Filters continued .

How do these filters work? That is, what does multiplication of two signals in the frequency domain imply in the time domain?

If we multiply two Fourier transforms X(f) and H(f), let us see what the Inverse Fourier transform of this product is.

Consider the integral

Let us replace H(f) by

This makes the integral,   

We can interchange the order of integration, so long as the new double integral converges

we note that the term inside the bracket is just the inverse Fourier transform of X(f) evaluated at

Thus the integral simplifies to which is simply the convolution of h(t) with x(t) !