Loading [MathJax]/jax/output/HTML-CSS/jax.js
Ask Your Question
0

Implementing the basic Fourier-Transformation

asked 12 years ago

Jakob Holderbaum gravatar image

updated 12 years ago

Hi there!

I'm currently plaing around with sage and I'm really excited about it.

I'd love to do my computations at university and home with a neat opensource-tool instead of the higly prices closed competitors.

Now, the problem I am facing is the symbolif computation of a fourier transformation.

Below is my current naive approach (I'm still learning fourier and complex mathematics, but with large interest!)

x,w,f_0,t = var("x,w,f_0,t") 

w = 2*pi*f_0 
x(t) = sin(w*t) 

integrate(x*exp(-I*w*t),t, -oo, oo)

which results in the following (obviously equal) result:

integrate(e^(-2*I*pi*f_0*t)*sin(2*pi*f_0*t), t, -Infinity, +Infinity)

My expectiation would be an equation without the t (since it has been substituted through integration) and an floating f_0 which I can set according to my desired sine frequency.

Please, could someone tell me, what exactly I am missing here?

Greetings Jakob

Preview: (hide)

3 Answers

Sort by » oldest newest most voted
1

answered 12 years ago

Dirk Danckaert gravatar image

Not every function has a wel defined fourier transform, sin(omegat) being an example of those. Try transforming a more 'well-behaved' function, such as exp(-at^2), or 1/(1+t^2). I didn't check, but I bet SAGE will return a correct answer for those.

Preview: (hide)
link
0

answered 12 years ago

koukourikos gravatar image

The integral that you want to calculate it is not so simple because as you can see it evolves the imaginary unit.

I would suggest to define f_0 first and then calculate the integral numerically .

Preview: (hide)
link
0

answered 12 years ago

Jakob Holderbaum gravatar image

So, as you suggested, I added a numerical Definition for f_0, now I'm getting an error:

x,w,f_0,t = var("x,w,f_0,t")

f_0 = 1
w = 2*pi*f_0

x(t) = sin(w*t)

integrate(x*exp(-I*w*t),t, -oo, oo)

This is the actual result:

Traceback (click to the left of this block for traceback)
...
ValueError: Integral is divergent.
Preview: (hide)
link

Comments

numerical_integral returns: TypeError: unable to coerce to a real number

Jakob Holderbaum gravatar imageJakob Holderbaum ( 12 years ago )
1

That's because the integral is actually divergent. The function sin(wt) is not integrable on (,).

benjaminfjones gravatar imagebenjaminfjones ( 12 years ago )

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

Stats

Asked: 12 years ago

Seen: 2,072 times

Last updated: Oct 25 '12