Re: [music-dsp] Did anybody here think about signal integrity
On 2015-06-08, Ethan Duni wrote: But both of these are overkill for day-to-day engineering practice. Especially so, because you can treat the distributional framework as a black box. It has a certain number of rules. If you follow the rules, you'll land with a nice calculus which fully encompasses all of the useful stuff like Dirac impulses. So, as an engineer, you should probably just learn the rules and utilize the stuff; the math folks did the heavy lifting ages ago, and assured you're safe, so why tinker with the internals? 'Cause the stuff *just* *works* *beautifully*. Take it as manna from heaven. In fact in DSP work I'd even say ditch the continuous time wonkiness as a whole. If you do everything you do by the bandlimited recipe, that works as well. Once you know what a sinc(x) pulse looks like, you can freely substitute it within the bandlimit for the Dirac one; the strict mathematical homology from the distributional framework to the theory within a bandlimit *is* there, in the background, so that you don't have to worry. Your math won't blow up. The *only* time you actually have to worry about the continuous time spectrum of a signal are those individual special functions, like FM, usually nonlinear. But even those have standard texts which deal with their approximation theory within the bandlimited framework. If you aim to get things done, as I think an engineers usually does, why the hell bother with more? -- Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
Now the assignment is as follows: can we, given the output signal coming from our filter which was fed the input signal, and the filter coefficients, compute the input signal ? Invertible digital filters are invertible, up to numerical precision. Are you wanting to talk about finite word length effects? Otherwise I don't see what you're trying to get at. Nobody is going to take up homework assignments from you, so I suggest that you simply try to state your thoughts directly. I will not answer to this thread further, unless directly asked to, because I think I've made my point clear enough. It seems somewhat bad form to exit a thread you started after your second post, but regardless, it remains unclear what your point is supposed to be. Also, I'd appreciate it if you'd refrain from casting aspersions on the educations and intellects of others (even though it's not clear who you are insulting, exactly). That's not a pleasant, constructive way to interact, and the music dsp list has an admirable history of being a welcoming, productive place. E On Mon, Jun 8, 2015 at 6:06 AM, Theo Verelst theo...@theover.org wrote: Clearly, there's very little knowledge around the basic mathematical proofs underpinning a decent undergrad engineering course. Prisms understand fine what the Fourier transform is, and isn't. Maybe there's an interest in this: http://mathworld.wolfram.com/FourierTransformExponentialFunction.html . Some people language is so full of mistakes an misinterpretations, I suggest a good university undergrad might solve that to understand the difference of endless hinein interpretations, and decent, usable theory. So that's a big njet on that stream of incoherent attempts to lay claim on certain long and well known theoretics that range from 19th century mathematics, physics, to in slightly different form, electronics. Also, it's not fair to change the direction of the question. I understand as a practical working software employee, there's a sort of conservative thought about one's credibility, but these theories aren't saying that the course that this place an many other modernistic DSP tracks took is a particularly good track to solve the problems at hand. A simple example of *my* point, not going into a quantification of the the rather basic errors I've tried to bring to the attention (even though that is an interesting exercise), how about we make a digital filter, let's say applied to a pretty decent input signal (for instance a number of added sine waves, possibly, if that satisfies some people more, including a step function at t=0) that simply approximates a first order high-cut filter. We could take a FIR or IIR implementation, and we could chose a shelving filter (so at inf frequency, there's still signal) or an approximation of a first order low pass in electronics (which at infinite frequency lets no signal though). Now the assignment is as follows: can we, given the output signal coming from our filter which was fed the input signal, and the filter coefficients, compute the input signal ? (Part of the answer is you have to contemplate on the full problem, so that preferably we would get (minus a signal shift we can ignore) the *exact* input signal. This might include an huge matrix inversion problem, as one of the possible solutions. Just taking the opposite filter leads to an understanding of why I do not like any digital processing unless the sampling frequency is pretty high, and a lot of signal and DSP precautions are taken, and a lot of signal integrity issues are involved in the big picture). I will not answer to this thread further, unless directly asked to, because I think I've made my point clear enough. T. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
