Science > Physics > Without solving ODE, is there a way to see the spectrum of a signal?
| Topic: |
Science > Physics |
| User: |
"Mike" |
| Date: |
19 May 2007 01:26:32 AM |
| Object: |
Without solving ODE, is there a way to see the spectrum of a signal? |
Hi all,
I have a signal which is embedded in a complicated ODE, non-linear, first
order.
I want to determine its bandwidth(end goal).
If I want to see the spectrum of the signal, I have to solve the ODE
numerically. I am worried about the possible numerical instability of the
numerical solution of ODE, let's say, using Matlab. There is no closed-form
solution, so there is no way to test if the solution has a large
approximation error or not. How do people handle this problem in practice?
Moreover, is there a way to see the spectrum of the signal without solving
the ODE?
Further, is it possible to determine the bandwidth without solving the ODE
and finding the spectrum first?
I hope there are some theorems around that can link a signal represented by
exp(f(t)),
where f(t) is represented by an ODE with initial value.
Given such an "exp(f(t))", I want to determine its bandwidth(or a reasonably
good approximation).
Is it possible, and how to do that?
Thanks a lot!
.
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| User: "Fred Marshall" |
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| Title: Re: Without solving ODE, is there a way to see the spectrum of a signal? |
19 May 2007 10:11:42 AM |
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"Mike" <meatheadIV@gmail.com> wrote in message
news:f2m5aj$f4g$1@news.Stanford.EDU...
Hi all,
I have a signal which is embedded in a complicated ODE, non-linear, first
order.
I want to determine its bandwidth(end goal).
If I want to see the spectrum of the signal, I have to solve the ODE
numerically. I am worried about the possible numerical instability of the
numerical solution of ODE, let's say, using Matlab. There is no
closed-form solution, so there is no way to test if the solution has a
large approximation error or not. How do people handle this problem in
practice?
Moreover, is there a way to see the spectrum of the signal without solving
the ODE?
Further, is it possible to determine the bandwidth without solving the ODE
and finding the spectrum first?
I hope there are some theorems around that can link a signal represented
by exp(f(t)),
where f(t) is represented by an ODE with initial value.
Given such an "exp(f(t))", I want to determine its bandwidth(or a
reasonably good approximation).
Is it possible, and how to do that?
Thanks a lot!
Saying that it's first order AND complicated tells me that the nonlinearity
is complicated. The solution to the ODE isn't I should think. So, I'm
puzzled as to why you're concerned about "stability".
I don't know what you mean that a signal is "embedded". I think you mean
"applied" to the system as a forcing function, right?
There are probably some things you could consider. One is the bandwidth of
the forcing function (the "signal"). The resulting output of the system
won't be less that that except to the degree that the system is something
like, say, a lowpass filter.
Here's a simple model or two:
An RC lowpass followed by a diode will eventually act like a linear system
as the input frequency gets higher.
An RC lowpass following a diode will likely have a higher bandwidth output
because the nonlinearity applies to all signals first and increases the
higher frequency energy going into the "filter".
So, the bandwidth depends on how the nonlinearity applies to the input.
You might do something like this:
Assume a very high level input amplitude with the objective of simplifying
the nonlinearity if possible. Then solve for that simpler case maybe in
closed form. That might be the worst case in terms of bandwidth.
I'd just go solve the thing numerically and be done with it. You've not
explained the reason for your concerns. Just remember that the result will
be dependent on the amplitude of the forcing function - unlike for linear
systems.
Fred
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| User: "Mike" |
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| Title: Re: Without solving ODE, is there a way to see the spectrum of a signal? |
19 May 2007 12:56:09 PM |
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Hi Fred,
"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:uoGdnV-iHqo5jtLbnZ2dnUVZ_jadnZ2d@centurytel.net...
Saying that it's first order AND complicated tells me that the
nonlinearity is complicated. The solution to the ODE isn't I should
think. So, I'm puzzled as to why you're concerned about "stability".
Let's say the signal is f(t).
I don't know what you mean that a signal is "embedded". I think you mean
"applied" to the system as a forcing function, right?
By "embedded" I meant f(t) satisfies, or f(t) is mandated by the ODE:
f'(t) =c1* f(t) + c2*F(f(t)) + c3
where F(x) = exp(c4+c5*x+c6*x^2 + exp(x)), just give an example.
What is the filter of this system?
I'd just go solve the thing numerically and be done with it. You've not
explained the reason for your concerns. Just remember that the result
will be dependent on the amplitude of the forcing function - unlike for
linear systems.
Of course I can just rest with a numerical solution, however, I have many
past experience that depending on the parameters, the numerical results were
errorneous...
I guess my first question is given a numerical solution, and knowing that we
won't be able to obtain closed-form solution to do the sanity-check, how do
I make sure the numerical solution is reliable?
.
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| User: "Rune Allnor" |
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| Title: Re: Without solving ODE, is there a way to see the spectrum of a signal? |
19 May 2007 11:09:41 AM |
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On 19 May, 08:26, "Mike" <meathea...@gmail.com> wrote:
Hi all,
I have a signal which is embedded in a complicated ODE, non-linear, first
order.
What does it mean that a signal is "embedded in an ODE"?
I know what a signal embedded in noise is, but "embedded
in an equation" needs further explanation.
I want to determine its bandwidth(end goal).
You can't. The term "bandwidth" only applies to systems
which can be expressed in terms of Helmholtz' equation,
which is linear. If your system is nonlinear, it can't be
expressed in terms of Helmholtz' equation.
If I want to see the spectrum of the signal, I have to solve the ODE
numerically. I am worried about the possible numerical instability of the
numerical solution of ODE, let's say, using Matlab. There is no closed-form
solution, so there is no way to test if the solution has a large
approximation error or not. How do people handle this problem in practice?
