Music by Bill Sethares

Integrate your function times a complex
exponential

It's really not so hard you can do it with your pencil

And when you're done with this calculation

You've got a brand new function - the Fourier Transformation

What a prism does to sunlight, what the ear does to sound

Fourier does to signals, it's the coolest trick around

Now filtering is easy, you don't need to convolve

All you do is multiply in order to solve.

From time into frequency - from frequency to time

Every operation in the time domain

Has a Fourier analog - that's what I claim

Think of a delay, a simple shift in time

It becomes a phase rotation - now that's truly sublime!

And to differentiate, here's a simple trick

Just multiply by J omega, ain't that slick?

Integration is the inverse, what you gonna do?

Divide instead of multiply - you can do it too.

From time into frequency - from frequency to time

Let's do some examples... consider a sine

It's mapped to a delta, in frequency - not time

Now take that same delta as a function of time

Mapped into frequency - of course - it's a sine!

Its Fourier Transform is simpler than you think.

You get a pulse that's shaped just like a top hat...

Squeeze the pulse thin, and the sinc grows fat.

Or make the pulse wide, and the sinc grows dense,

The uncertainty principle is just common sense.