If you try to take the Fourier transform integral of a exp(j*omega_0*t), it will not converge in the sense, how an improper integral's convergence is usually understood. You will need to employ something like Cauchy principal value or Cesaro convergence to make it converge to zero at omega!=omega_0. At omega=omega_0 the integral diverges no matter in which sense you take it. So, strictly speaking, Fourier transform of a sine doesn't exist. Usually understood by who? If you are interested in mathematical rigor, you define the Fourier transform in terms of distributions, and it has no trouble at all handling sinusoids. This is the Schwarz stuff that Sampo mentioned. Another option is to use non-standard analysis and work in terms of hyperreal numbers. Then you can have your Dirac delta functions and your rigor! But both of these are overkill for day-to-day engineering practice. E On Mon, Jun 8, 2015 at 1:50 AM, Vadim Zavalishin vadim.zavalis...@native-instruments.de wrote: If you try to take the Fourier transform integral of a exp(j*omega_0*t), it will not converge in the sense, how an improper integral's convergence is usually understood. You will need to employ something like Cauchy principal value or Cesaro convergence to make it converge to zero at omega!=omega_0. At omega=omega_0 the integral diverges no matter in which sense you take it. So, strictly speaking, Fourier transform of a sine doesn't exist. An equivalent look to this from the inverse transform's side is that the spectrum of the sine is a Dirac delta function, which is not a function in the normal sense. So, none of the statements of the Fourier transform theory (including the sampling theorem, which assumes the existence of the Fourier transform), taken rigorously, seem to apply to the sinusoidal signals. Regards, Vadim On 08-Jun-15 10:35, Victor Lazzarini wrote: Not sure I understand this sentence. As far as I know the FT is defined as an integral between -inf and +inf, so I am not quite sure how it cannot capture infinite-lenght sinusoidal signals. Maybe you meant something else? (I am not being difficult, just trying to understand what you are trying to say). Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 On 8 Jun 2015, at 08:19, vadim.zavalishin vadim.zavalis...@native-instruments.de wrote: It might seem that such signals are unimportant, however even the infinite sinusoidal signals, including DC, cannot be treated by the sampling theorem, since the Dirac delta (which is considered as their Fourier transform) is not a function in a normal sense and strictly speaking Fourier transform doesn't exist for these signals. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- Vadim Zavalishin Reaktor Application Architect | RD Native Instruments GmbH +49-30-611035-0 www.native-instruments.com -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
Hi,
I was asking a very similar question about one year ago on this list in
the context of the BLEP theory. The point is that the samping theorem is
incomplete compared to how we would like to (and do) use it in the
signal processing. The sampling theorem applies only to the class
signals which do have Fourier transform, separating this class further
into bandlimited and non-bandlimited. However, it doesn't say anything
about the signals which do not have Fourier transform.
It might seem that such signals are unimportant, however even the
infinite sinusoidal signals, including DC, cannot be treated by the
sampling theorem, since the Dirac delta (which is considered as their
Fourier transform) is not a function in a normal sense and strictly
speaking Fourier transform doesn't exist for these signals.
This can be further extrapolated to the polynomial signal case (which
are implicitly assumed bandlimited in the BLEP, BLAMP and higher order
derivative discontinuity approaches). Other signals (such as exponential
function, FM'ed sine etc) are also interesting.
So, the sampling rate theorem doesn't work for this kind of signals, in
the sense of giving no answer whether they are bandlimited or not.
Therefore we would like to have a generalization of the sampling
theorem, which includes these signals.
The question which I was asking didn't deal with all possible
imperfections of DACs, but this would be a reasonable generalization of
my question. We would be looking for a generalization of the sampling
theorem in a form of something like the following statement.
Let f(t) be a signal such that the statement A(f) holds, let F[n] be its
naively sampled counterpart and and let {D_n} be a sequence of DACs such
that the statement B({D_n}) holds. Then the sequence D_n(F) converges to
f.
