| Author |
Message |
porterboy
Guest
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 Posted:
Wed Dec 15, 2004 1:37 pm Post subject:
Analytic Functions and Single Side Band Signals |
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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:
"A complex function is said to be analytic on a region R if it is
complex differentiable at every point in R. The terms holomorphic
function, differentiable function, complex differentiable function,
and regular function are sometimes used interchangeably with "analytic
function" (Krantz 1999, p. 16). Many mathematicians prefer the term
"holomorphic function" (or "holomorphic map") to "analytic function"
(Krantz 1999, p. 16), while "analytic" appears to be in widespread use
among physicists, engineers, and in some older texts (e.g., Morse and
Feshbach 1953, pp. 356-374; Knopp 1996, pp.
83-111; Whittaker and Watson 1990, p. 83)."
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?
Cheers
Porterboy |
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Randy Yates
Guest
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| Posted:
Wed Dec 15, 2004 5:45 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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porterboy76@yahoo.com (porterboy) writes:
| Quote: | [...]
Does an analytic function necessarily
produce a single sideband signal?
|
Definitely not! F(z) = cos(x), where z = x + i*y,
is analytic over all C but it is not an analytic
signal.
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124 |
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Clay S. Turner
Guest
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| Posted:
Wed Dec 15, 2004 8:23 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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"porterboy" <porterboy76@yahoo.com> wrote in message
news:c4b57fd0.0412150037.5615c7c0@posting.google.com...
| Quote: | 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?
|
Hello Porterboy,
Yes analytic functions and analytic signals are related.
I.e.,
An analytic function may be written as
f(t,tau) + j*g(t,tau) where it obeys the C-R relations.
If you let tau==0 in an analytic function you get an analytic signal. And in
this case g(t,0) is the Hilbert transform of f(t,0). And using this you can
easily show that an analytic signal has a one sided transform. (Use the
sigum function in the fourier domain) One of the simplest analytic signals
is cos(t)+j*sin(t).
IHTH,
Clay |
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robert bristow-johnson
Guest
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| Posted:
Wed Dec 15, 2004 9:39 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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in article BCYvd.640$Z27.454@bignews5.bellsouth.net, Clay S. Turner at
Physics@Bellsouth.net wrote on 12/15/2004 10:23:
| Quote: |
"porterboy" <porterboy76@yahoo.com> wrote in message
news:c4b57fd0.0412150037.5615c7c0@posting.google.com...
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?
Hello Porterboy,
Yes analytic functions and analytic signals are related.
I.e.,
An analytic function may be written as
f(t,tau) + j*g(t,tau) where it obeys the C-R relations.
If you let tau==0 in an analytic function you get an analytic signal. And in
this case g(t,0) is the Hilbert transform of f(t,0). And using this you can
easily show that an analytic signal has a one sided transform. (Use the
sigum function in the fourier domain) One of the simplest analytic signals
is cos(t)+j*sin(t).
|
okay, Clay, now i'm confused. maybe it's semantic. first of all, what are
f() and g()? are they complex functions where the complex argument is
z = t + j*tau ? if so, by "C-R", do you mean that df/dtau = dg/dt ? when i
crack a complex variables book, that what i think they mean by "analytic
function". checking wikipedia, it looks like "holomorphic" is the new word
for the same thing, but my Levison and Redheffer book doesn't say
"holomorpic". i can be 25 - 30 years behind the times.
now, assuming i got the semantics right, *why* should that mean that
g(t,0) = Hilbert{ f(t,0) } ?
i don't see the two as related.
Of course the last thing you said is fundamentally true. if
g(t) = Hilbert{ f(t) } ,
then
a(t) = f(t) + j*g(t)
is one-sided.
i've always thought that this term "analytic" has been the victim of
semantics between two related fields: complex variables and
communications/signal_processing . i just accepted the semantic difference
in the same way that i accept the word "sample" means different things
within the audio DSP community, depending on who you're talking to.
so, Clay, please clear this up for me.
thnx.
