Fourier's Song (Table 4.1: Properties of the Fourier Transform)

Bill Sethares
Music by Bill Sethares Lyrics by Bob Williamson

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!

Sine x on x is handy, let's call it a sinc.
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.