If realistic is taken as meaning natural sound, we should get an idea of the steepest transients it by looking at the output voltage delivered by very high quality microphones when submitted to very high transients of sound pressure; and then multiplying it by the gain of the following amplifying devices up to the power amp output.
1Hz fundamental with H3, H5, H7, H9, H11, H13, H15, H17, H19.
That s 10 sines, so k = 10 and f = 1
Max SR = 8.f.k = 8 x 1 x 10 = 80.
With 4.f_max = 76, so that s rather 4 x (f + f_max) but the correct formalisation is 8.f.k.
If we look at the slope of a square wave, that is its derivative :
U'(t) = (4/pi)(w.cos(wt) + w.cos(3wt) + w.cos(5wt) + ... + w.cos((2k+1)wt))
At the point of maximum slope (t = 0) all the bolded parameters are equal to 1, what is left of the formulae is :
U'(t) = (4/pi)(w + w + w + ... + w) with w being in k quantity in the bracket.
U'(t) = (4/pi)(k.w) with w = 2.pi.f
Hi Guys
Where most of the posts here talk about the rise time of the highest single frequency anticipated, it is good to see an allusion to transient response made by maty. But there needs to be a more pointed statement made.
Music is not single frequencies, rather many frequencies present at once with their relative amplitudes providing the timbre of the sound which makes it unique and recognizable. The initial striking of a note on most instruments causes several frequencies to begin all at once. This creates the fast-rising transient attack of most sounds. The steepness of this transient depends on the superposition of all of the frequencies present and is therefore much faster than the highest frequency of the pass band.
We can make approximations as to how fast the fastest transient might be, based either on the audio band we can hear, or based on simple graphic addition of the rise times, of a simulated signal, or on square waves. The latter do not have instantaneous zero-cross times, and will be altered by the input filtering most properly designed amplifiers have. Any approximation we make will likely be generous inasmuch as the real amplifier will exceed the calculated requirement quite easily.
Measurements made of actual recordings pepper the audio journals and no fast transient was ever found that could cause a problem for a decent amp. Yes, there have been some lousy amps designed that have shown up things like slew-induced distortion, but in our modern times you almost have to deliberately design the amp to be that poor in performance.
The profile of energy per octave for most music and for natural sounds tapers off towards the treble, so very high amplitude 10kHz or 20kHz tones are not a reality (except maybe for live PAs where the sound is quite corrupted and way beyond the Human Scale of loudness).
None of the above is meant to suggest that striving for high slew rates is without reason, rather to suggest that the minimum slew rate required is surprisingly low. It is always good to have a margin for the unexpected. As stated by others, modern digital recording protocols place a hard limit to passband and to rise time. Low-power amps of 20V/us, medium-power of 40V/us and high-power of 60V/us have no issues in this regard.
Have fun
Where most of the posts here talk about the rise time of the highest single frequency anticipated, it is good to see an allusion to transient response made by maty. But there needs to be a more pointed statement made.
Music is not single frequencies, rather many frequencies present at once with their relative amplitudes providing the timbre of the sound which makes it unique and recognizable. The initial striking of a note on most instruments causes several frequencies to begin all at once. This creates the fast-rising transient attack of most sounds. The steepness of this transient depends on the superposition of all of the frequencies present and is therefore much faster than the highest frequency of the pass band.
We can make approximations as to how fast the fastest transient might be, based either on the audio band we can hear, or based on simple graphic addition of the rise times, of a simulated signal, or on square waves. The latter do not have instantaneous zero-cross times, and will be altered by the input filtering most properly designed amplifiers have. Any approximation we make will likely be generous inasmuch as the real amplifier will exceed the calculated requirement quite easily.
Measurements made of actual recordings pepper the audio journals and no fast transient was ever found that could cause a problem for a decent amp. Yes, there have been some lousy amps designed that have shown up things like slew-induced distortion, but in our modern times you almost have to deliberately design the amp to be that poor in performance.
The profile of energy per octave for most music and for natural sounds tapers off towards the treble, so very high amplitude 10kHz or 20kHz tones are not a reality (except maybe for live PAs where the sound is quite corrupted and way beyond the Human Scale of loudness).
None of the above is meant to suggest that striving for high slew rates is without reason, rather to suggest that the minimum slew rate required is surprisingly low. It is always good to have a margin for the unexpected. As stated by others, modern digital recording protocols place a hard limit to passband and to rise time. Low-power amps of 20V/us, medium-power of 40V/us and high-power of 60V/us have no issues in this regard.
Have fun
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Ahh but this is not true is the signal is bandlimited, which is always the case.
As I said earlier, you can add all kinds of 20kHz signal components together, but you will never get a transient that is faster than a full range 20kHz signal!
Jan
Hi Guys
The graphical approach suggests that you are mistaken, Jan. I'll leave it at that.
Have fun
The graphical approach suggests that you are mistaken, Jan. I'll leave it at that.
Have fun
Yes, the peak voltage must be factored, it s obviously normalized to a 1 magnitude, so it s rather A.8.f.k with A being the peak level, i used a magnitude of one since the factor A has no say in the moment at wich the slew rate is at its peak value.
