Differentials

Cool Things You Can Do With Derivatives

I mentioned earlier that f'(t) transforms to jω F(ω). This allows us to do Fourier transforms on differential equations in much the same way as we do for Laplace transformations. First, however, let's look at what we can do with differentials.

Take the triangle function from before:

Now, if we differentiate it, we get a strange-looking equation with unit steps in it:

Differentiating again gives us something stranger still: a function with Dirac impulse functions in it. These are infinitely narrow, infinitely high spikes.

The area of each δ(t) is 1, and hence they Fourier transform to 1. This allows us to Fourier transform this function.

Wow! That was really easy, wasn't it?