↩️

The last two words in the above recorded sentence are inaudible, part of the ultrasound. They have been encoded using ring modulation.

We modulate the amplitude of the audio by a sinusoid at the Nyquist frequency. The phase is ¼π, in order that the signal doesn't disappear (i.e alias to zero) when we sample it at our sample rate.

You can be absolutely sure this is possible by selecting a sample rate over 80kHz for export: that's enough room for 20kHz in the audio band, and another 20kHz in the ultrasound band.

We can then demodulate using the exact same operation. Here is code to do it in Audacity, using the Nyquist console:

(mult *TRACK* (hzosc (/ *sound-srate* 2.0) *table* 90.0))