The sum, the difference, and both original signals

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With frequency mixers in radios they multiply frequencies to create an intermediate frequency, but where in the wave are those? Every picture I’ve seen makes the two original signals look pretty obvious, but I don’t get where the sum and the difference are.

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The sun and difference signals (and sometimes one or both of the inputs) exist within the output. It isn’t a clean sinusoid, so that’s why the waves aren’t drawn. It’s typically drawn in frequency space instead

The result of multiplication is the sum of 2 sinusoidal waves with equal amplitude, one with sum frequency and one with difference frequency, so it’s not easy to see the frequency from the waves. But you can definitely see them if you look carefully. If the lower frequency is much lower than the higher frequency, then the signal will have much longer wave length, so the overall shape will look like something with an overall sinusoidal wave trend with a lot of rapid oscillation. But if the 2 frequencies are too similar, you end up with a “beat” which has very misleading frequency and it’s really hard to tell what are the component frequencies.

When you multiply two sinusoids, the result in general is NOT a sinusoid. It will be obviously repetative and may superficially resemble one sinusoid modulated by the other, but it is not.

It is tough for us to see frequencies in waveforms that are not a pure sinusoid even if they are there.

A square wave contain all odd harmonics of a fundemental sine wave, and yet it looks square. Those other frequencies are there, but your eye cannot see them.

You can run the multiplied signals through a fourrier transform to reveal the presence of the sum and difference frequencies.

Alternatively you can pass the multiplied signal through a bandpass filter at the sum/difference frequency. You should see a nice clean sinusoid.