--
r b-j rbj@audioimagination.com
"Imagination is more important than knowledge." |
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Richard Owlett
Guest
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| Posted:
Wed Dec 15, 2004 10:59 pm Post subject:
Cofused by math notation was [Re: Analytic Functions and |
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Randy Yates wrote:
| Quote: | [snip]
Definitely not! F(z) = cos(x), where z = x + i*y,
is analytic over all C but it is not an analytic
signal.
|
I think I'm missing something. Then again I tended to have to retake my
college math courses ~40 years ago ;{
I can see writing something of the *form*
[ That's not to say I would know what to do with it. ]
F(z) = cos(z)
z = x + i*y
But I don't understand writing
F(z) = cos(x)
z = x + i*y
I think I would write something like
F(x,y) = cos( x + i*y )
Thanks |
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Clay S. Turner
Guest
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| Posted:
Wed Dec 15, 2004 11:28 pm Post subject:
Re: Analytic Functions and Single Side Band Signals |
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Hello Robert,
Comments below:
"robert bristow-johnson" <rbj@audioimagination.com> wrote in message
news:BDE5D304.31EB%rbj@audioimagination.com...
| Quote: | in article BCYvd.640$Z27.454@bignews5.bellsouth.net, Clay S. Turner at
Physics@Bellsouth.net wrote on 12/15/2004 10:23:
"porterboy" <porterboy76@yahoo.com> wrote in message
news:c4b57fd0.0412150037.5615c7c0@posting.google.com...
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?
Hello Porterboy,
Yes analytic functions and analytic signals are related.
I.e.,
An analytic function may be written as
f(t,tau) + j*g(t,tau) where it obeys the C-R relations.
If you let tau==0 in an analytic function you get an analytic signal. And
in
this case g(t,0) is the Hilbert transform of f(t,0). And using this you
can
easily show that an analytic signal has a one sided transform. (Use the
sigum function in the fourier domain) One of the simplest analytic
signals
is cos(t)+j*sin(t).
okay, Clay, now i'm confused. maybe it's semantic. first of all, what
are
f() and g()? are they complex functions where the complex argument is
z = t + j*tau ? if so, by "C-R", do you mean that df/dtau = dg/dt ? when
i
crack a complex variables book, that what i think they mean by "analytic
function". checking wikipedia, it looks like "holomorphic" is the new
word
for the same thing, but my Levison and Redheffer book doesn't say
"holomorpic". i can be 25 - 30 years behind the times.
|
f(t,tau) and g(t,tau) are purely real functions. So the complex function is
built by
psi(t,tau) = f(t,tau) + j*g(t,tau)
The CR relations are:
df/dt = dg/d tau
and
df/d tau = - dg/dt
where the derivatives are partial derivatives.
| Quote: |
now, assuming i got the semantics right, *why* should that mean that
g(t,0) = Hilbert{ f(t,0) } ?
i don't see the two as related.
|
This is not immediately obvious but it may be derived from a neat
application of the Cauchy integral formula that says
f(z_0) = (1/(2pi*j)) * path integral f(z)/(z-z_0) dz
where the path is closed and the singular point is inside of the path. Also
f(z) needs to be analytic (obeys the C-R relations) both on and inside of
the path. You probably recall this when integrating Laurent series.
So now pick a path that has a straight segment along the real axis going
from -R to R and close the path with a semicircle of radius R. However the
straight part also has a little semicircle to hop around the singular point
z_0. (also the singular point is chosen to be on the real axis) When R is
allowed to grow towards infinity, it is helpful to think of the three parts
of the path and their contribution to the overall integral. The outer
semicircle will force its integral's contribution to go to zero. You may
wish to look up Jordan's lemma. So in the limit of large R
There is also a correlary to Cauchy's theorem that concerns using paths that
don't completely enclose the singular point. Basically you just use the
portion of a circle that a path uses around the point to proportion Cauchy'
standard result. For example if a semicircle is used, then instead of the
2pi*j, you just use pi*j.
So using the above path in a way analogous to the Cauchy integral formula we
find (I know many details skipped)
psi(t,0) = (1/(2pi*j)) * principal value integral psi(s)/(s-t) ds
The term "principal value" refers to the nature of the limiting process so
exact cancelation may occur. This type of integral just shows up in Hilbert
transforms.
