| Author |
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Rune Allnor
Guest
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 Posted:
Wed Dec 15, 2004 3:18 pm Post subject:
Analytic Functions and Single Side Band Signals |
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| Quote: | When I was in third year of Uni, we did a course on complex maths that
really threw me. It was all about analytic functions, the
Cauchy-Riemann equations, etc. and involved all sorts of integrations
of curves in the complex plane. Now I'd always been good at maths, but
what with all the other stuff going on, and the fact that this stuff
was quite difficult, I never really conquered the subject, which I
consider a serious hole in my knowledge. This is what Wolfram says
about analytic functions:
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[- snip -]
| Quote: | I hadn't thought of most of this stuff in ages until last night
something clicked in my head. I have been working for a few months
with Single-Sideband Signals generated by Hilbert Transforms. These
are of course analytic signals, since they contain only positive
frequencies. My question is this: Are analytic signals related in
anyway to analytic functions? Does an analytic function necessarily
produce a single sideband signal? Is there no connection at all?
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You have spotted the very essence of DSP. Have you ever wondered
why you use those _rational_ transfer functions in z domain, and
not arbitrary experssions of z, like Log(z) or exp(Z)? It's
because rational complex-valued functions are analytic.
Have you ever wondered why you use the tabulated inverse Laplace
or inverse Fourier transforms instead of computing an integral?
It's becuase the Cauchy residual theorem applies with analystic
functions.
To my knowledge, there is only one book that shows these things
in all their gory mathemathical details:
Oppenheim & Schafer: "Digital Signal Processing"
Prentice Hall, 1975.
(The Rabiner & Gold book from about the same time could very well
be close, but I have never actually seen it, so I don't know.)
Oppenheim & Schafer have written lots of books, so make sure you get
the one from 1975. Their discussion of the Hilbert transform is
basically a full semester course on complex functions, much like the
course you took at university, condensed to 10-pages.
DSP is applied maths. You have spotted the essence of it.
Rune |
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Andor Bariska
Guest
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| Posted:
Wed Dec 15, 2004 4:08 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Rune Allnor wrote:
....
| Quote: |
You have spotted the very essence of DSP. Have you ever wondered
why you use those _rational_ transfer functions in z domain, and
not arbitrary experssions of z, like Log(z) or exp(Z)? It's
because rational complex-valued functions are analytic.
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They are not, Rune. Rational functions are meromorphic, which basically
means they are analytic on the complex numbers C minus the set of their
poles.
If you include a complex infinity into the set of the complex numbers,
which is a one-point compactification of C, sometimes denoted as C-bar,
then one can say that the meromorphic fucntions are analytic on C-bar. A
nice model of C-bar is the complex number sphere (where 0 is the
south-pole and complex infinity is the north-pole and the unit circle is
the equator).
The meromorphic functions constitute a field which naturally embedds the
complex numbers, similar to the way the complex number field embedds the
real numbers.
| Quote: | Have you ever wondered why you use the tabulated inverse Laplace
or inverse Fourier transforms instead of computing an integral?
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To avoid having to teach complex integration?
:-)
Regards,
Andor |
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Randy Yates
Guest
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| Posted:
Wed Dec 15, 2004 5:41 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Andor Bariska <an2or@nospam.net> writes:
| Quote: | Rune Allnor wrote:
...
You have spotted the very essence of DSP. Have you ever wondered why
you use those _rational_ transfer functions in z domain, and not
arbitrary experssions of z, like Log(z) or exp(Z)? It's because
rational complex-valued functions are analytic.
They are not, Rune. Rational functions are meromorphic, which
basically means they are analytic on the complex numbers C minus the
set of their poles.
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Andor,
It's good that you point out this fact for the OP, but don't you
think Rune probably already knows this? He probably just posted in
a hurry and/or without feeling it necessary to spell out all the
details. As you imply, "analytic" is meaningless without a domain.
Note also that in my old Complex Variables class, we used to call a
function that was analytic over all C "entire." (That class was based
on the Churchill and Brown text.)
| Quote: | The meromorphic functions constitute a field which naturally embedds
the complex numbers, similar to the way the complex number field
embedds the real numbers.