The question is what are the statements A and B and how to understand
the convergence of D_n(F) to f (where particulary some nonzero DC offset
seem to be emerging in certain cases, but we probably woudln't care)
I made some initial conjectures in this regard in the mentioned thread,
where A had to do with the rolloff speed of the function's derivatives
and D_n was just a sequence of time-windowed sincs with increasing
window size.
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect | RD
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
Theo Verelst писал 2015-06-03 22:47:
Hi,
Playing with analog and digital processing, I came to the conclusion
I'd like to contemplate about certain digital signal processing
considerations, I'm sure have been in the minds of pioneering people
quite a while ago, concerning let's say how accurate theoretically and
practically all kinds of basic DSP subjects really are.
For instance, I care about what happens with a perfect sine wave
getting either digitized or mathematically and with an accurate
computer program put into a sequence of signal samples. When a close
to perfect sample (in the sense of a list of signal samples) gets
played over a Digital to Analog Converter, how perfect is the analog
signal getting out of there? And if it isn't all perfect, where are
the errors?
As a very crude thinking example, suppose a square wave oscillator
like in a synthesizer or an electronic circuit test generator is
creating a near perfect square wave, and it is also digitized or an
attempt is made in software to somehow turn the two voltages of the
square wave into samples.
Maybe a more reasonable idea is to take into account what a DAC will
do with the signal represented in the samples that are taken as music,
speech, a musical instrument's tones, or sound effects. For instance,
what does the digital reconstruction window and the build in
oversampling make of a exponential curve (like the part of an
envelope could easily be) with it's given (usually FIR) filter length.
In that context, you could wonder what happens if we shift a given
exponential signal (or signal component) by half a sample ? Add to
the consideration that a function a*exp(b*x+c) defines a unique
function for each a,b and c.
Anyone here think and/or work on these kinds of subjects, I'd like to
hear. (I think it's an interesting subject, so I'm serious about it)
T. Verelst
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive, list archive, book
reviews, dsp links
http://music.columbia.edu/cmc/music-dsp
http://music.columbia.edu/mailman/listinfo/music-dsp
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive, list archive, book reviews, dsp
links
http://music.columbia.edu/cmc/music-dsp
http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
IIRC, the discussion back then covered some topics like distortions created with polynomial functions, etc. Although DC isn’t a real problem in practical applications, there are many cases, which are hard to predict, if they cause aliasing. A good example is FM, which spectra can be predicted using Bessel functions, but who wants to do that? Or wave shaping using atan() or other transcendent functions. I think, there was a conclusion, that there are cases, which aren’t covered so well by theory, but the impact on the application can be overseen. Steffan On 08.06.2015|KW24, at 10:35, Victor Lazzarini victor.lazzar...@nuim.ie wrote: Not sure I understand this sentence. As far as I know the FT is defined as an integral between -inf and +inf, so I am not quite sure how it cannot capture infinite-lenght sinusoidal signals. Maybe you meant something else? (I am not being difficult, just trying to understand what you are trying to say). Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 On 8 Jun 2015, at 08:19, vadim.zavalishin vadim.zavalis...@native-instruments.de mailto:vadim.zavalis...@native-instruments.de wrote: It might seem that such signals are unimportant, however even the infinite sinusoidal signals, including DC, cannot be treated by the sampling theorem, since the Dirac delta (which is considered as their Fourier transform) is not a function in a normal sense and strictly speaking Fourier transform doesn't exist for these signals. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
If you try to take the Fourier transform integral of a exp(j*omega_0*t), it will not converge in the sense, how an improper integral's convergence is usually understood. You will need to employ something like Cauchy principal value or Cesaro convergence to make it converge to zero at omega!