Notice that the peak slew rate occur when the value of the signal is 0V.
Oh... lord
In many regards, I probably shouldn't be sticking my snoot in here. Clearly there are a lot of contentious ideas in play.
But here's the thing.
This is a bit beyond what most people like to flail, mathematically.
…
The “slew rate” is the first derivative of the wave shape.
For sinusoidal signals (pure), if
Vo = AoVisin( 2πFt ) … then
dVo/dt = Ao Vi 2πf cos( 2 π F t )
To simplify, that becomes
max[ Vo 2πF cos( 2πFt )… ] which reduces to
Vo 2πF
as the maximum slew rate for the sinusoidal signal. Frequency based.
_______
Now on the other hand, there are three signals that can have significantly higher dV/dt slew rates.
(1) transient pulses
(2) square waves and their sister, sawtooth waveforms
(3) complex spectrum waveforms.
Number 3 has two cases worth thinking about separately. (3a) is noise spectrum waveforms, where for the most part humans simply can't detect slew-rate limited reproduction errors. Noise is noise, unless egregiously limited in reproduction. The other (3b) is sonically complex sources such as symphony orchestras and much music in general. The overlay of various instruments and sound source can cause slew rates during some parts of the waveform that exceed that of a pure sine source at maximum amplitude.
Technically … and I'm sorry if this is so obviously that it sounds preachy … the maximum slew rate would be between the -peak and +peak voltage swings, in the shortest meaningful interval imposed by the harmonics that one is willing to reproduce. Square waves and sawtooth waves in particular have “infinite slew rates” theoretically. Purely pragmatically, when an amplifier can 2× the slew rate of the highest full-amplitude sinusoidal frequency, it is “enough”. Further improvements, no matter how technically ideal, simply do not impact hearing a difference, to the most sensitive listeners.
The problem usually is in the last stage, the drivers in the speakers and their sound fields. All the slew rate in the world can't make up for the inertia and momentum of the flying speaker cones and elements, and the inability of any real-world amplifier to entirely compensate for the same.
So…
Net…
Its almost exactly as DF96 says.
Theoretical practical maximum slew rate is Ao F 2 π Vin, or, Vo F 2 π for the sinusoid of highest practical regard (F). If one is aiming to reproduce the most acute step-functions, use 2× that slew rate. If one takes the very much sidelined research about the actual spectrum of energy of various musical sources, rarely is there appreciable energy above 12,000 Hz or so. So, indeed… one very often is quite well served with an amplifier whose maximum slew rate is between Vo F π and Vo 2 F π volts per second.
Just saying…
GoatGuy
In many regards, I probably shouldn't be sticking my snoot in here. Clearly there are a lot of contentious ideas in play.
But here's the thing.
This is a bit beyond what most people like to flail, mathematically.
…
The “slew rate” is the first derivative of the wave shape.
For sinusoidal signals (pure), if
Vo = AoVisin( 2πFt ) … then
dVo/dt = Ao Vi 2πf cos( 2 π F t )
To simplify, that becomes
max[ Vo 2πF cos( 2πFt )… ] which reduces to
Vo 2πF
as the maximum slew rate for the sinusoidal signal. Frequency based.
_______
Now on the other hand, there are three signals that can have significantly higher dV/dt slew rates.
(1) transient pulses
(2) square waves and their sister, sawtooth waveforms
(3) complex spectrum waveforms.
Number 3 has two cases worth thinking about separately. (3a) is noise spectrum waveforms, where for the most part humans simply can't detect slew-rate limited reproduction errors. Noise is noise, unless egregiously limited in reproduction. The other (3b) is sonically complex sources such as symphony orchestras and much music in general. The overlay of various instruments and sound source can cause slew rates during some parts of the waveform that exceed that of a pure sine source at maximum amplitude.
Technically … and I'm sorry if this is so obviously that it sounds preachy … the maximum slew rate would be between the -peak and +peak voltage swings, in the shortest meaningful interval imposed by the harmonics that one is willing to reproduce. Square waves and sawtooth waves in particular have “infinite slew rates” theoretically. Purely pragmatically, when an amplifier can 2× the slew rate of the highest full-amplitude sinusoidal frequency, it is “enough”. Further improvements, no matter how technically ideal, simply do not impact hearing a difference, to the most sensitive listeners.
The problem usually is in the last stage, the drivers in the speakers and their sound fields. All the slew rate in the world can't make up for the inertia and momentum of the flying speaker cones and elements, and the inability of any real-world amplifier to entirely compensate for the same.
So…
Net…
Its almost exactly as DF96 says.
Theoretical practical maximum slew rate is Ao F 2 π Vin, or, Vo F 2 π for the sinusoid of highest practical regard (F). If one is aiming to reproduce the most acute step-functions, use 2× that slew rate. If one takes the very much sidelined research about the actual spectrum of energy of various musical sources, rarely is there appreciable energy above 12,000 Hz or so. So, indeed… one very often is quite well served with an amplifier whose maximum slew rate is between Vo F π and Vo 2 F π volts per second.
