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Understanding Ultrasonic Heterodyning: EM Voltages and Human Hearing Explained
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[QUOTE="rude man, post: 4518703, member: 350494"] This is a very good question. Some time ago a writer wrote a paper in IEEE [I]Spectrum[/I] confusing addition with mixing. I pointed the error out but this guy stuck to his guns. It was quite an amusing exchange of published letters. What you did by applying the two signals to the spectrum analyzer was to [I]add[/I] two signals: cos(w[SUB]1[/SUB]t) + cos(w[SUB]2[/SUB]t). There are no sum & difference frequencies produced by this process. There is just w[SUB]1[/SUB] and w[SUB]2[/SUB]. A mixing process on the other hand is essentially [I]multiplicative[/I]: cos(w[SUB]1[/SUB]t)*cos(w[SUB]2[/SUB]t) which by simple trig gives you a signal at (w[SUB]1[/SUB] + w[SUB]2[/SUB]) and another at (w[SUB]1[/SUB] - w[SUB]2[/SUB]). Mixers usually are not simple multipliers so you also get other output frequencies but you will always get the sum & difference frequencies. All mixing is based on a nonlinear process. Addition is of course linear. The human ear may have some nonlinearity which would yield a small difference frequency, like the "diode" effect you suggested. I believe on a healthy ear it's extremely small. Two tuning forks detuned a few Hz from each other, i.e. (w[SUB]1[/SUB]-w[SUB]2[/SUB])/2π give a sound that sounds lile a low-frequency signal but it's just the average of the two frequencies varying in intensity at the difference frequency. You obviously could not hear the beats from two tuning forks separated by a few Hz if your ear produced only a mixing operation (you might hear the sum but not the "beat" frequency.) The beat signal for cos(w[SUB]1[/SUB]t) + cos(w[SUB]2[/SUB]t) = 2{cos[(w[SUB]2[/SUB] - w[SUB]1[/SUB])/2 t]}cos[(w[SUB]1[/SUB] + w[SUB]2[/SUB])/2 t]. This may look like new frequencies at w[SUB]1[/SUB] + w[SUB]2[/SUB] and w[SUB]1[/SUB] - w[SUB]2[/SUB] are produced, but by looking at the LHS of the equation there obvioulsy aren't. You can see though that it can be interpreted as a signal at w = (w[SUB]1[/SUB] + w[SUB]2[/SUB])/2 moduated in amplitude by a signal at the beat frequency (w[SUB]1[/SUB] - w[SUB]2[/SUB])/2. The standard superhet radio on the other hand [I]mixes[/I] the incoming signal (RF) with a local oscillator (LO) to produce a difference frequency called the intermediate frequency (IF). [/QUOTE]
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Understanding Ultrasonic Heterodyning: EM Voltages and Human Hearing Explained
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