# Fourier transformation

1. May 23, 2005

### Henk

For an experiment about Holograms I have to find a couple of Fourier transformations. The one I'm having troubles with is the following:

Find the Fouriertransform of a rectangular wave:
thus: f(x)=1 from (-5b,-3b), (-b,b) and (3b,5b) and (7b,9b) etc.
and f(x) = -1 from (-3b,-b), (b,3b), (5b,7b) etc.

Could someone give me a hint?

2. May 23, 2005

where A(x) is a the rectangular wave shifted up by one (so it goes from zero to two), and B(x) = -1. This helps because F(f(x)) = F(A(x)+B(x)) = F(A(x)) + F(B(x)), where F( ) is the Fourier transform. Now your rectangular wave is a train of boxes, which can also be thought of as a single box convolved with an impusle train, or A(x) = A1(x)$$\star$$A2(x). And since convlolution in one domain is multiplication in the other, you now have
F(f(x)) = F( A1(x)$$\star$$A2(x) ) + F(B(x)) = F(A1(x))F(A2(x)) + F(B(x))