Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions)

manpreet Best Answer 2 years ago

I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting question to me that I doubt the professor will answer (if I wasn't ashamed to ask).

I am asked to calculate the Fourier transform of the convolution of two signals, for generality:

F{sin3(at+b)∗cos3(ct+d)}F{sin3⁡(at+b)∗cos3⁡(ct+d)}.

I have tried two approaches.

First, take the product of the Fourier transforms of the sinusoids. This leads to an expression that contains terms of the form δ(ω−a)δ(ω−b)δ(ω−a)δ(ω−b). According to [1] and unless I missed it, the product of two distributions, unlike other operations, is not defined.

Secondly calculate the convolution directly. This leads me to an integral of the form:

∫∞−∞sin3(aτ+b)cos3(c(τ−t)+d)dτ∫−∞∞sin3⁡(aτ+b)cos3⁡(c(τ−t)+d)dτ

This also I think is non-convergent.

So am I right to think that this convolution and its Fourier transform are not defined?

[1] Zemanian: Distribution Theory and Transform Analysis

manpreet 2 years ago