Thanks for the correction
What other technical errors do you mean?
(1) The amplitude response and magnitude response are not interchangeable terms and (2) the frequency response and the time response are known exactly from each other regardless of whether the system is minimum phase or not.
Agreed. Mathematically speaking, either the frequency response, or the impulse response, completely defines the other.
It's been a few years since I last worked through the chapter on Fourier and impulse transforms in a math textbook, but Wikipedia confirms what I remember, and what Soundbloke has been saying ( Impulse response - Wikipedia ):
I think it's entirely reasonable to apply this to a sealed-box woofer, where there is one source of sound (the woofer cone), and the woofer cone remains a piston through the entire woofer bandwidth (i.e., we don't use the woofer at frequencies high enough for cone breakup to be occurring.)
I am less sure when it comes to bass-reflex or transmission-line speakers. There are now multiple sound sources (woofer cone, and the output from the tuned port or far end of the transmission line), and these two outputs have to be pieced or summed together in some way to generate a single frequency response plot.
Sometimes measurements on these systems are done with more than one microphone, and the summed plot is fiddled with in order to generate the best-looking frequency response plot. This usually involves close-miking both port and woofer cone, and then manually merging the two measured frequency responses. But where do you put the mic to measure the impulse response? And will it match the pieced-together frequency response?
Mathematically, the Fourier transform is a linear transform, so in principle, you can sum two impulse responses, take the Fourier transform of the sum, and get the right answer. And vice-versa, you can sum two frequency responses, take the inverse Fourier transform, and get back the summed impulse response.
But the physical reality of summing direct woofer output + port output is that the sum varies dramatically based on where your microphone (or ears) are, and when it's done with two microphones and tweaked to look as good as possible, I don't know if that on-paper summed response will actually correspond exactly to anything real.
I think there is yet another consideration regarding so-called "slow" bass. In a typical listening room there are high-Q acoustic room modes that will exhibit the textbook forced damped harmonic oscillator time response - vibrational amplitude will "swell" slowly to full intensity if driven at the right frequency, and fade slowly to zero when the drive signal is removed. This really will give you "slow" bass - but it's not from the speaker, it's from the slow response times of the high-Q acoustic eigenmodes of the listening space!
-Gnobuddy
It is very true that when we have more than one source that things get more complicated. If we ignore for the time being having two spatially separated ears too, the system is still linear and everything that I have said before applies: All the audible artefacts are distributed in the time-frequency plane.
The issue of "close-miking" is something of a different subject relating to the differences in the near and far fields or a sound source (although not entirely irrelevant when considering loudspeakers that form large line arrays, for example).
Adding the room response into the model is not particularly different either, however, just it is more difficult to separate out the information we seek: Resonances are resonances whether they are viewed in time or in frequency.
Once proviso here is that pre-ringing is distinctly more audible than post-ringing (to which most references elude): Advocates of arbitrarily applied linear phase compensation need beware!
But we also need remember that distributed in the time-frequency plane we will also have the energy from non-linear artefacts too. So we either need minimise them or hope they appear as noise.
Where we might look for something more abstract is in considering that real sound sources have harmonics - that is, they are non-linear generators where the non-linear parts are a good thing (that make different instruments playing the same note sound different, for example). So here the non-linear artefacts might have an audible significance if we could discern them appropriately.
The goal of audio recording and reproduction is to replay those harmonics components but without adding any more non-linearities. That is, we seek to linearly represent non-linear information, but seldom is it achieved and additional non-linearity is inevitable.
And maybe to distinguish the perceptual differences in such circumstances we need go beyond the second order spectrum to higher order mappings such as the bispectra. But here we run put of spatial dimensions to usefully portray the results - unless anyone has any good ideas? Nevertheless characterising the perceived "speed" of a kick drum, for example, needs more than just a glance at low frequency ringing.
The issue of "close-miking" is something of a different subject relating to the differences in the near and far fields or a sound source (although not entirely irrelevant when considering loudspeakers that form large line arrays, for example).
Adding the room response into the model is not particularly different either, however, just it is more difficult to separate out the information we seek: Resonances are resonances whether they are viewed in time or in frequency.
Once proviso here is that pre-ringing is distinctly more audible than post-ringing (to which most references elude): Advocates of arbitrarily applied linear phase compensation need beware!
But we also need remember that distributed in the time-frequency plane we will also have the energy from non-linear artefacts too. So we either need minimise them or hope they appear as noise.
Where we might look for something more abstract is in considering that real sound sources have harmonics - that is, they are non-linear generators where the non-linear parts are a good thing (that make different instruments playing the same note sound different, for example). So here the non-linear artefacts might have an audible significance if we could discern them appropriately.
The goal of audio recording and reproduction is to replay those harmonics components but without adding any more non-linearities. That is, we seek to linearly represent non-linear information, but seldom is it achieved and additional non-linearity is inevitable.
