In radio electronics, one of the first concept to learn is frequency conversion by using a mixer. And it's strongly emphasized that the mixer is a multiplier, not an adder used in audio circuitry. The description is usually something like the following…
Heterodyning
Adding or summing two sine waves of different frequencies (f1 and f2) combines their amplitudes without affecting their frequencies. Viewed with an oscilloscope, adding two signals appears as a simple superimposition of one signal on the other. Viewed with a spectrum analyzer, adding two signals just sums their spectra.
Multiplying two sine waves of different frequencies produces a new output spectrum. Viewed with an oscilloscope, the result of multiplying two signals is a composite wave that seems to have little in common with its components. A spectrum-analyzer view of the same wave reveals why: The original signals disappear entirely and are replaced by two new signals — at the sum and difference of the original signals’ frequencies. – ARRL Handbook for Radio Communication
And we have,
$$ \sin (2 \pi f_1 t)\sin (2 \pi f_2 t) = \frac{1}{2}\cos [2 \pi (f_1 – f_2) t] – \frac{1}{2}\cos [2 \pi (f_1 + f_2) t] \ $$
Most books also emphasize that
Frequency conversion (or heterodyning, or amplitude modulation) is impossible in a LTI system.
These made perfect sense to me, until I revisit the description of acoustic beats and beating frequency in most physics textbooks… It is said that if you add two sound wave together, the result is an amplitude modulated sound wave due to constructive and destructive interference of waves.
Beating
Suppose […] that we have two waves, […] From one source, let us say, we would have \$ \cos{\omega_{1} t} \$, and from the other source, \$ \cos{\omega_{2} t} \$, where the two ω’s are not exactly the same. […] Let us first take the case where the amplitudes are equal. Then the total amplitude at P is the sum of these two cosines. If we plot the amplitudes of the waves against the time, […] we see that where the crests coincide we get a strong wave, and where a trough and crest coincide we get practically zero, and then when the crests coincide again we get a strong wave again.
On this basis one could say that the amplitude varies at the frequency \$ \frac{1}{2}(\omega_1−\omega_2) \$, but if we are talking about the intensity of the wave we must think of it as having twice this frequency. That is, the modulation of the amplitude, in the sense of the strength of its intensity, is at frequency \$ \omega_{1}-\omega_{2} \$ – The Feynman Lectures on Physics
And we have,
$$ { \cos(2\pi f_1t)+\cos(2\pi f_2t) } = { 2\cos\left(2\pi\frac{f_1+f_2}{2}t\right)\cos\left(2\pi\frac{f_1-f_2}{2}t\right) } $$
Why is this even possible in the case of beating?
I initially suspect this is the result of being a non-LTI system, the system somehow contains a time-varying element, but I cannot identify it.
On my second thought, it seems that \$ \frac{1}{2}(\omega_1−\omega_2) \$ (or \$ \omega_{1}-\omega_{2} \$) is only a human perception of the change in intensity of two frequencies, and it's not an actual frequency component created by a modulation process, is it the correct interpretation?
Best Answer
If you consider the human ear as a part of the system then a modulation process is involved. If it wasn't we would not be able to hear the 'beat' frequency.
In a perfectly linear medium any number of waves can coexist independently, and a time-invariant system is insensitive to instantaneous amplitude so 'beats' have no effect. But the human ear - like an rf 'mixer' or 'detector' - is nonlinear.
The generic definition of 'mixer' is 'a machine used for mixing things together'. The mixing process may combine those things to make a different product (eg. cake mixer) or simply so they can be transported together (eg. mixed sweets). In audio equipment a 'mixer' is the name for a circuit that combines signals without distorting them - ie. an adder. In rf design it is the name for a circuit which combines signals with distortion, ie. a multiplier.