What is the Cross-Correlation Theorem for Continuous Fourier Transform?

[T.Engineer]

can anybody find the result for the following equation:

F(d)= $$\int^{T_f}_{0}p(t)p(t-d) dt$$

where
916; = d but it doesn't appears very well
and
p(t) = (-1)^n * e^(t^2) * d/dt * e^(-t^2)

thanks a lot!

 

[dextercioby]

The way you've written, it appears that

$$p(t)=-2 (-1)^n t$$

I believe that in this case the integral would be a formality.

 

[T.Engineer]

I am sorry I didnt write it correctly

p(t) = (-1)^n * e^(t^2) * d^n/dt^n * e^(-t^2)

where n= 1,2,...,N

I just want a general formula for the result

 

[EnumaElish]

Have you attempted at an answer? What technique(s) have you tried? Integration by parts comes to mind...

 

[T.Engineer]

my question is how to find a general formula for the following

F(d)= $$\int^{T_f}_{0}p(t)p(t-d) dt$$

where
p(t) = (-1)^n * e^(t^2) * d/dt * e^(-t^2)

and
n=1,2,...,N
Thanks a lot!

 

[EnumaElish]

And my response is, have you tried integration by parts? http://en.wikipedia.org/wiki/Integration_by_parts

 

[matt grime]

And my response is 'why do you keep putting n's in and then taking them out?' State the question precisely. I presume you're just supposed to do it for n an integer, rather than for n=1,2,3,...

 

[dextercioby]

That p_{n} is, up to a normalization constant, a Hermite polynomial of degree "n". So try looking that integral in mathematical tables of integrals or ask Mathematica software about the result.

 

[EnumaElish]

"There are many functions - called special functions - which fail to
have an anti-derivative expressible as a finite combination of
elementary functions. The so-called elliptic functions, the error
function, and the gamma function are a few examples. The error
function, which is extremely useful in both physics and statistics, is
defined as:

erf(x) = (2/sqrt(pi))integral from 0 to x of e^(-t^2)dt

Extensive tables of the error function would not exist if the
anti-derivative of e^(-t^2) were expressible as a finite combination
of elementary functions."
http://mathforum.org/library/drmath/view/53554.html

 

[Dick]

"There are many functions - called special functions - which fail to
have an anti-derivative expressible as a finite combination of
elementary functions. The so-called elliptic functions, the error
function, and the gamma function are a few examples. The error
function, which is extremely useful in both physics and statistics, is
defined as:

erf(x) = (2/sqrt(pi))integral from 0 to x of e^(-t^2)dt

Extensive tables of the error function would not exist if the
anti-derivative of e^(-t^2) were expressible as a finite combination
of elementary functions."
http://mathforum.org/library/drmath/view/53554.html

p(t) is not an exponential. It's a polynomial as dextercioby has already pointed out.

 

[EnumaElish]

I have realized the same because, as dextercioby had pointed out, there is a closed solution for n=1.

Here's how I'd approach the problem (and I communicated this to T.Engineer at least once before, under Statistics & Probability). I'd start with n=1 and calculate the closed solution, which is easy. Then move on to n=2, 3, ..., and see if there is a pattern.

 

[Dick]

I have realized the same because, as dextercioby had pointed out, there is a closed solution for n=1.

Here's how I'd approach the problem (and I communicated this to T.Engineer at least once before, under Statistics & Probability). I'd start with n=1 and calculate the closed solution, which is easy. Then move on to n=2, 3, ..., and see if there is a pattern.

There are also recurrence relations for the Hermite polynomials which could be exploited without rediscovering them empirically. It's clear there is no exponential in the final integral since exp(-t^2) comes out from the differentiation intact and cancels with the exp(t^2) without even knowing there are closed form solutions for particular n.

 

[EnumaElish]

There are also recurrence relations for the Hermite polynomials which could be exploited without rediscovering them empirically. It's clear there is no exponential in the final integral since exp(-t^2) comes out from the differentiation intact and cancels with the exp(t^2) without even knowing there are closed form solutions for particular n.

Right; the fact that a closed solution exists is sufficient but not necessary to see that it is not the error function, or any other special function.

 

[Dick]

If anybody is interested in pursuing this, here's a Maxima (free math software) program to compute the nth case. It should be pretty self explanatory.

p(t,n):=expand(exp(t^2)*(-1)^n*diff(exp(-t^2),t,n));
integ(n):=p(t,n)*subst((t-d),t,p(t,n));
final(n):=integrate(integ(n),t,0,Tf);

E.g. typing 'final(7);' after this is entered computes the n=7 case.

 

[T.Engineer]

If anybody is interested in pursuing this, here's a Maxima (free math software) program to compute the nth case.

Please, can you tell me from where can I download this program.
Thanks a lot!

 

[EnumaElish]

http://sourceforge.net/project/showfiles.php?group_id=4933

 

[Dick]

Please, can you tell me from where can I download this program.
Thanks a lot!

Try http://maxima.sourceforge.net/. If you are running a debian flavor linux it might be as simple doing 'apt-get install xmaxima'.

 

[T.Engineer]

Try http://maxima.sourceforge.net/. If you are running a debian flavor linux it might be as simple doing 'apt-get install xmaxima'.

I am runing windows Xp.

 

[Dick]

I am running windows Xp.

Then get, for example, maxima-5.12.0a.exe from the downloads section.

 

[T.Engineer]

If anybody is interested in pursuing this, here's a Maxima (free math software) program to compute the nth case. It should be pretty self explanatory.

can you help to find a general formula for the autocorrelation function Hermite polynomials.
Thanks a lot!

 

[EnumaElish]

See this:

http://en.wikipedia.org/wiki/Continuous_Fourier_transform#Cross-correlation_theorem