[lutusp]

Here's the first crossroad, where we diverge:

> Now define a maximum amplitude V > 0.

The problem being discussed is one in which the user cranks up the volume such that the maximum volume level is far exceeded, and the amplifier circuit, which cannot reproduce the higher level, instead clips the waveform at the supply voltage.

> The claim you made that sparked this conversation was this: "the resulting clipped waveform carries a lot more energy than an unclipped one."

And it does, as you can see from this diagram:

http://i.imgur.com/oE5NFZ9.png

The red trace is the 100% volume level for the system, and it is not clipped. The green, "clipped" trace is far above 100%. It's really very simple.

> But this claim seems to contradict my Claim 2.

Yes. Your claim 2 doesn't apply to a case in which the maximum output voltage of the amplifier cannot reproduce the input waveform. So instead the amplifier clips its input signal at its voltage limits, as the above diagram shows.

The interesting thing about this analysis is that two ways to analyze it produce the same result.

If we use a simple integration of the unclipped and clipped cases, we see a power increase of 2.46, as discussed earlier.

If instead we use Fourier analysis to examine the harmonics of a sine wave and a square wave, the square wave's higher harmonic energies sum to 2.46 of the original sine wave's energy level in the speaker (but pi/2 above the sinewave voltage levels):

Sine wave harmonic lines = 1f

Square wave harmonic lines = (4/pi) sum(sin(2 pi f t (2k-1) / (2k-1),k,1,oo) (http://en.wikipedia.org/wiki/Square_wave#Examining_the_squar...)

The outcomes obviously have to come out the same. The above shows how they do it.

[tmoertel]

OK. I think I see where our signals are getting crossed. I look at your diagram and think, The green signal g is not what you get when you try to output the red signal r but the output stage clips it at the supply voltage (my V). Rather, the green signal g is what you get when you take the red signal r, amplify it by A = Infinity, and then try to output that new signal rA, which the output stage then clips.

For all A > 1, |rA| > |r|. Therefore, I have no problem agreeing that a clipped rA carries more energy than the original r. (In fact, this is just saying that bigger signals carry more energy, regardless of clipping.)

To me, your original statement read as though it made the following claim: If your amplifier tries to output a signal r but clips off the peaks, what you get out is a signal that carries more energy than r. (This is the claim I found hard to believe.)

But what you really meant (I think) was that, if you take a sine wave as input and, as a human controlling the volume knob, keep turning up the volume until the sine wave's maximum amplitude so far exceeds what the amplifier can reproduce that it comes out looking like a square wave, then that square wave carries more energy than the original, un-volume-adjusted sine wave. Is that what you meant?

If that's what you meant, I have no problem believing your claim. It's just that I wouldn't call turning up the volume knob to be part of the "clipping" process, but rather amplifying. When you turn up the volume knob, what you get out is always a "bigger" signal, even if it's clipped.

[lutusp]

> It's just that I wouldn't call turning up the volume knob to be part of the "clipping" process, but rather amplifying.

But it is, indeed it's the most common example of clipping in engineering practice. Consider two alternatives -- one in which the amplifier is able to accommodate a great increase in volume without any distortion, and another in which it cannot.

In this image:

http://i.imgur.com/g8txTmC.png

The blue trace represents maximum normal volume (100%). The green trace represents a system in which the amplifier can accommodate a great increase in volume setting (200% of normal), simply because its power supply voltage is at least twice as high as that required for 100% volume. The red trace represents a system that cannot tolerate any volume settings greater than 100%.

In this diagram, the 200%-volume green trace has a higher subjective volume level than the 100%-volume blue trace (i.e. twice as high in voltage, four times as high in speaker power). The red trace, which would sound very distorted to the user, also has a higher power level than the 100% trace, but less than the green trace. The red trace is typical of systems that are limited to 100% volume by power supply voltage.

The red trace represents the most common example of "clipping" as it takes place in actual equipment and in engineering practice. Its output has more energy than the blue trace, and less than the green trace.

-- former NASA engineer

[tmoertel]

I agree with everything you wrote. Again, however, I think we're talking past one another. When you talk about clipping, you seem to mean turning up the volume until a signal clips. This both amplifies the signal and clips off its extremes. What I mean by clipping is to take a signal and clip off its extremes.

That sole difference explains our entire conversation. In your view, clipping involves making the signal bigger. That bigger signals carry more energy naturally leads to your claim. In my view, clipping a signal makes it smaller, leading to the opposite claim.

Thanks for sticking with me through the conversation.