Just saying…
GoatGuy
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My only (very minor) contention with that assertion is that average energy is low at audio-HF (let's get real here 😀), but tends to have high crest factor. I.e., we're right back to ensuring that our slew rate covers the audio bandwidth, as you say. 🙂
Hi Guys
The concept of superposition of sine waves is pretty old and most people see it in high school math, so why is it so "alien" here?
A prime example is a square wave. A square wave is a specific series of sine waves with specific amplitudes. The fundamental is just the same frequncy as the final square wave. The harmonics of this frequency are all odd-order and have an amplitude that is the reciprocal of their order. For example, the third harmonic is at one-third the amplitude of the fundamental; the fifth harmonic is one-fifth the amplitude,and so on.
As you superimpose each harmonic over the fundamental, the slope of the leading edge of the composite wave becomes steeper. In a properly scaled drawing of a sine wave the zero-cross slope is actually 45-degrees, but as you superimpose further frequencies the slope approaches 90-degrees. This is just high-school math, as I said. Not complicated enough for some around here, I guess.
Music superimposes many frequencies at a time. If it did not, the envelope of the sound would not be as it is - uncompressed sounds have transient attack that is up to four times the amplitude of the sustain part of the envelope, but more typically just twice as high. Because the superposition dramatically steepens the slope of the leading edge, it is pretty much a necessity that an audio circuit have fast response and that this response be much faster than simply passing the highest sine frequency demands.
Have fun
The concept of superposition of sine waves is pretty old and most people see it in high school math, so why is it so "alien" here?
A prime example is a square wave. A square wave is a specific series of sine waves with specific amplitudes. The fundamental is just the same frequncy as the final square wave. The harmonics of this frequency are all odd-order and have an amplitude that is the reciprocal of their order. For example, the third harmonic is at one-third the amplitude of the fundamental; the fifth harmonic is one-fifth the amplitude,and so on.
As you superimpose each harmonic over the fundamental, the slope of the leading edge of the composite wave becomes steeper. In a properly scaled drawing of a sine wave the zero-cross slope is actually 45-degrees, but as you superimpose further frequencies the slope approaches 90-degrees. This is just high-school math, as I said. Not complicated enough for some around here, I guess.
Music superimposes many frequencies at a time. If it did not, the envelope of the sound would not be as it is - uncompressed sounds have transient attack that is up to four times the amplitude of the sustain part of the envelope, but more typically just twice as high. Because the superposition dramatically steepens the slope of the leading edge, it is pretty much a necessity that an audio circuit have fast response and that this response be much faster than simply passing the highest sine frequency demands.
Have fun
How many of those harmonics to that slew-rate breaking 20 kHz square waves are audible? The ear itself is a band limiting element. So what's the point of hand wringing about whether we can pass square waves? Never mind so many of us listen to 16/44.1 media, which means (ignoring sample-to-sample overshoot) we're Nyquist limited to 22.05 kHz.
Perfect square waves or perfect Dirac deltas aren't audio.
Perfect square waves or perfect Dirac deltas aren't audio.
Mind. Blown. Six pages of posts following a question that was answered in Post #2. 🙂
The initial question was perfectly valid. Jan Didden provided the answer in Post #2. The equation he provided is correct. Sadly he was off by an order of magnitude in his calculation, but that was promptly corrected. The derivation of the equation in Post #2 is shown just a few posts back.
The highest frequency an average human can hear is 20 kHz. I forget at what age, but it's fairly young. Once you reach adulthood the hearing degrades, just as every other part of the body does.
Some will argue that transients require wider bandwidths to reproduce accurately. That's true. However, if the limit of human perception is 20 kHz, does the playback chain need to reproduce frequencies beyond that? It would be nice to see some science about this rather than the usual "someone on the Internet said, so it must be true". As many have pointed to earlier in this thread, many digital sources employ brick-wall filters that attenuate anything above 0.45 * fs by 100+ dB. I doubt the bandwidth of the vinyl-to-stylus interface support any higher bandwidth, though, that's my estimate. If anybody has hard data, I'm all ears.
Interesting tempest in this here teacup, though. 🙂
Tom
The initial question was perfectly valid. Jan Didden provided the answer in Post #2. The equation he provided is correct. Sadly he was off by an order of magnitude in his calculation, but that was promptly corrected. The derivation of the equation in Post #2 is shown just a few posts back.
The highest frequency an average human can hear is 20 kHz. I forget at what age, but it's fairly young. Once you reach adulthood the hearing degrades, just as every other part of the body does.
Some will argue that transients require wider bandwidths to reproduce accurately. That's true. However, if the limit of human perception is 20 kHz, does the playback chain need to reproduce frequencies beyond that? It would be nice to see some science about this rather than the usual "someone on the Internet said, so it must be true". As many have pointed to earlier in this thread, many digital sources employ brick-wall filters that attenuate anything above 0.45 * fs by 100+ dB. I doubt the bandwidth of the vinyl-to-stylus interface support any higher bandwidth, though, that's my estimate. If anybody has hard data, I'm all ears.
Interesting tempest in this here teacup, though. 🙂
Tom
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