=omega_0. At omega=omega_0 the integral diverges no matter in which sense you take it. So, strictly speaking, Fourier transform of a sine doesn't exist. An equivalent look to this from the inverse transform's side is that the spectrum of the sine is a Dirac delta function, which is not a function in the normal sense. So, none of the statements of the Fourier transform theory (including the sampling theorem, which assumes the existence of the Fourier transform), taken rigorously, seem to apply to the sinusoidal signals. Regards, Vadim On 08-Jun-15 10:35, Victor Lazzarini wrote: Not sure I understand this sentence. As far as I know the FT is defined as an integral between -inf and +inf, so I am not quite sure how it cannot capture infinite-lenght sinusoidal signals. Maybe you meant something else? (I am not being difficult, just trying to understand what you are trying to say). Dr Victor Lazzarini Dean of Arts, Celtic Studies and Philosophy, Maynooth University, Maynooth, Co Kildare, Ireland Tel: 00 353 7086936 Fax: 00 353 1 7086952 On 8 Jun 2015, at 08:19, vadim.zavalishin vadim.zavalis...@native-instruments.de wrote: It might seem that such signals are unimportant, however even the infinite sinusoidal signals, including DC, cannot be treated by the sampling theorem, since the Dirac delta (which is considered as their Fourier transform) is not a function in a normal sense and strictly speaking Fourier transform doesn't exist for these signals. -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- Vadim Zavalishin Reaktor Application Architect | RD Native Instruments GmbH +49-30-611035-0 www.native-instruments.com -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
On 08-Jun-15 15:06, Theo Verelst wrote: Clearly, there's very little knowledge around the basic mathematical proofs underpinning a decent undergrad engineering course. Prisms understand fine what the Fourier transform is, and isn't. Maybe there's an interest in this: http://mathworld.wolfram.com/FourierTransformExponentialFunction.html . Clearly this is not the exponential signal which I was referring to. This is also a clear indication of the limitation of the sampling theorem I was referring to: since it's not possible to take Fourier transform of an exponential function, people refer to some other function as the exponential function in the context of the Fourier transform :) Anyway, since Theo expressed the wish not to change the direction of the thread, let's not stick to the question that I brought up. Although, if there is a renewed interest to discuss this aspect, I guess, creating a new thread could be an appropriate way to go. Regards, Vadim -- Vadim Zavalishin Reaktor Application Architect | RD Native Instruments GmbH +49-30-611035-0 www.native-instruments.com -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
On 2015-06-08, vadim.zavalishin wrote: The sampling theorem applies only to the class signals which do have Fourier transform, separating this class further into bandlimited and non-bandlimited. However, it doesn't say anything about the signals which do not have Fourier transform. Correct. But the class of signals which do have a Fourier theory is much, *much* larger than you appear to think. Usually continuous time Fourier theory is set in the class of tempered distributions, which does include such things as Dirac impulses, trains of them with suitably regular supports, and of course shifts, sums, scalings and derivatives of all finite orders, and so on. Not everything you could possibly ever want -- products are iffy, for instance -- but everything you need in order to bandlimit the things and get the most usual forms of converence in norm we're interested in. After you get that theory, you'll notice that the class of bandlimited functions is in fact dense in the whole set, and a clean approximation theory surfaces when you just progressively raise the bandlimit. When you combine that with a bit of physics and information theory, you get that any realistic signal -- approximations of impulses included -- can be systematically reduced to a class of bandlimited, finite energy signals which are indistinguishable from each other, and indeed admit the Nyquist-Shannon theorem. It might seem that such signals are unimportant, however even the infinite sinusoidal signals, including DC, cannot be treated by the sampling theorem, since the Dirac delta (which is considered as their Fourier transform) is not a function in a normal sense and strictly speaking Fourier transform doesn't exist for these signals. Indeed it does exist, as the delta distribution. That result is exact, and furthermore fits within a perfectly formal mathematical theory which includes convolution with signals in the Schwartz class. That theory in fact admits a straight forward representation of sampling as a multiplication by the Dirac comb, which by itself has far nastier convergence properties than anything you're likely to have seen till now. That framework is *really* useful for deriving connections between common operations, like, for instance, the four common forms of Fourier theory (FT, DTFT, FS, DFT), bandlimiting, differentiation, Hilbert transform, odd-even decompositions, the uncertainty principle, the central limit theorem, and so on. They just kind of naturally drop out of the framework once you admit such monstrosities as convolutions with a Dirac comb or the derivative of an impulse. (So, e.g. convolution with the impulse is an identity operator, which leads convolution with the