Hi Dirk.
Dirk> Hi Mads, Tom had said that if you had minimum phase then you
Dirk> could avoid the costly phase unwrap. For a minimum phase
Dirk> system/signal the log-magnitude and phase are a Hilbert
Dirk> transform pair so the phase is completely determined by the
Dirk> magnitude, which would have been what Tom was referring to,
Dirk> although this would seem to be another calculation that had to
Dirk> be made instead of the phase unwrapping if you needed to
Dirk> determine the phase. My question
Dirk> "What signals do you normally encounter that are guaranteed to
Dirk> be minimum phase?"
Dirk> in response to Tom's comment
Dirk> "I think the phase unwrapping is only necessary for non-minimum
Dirk> phase systems?"
Dirk> was relating to how often would the signal/system be known to be
Dirk> minimum phase so the Hilbert relationship could be used instead
Dirk> of the phase unwrapping. I was thinking in terms of deconvolving
Dirk> signals, which had been brought up earlier in the thread.
Yeah, I know. I was just commenting on your question "what signals
[..]". My point was that not even speech, which is where you usually
see cepstrum (and much other minimum phase stuff) applied, is minimum
phase, so no disagreement. My background is in speech coding so I
have encountered a lot of the problems associated with the minimum
phase assumption. Just trying to share ... :-)
--
/Mads (http://kom.auc.dk/~mgc)
Reply by Dirk Bell●August 25, 20032003-08-25
Hi Mads,
Tom had said that if you had minimum phase then you could avoid the costly
phase unwrap. For a minimum phase system/signal the log-magnitude and phase
are a Hilbert transform pair so the phase is completely determined by the
magnitude, which would have been what Tom was referring to, although this
would seem to be another calculation that had to be made instead of the
phase unwrapping if you needed to determine the phase. My question
"What signals do you normally encounter that are guaranteed to be minimum
phase?"
in response to Tom's comment
"I think the phase unwrapping is only necessary for non-minimum phase
systems?"
was relating to how often would the signal/system be known to be minimum
phase so the Hilbert relationship could be used instead of the phase
unwrapping. I was thinking in terms of deconvolving signals, which had been
brought up earlier in the thread.
Dirk
Dirk A. Bell
DSP Consultant
"Mads G. Christensen" <christensen@nospam.ieee.org> wrote in message
news:wkyu186uitq.fsf@leo.kom.auc.dk...
> Hi Dirk.
>
> Dirk> What signals do you normally encounter that are guaranteed to be
> Dirk> minimum phase?
>
> As some have already mentioned in this thread, the cepstrum is (or
> was?) often used in speech processing. It is often argued, that an AR
> model is physically meaningful in modeling the vocal tract. However,
> for nasal sounds, you essentially have two ARs in parallel, the oral
> cavity and the nasal cavity, which leaves you with an ARMA model. The
> resulting model is not necessarily minimum phase, so even for speech,
> the fundamental assumptions of the cepstrum do not always hold.
>
> --
> /Mads (http://kom.auc.dk/~mgc)
Reply by Mads G. Christensen●August 25, 20032003-08-25
Hi Dirk.
Dirk> What signals do you normally encounter that are guaranteed to be
Dirk> minimum phase?
As some have already mentioned in this thread, the cepstrum is (or
was?) often used in speech processing. It is often argued, that an AR
model is physically meaningful in modeling the vocal tract. However,
for nasal sounds, you essentially have two ARs in parallel, the oral
cavity and the nasal cavity, which leaves you with an ARMA model. The
resulting model is not necessarily minimum phase, so even for speech,
the fundamental assumptions of the cepstrum do not always hold.
--
/Mads (http://kom.auc.dk/~mgc)
Reply by Tom●August 22, 20032003-08-22
Dirk Bell wrote:
> "Tom" <somebody@nOpam.com> wrote in message
> news:3F4422E1.B1638BD3@nOpam.com...
> >
> >
> > Dirk Bell wrote:
> >
> > > "Jerry Avins" <jya@ieee.org> wrote in message
> > > news:3F40E1D7.D57B5EC4@ieee.org...
> > > > Rune Allnor wrote:
> > > > >
> > > > ...
> > > > >
> > > > > While I have no problems to accept that the cepstrum in certain
> > > > > applications may have beneficial properties, I can't see that it can
> > > > > dispose of the need to exploit whatever properties are known/assumed
> > > > > about the input signal.
