Friday, February 28, 2020

Time Frequency Duality

An particularly interesting characteristic of Fourier transforms is time-frequency duality. This duality exposes a beautiful deep symmetry between the time and frequency domains of a signal.

For example, a sinusoid in the time domain is an impulse in the frequency domain, and vice versa.

Here's what a 1-second 20hz sine wave looks like. If you play this on your audio device, you'll hear a 20hz tone.

20hz Sine Wave

When you take the Fourier transform of the wave, and plot the frequency domain representation of the signal, you get an impulse in the bin representing the 20hz. (Ignore the tiny neighbours for now.)

Frequency Domain of 20hz Sine Wave

If you play this transformed representation out to your audio device, you'll hear a click, generated from the single impulse pushing the speaker's diaphragm. This is effectively an impulse signal.

Okay, let's create an impulse signal by hand -- a string of zeros, with a 1 somewhere in the middle. Play this on your speaker, and, again, you'll hear a click. This signal is no different from the the previous transformed signal, except for maybe the position of the impulse.

So, check this out. If you take the the FFT of the impulse and plot the frequency domain representation, you get... a sinusoid!

This works both ways. You can the the inverse FFT of a sine wave in the frequency domain, to produce an impulse in the time domain.

Inverse Fourier Transform of a Sine Wave

This is a wonderfully striking phenomenon, which I think reveals a lot about our perception of nature.

For example, here's another property of time-frequency duality -- convolutions in the time domain are multiplications in the frequency domain, and vice versa. Because multiplications require far fewer operations than convolutions, it's much simpler to operate on frequency domain representations of signals.

Your inner ear consists of lots of tiny hairs that vary in thickness and resonate at different frequencies sending frequency domain representations of sound to your brain -- i.e., your ear evolved a little DSP chip in it to make it easier on your brain.

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