How Do You Calculate the Fourier Transform of a Rectangular Wave?

[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?

 

[LeBrad]

I would do this by breaking it up into simpler pieces that you know the transforms of. Let your rectangular wave centered at zero be f(x), and say

f(x) = A(x) + B(x)

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))

Now you are left with all simple transforms, and it looks like you will end up with a sampled sinc function with an impulse somewhere.

 

[thepatientmental]

Sure, I can provide some guidance for finding the Fourier transform of a rectangular wave. First, it's important to understand that the Fourier transform is a mathematical tool used to convert a function from its original domain (in this case, position or space) to its frequency domain. This means that instead of representing the function in terms of position, it will be represented in terms of frequency.

To find the Fourier transform of a rectangular wave, you can start by writing out the definition of the Fourier transform:

F(k) = ∫f(x)e^(-2πikx)dx

Where F(k) is the Fourier transform of f(x), k is the frequency, and f(x) is the function in the original domain (in this case, the rectangular wave function).

Next, you can plug in the given function of the rectangular wave into this definition and solve the integral. Since the function is defined piecewise (1 for certain intervals, -1 for others), you will need to break up the integral into smaller intervals and solve them separately. For example, for the first interval of (x=-5b to x=-3b), the integral would be:

∫1*e^(-2πikx)dx from -5b to -3b

You can use basic integration techniques to solve this integral, and then do the same for the other intervals. Once you have solved the integral for all the intervals, you can combine them to get the final Fourier transform of the rectangular wave function.

I hope this helps and gives you a good starting point for finding the Fourier transform of a rectangular wave. Remember, practice makes perfect, so don't be discouraged if it takes some time to get the hang of it. Good luck with your experiment on holograms!