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In Reply to: RE: Phase Distortion, dv/dt, and Slew Rate posted by Lee of Omaha on December 17, 2015 at 09:55:23
You are correct that some sound waves can line up in phase at certain times, but aside from the harmonics resulting from a single voice or instrument, most won't. As Geoff mentioned, you need some luck. I will try to show that via math at the bottom of this post.
I think what we really care about is crest factor, the ratio of the highest peak level of the music over the RMS (average) level of the music. I've heard that the crest factor of the human voice is around 12 dB, whereas the crest factor for percussion could be over 24 dB. So I would not expect choral music to be the most stressing. Back in the days before 24bit digital recording, it was typical to use a compressor in the recording chain, at least for percussion. These days, the compression is usually applied after the fact in software. Either way, the end product usually has some compression in it. So the recordings we play back through our systems have crest factors of no more than about 1618 dB at the high end, in the case of a relatively uncompressed recording from before the loudness wars.
Given the crest factor, the average level you would like to listen at, loudspeaker sensitivity and impedance, how far away you sit from the speakers, and an approximate room gain, you can calculate how much peak amplifier power you need and what the peak voltage will be. Suppose it's 200W and 40V.
Then you pick a maximum frequency. For CD, 22.05 KHz. The formula from my previous post gives a maximum slew rate requirement of 2 * pi * 40 * 22050 = 5.54V/usec. This calculation is conservative because the power spectra of music falls off at high frequency so you would never get a 20KHz component at 0dBFS.
But suppose you want to be ultra conservative. Let's say the amplifier has a wide 100KHz bandwidth and its voltage rails are at 50V. In order to make sure there is no possibility whatsoever of slew rate limiting regardless of what input signal you feed it, you would need a maximum slew rate of 2 * pi * 50 * 100000 = 31.4V/usec. For modern solid state amplifiers where a maximum slew rate specification is available, a typical value is around 40V/usec.

Here is the math bit I promised:
Start with the formula for a sine wave:
V(t) = A * sin(2*pi*f*(tt0))
V is volts, A is the amplitude, and 2*pi*f*(tt0) is the phase. The phase has a time varying term 2*pi*f*t and a constant term 2*pi*f*t0 where t0 is the time offset and f is the frequency of the sine wave.
If you have two sine waves:
V1(t) = A1 * sin(2*pi*f1*(tt1))
V2(t) = A2 * sin(2*pi*f2*(tt2))
The sum V(t) = V1(t) + V2(t)
The derivative dV/dt = dV1/dt + dV2/dt
dV1/dt = A1 * cos(2*pi*f1*(tt1))
dV2/dt = A2 * cos(2*pi*f2*(tt2))
Now we're concerned with the maximum slew rate, i.e. the maximum absolute value of dV/dt. You can see that IF the phase of the two sine waves lines up, the maximum will be A1 + A2. So the important question is under what conditions will the phase line up.
The maximum of value of cos(x) is 1 and it occurs at values of x=0, x=2*pi, x=4*pi, ... The minimum of value is 1 and it occurs at values of x=pi, x=3*pi, x=5*pi, ...
So the maximum value of dV1/dt occurs at t = t1 + n * 1/f1 where n=0,1,2,3...,
and the maximum value of dV2/dt occurs at t = t2 + m * 1/f2 where m=0,1,2,3...
Likewise, the minimum value of dV1/dt occurs at t = t1 + pi + n * 1/f1 where n=0,1,2,3...,
and the minimum value of dV2/dt occurs at t = t2 + pi + m * 1/f2 where m=0,1,2,3...
In order for a maximum of dV1/dt to line up in time with a maximum of dV2/dt, or alternatively for two minima to line up, there has to be a pair of integers m,n where t1 + n * 1/f1 = t2 + m * 1/f2. Rearrange this to be n = m * f1/f2 + f1*(t2t1). Since n and m have to be integers, the sum of m * f1/f2 + constant has to result in an integer value.
