., Wiley, NY, 1963, pp 209-243


original definition - real cepstrum

Inverse Fourier transfrom of the logarithm of themagnitude of the Fourier transform:

[ ] ( ) ωπ

ωπ

π

ω GHH;QF QMM

[ ∫−

= log2

1

magnitude is real and nonnegative

not invertible as the phase is missing


more general definitioncomplex cepstrum

∫−

=π

π

ωω ωπ

GHH;Q[ QMM )](log[2

1][ˆ

∫−

+=π

π

ωωω ωπ

GHH;MH; QMMM ))](arg()(log[2

1

arg is the continuous (unwrapped) phase function.

The term complex cepstrum refers to the use of thecomplex logarithm, not to the sequence.

The complex cepstrum of a real sequence is also a real sequence.


complex logarithm

The complex cepstrum exists if the complexlogarithm has a convergent power seriesrepresentation of the form :

∑∞

−∞=

− ===Q

Q ]]Q[];]; 1,][ˆ)](log[)(ˆ

So log[X(z)] must have the properties of the z-transform of a stable sequence.


using the inverse z-transform integral:

∫ −=&

Q G]]];M

Q[ 1)](log[2

1][ˆ

π

The contour of integration C is within the region of convergence.

The region of convergence must include the unit cyclefor stability.

That‘s why we can use the inverse Fourier transform.


The real cepstrum is the inverse transform of the real partof and therefore is equal to the conjugate-symmetric part of :

)(ˆ ωMH;][ˆ Q[

2

][ˆ][ˆ][

* Q[Q[QF[

−+=


Alternative Expressions

We can use the logarithmic derivative to avoid thecomplex logarithm.Assuming log[X(z)] is analytic, then

)(

)(’)(’ˆ

];

];]; =

Using the z-transform formula we get

∫ −=−&

Q G]]];

];]

MQ[Q 1

)(

)(’

2

1][ˆ

π


Dividing both sides by (-n) yields

∫ ≠−= −

&

Q QG]]];

];]

QMQ[ 0,

)(

)(’

2

1][ˆ 1

π

and per definition

∫−

=π

π

ω ωπ

GH;[ M )(ˆ2

1]0[ˆ

A (nonlinear) difference equation for x[n] is :

0,][][ˆ][ ≠−= ∑∞

−∞=

QNQ[N[Q

NQ[

N

(for computation with recursive algorithms)


Complex cepstrum of exponential sequences

If a sequence x[n] consists of a sum of complexexponential sequences, it‘s z-transform is a rational function of z.

( ) ( )

( ) ( )∏∏

∏∏

==

−

==

−

−−

−−=

RL

RL

1

N

N

1

N

N

0

N

N

0

N

N

U

]G]F

]E]D]$

];

11

1

11

1

11

11)(

i…inside, o…outside of the unit circle


then log[X(z)] is :

( ) ( )

( ) ( )∑∑

∑∑

==

−

==

−

−−−−

−+−++=

RL

RL

1

N

N

1

N

N

0

N

N

0

N

N

U

]G]F

]E]D]$];

11

1

11

1

1log1log

1log1log)log()log()(ˆ

A is real for real sequences but could be negative.

A nonzero r will cause a discontinuity in arg[X].

In practice this is avoided by determining A and r and altering the input so, that A _$_��U� ��


with the power series expansions

∑

∑∞

=

−

∞

=

−−

<−=−

<−=−

1

1

1

1

,)1log(

,)1log(

Q

QQ

Q

QQ

]]Q

]

D]]Q

]

βββ

αα

we obtain

<+

>+−

=

=

∑∑

∑∑

=

−

=

−

==

��

��

1

N

Q

N

0

N

Q

N

1

N

Q

N

0

N

Q

N

QQ

G

Q

E

QQ

F

Q

D

Q$

Q[

11

11

.0,

,0,

,0,log

][ˆ


From this we can derive the following general

properties:

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)


minimum-phase and maximum-phase sequences

A minimum-phase sequence is a real, causaland stable sequence whose poles and zerosare all inside the unit cycle.That means that all the singularities of log[X(z)] are inside the unit cycle. So we have the property

4) The complex cepstrum is causal (0 for all n<0) if and only if x[n] is minimum phase.


similarily we conclude, that the complexcepstrum for maximum-phase (=left sided) systems is also left-sided

5) The complex cepstrum is 0 for all n>0 ifand only if x[n] is maximum phase, i.e. X(z) has all poles and zeros outside the unitcircle


Hilbert Transform Relations

Causality (or anticausality) of a sequenceplaces powerful constraints on the FourierTransform.

The Fourier Transform of any real, causalsequence is almost completely determinedby either the real or the imaginary part of the Fourier Transform.


Thus, if is causal (x[n] is minimum phase), wecan obtain Hilbert transform relations between thereal ( ) and imaginary ( ) partsof the complex cepstrum.

As noted earlier, the cepstrum is the even part of thecomplex cepstrum:

][ˆ Q[

)(log ωMH; )](arg[ ωMH;

2

][ˆ][ˆ][

* Q[Q[QF[

−+=

So if the complex cepstrum is known to be causal, it can berecovered from the real cepstrum by frequency-invariantlinear filtering.

][][2][

][][][ˆ

min

min

QQXQ

QQFQ[ [

δ−==

A

A


Homomorphic deconvolution

• Generalized superposition

T{x1[n]+x2[n]}= T{x1[n]}+ T{x2[n]}

T{c x1[n]}=c T{x1[n]}

becomes

T{x1[n] �[2[n]}= T{x1[n]} �7^[2[n]}

T{c �[1[n]}=c �7^[1[n]}

with different input and output operations


Systems satisfying this generalizedprinciple of superposition are calledhomomorphic systems since they can berepresented by algebraically linear (homomorphic) mappings between inputand output signal spaces.

Linear systems are obviously a special casewhere �DQG� �are addition and �DQG� �aremultiplication.


Now suppose the input of a system is the

convolution of two signals :

x[n]=x1[n]*x2[n]

Their Fourier transforms would therefore be

multiplied :

X(z)=X1(z) X2(z)

applying the logarithm :

log[X(z)]=log[X1(z)] + log[X2(z)]

So the cepstrum is:

][ˆ][ˆ][ˆ 21 Q[Q[Q[ +=


Thus, the cepstrum can be interpreted as a System that satisfies the generalizedprinciple of superposition with convolutionas the input function of superposition and addition as the output function of superposition.

This system is called the CharacteristicSystem for Convolution D*.


D* L D*-1

L is a linear System in the usual sense.

Now if we choose a system L that removes the additivecomponent , then x2[n] will be removed from theconvolutional combination.

In practical applications, L is a linear frequency-invariantsystem.

Such L are called ‚lifters‘ because of their similarity to filters in the frequency domain.

][ˆ2 Q[


Applications

• processing signals containing echoesseismology, measuring properties of reflecting surfaces, loudspeaker design, dereverberation, restoration of acousticrecordings

• speech processingestimating parameters of the speech model


• machine diagnostics

detection of families of harmonics and sidebands, for example in gearbox and turbine vibrations

• calculating the minimum phase spectrumcorresponding to a given log amplitudespectrum with Hilbert transform


example: delay time detection


Example: echo removal

cepstrum analysis - mario bonusing the inverse z-transform integral: = ∫ − & q; ] ] g] m [ q...

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