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[jimmyjazz]

Gideon wrote on Tue, 19 July 2005 19:12

Linear superposition is a property of a linear system. In a perfectly linear system waves will just add together and there will not be any beats (sum and difference terms).

Not true.  Plot (as I just did here in MathCad)

f(t) = sin(100*t) + sin(101*t)

and you will see clear evidence of a "beat frequency" that manifests as an envelope modulator which repeats every 6.28 secons (i.e., 2*pi).  It's true that if you ran an FFT on that signal you would not see the beat frequency, but rather only the two inputs (100 rad/sec & 102 rad/sec).  That doesn't mean you wouldn't hear the envelope effect, though.

[Eric Bridenbaker]

I always thought that Heterodyning was one of the possible outcome of a guy asking a gal out to dinner.

(So bad I had to put it up:)

[Gideon]

jimmyjazz wrote on Wed, 20 July 2005 11:22

Gideon wrote on Tue, 19 July 2005 19:12 Linear superposition is a property of a linear system. In a perfectly linear system waves will just add together and there will not be any beats (sum and difference terms). ... It's true that if you ran an FFT on that signal you would not see the beat frequency, but rather only the two inputs (100 rad/sec & 102 rad/sec).  That doesn't mean you wouldn't hear the envelope effect, though.

There is no argument between us--there is just a difference in semantics; the two statements above (yours and mine) are essentially equivelent. A Fast Fourier Transform will not show a difference tone: i.e. the system is linear.
Our ear willhear a difference tone: i.e. the ear is nonlinear.

Also, note that there is an important difference in the definition of a beat percieved as a tone and a beat that is percieved as a pulsing of amplitude (such as is used in piano tuning). This definition of "beat," as a modulation of amplitude, corresponds to your "envelope" and would indeed be heard even if the ear were perfectly linear. That is why I put "sum and difference terms" in parentheses after "beat."
But you are right that the term "beat" is perhaps too ambiguous for this discussion; I should just say "sum and difference terms."

[JDSStudios]

Eric Bridenbaker wrote on Wed, 20 July 2005 16:29

I always thought that Heterodyning was one of the possible outcome of a guy asking a gal out to dinner. (So bad I had to put it up:)

And if they decide to marry, and if the Bride decides to bake a
wedding cake, you would get a Bridenbaker.!


.

[sdevino]

Heterodyning requires combining the signals in a non-linear mixer. An audio mixer is a linear device, an rf mixer  is a NON-LiNEAR device (like your ear). The summing and difference signals are created as a by product of the non-linearity.

Here is a link with a nice discussion about the subject:
http://www.davidbridgen.com/mixers.htm

Steve

[SteveBoker]

This has been an interesting convergence on a set of mutually acceptable terms.

So, now, suppose that you have two signals that produce a modulated amplitude envelope.  But further suppose that you resample this signal at a rate such that the original signals are above Nyquist limit of the sampling rate but the amplitude envelope is below the Nyquist limit of the sampling rate.  The original frequencies would be lost since they were above the Nyquist limit of the sampling rate.   But the amplitude modulation would be retained.  Correct?  All still linear?

Now, consider performing an FFT on the linear superposition of two oscillators with frequencies above 20k, but whose superposition creates an amplitude modulation below 20k and where you do not allow Fourier components above 20k.   If you had allowed all components, of course all of the power would be back in the two original components.  But since you've restricted the range of the Fourier components, you would find some power in the frequency of the amplitude modulation.

If the nonlinearity that people are speaking of is the nonlinearity imposed by a frequency threshold, then I think we all agree.  However, may I suggest that we do not need to throw out linear superposition in order to find power in an amplitude envelope modulation if there is a low pass filter (such as our ears or such as our speakers) with a threshold in between the original frequencies and the frequency of the amplitude modulation.

Would this explain the reported auditory illusion?

[jimmyjazz]

SteveBoker wrote on Mon, 01 August 2005 12:44

So, now, suppose that you have two signals that produce a modulated amplitude envelope.  But further suppose that you resample this signal at a rate such that the original signals are above Nyquist limit of the sampling rate but the amplitude envelope is below the Nyquist limit of the sampling rate.  The original frequencies would be lost since they were above the Nyquist limit of the sampling rate.   But the amplitude modulation would be retained.  Correct?  All still linear?

Ummm . . . I don't think so.  First of all, if you sample the signal at an inadequate sample rate, you will either

a) see aliasing if you haven't properly filtered the combined signal . . . garbage in, garbage out.
b) see nothing if you HAVE properly filtered the combined signal, because the original two signals are now gone.

That having been said, the linear superposition of two closely-matched signals which leads to amplitude modulation will have NO energy at the frequency of the modulation.  There's nothing there.  You can hear it, but it's not "there".

[Eric Bridenbaker]

thanks for all the great replies on this one... True there is definitely something going on in the audible spectrum, which may, as suggested earlier be a result of aliasing or reconstruction artifacts. I like Jimmyjazz's post that referred to this as the "envelope effect", as this is a good description of the phenomenon.

I'm interested about how Jimmyjazz plotted out a scenario in Mathcad, showing how these beats will occur in a linear system.

The question I have about it is this:

What is the amplitude value for the envelope ripple, relative to the that of the input tones?

If there is no energy at the beat frequency, how can the envelope effect be visible in a Mathcad plot (at zero energy the ripples should have amplitude zero and NOT be visible in the simulation).

I'd hazard a guess that calculating the total area under each curve might shed more light on the subject...

Cheers,
Eric

[jimmyjazz]

Eric, the amplitude of the combined signals is just the sum of the amplitudes of the two component signals.  In other words, at the envelope's "peak" the two signals are combining "in phase" (for lack of a better word), and at the envelope's "anti-peak", they're combining "out of phase" (ditto).  To illustrate:

A1 = 1, A2 = 1 . . . Envelope peak = 2, Envelope minimum = 0

A1 = 1, A2 = 2 . . . Envelope peak = 3, Envelope minimum = 1

A1 = 2, A2 = 1 . . . Envelope peak = 3, Envelope minimum = 1


It's just signal interaction.  The energy is in the two signals themselves.  An FFT would show the two component signal amplitudes at their correct frequencies and nothing else.  In the time domain, though, the envelope shows up as the signals constinuously slide between constructive and destructive interference.