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I believe you've got some detail wrong here because if what you say
is true, the Fundamental Theorem of Algebra is wrong. There are no
extension fields (splitting fields?) of the complex.
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124 |
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Andor Bariska
Guest
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| Posted:
Wed Dec 15, 2004 6:33 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Randy Yates wrote:
| Quote: | Andor,
It's good that you point out this fact for the OP, but don't you
think Rune probably already knows this?
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If he does, then he forgot to mention it. I thought I'd jump in and jog
everybody's memories.
| Quote: | He probably just posted in
a hurry and/or without feeling it necessary to spell out all the
details. As you imply, "analytic" is meaningless without a domain.
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Indeed.
| Quote: | The meromorphic functions constitute a field which naturally embedds
the complex numbers, similar to the way the complex number field
embedds the real numbers.
I believe you've got some detail wrong here because if what you say
is true, the Fundamental Theorem of Algebra is wrong. There are no
extension fields (splitting fields?) of the complex.
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I meant it in the following sense:
Def: A subset G of a set F is called a subfield, if (F,+,*) and (G,+,*)
are both fields.
Examples:
- Q is a subfield of R
- R is a subfield of C
- C is a subfield of M
(M is the set of meromorphic functions)
Notice my choice of "embed" rather than "extend".
Also, I believe the Fundamental Theorem of Algebra was concerned with
the number of zeroes of a polynomial? The fact that there are no
commutative division algebras with order larger than 2 is known to me as
Theorem of Frobenius (well, at least it is a direct corollary thereof).
Perhaps you got confused by the fact that M is not, unlike R and C, an
algebra.
Regards,
Andor |
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Rune Allnor
Guest
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| Posted:
Thu Dec 16, 2004 12:22 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Andor Bariska wrote:
| Quote: | Rune Allnor wrote:
...
You have spotted the very essence of DSP. Have you ever wondered
why you use those _rational_ transfer functions in z domain, and
not arbitrary experssions of z, like Log(z) or exp(Z)? It's
because rational complex-valued functions are analytic.
They are not, Rune. Rational functions are meromorphic, which
basically
means they are analytic on the complex numbers C minus the set of
their
poles.
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If I ever heard the term "meromorphic" in use before, I've forgotten
about it. The "analytic everywhere except at the poles and branch
cuts" goes without saying, particularly since the OP both works in
DSP and has the complex maths course.
What I forgot to mention was that the Exp(z) function probably is
analytic (almost) everywhere, but has a Laurent series representation
with infinitely many terms. Which is a bit awakward. And the Log(z)
function has branch cuts as well. Which is even more awkward.
Interestingly, the branch cuts of the Log(z) function appear to be
the main obstacle for getting the complex cepstra to work.
| Quote: | If you include a complex infinity into the set of the complex
numbers,
which is a one-point compactification of C, sometimes denoted as
C-bar,
then one can say that the meromorphic fucntions are analytic on
C-bar. A
nice model of C-bar is the complex number sphere (where 0 is the
south-pole and complex infinity is the north-pole and the unit circle
is
the equator).
The meromorphic functions constitute a field which naturally embedds
the
complex numbers, similar to the way the complex number field embedds
the
real numbers.
|
Interesting. I didn't know that.
| Quote: | Have you ever wondered why you use the tabulated inverse Laplace
or inverse Fourier transforms instead of computing an integral?
To avoid having to teach complex integration?
|
Yes. My main reaction last year when I saw the Oppenheim & Schafer
book from 1975 for the first time, was that that book had little to
do with DSP as we know it today. Their book was more about applied
maths. If one had to compute the IFT in the "algebraically correct"
way, no electrical engineer would ever qualify for the DSP intro
class, let alone get any work done.
Those who have a copy available, check out section 2.2 in O&S, 1975.
What saves the day is Cauchy's residue theorem that ensures that
given a H(z) on the form of a ratio of two finite polynomials,
all we need is to know the zeros and poles and then we can set up
a table of formulas that takes from z domain to time domain.
All of that saves ridiculous amounts of work, both in teaching DSP
and in working with DSP, and it works only because it is based
on the theory of analytic functions.