derivative of an impulse being the derivative, which you can then project to a bandlimited subspace, (roughly) odd-even decompose and sample, leading to discrete derivation being the composition of a Hilbert transformer and a linear phase 3dB/octave highpass. That shit just goes on, and on, and on, especially when you plug in some complex analysis for your bandlimited (and so necessarily real-analytic) signals.) This can be further extrapolated to the polynomial signal case (which are implicitly assumed bandlimited in the BLEP, BLAMP and higher order derivative discontinuity approaches). They are not implicitly assumed anything, but instead a) the part of them which lands outside of your bandlimit are carefully bounded to where aliasing is perceptually harmless, or b) they're used so that they're fed bandlimited signals, which via the theory of Chebyshev polynomials and the usual properties of the full Fourier transform lead to exactly as many fold band expansion as the order of the polynomial used has. That stuff then works perfectly fine with oversampling. Other signals (such as exponential function, FM'ed sine etc) are also interesting. Sure. And all that is well understood. The exponential function is easy because of its connection via the Euler equation to sinusoids, and via the usual energy norm, to the second order theory of the Gaussian. FM, that's usually expanded in the terms of Bessel functions -- nasty, but in the narrow band, small modulation index limit, again well understood from decade of radio work. So, the sampling rate theorem doesn't work for this kind of signals, in the sense of giving no answer whether they are bandlimited or not. The latter two aren't. But they do possess Fourier transforms and once we have that, we can study which portion of their spectrum goes outside of any given bandlimit. Therefore we would like to have a generalization of the sampling theorem, which includes these signals. You can't really have any general version of such, because the class of general signals is just too big to admit such a discrete basis. But certain special cases can in fact be handled. If you want to see about that stuff, the compressed sampling
Re: [music-dsp] Did anybody here think about signal integrity
Theo, Any continuous function bandlimited to frequencies fs/2 is completely determined by its samples. That’s the essence of the sampling theorem, which answers all your questions. Stefan On 03 Jun 2015, at 22:47 , Theo Verelst theo...@theover.org wrote: Hi, Playing with analog and digital processing, I came to the conclusion I'd like to contemplate about certain digital signal processing considerations, I'm sure have been in the minds of pioneering people quite a while ago, concerning let's say how accurate theoretically and practically all kinds of basic DSP subjects really are. For instance, I care about what happens with a perfect sine wave getting either digitized or mathematically and with an accurate computer program put into a sequence of signal samples. When a close to perfect sample (in the sense of a list of signal samples) gets played over a Digital to Analog Converter, how perfect is the analog signal getting out of there? And if it isn't all perfect, where are the errors? As a very crude thinking example, suppose a square wave oscillator like in a synthesizer or an electronic circuit test generator is creating a near perfect square wave, and it is also digitized or an attempt is made in software to somehow turn the two voltages of the square wave into samples. Maybe a more reasonable idea is to take into account what a DAC will do with the signal represented in the samples that are taken as music, speech, a musical instrument's tones, or sound effects. For instance, what does the digital reconstruction window and the build in oversampling make of a exponential curve (like the part of an envelope could easily be) with it's given (usually FIR) filter length. In that context, you could wonder what happens if we shift a given exponential signal (or signal component) by half a sample ? Add to the consideration that a function a*exp(b*x+c) defines a unique function for each a,b and c. Anyone here think and/or work on these kinds of subjects, I'd like to hear. (I think it's an interesting subject, so I'm serious about it) T. Verelst -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
