Fourier transform of a compicated function

[sphys]

Hi

Could someone help me to calculate the Fourier transform of the following function:

rect(x/d)exp(2ipia|x|)

 

[mathman]

What is rect(x/d)?

 

[sphys]

rect(x/d) is a rectangle function.
rect(x/d)=1 if -d/2<x<d/2;
rect(x/d)=0 if x<-d/2 or x>d/2.

 

[mathman]

It looks like a straightforward integration.
∫(x=-d/2,0)exp(-2πiax+itx) dx + ∫(x=0,d/2)exp(2πiax+itx) dx

 

[sphys]

Instead of numerical solution, is there an analytical solution for this problem?

 

[vela]

That's not a numerical solution.

You could try using the convolution theorem to find the Fourier transform, but that seems like even more work.

 

[sphys]

Can you please give me the solution using the convolution theorem?

 

[vela]

No, that's against the forum rules. Conceptually, it's straightforward. The convolution theorem tells you

$$\mathcal{F}[f(x)g(x)]=\mathcal{F}[f(x)]*\mathcal{F}[g(x)]$$

so you just have to find the transforms of the rectangle and exponential functions individually and convolve the results.