> > > > >
> > > > > Even if the cepstrum transforms deconvolution to a mere subtraction,
> > > > > you do need to know what to subtract.
> > > > >
> > > > Moreover, it can be only approximate. The logs of negative and complex
> > > > numbers are messy, so cepstrum deals with logs of magnitudes*. The
> loss
> > > > of phase information is just that: a loss of information.
> > > >
> > > > Jerry
> > > > _________________________
> > > > * Or squared magnitudes, which are easier to compute if the quantity
> is
> > > > complex, and differ only by a factor of 2.
> > > > --
> > > > Engineering is the art of making what you want from things you can
> get.
> > > > ����������������������������������������������������������������
> > >
> > > Jerry,
> > >
> > > The 'real' cepstrum deals with logs of magnitudes and no phase.
> > >
> > > To convert convolution to addition, you have to use the 'complex'
> cepstrum,
> > > rather than the 'real' cepstrum that the OP asked about. For the complex
> > > cepstrum you will have to properly unwrap the phase of the first FFT
> since
> > > you will be using the complex logarithm. To minimize errors, a
> significant
> > > amount of additional processing may be required to do a good job on the
> > > unwrapping.
> > >
> > > Dirk
> > >
> > > Dirk A. Bell
> > > DSP Consultant
> >
> > I think the phase unrapping is only necessary for non-minimum phase
> systems?
> >
> > Tom
> >
> >
>
> What signals do you normally encounter that are guaranteed to be minimum
> phase?
>
> Dirk
I was not thinking of signals but rather systems. You can identify the TFs of
certain systems with cepstrum methods. You can also for example perform spectral
factorisation.(for Wiener filters) The spectral factor is by definition always
minimum phase.
See the paper by Silvia and Robinson, 1978 USe of the kepstrum in signal
analysis,
Geoexploration 16,55-73.
for a start off.
Tom
Reply by Dirk Bell●August 21, 20032003-08-21
"Tom" <somebody@nOpam.com> wrote in message
news:3F4422E1.B1638BD3@nOpam.com...
>
>
> Dirk Bell wrote:
>
> > "Jerry Avins" <jya@ieee.org> wrote in message
> > news:3F40E1D7.D57B5EC4@ieee.org...
> > > Rune Allnor wrote:
> > > >
> > > ...
> > > >
> > > > While I have no problems to accept that the cepstrum in certain
> > > > applications may have beneficial properties, I can't see that it can
> > > > dispose of the need to exploit whatever properties are known/assumed
> > > > about the input signal.
> > > >
> > > > Even if the cepstrum transforms deconvolution to a mere subtraction,
> > > > you do need to know what to subtract.
> > > >
> > > Moreover, it can be only approximate. The logs of negative and complex
> > > numbers are messy, so cepstrum deals with logs of magnitudes*. The
loss
> > > of phase information is just that: a loss of information.
> > >
> > > Jerry
> > > _________________________
> > > * Or squared magnitudes, which are easier to compute if the quantity
is
> > > complex, and differ only by a factor of 2.
> > > --
> > > Engineering is the art of making what you want from things you can
get.
> > > ����������������������������������������������������������������
> >
> > Jerry,
> >
> > The 'real' cepstrum deals with logs of magnitudes and no phase.
> >
> > To convert convolution to addition, you have to use the 'complex'
cepstrum,
> > rather than the 'real' cepstrum that the OP asked about. For the complex
> > cepstrum you will have to properly unwrap the phase of the first FFT
since
> > you will be using the complex logarithm. To minimize errors, a
significant
> > amount of additional processing may be required to do a good job on the
> > unwrapping.
> >
> > Dirk
> >
> > Dirk A. Bell
> > DSP Consultant
>
> I think the phase unrapping is only necessary for non-minimum phase
systems?
>
> Tom
>
>
What signals do you normally encounter that are guaranteed to be minimum
phase?
Dirk
Reply by Fred Marshall●August 21, 20032003-08-21
"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0308180042.55e51cbe@posting.google.com...
> Jerry Avins <jya@ieee.org> wrote in message
news:<3F40357F.C6DA6E5E@ieee.org>...
> Which means that the computation of the output signal of whatever system
> is reduced to a mere addition of cepstra if the input signal and system
> impulse response [*].
