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Ross Clement (Email addre · Guest

Hi. I need to calculate the cepstrum of a signal. I understand that the
cepstrum of a signal is the fourier transform of the log of the fourier
transform of a signal. I can easily calculate each of these operations
individually, using the fftw3 library. However I have to unwrap the
phase of the first FT, and I don't yet know how to do this.

Can anyone either (i) point me in the direction of a freely available
cepstrum calculating library, or phase unwrapping library usable on
Linux, or (ii) convince me that phase unwrapping of a fourier transform
of a signal is easy and with a bit of study I'll have no trouble doing
it?

Cheers,

Ross-c

Jerry Wolf · Guest

Ross -- it's the complex cepstrum that you need, right?

I ask because "cepstrum" is commonly used to refer to the FT of the log
of the power spectrum, which is a bit simpler to compute -- no phase
involved.

cheers,
jerry wolf

Jerry Wolf · Guest

As I suspected. One doesn't need the complex cepstrum unless one is
going to reconstitute the signal. For pitch estimation, you want the
real cepstrum, commonly called just the cepstrum. See Rabiner &
Schafer, secs. 7.1.2, 7.3.

Ross Clement (Email addre · Guest

Thanks for the answer.

(i) Hmmm..... I have to review some sources on the use of the cepstrum
in pitch detection before I can say whether I need the complex cepstrum
or not. The only source I have within arm's reach is Hess's 1983 book
on Pitch Detection. This sort of suggests that the FT of the log of the
power spectrum is what is used. However I'm confused in that the
Wikipedia article on the cepstrum says that the cepstrum is the FT of
the log of the FT, and that it is a common mistake to claim that the
cepstrum is the *INVERSE* FT of the log of the FT. Hess describes the
cepstrum as being the INVERSE FT of the power spectrum. So, I need to
look into this further. But in reality since I'm currently in an
"implement *everything*" mode, I'll probably end up implementing both.

(ii) I was presuming that unwrapping the phase of a FT was a much more
complex operation than unwrapping phase shift in FIR filters. After
running into a DSP specialist colleague in the corridor, it appears
that it's easy. I didn't realise that a discontinuity in the phase
spectrum of a FT was just a discontinuity in phase between two adjacent
FT (frequency) bins. So, I'm not sure if I need to unwrap phase or not.
But if I need to, it looks like I can with no problem.

Cheers,

Ross-c

Ingeniur · Guest

"Jerry Wolf" <sl_jerry1@hotmail.com> wrote in message
news:1134570017.351854.16430@g49g2000cwa.googlegroups.com...

Guest · Posted: Thu Dec 15, 2005 1:17 am Post subject: Re: Code for cepstrum

For pitch estimation it is common to take the log of the power spectrum
which is obtained by summing the squares of the real and imaginary
outputs of the first FFT. There is no need to square root the sum of
squares, as it is more efficient to adjust after the log.

John

Ron N. · Guest

Jerry Wolf wrote:

Ron N. · Guest

Ross Clement (Email address invalid - do not use) wrote:

Fred Marshall · Guest

I don't know if this helps at all. Here is an older message I wrote to Rune
Allnor:

Fred

***************************

I made some diagrams out of the O&S first ed. book and substituted "s" for
the "square" operator.

The "Convolutional System" is what you're dealing with. What interests me
below is that the Z Transforms go forward/inverse (in that order) in both
Figures 5 and 6. It's because in the input case the first transformation is
from convolutional to multiplicative and in the output case the second
transformation is from multiplicative to convolutional.

Of course an L linear stage is implied between Figures 5 and 6.

I guess this simply says that a multiplicative system can directly be dealt
with using complex log and complex exponential but a convolutional system
needs to be:
1) transformed first so that it becomes a multiplicative system. Then we
can do the log operation.
2) transformed back so that it becomes an additive/linear system. Then we
can apply a linear filter.
3) transformed next so that it becomes a multiplicative system again. Then
we can apply the exponent operation.
4) transformed back so that it becomes a convolutional system once more - in
other words getting back to the domain of interest.

I'm used to doing linear filtering in the frequency domain using
multiplication just as well as doing it in the time domain with convolution.
So, it appears that the middle transform operation pair is unnecessary if
you do the filtering in the frequency domain. Why bother computing the
complex cepstrum at all? Am I missing something here. Oh, they probably
call that a "trick" later on! I just haven't read enough. Note in Fig 1
that they are applying a linear filter in the time domain and are using the
operational symbols "+" for input and output even though it's a
convolutional process in the time domain. The "+"s mean that the inputs add
and the outputs add as in superposition.

I'm sure you've already gone through all this and that's why you're looking
at phase unwrapping programs.

s + + + + o
+-----------+ +-------+ +-----------+
x(n) | | x^(n)| | y^(n)| | y(n)
----->| Ds |----->| L |----->| Ds^-1 |----->
| | | | | |
+-----------+ +-------+ +-----------+

Fig 1 Canonic Representation of homomorphic systems

x + + + + x
+-----------+ +-------+ +-----------+
x(n) | complex | | | | complex | y(n)
----->| log |----->| L |----->|exponential|----->
| | | | | |
+-----------+ +-------+ +-----------+

Fig 2 Multiplicative System with Ds = Complex Log

* + + + + *
+-----------+ +-------+ +-----------+
x(n) | | x^(n)| |y^(n) | | y(n)
----->| D* |----->| L |----->| D*^-1 |----->
| | | | | |
+-----------+ +-------+ +-----------+

Fig 3 Convolutional System Abstract Representation

* + + + + *
+-----------+ +-------+ +-----------+
X(z) | complex | X^(z)| |Y^(z) | complex | Y(z)
----->| log |----->| L |----->| exp |----->
| | | | | |
+-----------+ +-------+ +-----------+

Fig 4 Convolutional System Z Transform Domain Representation
as a Canonic Homomorphic System

* x x + + +
+-----------+ +-------+ +-----------+
x(n) | | X(z) |complex|X^(z) | | x^(n)
----->| z[] |----->| log |----->| z^-1[] |-----> complex
cepstrum > to "L"
| | | | | |
+-----------+ +-------+ +-----------+

Fig 5 Convolutional System Representation for D* as in Fig 3

+ + + x x *
+-----------+ +-------+ +-----------+
y^(n) | | Y^(z)|complex| Y(z) | | y(n)
----->| z[] |----->| exp |----->| z^-1[] |----->
| | | | | |
+-----------+ +-------+ +-----------+

Fig 6 Convolutional System Representation for D*^-1 as in Fig 3

Ross Clement (Email addre · Guest

Thanks to everyone who replied on this thread. Given the time being
just before 5pm on a Friday, I won't be trying to write any cepstrum
code until Monday. But your postings will be very helpful when I get
around to doing it.

Cheers,

Ross-c

Rune Allnor · Guest

Ross Clement (Email address invalid - do not use) wrote:

John Herman · Guest

In article <1135010286.551638.310400@z14g2000cwz.googlegroups.com>, "Rune Allnor" <allnor@tele.ntnu.no> wrote:

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