Fourier Transform of continued .

Now,

But then, we have seen that can be non-zero only when k is a multiple of n, and can be non-zero only when k is a multiple of m. Their product can clearly be non-zero only when k is a multiple of m and n. Thus if p is the LCM (least common multiple) of m and n, we have:

What can we make out of this?

The Fourier Transform of the circular convolution has impulses at all (common) frequencies where the Fourier transforms of x(t) and h(t) have impulses. The circular convolution therefore "picks out" common frequencies, at which the spectra of x(t) and h(t) are non-zero and the strength of the impulse at that frequency is the product of the strengths of the impulses at that frequency in the original two spectra.

This result is the equivalent of the Convolution theorem in the context of periodic convolution.