Modulation

Modulating, or, A Multiplication By Any Other Name

From the previous section, you may remember that:

• The Fourier transform of a product x(t) y(t) is (1 / 2π) X(ω) * Y(ω).

This is an important result, as it allows us to find the Fourier transform of any product (assuming you can convolve the individual transforms!). We call this process modulation.

Let's Go Over That Again

Let's take our important frequency-shifting example and do it with modulation instead.

If we take the product of our function f (t) and an infinite series of Dirac impulse functions, we get the expression

Now, we should be able to transform this directly from the property of modulation:

This is a rather nasty convolution at first sight, since we don't really have a clue what f (t) (or F(ω)) is yet... However, all is not lost, as the convolution integral is only defined for ω = k ω_s, where k is integer. Therefore, we get a sum of F(ω ± k ω_s ), which is easy to represent graphically:

 

And you should be able to see that this is the same as the previous example (it's the same function, after all!).

I still haven't told you why this is important. Don't worry, I'll get to that...