Shifting and Scaling

Building A Time [Reversal] Machine

One of the more interesting properties of the Fourier transform is what happens when you go backwards in time (cue BBC Radiophonic Workshop). Time reversal is indicated by a # superscript; therefore

The Fourier transform of this is simply

being the complex conjugate of F. Now, that was easy, wasn't it?

Oh, so you want to know why? Well, if you reverse time, all that happens in the Fourier integral is the e ^-jωt term becomes e ^{-jω (-t)} = e ^jωt, which is merely the complex conjugate of e ^-jωt. Therefore, provided f (t) is real, the Fourier integral will be the complex conjugate.

Scaling Time

Surely, you say, we can extend this idea to scaling time by anything at all? Well, we can.

All that we need to do is make the substitution u = k t, and du = k dt, in the Fourier integral:

That was easy, wasn't it?

Frequency Shifting: Take Me Higher

Going the reverse way - scaling frequency - is equally simple:

A Frequency Shifting Example

An example would be cool here, and in fact this example has great importance (but not just yet...)

Let's multiply an arbitrary function f (t) by an infinite series of Dirac impulse functions. (Why? I hear you ask. Well, be patient and all will become clear.) If we now replace the impulse train by its Fourier series:

What we've done is replace the function with its Fourier series expansion. If you work out the expansion of that infinite series of delta functions (remember that that function is periodic), you get that function. Note that it is not replacing each delta function with an exponential - it is merely replacing a sum to infinity over time of delta functions, with a sum to infinity over frequency of exponentials.

Now, each of these terms is of the form e^-jωt f(t), which is a frequency shift giving F(ω - W ):

We therefore have F(ω) repeated to infinity at intervals of 2π / T_s, and hence the frequency spectrum is repeated at intervals of 2π / T_s on the frequency axis.

Later on, we'll do this with modulation, just to compare the two approaches.