A long time ago I read a paper in the IEEEProceedings recounting the history of the superheterodyne receiver. Overall it was a very interesting and informative article, with one exception: in it the author remarked that the modulation (or mixing) principle was really nothing new, being already known to piano tuners who traditionally used a tuning fork to beat against the piano string’s vibrations. I did a double-take on this assertion and wrote the IEEE a letter to that effect, which was published, explaining why this interpretation was fallacious. Most unexpectedly, the author replied by contesting my explanation, maintaining that the beat frequency was indeed equivalent to the intermediate frequency (IF) formed when the radio frequency (RF) and local oscillator (LO) signals are mixed to form a difference-frequency signal. With no further letters supporting (or opposing) my viewpoint, it was left to the Proceedings readers to decide who was right. (I did get one private letter of support).
I thought maybe it was time to exhume the argument and so include those in the the PF community who might have an interest and/or opinion in the subject.
Let me start by stating that modulation (a.k.a. mixing) is a nonlinear process while beating is a linear process. In physics these are distinctly different processes.
The term “mixing” has a very specific meaning in radio parlance. Mixing two signals of differing frequencies f1 and f2 results in sidebands (f1 + f2) and |f1 – f2|. These are new signals at new frequencies. Depending on the mixing circuit there can be many higher sidebands as well but let’s assume a simple multiplier as the mixer:
Mixed signal = sin(ω1t)*sin(ω2t) = ½ cos(ω1 – ω2)t – ½ cos(ω1 + ω2)t, ω= 2πf.
Note that two new frequencies are produced. (The original frequencies are lost in this case but that does not always obtain; cf. below).Now consider two signals beating against each other. Let the tuning fork be at f1 and the piano string at f2; then:
Beat signal = sin(ω1t) + sin(ω2t). This can be rewritten as
Beat signal = [cos(ω1 – ω2)t/2][cos (ω1 + ω2)t/2].
Now, this may look like the same two new signals generated by mixing. But that would be wrong. There are no new signals of frequencies (ω1+ω2) and |ω1 – ω2| generated. A look at a spectrum analyzer would quickly confirm this. (I am aware that the human ear does produce some distortion-generated higher harmonics, but these are small in a normal ear and certainly not what the piano tuner is listening to).
The beat signal is just the superposition of two signals of close-together frequencies. Assuming |f1 – f2| << f1, f2, the “carrier” frequency of the beat signal is at (f1 + f2)/2 and so approaches f1 = f2 when the piano string is perfectly strung, and the beat signal amlitude varies with a frequency of |f1 – f2|. |f1 – f2| can be extremely small before essentially disappearing altogether to the piano tuner, certainly on the order of a fraction of 1 Hz It should be obvious that no human ear could detect a sound at that low a frequency (the typical human ear lower cutoff frequency is around 15-20 Hz).
This confusion is not helped by other authors of some repute. For example, my Resnick & Halliday introductory physics textbook describes the beating process as follows:
“This phenomenon is a form of amplitude modulation which has a counterpart (side bands) in AM radio receivers”.
Most inapposite in a physics text! Beating produces no sidebands. A 550-1600 KHz AM signal is amplitude-modulated and of the form
[1 + a sin(ωmt)]sin(ωct)
which produces sidebands at (fc + fm) and (fc – fm) in addition to retention of the carrier signal at fc. Here fc is the carrier (say 1 MHz “in the middle of your dial”), ωm is the modulating signal,and a is the modulation index, |a| < 1 . Of course, in a radio signal, asin(ωmt) is really a linear superposition of sinusoids, typically music and speech, in the range 50 – 2000 Hz, .