Rune |
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Andor
Guest
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| Posted:
Thu Dec 16, 2004 3:07 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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The Google guys still seem to be experimenting with an optimal user
interface. Right now, they don't give me the original message when
replying, so please excuse the odd quote:
Rune wrote:
"
What I forgot to mention was that the Exp(z) function probably is
analytic (almost) everywhere, but has a Laurent series representation
with infinitely many terms. Which is a bit awakward.
"
The Laurent series expansion of exp(z) stops dead at 0. Exp(z) is about
as analytic as they come. Did you mean exp(1/z)? This has a "major"
singularity (what is the correct term in English?) at zero, meaning
exactly that the Laurent series expansion around zero has infinitely
many (negative power) terms.
"
And the Log(z) function has branch cuts as well. Which is even more
awkward.
Interestingly, the branch cuts of the Log(z) function appear to be
the main obstacle for getting the complex cepstra to work.
"
This was something I never quite understood. Is the complex cepstrum of
s(t) not defined as FT^(-1) ( log( Abs( FT( s(t) ) ) ) )? What kind of
information can one possibly gather from that? What do you mean by
"getting the complex cepstra to work"?
Regards,
Andor |
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Rune Allnor
Guest
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| Posted:
Thu Dec 16, 2004 4:31 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Andor wrote:
| Quote: | The Google guys still seem to be experimenting with an optimal user
interface. Right now, they don't give me the original message when
replying, so please excuse the odd quote:
Rune wrote:
"
What I forgot to mention was that the Exp(z) function probably is
analytic (almost) everywhere, but has a Laurent series representation
with infinitely many terms. Which is a bit awakward.
"
The Laurent series expansion of exp(z) stops dead at 0. Exp(z) is
about
as analytic as they come. Did you mean exp(1/z)? This has a "major"
singularity (what is the correct term in English?) at zero, meaning
exactly that the Laurent series expansion around zero has infinitely
many (negative power) terms.
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I am on thin ice here, and I may very well have messed up the very
precise terminology. I took a basic course on maths where complex
numbers were taught, but I have forgotten most of it. At least the
subtle details in the levels slightly above basic, that we discuss
here.
| Quote: | "
And the Log(z) function has branch cuts as well. Which is even more
awkward.
Interestingly, the branch cuts of the Log(z) function appear to be
the main obstacle for getting the complex cepstra to work.
"
This was something I never quite understood. Is the complex cepstrum
of
s(t) not defined as FT^(-1) ( log( Abs( FT( s(t) ) ) ) )? What kind
of
information can one possibly gather from that? What do you mean by
"getting the complex cepstra to work"?
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What you cite is the *real* cepstrum. The complex cepstrum is
defined as
S(t) = IFT{Log{FT{x(t)}}} [1]
where X(w) = FT{x(t)} is complex-valued and Log is the complex
logarithm,
Log(z) = Log{|z|*exp(j*phi)} = log(|z|) +j(phi + n*2*pi). [2]
The requirement is that the complex cepstrum is real-valued and
causal (the term "complex" refers to the complex Log function
being used) so the real and imaginary parts in [2] must be Hilbert
transform pairs. Which in turn means that the real and imaginary
parts must be continuous on the unit circle in complex z domain.
The real part, log(|z|), is no problem except for z=0. The imaginary
part is a problem, since the periodicity in phi introduces a
discontinuity, a branch cut. Basically, it's very awkward to unwrap
the phase of the spectrum in a way that is both continuous on the
unit circle, satisfies the Hilbert tranform relation to the
log(|z|) function, and also results in a causal, real-valued
cepstrum S(t). Check out chapters 7 and 10 in Oppenheim and
Schafer's 1975 book for details of the statement of the problem.
Rune |
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Andor
Guest
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| Posted:
Thu Dec 16, 2004 5:07 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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I wrote:
"
This was something I never quite understood. Is the complex cepstrum of
s(t) not defined as FT^(-1) ( log( Abs( FT( s(t) ) ) ) )? What kind of
information can one possibly gather from that?