So from here with a little work one can show that
f(t,0) = (1/pi) principal value integral g(s,0)/(s-t) ds
And apart from the ,0 in the f(t,0) and g(s,0) terms - this is the
definition for a Hilbert transform.
| Quote: |
Of course the last thing you said is fundamentally true. if
g(t) = Hilbert{ f(t) } ,
then
a(t) = f(t) + j*g(t)
is one-sided.
i've always thought that this term "analytic" has been the victim of
semantics between two related fields: complex variables and
communications/signal_processing . i just accepted the semantic
difference
in the same way that i accept the word "sample" means different things
within the audio DSP community, depending on who you're talking to.
so, Clay, please clear this up for me.
|
Having come from a Math background, I learned to associate the term
"analytic function" as obeying the C-R relations, with the usage of saying a
function is "analytic" also meaning the same thing in the context we are
talking about a complex function. Dennis Gabor in 1946 extends the analytic
concept of evaluating the analytic function along just one axis to create a
"analytic signal." Even though I showed the mapping along the t axis, one
may also use the tau axis. So I'm just careful to use the term "analytic
signal" when referring to functions with one-sided Fourier transforms. Using
"analytic" by itself is where one may get into trouble without ensuring
proper clarification.
I hope this helps some. If I get a chance, I may work out all of the gory
details on this and post it on my site. It has been a long time since I
trudged through this. However the result is quite neat and has practical
applications as we all know.
Clay
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Randy Yates
Guest
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| Posted:
Wed Dec 15, 2004 11:32 pm Post subject:
Re: Cofused by math notation was [Re: Analytic Functions |
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Richard Owlett <rowlett@atlascomm.net> writes:
| Quote: | Randy Yates wrote:
[snip]
Definitely not! F(z) = cos(x), where z = x + i*y, is analytic over
all C but it is not an analytic
signal.
I think I'm missing something. Then again I tended to have to retake
my college math courses ~40 years ago ;{
I can see writing something of the *form*
[ That's not to say I would know what to do with it. ]
F(z) = cos(z)
z = x + i*y
But I don't understand writing
F(z) = cos(x)
z = x + i*y
I think I would write something like
F(x,y) = cos( x + i*y )
|
Hey Richard,
It does look odd, doesn't it? But I believe it's
correct.
Note that a complex function F is a mapping from the complex to the
complex, F: A \in C --> B \in C. That is, it maps the set of complex
numbers in its domain, A, to the set of complex numbers in its range,
B (remember domain and range from high school?). So in general a
complex function F(z) = f(x,y) + i*g(x,y), where z = x + i*y.
In my function, f(x,y) = cos(x) and g(x,y) = 0. In your function,
f(x,y) = cosh(y)*sin(x) and g(x,y) = sinh(y)*sin(x) - quite a
different function.
Does that help?
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124 |
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Randy Yates
Guest
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| Posted:
Wed Dec 15, 2004 11:36 pm Post subject:
Re: Cofused by math notation was [Re: Analytic Functions |
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Randy Yates <randy.yates@sonyericsson.com> writes:
| Quote: | [...]
In your function,
f(x,y) = cosh(y)*sin(x) and g(x,y) = sinh(y)*sin(x)
|
Typo: should be
f(x,y) = cosh(y)*cos(x) and g(x,y) = sinh(y)*sin(x)
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124 |
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Richard Owlett
Guest
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| Posted:
Thu Dec 16, 2004 12:02 am Post subject:
Re: Cofused by math notation was [Re: Analytic Functions |
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Randy Yates wrote:
| Quote: | Richard Owlett <rowlett@atlascomm.net> writes:
Randy Yates wrote:
[snip]
Definitely not! F(z) = cos(x), where z = x + i*y, is analytic over
all C but it is not an analytic
signal.
I think I'm missing something. Then again I tended to have to retake
my college math courses ~40 years ago ;{
I can see writing something of the *form*
[ That's not to say I would know what to do with it. ]
F(z) = cos(z)
z = x + i*y
But I don't understand writing
F(z) = cos(x)
z = x + i*y
I think I would write something like
F(x,y) = cos( x + i*y )
Hey Richard,
It does look odd, doesn't it? But I believe it's
correct.
|
I'll accept you as an expert witness -- see below ;)
| Quote: | Note that a complex function F is a mapping from the complex to the
complex, F: A \in C --> B \in C. That is, it maps the set of complex
numbers in its domain, A, to the set of complex numbers in its range,
B (remember domain and range from high school?). So in general a
complex function F(z) = f(x,y) + i*g(x,y), where z = x + i*y.