In the lucky case of t1=t2, there is at least one solution: m=n=0. That is, if t1=t2 then the two maxima are guaranteed to line up at least once at time t=t1=t2 but not necessarily before or after. In the similarly lucky case where f1*(t2t1) happens to fall on an integer value other than 0, then there are certain combinations of f1 and f2 which will also produce a solution at one time but not necessarily before or after. These lucky occurrences aren't very interesting solutions because they only produce the maximum of A1+A2 at one time value. I'm more interested in knowing under what conditions this can recur, and the answer is time/phase aligned harmonics.
If t1=t2 and f1 is an integer multiple of f2 or vice versa, then you will get a periodic series of solutions. This would occur if f1 is a harmonic of f2 or vice versa. Suppose you have the case of t1=t2 and f1 = 3*f2. Then there will be solutions at t=0, t=3/f1, t=6/f1, and so on.
Any time you are adding harmonics, you will get a pattern repeating at the frequency of the fundamental. But in order for the pattern to include maximum values of A1+A2, the harmonics have to align in time/phase. That itself is a lucky occurence given that the phase response of a microphone is generally not a constant vs. frequency.
Also, this is an analysis of just two sine waves. It can be extended to account for additional frequency components, but as you increase the number of components, the number of solutions which can produce a pathological maximum of A1+A2+A3+... become significantly more rare. Which explains why we don't see these in music.
Thanks for the analysis. I followed it (but am too rusty to do it).
I think you reinforced my argument. If you picture a bunch of voices (or instruments, or a mix) playing vibrato (or even just a little out of tune) then, while the probability of most of them lining up at a peak or valley is low at any one time, there are a hell of a lot of temporal opportunities (cycles) for that to happen. So I think it probably does happen, some, and the distribution of "slewinduced distortion events" would be predictable, and the intensity of such events would follow some kind of normal curve.
Thanks again for the insight and analysis. It helps.
The longer you sample, the greater the chance of a random collection of waves adding up and creating a large peak. Most recordings that are made cleanly (uncompressed and not subject to tape compression from loud recordings) will have a crest factor around 18 dB. With mild compression the crest factor will be around 15 dB. Percussive recordings can have much greater crest ratios, of course. Also, the recording style affects crest ratios, due to reverberations, etc.
These figures relate to short term period of averaging (0.5 to 2 seconds) during which the music is homogeneous. If you average over longer periods there will be occasional larger peaks and the possibility depends on the number of individual waveform sources. If musical dynamics come into play, then the crest factors can easily get to 25 dB or more.
If there is a rare peak over the ability of the recording or playback this may or may not be audible, depending on the technology. For example, momentary clipping of a stable amplifier on a percussive transient will not be audible if this happens infrequently in the course of a track. However, if the amplifier is unstable (i.e. incompetently designed) then a temporary overload on one peak may cause the amplifier to misbehave for an extended period. Similar problems happen with clipping in digital recording or playback. With LP playback peaks can cause needles to jump out of the groove, etc...
These are all factors well familiar to recording engineers, especially mastering engineers.
Tony Lauck
"Diversity is the law of nature; no two entities in this universe are uniform."  P.R. Sarkar
The math seems to indicate that the only waves which will add coherently are harmonics that are time aligned, i.e. harmonics from a single instrument or voice. The voices in a chorus will add incoherently, so the total acoustic power will be the sum of their individual acoustic powers.
Measurements of crest factor indicate that the pathological cases you're thinking of don't really happen, or at least they are so rare that they aren't a factor in recording. The highest observed crest factors are from impulses generated by percussive instruments, not from massed voices.
Anyway it's a moot point with modern solid state amplifiers because most are capable of slew rates faster than the product of their rail voltage and max frequency, which guarantees that they can't be driven into slew rate limiting with any input. That was not the case back in the 1970s when Walt Jung was writing about slew rate distortion, when amplifier slew rates were 12 orders of magnitude slower than today.
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