And maybe to distinguish the perceptual differences in such circumstances we need go beyond the second order spectrum to higher order mappings such as the bispectra. But here we run put of spatial dimensions to usefully portray the results - unless anyone has any good ideas? Nevertheless characterising the perceived "speed" of a kick drum, for example, needs more than just a glance at low frequency ringing.
My point was a little different: when you close-mic two sources and mathematically combine the two frequency response curves, you are doing a different sum than you'd get by using a single microphone at the usual 1-metre distance; firstly, there will be phase differences from path length differences, and secondly, the linear coefficients in the sum (A1*source1 + A2* source2) may not be the same.
What's more, as you move that single microphone around, the acoustically summed frequency response changes continually. None of these frequency responses may match the summed-on-paper, combined close-miked response.
In this situation, there is no actual acoustic impulse response that corresponds to the pretty frequency response plot published by the manufacturer!
This is a misunderstanding. Two different instruments sound different, even when played very lightly - i.e. very much in the linear region of operation - because they inherently have different harmonic spectra. (In some cases, this is also, to varying degrees, under the control of the musician.)
The effect of non-linearity in the instrument is to cause something else entirely - it causes the instruments timbre (harmonic spectrum) to change as the instrument is played harder.In other words, it causes timbre to change with playing dynamics. An acoustic piano, for instance, gradually becomes brighter and harsher sounding as you hit the keys harder.
Manufacturers of electronic pianos struggle to reproduce this - the usual way is to make an audio recording of every key on an acoustic piano, then repeat the entire process with the keys struck a little harder, then repeat the entire process one more time with the keys struck even harder, and so on.
Typical Yamaha electronic pianos, for example, may have four of these "layers", each with a different timbre, so that as you progressively play harder and harder, the timbre goes through four different steps.
In the days when onboard read-only memory was expensive, this was an expensive luxury, only found on expensive instruments, as you had to store a much larger set of sound-fonts in your instrument's firmware.
-Gnobuddy
It is all still linear and all audible artefacts are discernible.
This is exactly what I said, only I was getting at how timbre is perceived and how it might be related to measurement displays in a what is normally conceived to be a linear time-frequency plane.
"Frequency Response" in this circle is commonly understood as the magnitude of the complex frequency domain response. I believe that is what the OP meant by the question, so I answered it in those terms. I probably should have chosen more precise wording to avoid confusion.
Maybe an electronic instrument that uses separate linear frequency generators and then adds them up linearly to apporoximate a real instrument is a linear system. But the real instrument that it is trying to emulate is a non-linear system precisely because the its "harmonic spectra" change with level - which as you say, requires that your linear electronic instrument needs to somehow model separately. But this is off-topic I would suggest.
It mis-usage has unfortunately become more common than the correct usage, so maybe we can start on putting it right here
Well, no, it isn't. Here is exactly what you wrote:
In short, you wrote "real sound sources...are non-linear generators where the non-linear parts make different instruments playing the same note sound different".
This is incorrect. Different instruments do not sound different because they are nonlinear. They sound different because they produce different frequency spectra, different waveforms, because they have different starting transients, because they sustain notes for different durations, because they have different decay times. This is true even when all instruments are played in their linear region.
However, the timbre of one single instrument playing one single note changes with playing dynamics, due to instrument nonlinearity. This is not at all the same thing at all as saying that nonlinearity makes different instruments sound different.
I have no way of knowing what you intended to write. But what you actually wrote is incorrect, and doesn't mean the same thing as what I wrote in post #24.
-Gnobuddy
You're talking about synthesizers. I was not - most electronic pianos do not synthesize the sound, they play back a pre-recorded sample (sound clip) of an actual recording of an actual acoustic piano. This is quite clear if you re-read my post.
Of course it is. Nobody disputes this in the slightest.
However, it is not instrument non-linearity that causes different instruments playing the same note to sound different, as you wrote earlier. That's the point of contention, not whether real-world vibrating systems are nonlinear or not.
-Gnobuddy
It mis-usage has unfortunately become more common than the correct usage, so maybe we can start on putting it right here
I prefer terminology like:
"the response function in the frequency domain"
"the response function in the time domain"
They are both describing the exact same response of a/the system, e.g. H(w) <=> h(t) via fFF and iFT
Kudos for pointing out the language, but I think you will be fighting an uphill battle. Frequency response has long been appropriated to mean the magnitude of the response function in the frequency domain. And when in Rome...
No it is not incorrect. The "linear region" of a non-linear system is an approximation: It is still a non-linear system. The factors you cite above that make instruments sound different are all non-linear elements precisely because the differences they produce cannot be modelled linearly. There is no linear transformation I can think of that can turn a plucked string waveform into one produced by a bowed string vibrating at the same fundamental frequency. It needs instead a switch, which is a good example of a non-linear element.
To hopefully better explain the gist of my thoughts here...
Using a maximum length sequence for measuring a system's impulse response is relatively straightforward. One of its advantages has been thought to be its immunity to non-linear elements, where essentially the pseudo random nature of the maximum length sequence randomises the phases of the harmonic components and what is left of them in a magnitude response appears largely as noise.
