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scholzie
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Homework Statement
Take the Fourier Transform of f(x)=rect(x/2)*comb(x) where rect is the rectangle function and comb is the Dirac comb. Sketch the results.
Homework Equations
The FT of a convolution is the product of the individual FTs.
The Attempt at a Solution
Taking the FT is pretty simple, but it's the graph that I'm a little confused about. If we take the FT of f(x)<->F(u)
F(u)=2sinc(2u)comb(u), if we define sinc(u)=sin(pi u)/(pi u), rather than the other way which leaves out pi.
The graph of such a function is simply the sampling of points along 2sinc(2u) at the period of comb(u), which should just be 1. Therefore at u=0, you'd have a point of 2 (although 2sinc(2u) isn't defined at u=0, though I believe it's commonly considered to be defined as its limit, which in this case is 2). All of the other integer values of 2sinc(2u) are simply 0.
Am I correct in assuming that the graph of 2sinc(2u)comb(u) is just {2,0,0,...,0} for all n>=0?
This seems like a silly answer for a problem, but maybe that's the point. I just have a tendency to second guess myself when given a stupid problem with a simple answer.