>
> At least what the time/frequency duality is concerned, the preferable
> domain is a matter of computational algorithms. Whether one chooses this
> or that domain, if you want to do some sort of deconvolution, one needs
> information or assumptions on at least one of the factors (input signal
> or impulse response).
>
> While I have no problems to accept that the cepstrum in certain
> applications may have beneficial properties, I can't see that it can
> dispose of the need to exploit whatever properties are known/assumed
> about the input signal.
>
> Even if the cepstrum transforms deconvolution to a mere subtraction,
> you do need to know what to subtract.
Rune,
Underwater applications should have been thought of by now I should think.
I wonder why there aren't more in evidence? Perhaps the nonstationary
nature of the medium in most applications? I don't know all that much about
this area but am willing to jump in.
Fred
Reply by Tom●August 20, 20032003-08-20
Dirk Bell wrote:
> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F40E1D7.D57B5EC4@ieee.org...
> > Rune Allnor wrote:
> > >
> > ...
> > >
> > > While I have no problems to accept that the cepstrum in certain
> > > applications may have beneficial properties, I can't see that it can
> > > dispose of the need to exploit whatever properties are known/assumed
> > > about the input signal.
> > >
> > > Even if the cepstrum transforms deconvolution to a mere subtraction,
> > > you do need to know what to subtract.
> > >
> > Moreover, it can be only approximate. The logs of negative and complex
> > numbers are messy, so cepstrum deals with logs of magnitudes*. The loss
> > of phase information is just that: a loss of information.
> >
> > Jerry
> > _________________________
> > * Or squared magnitudes, which are easier to compute if the quantity is
> > complex, and differ only by a factor of 2.
> > --
> > Engineering is the art of making what you want from things you can get.
> > ����������������������������������������������������������������
>
> Jerry,
>
> The 'real' cepstrum deals with logs of magnitudes and no phase.
>
> To convert convolution to addition, you have to use the 'complex' cepstrum,
> rather than the 'real' cepstrum that the OP asked about. For the complex
> cepstrum you will have to properly unwrap the phase of the first FFT since
> you will be using the complex logarithm. To minimize errors, a significant
> amount of additional processing may be required to do a good job on the
> unwrapping.
>
> Dirk
>
> Dirk A. Bell
> DSP Consultant
I think the phase unrapping is only necessary for non-minimum phase systems?
Tom
Reply by Jerry Avins●August 19, 20032003-08-19
Rune Allnor wrote:
>
> ... I don't
> know why I thought of the white noise model, but it did not come from
> Jackson's discussion of the cepstrum.
>
...
Noise for sibilants. Periodic pulses for vowels. Formants for both.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Rune Allnor●August 18, 20032003-08-18
It's 2 AM and I ought to be sleeping. I'm not. I've been thinking about
cepstra for hours. And here's what came out of it, apart from the hope
that I, if nothing else, can get a half nights sleep after having posted
this...
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<iPU%a.4102$Jk5.3908466@feed2.centurytel.net>...
> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> > As for the pitfalls, it appears to me (after a less than 5 min review of
> > Jackson's breaf treatment) that the cepstrum is used for deconvolution
> > of speech signals to extract the impulse response of the human vocal
> tract.
> > The vocal tract appears to be modeled as a broad-band linear filter that
> > is excited by some stationary white noise signal.
I think I may have been wrong in that assumption. After a somewhat more
through reading of the text (amounting to some 15 min) it seems as the
speech model amounts to a resonant system that is excited by a regular
pulse train, not entirely dissimilar to one model that was discussed
in conjunction with Jerry's woodpecker a couple of months ago. I don't
know why I thought of the white noise model, but it did not come from
Jackson's discussion of the cepstrum.
> > Of course, it's impossible to grasp the obscure details of a method just
> > like that, but it *may* be that the properties of the cepstrum processing
> > are such that speech processing is the only application that meet the
> > required conditions. If that's correct, cepstrum processing is best left
> > to books that treat speech processing in particular, and deleted from
> > general DSP books.
This is the point that has been bothering me. Explanations and elaborations
below.
> > Quite a few echographic remote sensing techniques
> > (sonar, seismics, ultrasound) would benefit significantly if the impulse
> > response of the medium could be estimated. However, the input signal in
> > those kinds of systems are impulses, not white noise.
