Solving Differential Equations

The Continuous Solution

You might remember using Laplace transforms to solve differential equations; well, now we're going to do something similar with Fourier transforms to solve them.

If I have a differential equation of the form

then this will transform to

which we can then factorise to

Oh look. We now have a transfer function in terms of jω that is very similar to what we would get from Laplace transforms. We can now apply the standard techniques we used in the Electronics lectures last year to work out the gain and phase response of the system. This is nothing new - except that now we can apply it to functions other than simple sinusoidal waves.

Next, we will look at a discrete-time variant of Fourier transforms - the discrete Fourier transform (DFT).