They spend huge amounts of time and resources. This is
specialist's work.
Moreover, is there a way to see the spectrum of the signal without solving
the ODE?
No. Once you have the spectrum of a linear ODE, you are
just one Fourier transform away from the full solution.
Finding the spectrum of the linear ODE is equivalent to
solving the ODE.
Further, is it possible to determine the bandwidth without solving the ODE
and finding the spectrum first?
No. And again, the ODE has to be linear for the term
"bandwidth" to make sense.
Given such an "exp(f(t))", I want to determine its bandwidth(or a reasonably
good approximation).
This wouldn't be a Frequency Modulation problem? The methods
for analysing such problems are well-known (at least under
certain constraints), but hardly give the closed-form answers
you apparently want.
Why don't you state the task you have been assigned in
as exact terms as possible? State what your starting
point and objective are, and explain the context where
this problem has appeared.
Rune
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| User: "Mike" |
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| Title: Re: Without solving ODE, is there a way to see the spectrum of a signal? |
19 May 2007 01:07:22 PM |
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"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:1179590981.771741.120410@l77g2000hsb.googlegroups.com...
What does it mean that a signal is "embedded in an ODE"?
I know what a signal embedded in noise is, but "embedded
in an equation" needs further explanation.
By "embedded" I meant f(t) satisfies, or f(t) is mandated by the ODE:
f'(t) =c1* f(t) + c2*F(f(t)) + c3
where F(x) = exp(c4+c5*x+c6*x^2 + exp(x)), just give an example.
What is the filter of this system?
I want to determine its bandwidth(end goal).
You can't. The term "bandwidth" only applies to systems
which can be expressed in terms of Helmholtz' equation,
which is linear. If your system is nonlinear, it can't be
expressed in terms of Helmholtz' equation.
Why? Any reasonably well-behaved signal should have a bandwidth. We are not
talking about system, we are talking about signal. The ODE is non-linear,
but I am not sure about the system filter yet. Are you saying that a singal
filtered by a non-linear system doesn't have a bandwidth?
If I want to see the spectrum of the signal, I have to solve the ODE
numerically. I am worried about the possible numerical instability of the
numerical solution of ODE, let's say, using Matlab. There is no
closed-form
solution, so there is no way to test if the solution has a large
approximation error or not. How do people handle this problem in
practice?
They spend huge amounts of time and resources. This is
specialist's work.
That's why I am asking...
Moreover, is there a way to see the spectrum of the signal without
solving
the ODE?
No. Once you have the spectrum of a linear ODE, you are
just one Fourier transform away from the full solution.
Finding the spectrum of the linear ODE is equivalent to
solving the ODE.
I don't have the linear ODE, as I said... That's too bad...
Further, is it possible to determine the bandwidth without solving the
ODE
and finding the spectrum first?
No. And again, the ODE has to be linear for the term
"bandwidth" to make sense.
Given such an "exp(f(t))", I want to determine its bandwidth(or a
reasonably
good approximation).
This wouldn't be a Frequency Modulation problem? The methods
for analysing such problems are well-known (at least under
certain constraints), but hardly give the closed-form answers
you apparently want.
But the signal f(t) is mandated by the complicated non-linear ODE... Even I
have the solution to the ODE, I still don't see how is it related to the
Frequency Modulation problem.
--------------------
Let's say I simplify the problem a little bit for the moment:
f'(t) =c1* f(t) + c2*F(f(t)) + c3
where F(x) = exp(c4+c5*x),
For this signal f(t), if I want to look at it from the perspective of a
non-linear system, what is the input signal, and what is the system?
There is no way for me to estimate the bandwidth of f(t), I just need a
upperbound?
I am thinking of doing some Taylor expansion to get an infinite series of
F(x) and then it becomes a linear system I guess...?
Thanks!
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| User: "Rune Allnor" |
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| Title: Re: Without solving ODE, is there a way to see the spectrum of a signal? |
19 May 2007 01:20:20 PM |
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On 19 May, 20:07, "Mike" <meathea...@gmail.com> wrote:
"Rune Allnor" <all...@tele.ntnu.no> wrote in message
news:1179590981.771741.120410@l77g2000hsb.googlegroups.com...
What does it mean that a signal is "embedded in an ODE"?
I know what a signal embedded in noise is, but "embedded
in an equation" needs further explanation.
By "embedded" I meant f(t) satisfies, or f(t) is mandated by the ODE:
f'(t) =c1* f(t) + c2*F(f(t)) + c3
where F(x) = exp(c4+c5*x+c6*x^2 + exp(x)), just give an example.
What is the filter of this system?
How long is a string? The question is impossible to answer
without you stating where this signal originates and what
you try to use it for.
I want to determine its bandwidth(end goal).
You can't. The term "bandwidth" only applies to systems
which can be expressed in terms of Helmholtz' equation,
which is linear. If your system is nonlinear, it can't be
expressed in terms of Helmholtz' equation.
Why? Any reasonably well-behaved signal should have a bandwidth.
Yes. The key is the term "reasonably well-behaved".
For all intents and purposes of practical signal analysis,
is a synonym for "linear."
We are not
talking about system, we are talking about signal. The ODE is non-linear,
but I am not sure about the system filter yet. Are you saying that a singal
filtered by a non-linear system doesn't have a bandwidth?
A "signal" is nothing more than a sequence of numbers,
and you can do what you want with them: Compute the DFT
or fill them in your tax return form.
The terms "linear" and "nonlinear" become important when
you connect the "signal" and the "system", which you say
is nonlinear. If you want to use the signal to learn
something aboout the system, then the usual tools of the
trade are based on the assumption that the system is
linear. If it isn't, you basically are on your own.
Rune
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