>
> Rune,
>
> I heard a lecture by Tom Stockham who had done work at the U. of Utah in the
> early days. They did a number of things with old Caruso recordings:
> - they removed the echo structure of the recording horn
> - they removed the orchestra
> As I understood it, the spectrum was smoothed to get rid of the multipath
> effects. But I haven't followed the process to the point I understand it.
Here's another "�re" (1/100 of a Norwegian "krone", just more than 0.1 cent
according to current exchange rates) from me:
*IF* the signal model consisting of a "clean" impulse response that's
excited by a regular impulse train is correct (and "regular" may be a very
important constraint), I think the spectrum of the total signal will look
as if the broad-band spectrum of the system impulse response has been
"sliced up" along the frequency axis by an "inverse comb filter". Now,
I haven't worked out the maths so I may be wrong!
The period of the notches in this "inverse comb filter" is somehow
related to the pulse rate in the regular impulse excitation signal.
Which is why I believe "regular" is important in the problem specification.
If we forget that the spectrum is a spectrum and regard it as an arbitrary
data series, this new data series shows a dual behaviour in that it has
a low-frequency "background trend" (due to the broadband component of the
measured signal) and a high-frequency component (due to the notches).
Taking the log of the (positive) data essentially "sharpenes up" this
general behaviour.
OK, at this point we have an almost flat data series with a set of very
narrow spikes superimposed. Computing the DFT of this series produces a
bimodal spectrum where the low-freq parts represent the general trend
(the broad-band pulse) and the high-freq parts represent the spike train.
If I haven't messed up so far, the cepstrum somehow "inverts" the contents
of the spectrum, in that the "usual" broad-band parts of the spectrum is
mapped into "almost DC" in the cepstrum while the "almost DC" parts (impulse
rate) of the spectrum are mapped out to the high-freq parts of the
cepstrum... hey, wait a minute... *spec*trum vs *ceps*trum... is that's
what's hidden in the name?
Anyway, applying a low-freq filter in this domain and inverse-transforming
the result should produce a smoothed, broad-band spectrum that is only
related to the impulse response of the system. With some more care
(i.e. using the phase cepstrum), it should even be possible to reproduce
the impulse response itself.
If all this isn't way off target, it appears that the cepstrum may be
useful in applications that process data that comprise sets narrow,
harmonic lines superimposed over low background noise. In this case the
cepstrum probably would not show much energy in the broad-band segments
(near the cepstrum "DC"), but the sets of harmonic contents in the spectrum
should show clearly up in the "low-freq" parts of the cepstrum (away from
the cepstrum "DC").
Actually, it appears that the cepstrum actually could have a couple of
applications outside speech processing as well... Oh no! Now I'll be
thinking about that for the rest of the night... well, I'll just find
my telescope and take a look at Mars. It was astonishing earlier tonight.
Rune
Reply by Dirk Bell●August 18, 20032003-08-18
"Jerry Avins" <jya@ieee.org> wrote in message
news:3F40E1D7.D57B5EC4@ieee.org...
> Rune Allnor wrote:
> >
> ...
> >
> > While I have no problems to accept that the cepstrum in certain
> > applications may have beneficial properties, I can't see that it can
> > dispose of the need to exploit whatever properties are known/assumed
> > about the input signal.
> >
> > Even if the cepstrum transforms deconvolution to a mere subtraction,
> > you do need to know what to subtract.
> >
> Moreover, it can be only approximate. The logs of negative and complex
> numbers are messy, so cepstrum deals with logs of magnitudes*. The loss
> of phase information is just that: a loss of information.
>
> Jerry
> _________________________
> * Or squared magnitudes, which are easier to compute if the quantity is
> complex, and differ only by a factor of 2.
> --
> Engineering is the art of making what you want from things you can get.
> ����������������������������������������������������������������
Jerry,
The 'real' cepstrum deals with logs of magnitudes and no phase.
To convert convolution to addition, you have to use the 'complex' cepstrum,
rather than the 'real' cepstrum that the OP asked about. For the complex
cepstrum you will have to properly unwrap the phase of the first FFT since
you will be using the complex logarithm. To minimize errors, a significant
amount of additional processing may be required to do a good job on the
unwrapping.
Dirk
Dirk A. Bell
DSP Consultant