Sunday, May 17, 2020

Heterodyne

Frequency modulation (FM) is one of the techniques used to transmit audio signals at radio frequencies. If we "multiply" a sinusoidal signal wave with a sinusoidal carrier wave, the result is two waves at frequencies just above and just below the original carrier.

By way of an example let's try and transmit a 440 Hz tone using a 10 kHz carrier.

\(signal(t) * carrier(t) = mixed(t)\)

The mixed result will then be the superposition of a 10,000 Hz - 440 Hz and a 10,000 Hz + 440 Hz wave.



We can un-mix the mixed wave by applying the same function to our pair of tones at 9,560 Hz and 10,440 Hz; doing so will create another pair of tones: one for each input. Two of those tones will be very high frequency and almost inaudible; the other two will be centered around 0 Hz. Originally, the tone at 9,560 Hz would be a kind of "mirror image" of our original tone (though this distinction isn't readily apparent for a single frequency), but after the second multiplication, it's now the "mirror image of a mirror image" so it's the right way around. The tone that was at 10,440 Hz when modulated will now be at -440 Hz and inaudible.

As a second example, let's say the signal is an audio file containing many frequencies between 20 Hz and 20 kHz. We know from the superposition property of waves that we can decompose our song into 19,980 different signals (yes, there will be some rounding errors if we only allow integer values, but hear me out). And let's chose a slightly higher frequency carrier of 100 kHz. Our lowest frequency of 20 Hz will now be mixed to 99,980 Hz and 100,020 Hz while our highest frequency will be mixed to 80,000 Hz and 120,000 Hz. The signal frequencies in between the lowest and highest will then fall into the two bands when mixed. One band will appear to have the "mirror image" of our original signal: that is to say the higher signal frequencies will now occupy lower mixed frequencies and the lower signal frequencies will now occupy higher mixed frequencies. The other band will contain the same information as the signal but simply shifted to a higher base frequency.