In my function, f(x,y) = cos(x) and g(x,y) = 0. In your function,
f(x,y) = cosh(y)*sin(x) and g(x,y) = sinh(y)*sin(x) - quite a
different function.
Does that help?
|
In that it seems to prove my lack of comprehension of maths in general.
The next to last time I flunked out of an Ivy League BSEE program, the
"Testing and Guidance Center" said my aptitude/interest profile
resembled 'law' rather than 'engineering' students ;}
So my *INTERESTS* are in one area
my *APTITUDE* in another
Thanks for the feedback.
[ PS once got a job interview solely on the basis that owner of company
wanted to meet someone who had a resume (with verifiable references) as
strange as mine ;] |
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Clay S. Turner
Guest
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| Posted:
Thu Dec 16, 2004 12:19 am Post subject:
Re: Cofused by math notation was [Re: Analytic Functions |
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"Richard Owlett" <rowlett@atlascomm.net> wrote in message
news:10s0uu14q3r1uac@corp.supernews.com...
| Quote: | Randy Yates wrote:
[snip]
Definitely not! F(z) = cos(x), where z = x + i*y, is analytic over all C
but it is not an analytic
signal.
I think I'm missing something. Then again I tended to have to retake my
college math courses ~40 years ago ;{
I can see writing something of the *form*
[ That's not to say I would know what to do with it. ]
F(z) = cos(z)
z = x + i*y
But I don't understand writing
F(z) = cos(x)
z = x + i*y
|
I believe Randy meant to say
F(z) = cos(z)
To show it is analytic (function)
so cos(z) = cos(x+iy)
= cos(x)*cos(iy) - sin(x)*sin(iy)
recall trig functions with imaginary arguments go into hyperspace.
= cos(x)*cosh(y)-i*sin(x)*sinh(y)
so when put into the u(x,y) + j*v(x,y) form
u(x,y) = cos(x)*cosh(y)
and
v(x,y) = -sin(x)*sinh(y)
Now test to see if the Cauchy Riemann relations hold.
du/dx = -sin(x)*cosh(y)
and
dv/dy = -sin(x)*cosh(y)
so du/dx = dv/dy so far so good (CR relation #1 holds)
now
du/dy = cos(x)*sinh(y)
and
dv/dx = -cos(x)*sinh(y)
thus
du/dy = - dv/dx and CR relation #2 holds
so cos(z) is an analytic function.
Clay
| Quote: |
I think I would write something like
F(x,y) = cos( x + i*y )
Thanks |
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Randy Yates
Guest
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| Posted:
Thu Dec 16, 2004 12:30 am Post subject:
Re: Cofused by math notation was [Re: Analytic Functions |
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|
"Clay S. Turner" <Physics@Bellsouth.net> writes:
| Quote: | "Richard Owlett" <rowlett@atlascomm.net> wrote in message
news:10s0uu14q3r1uac@corp.supernews.com...
Randy Yates wrote:
[snip]
Definitely not! F(z) = cos(x), where z = x + i*y, is analytic over all C
but it is not an analytic
signal.
I think I'm missing something. Then again I tended to have to retake my
college math courses ~40 years ago ;{
I can see writing something of the *form*
[ That's not to say I would know what to do with it. ]
F(z) = cos(z)
z = x + i*y
But I don't understand writing
F(z) = cos(x)
z = x + i*y
I believe Randy meant to say
F(z) = cos(z)
|
Hi Clay,
No, I meant to say F(z) = cos(x). The problem is, I
am flat-out wrong. That function is not analytic! I
made the same mistake I've made several times before:
differentiable at all points is not the same as
analytic!
Have mercy on me - I was going from memory: my Churchill
and Brown text is at home. I was able to see my error
from the Cauchy-Riemann conditions you gave in another
post of yours today, Clay.
To Porterboy, you have my sincere apologizes for giving
you bad information.
--
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124 |
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Matt Timmermans
Guest
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| Posted:
Fri Dec 17, 2004 4:44 am Post subject:
Re: Analytic Functions and Single Side Band Signals |
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"Clay S. Turner" <Physics@Bellsouth.net> wrote in message
news:BCYvd.640$Z27.454@bignews5.bellsouth.net...
| Quote: | Yes analytic functions and analytic signals are related.
|
And yet, the word "analytic" means two entirely different things in those
two contexts.
Shame, that.
--
Matt |
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