Modulation vs. Beating Confusion

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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, .

Comments welcome!

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AB Engineering and Applied Physics
Aerospace electronics career
Used to hike; classical music, esp. contemporary; Agatha Christie mysteries.

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  1. rude man
    rude man says:

    Yes, that seems to be the problem here.  I thought I made it pretty clear what kind of "beats" I was referring to, but I also acknowledge that the term "beat" can include mixing.   A clear example was already mentioned, to wit, the BFO, which of course is a mixing operation.I do disagree totally with whoever thinks mixing is done in the ear to any audible extent. The lowest audible sound would have to be at the sum frequency, i.e. at twice the t-f frequency, which it clearly isn't; or it would have to be a very high harmonic of the difference frequency, which still would be at a very low frequency, near the lower end of audibility, which again is not at all what the tuner hears.  So please, folks, forget about nonlinear ear response!  :smile:

  2. Baluncore
    Baluncore says:

    Maybe we need to stop using the term “mixer”. To an audio engineer, mixing involves adding signals in a linear device, so as to prevent energy appearing at new frequencies. To a radio engineer, mixing involves multiplying signals in a non-linear device, so as to cause energy to appear at new frequencies.

  3. nsaspook
    nsaspook says:

    The heterodyne process is non-linear (mixing) but modulation can be both.
    Non-linear Ring Rodulator.

    or linear as with a AM modulator where we have a bandwidth amplitude that’s equal to the modulation signal that obeys the principle of superposition.

    Yes, I agree that an audio ‘beat’ like when using a BFO on a Morse code receiver is not the same as a ‘mixed’ signal but ‘modulation’ in general is not restricted to superposition (or the lack of superposition) of signals.

  4. Averagesupernova
    Averagesupernova says:

    This subject has been beat to death with disagreement in each related thread here on PF. meBigGuy pretty much hit the nail on the head. Mixing and modulation are both multiplication. Depending on who you ask they are linear or non-linear. A true linear amplifier can have many signal input to it and will not generate new frequencies. So this implies to me that mixing and modulation are non-linear processes.

    I think that the word beat was originally used interchangeably with mixing. A BFO used in a SSB or CW (morse) receiver is in fact mixed with the IF in order to generate an audio signal. It is NOT linear. The ear is in fact non-linear but we don’t beat a couple of MHz signals together in our ear to get an audible signal. The non-linear process has to occur in the radio, not the ear.

  5. nsaspook
    nsaspook says:

    I think that the word beat was originally used interchangeably with mixing. A BFO used in a SSB or CW (morse) receiver is in fact mixed with the IF in order to generate an audio signal. It is NOT linear. The ear is in fact non-linear but we don’t beat a couple of MHz signals together in our ear to get an audible signal. The non-linear process has to occur in the radio, not the ear.

    The signal AM demodulation process (envelope detector diode in this circuit) is non-linear but the actual BFO injection circuit is usually a simple linear signal injection (added to the antenna signal here) like in this simple crystal radio circuit.

  6. meBigGuy
    meBigGuy says:

    Let’s talk about beating from the frequency domain perspective. If beating is like modulation, then there must be a carrier (real or suppressed) and sidebands. (and it turns out there are such, in a crazy sort of way)


    If you look at the trig function for adding two sinewaves (of equal amplitude), the right side represents a carrier of frequency (x+y)/2 being multiplied by a modulation function at (x-y)/2.

    If you look at the summed signals (x and y) in the frequency domain, there has to be those two signals at x and y, and nothing else (because I am just linearly adding two sine waves). So, where are the carrier and sidebands?

    Well, it turns out the two signals, x and y, ARE the sidebands, and the suppressed carrier (x+y/2) is halfway between them. It’s strange to think about it that way, but it is an accurate portrayal of what is actually happening.

    That illustrates that beating causes no new frequencies, and is therefor totally useless as (and is totally distinct from) a mixing function. It does cause modulation. Beating is analogous to what happens when you create standing waves. When the waves are 180 out, they cancel, but no new frequencies are created. Think of what is happening as you walk through a room with a tone playing. (and think dopplar)

    Therefor, any conclusion that beating caused by linearly adding two sinewaves is the same as, or even similar to, hetrodyning is totally incorrect.

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