"
I did a quick search, and the first link churned out by google was
this:
http://www.lsv.uni-saarland.de/teaching/pattern_and_speech_recognition/ws0405/Pattern_and_Speech_Recognition_Ch4.pdf
It seems that the cepstrum (at least in that article) is used to find
formants for speech. This does make sense, as the regular spacing of
the formant harmonics in the log magnitude of the spectrum will show up
as a peak if you take the fourier transform again. Quite clever.
Any other uses?
Regards,
Andor |
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David Kirkland
Guest
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| Posted:
Thu Dec 16, 2004 11:32 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Andor wrote:
| Quote: | I wrote:
"
This was something I never quite understood. Is the complex cepstrum of
s(t) not defined as FT^(-1) ( log( Abs( FT( s(t) ) ) ) )? What kind of
information can one possibly gather from that?
"
I did a quick search, and the first link churned out by google was
this:
http://www.lsv.uni-saarland.de/teaching/pattern_and_speech_recognition/ws0405/Pattern_and_Speech_Recognition_Ch4.pdf
It seems that the cepstrum (at least in that article) is used to find
formants for speech. This does make sense, as the regular spacing of
the formant harmonics in the log magnitude of the spectrum will show up
as a peak if you take the fourier transform again. Quite clever.
Any other uses?
Regards,
Andor
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Yup, a couple:
1) convolution in time = Multiplication in frequency = Addition in
Ceptstrum (quefrency ?). It has been used for deconvolution.
2) It can also be used to form the minimum phase decomposition of a
frequency response.
I believe one of the other problems is that to get back you must take
the exponential - thus any error can be hugely magnified.
Cheers,
David |
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Andor Bariska
Guest
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| Posted:
Fri Dec 17, 2004 5:03 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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David Kirkland wrote:
....
| Quote: | Yup, a couple:
1) convolution in time = Multiplication in frequency = Addition in
Ceptstrum (quefrency ?).
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David,
do you know who invented these ridiculous terms? Cepstrum, Quefrency,
.... they produce knots in my tongue!
Regards,
Andor |
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David Kirkland
Guest
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| Posted:
Fri Dec 17, 2004 6:21 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Andor Bariska wrote:
| Quote: | David Kirkland wrote:
...
Yup, a couple:
1) convolution in time = Multiplication in frequency = Addition in
Ceptstrum (quefrency ?).
David,
do you know who invented these ridiculous terms? Cepstrum, Quefrency,
... they produce knots in my tongue!
Regards,
Andor
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I know what you mean :)
Sorry, but no, I don't know who came up with those terms. I have a hard
enough time coming up with good variable names in my code.
Quefrency is just a block reversal of the first few letters of Frequency.
Cheers,
David |
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Clay S. Turner
Guest
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| Posted:
Fri Dec 17, 2004 7:43 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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"Andor Bariska" <an2or@nospam.net> wrote in message
news:41c2cb13$1@news1.ethz.ch...
| Quote: | David Kirkland wrote:
...
Yup, a couple:
1) convolution in time = Multiplication in frequency = Addition in
Ceptstrum (quefrency ?).
David,
do you know who invented these ridiculous terms? Cepstrum, Quefrency, ...
they produce knots in my tongue!
Regards,
Andor
|
It was these guys:
B.P. Bogert, M.J.R. Healey, and J.W. Tukey. The quefrency analysis of time
series for echoes: Cepstrum, pseudo-autocovariance, cross-cepstrum and saphe
cracking. In M. Rosenblatt, editor, Proceedings of the Symposium on Time
Series Analysis, pages 209--243. John Wiley and Sons,New York, 1963.
Clay |
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Jim Thomas
Guest
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| Posted:
Fri Dec 17, 2004 8:56 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Clay S. Turner wrote:
| Quote: | It was these guys:
B.P. Bogert, M.J.R. Healey, and J.W. Tukey.
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Maybe we should start calling them Gobert, Leahey, and Kutey ;-)
--
Jim Thomas Principal Applications Engineer Bittware, Inc
jthomas@bittware.com http://www.bittware.com (603) 226-0404 x536
Getting an inch of snow is like winning ten cents in the lottery - Calvin |
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