But when using an impulse instead as the stimulus, the harmonics and non-linear interactions will not appear as noise. So for helping with the kick drum example, we might have something in the measure that correlates with the combination of low frequency ringing and the higher frequency strike of the pedal on the drum. And in such a measure we might then find a way to better model audible transient phenomena such as the perception of "speed" that goes a little deeper than just looking at resonances ringing.
Quite possibly it is a nonsense idea, however. Quite possibly identifying all the ringing resonances is sufficient
Using a maximum length sequence for measuring a system's impulse response is relatively straightforward. One of its advantages has been thought to be its immunity to non-linear elements, where essentially the pseudo random nature of the maximum length sequence randomises the phases of the harmonic components and what is left of them in a magnitude response appears largely as noise.
But when using an impulse instead as the stimulus, the harmonics and non-linear interactions will not appear as noise. So for helping with the kick drum example, we might have something in the measure that correlates with the combination of low frequency ringing and the higher frequency strike of the pedal on the drum. And in such a measure we might then find a way to better model audible transient phenomena such as the perception of "speed" that goes a little deeper than just looking at resonances ringing.
Quite possibly it is a nonsense idea, however. Quite possibly identifying all the ringing resonances is sufficient
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After some digging around in old text books today, I think we have stumbled upon a world of abundant confusion - my own included...
Musical instruments are deemed to have overtones that are called harmonics in the frequency domain.
A non-linear mathematical function exhibits elements in the frequency domain that are also called harmonics.
Non-linearities in audio systems often describe distortion mechanisms where harmonics are uniformly distributed in frequency. But for instruments, say a circular drum, the harmonics are not so conveniently distributed.
Nevertheless, if we beat a drum, we have a transient that causes the drum to vibrate (and sound). But different harmonics of the drum are excited to different extents depending on, for examples, if we use a stick or a brush, where we hit the skin, how hard we hit it and if the drum has a pillow inside it.
As such characterising the drum will need a non-linear mathematical description.
For a musician, however, each instance might be regarded as a different expression of their art wherein overtones are selected by the player (who is then the non-linear element?).
I also need say that my post was also rather slack as it invites one to think of two instruments that are nominally the same, say two cellos, but sound different because they produce different overtones when played identically. That was not my intended meaning, so my apologies.
But as my last post hinted, the non-linear elements might be irrelevant to the audible transient phenomena that this thread is about anyway...
That was my take on the conversation, that the nature of the musical source for the recording doesn't have much, if anything, to do with what we want our playback system to do. Different animals imho.
Give me impulse perfect, ie mag and phase perfect, all the time as a start....
...so i can move on to the 'easy stuff' like pattern control and room integration LoL
Good stuff, Soundbloke
Sadly such an aspiration is not possible. Whilst equalisation can (with care) provide for a flat magnitude reponse over a sufficiently wide bandwidth, the non-infinite nature of that bandwidth necessitates a non-zero phase response. But all is not lost if we realise that we don't need a "flat" zero phase response, just a uniform one - that is, an angled straight line in the phase response that represents a "linear phase" response - or in better words, a time delay that is the same for all frequencies.
In essence then, aspiring instead to a delayed perfect-but-band-limited impulse seems to be a good idea. Sadly again, however, more compromise is inevitable...
I repeat my previous assertion that phase compensation of the low frequency roll-off is beneficial, but seldom done for reasons of practicality and because of more dominant issues with room modes. And the phase properties of room responses often prevent their beneficial compensation too. There is also a further complication that arises with stereo. Even linear phase crossovers can make matters worse rather than better in certain applications.
So we are left with an engineering compromise where in certain aspects, a minimum phase response might be audibly superior to the linear phase one and our aspirational impulse suffers further - albeit for our audible benefit.
I totally agree all we need is a smoothly transitioning phase trace. Prosound high $$$ Meyer has been doing that for decades, making all their products have the same signature smooth sloping trace. Using analog too, btw.That said, I've been having great success achieving near perfect mag and phase, and hence impulse, certainly on-axis and for the most part holding up off-axis, with today's FIR tools.
I think if there is such as thing as a consensus about flat phase audibility, no doubt not haha, it is that it helps the most down low...which only makes sense because that is where group delay rises so dramatically.
Anyway, with 43 ms worth of FIR delay, FIR has let me stay flat phase until the bottom 1 or 1.5 octaves. I run high power stuff, and so help me, sometimes it sounds like it can literally drive nails.
I am not sure that is possible - it would certainly be interesting to see how it was done if so. And I am not sure the delays we are talking of would be appropriate in any live monitoring application.
All discussion I know of has concerned only linear phase crossover implementations - which, for good reasons, are not always as good as their minimum phase counterparts. Those reasons do not apply to the low frequency roll-off.
There are also applications at much higher frequencies, such as in stereo shuffling, where linear phase filters are clearly preferable. I would even suggest that a lack of such filtering is why what should be a fixture of stereo has been absent for so many decades.
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