Well, bandlimited to a bandwidth fs/2 (but the distinction isn't useful for audio), and given perfect reconstruction circuitry. But as far as I can gather, Theo's concern is what happens when, as is inevitable in practice, the reconstruction circuitry is imperfect? And that is an interesting question, academically speaking, if there really isn't a body of theory to cover reconstructive imperfection - if only to be certain that all the improvements made to DAC technology in the last three decades have actually been _improvements_. But I think all of this is probably covered by existing research anyway. Minimising phase disruption in digital filters is well understood; LSB errors and resistor chain nonlinearities are fairly obvious, sources of relatively predictable badness, and can be assessed in the same way as nonlinearity in general; clock jitter is easy to simulate... and then there are analogue reconstruction filters. Unless I've missed any, I don't think there's anything else to look at, unless he wants to disprove or augment Nyquist-Shannon. Which would be an achievement, true, but... I would humbly submit that there might be more fruitful avenues towards seeing Verelst theorem in the indices of 22nd-century audio textbooks. Still, I understand great white whales all too well; and if Theo _needs_ to harpoon this one, we should perhaps not stand in his way. On 05/06/2015, Stefan Stenzel stefan.sten...@waldorfmusic.de wrote: Theo, Any continuous function bandlimited to frequencies fs/2 is completely determined by its samples. That’s the essence of the sampling theorem, which answers all your questions. Stefan On 03 Jun 2015, at 22:47 , Theo Verelst theo...@theover.org wrote: Hi, Playing with analog and digital processing, I came to the conclusion I'd like to contemplate about certain digital signal processing considerations, I'm sure have been in the minds of pioneering people quite a while ago, concerning let's say how accurate theoretically and practically all kinds of basic DSP subjects really are. For instance, I care about what happens with a perfect sine wave getting either digitized or mathematically and with an accurate computer program put into a sequence of signal samples. When a close to perfect sample (in the sense of a list of signal samples) gets played over a Digital to Analog Converter, how perfect is the analog signal getting out of there? And if it isn't all perfect, where are the errors? As a very crude thinking example, suppose a square wave oscillator like in a synthesizer or an electronic circuit test generator is creating a near perfect square wave, and it is also digitized or an attempt is made in software to somehow turn the two voltages of the square wave into samples. Maybe a more reasonable idea is to take into account what a DAC will do with the signal represented in the samples that are taken as music, speech, a musical instrument's tones, or sound effects. For instance, what does the digital reconstruction window and the build in oversampling make of a exponential curve (like the part of an envelope could easily be) with it's given (usually FIR) filter length. In that context, you could wonder what happens if we shift a given exponential signal (or signal component) by half a sample ? Add to the consideration that a function a*exp(b*x+c) defines a unique function for each a,b and c. Anyone here think and/or work on these kinds of subjects, I'd like to hear. (I think it's an interesting subject, so I'm serious about it) T. Verelst -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
[music-dsp] Did anybody here think about signal integrity
Hi, Playing with analog and digital processing, I came to the conclusion I'd like to contemplate about certain digital signal processing considerations, I'm sure have been in the minds of pioneering people quite a while ago, concerning let's say how accurate theoretically and practically all kinds of basic DSP subjects really are. For instance, I care about what happens with a perfect sine wave getting either digitized or mathematically and with an accurate computer program put into a sequence of signal samples. When a close to perfect sample (in the sense of a list of signal samples) gets played over a Digital to Analog Converter, how perfect is the analog signal getting out of there? And if it isn't all perfect, where are the errors? As a very crude thinking example, suppose a square wave oscillator like in a synthesizer or an electronic circuit test generator is creating a near perfect square wave, and it is also digitized or an attempt is made in software to somehow turn the two voltages of the square wave into samples. Maybe a more reasonable idea is to take into account what a DAC will do with the signal represented in the samples that are taken as music, speech, a musical instrument's tones, or sound effects. For instance, what does the digital reconstruction window and the build in oversampling make of a exponential curve (like the part of an envelope could easily be) with it's given (usually FIR) filter length. In that context, you could wonder what happens if we shift a given exponential signal (or signal component) by half a sample ? Add to the consideration that a function a*exp(b*x+c) defines a unique function for each a,b and c. Anyone here think and/or work on these kinds of subjects, I'd like to hear. (I think it's an interesting subject, so I'm serious about it) T. Verelst -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
Also a good starting place for beginners are the xiph show-and-tell videos (probably been posted here before, but whatever): https://xiph.org/video/vid2.shtml E On Wed, Jun 3, 2015 at 3:05 PM, Ethan Duni ethan.d...@gmail.com wrote: Perfect sinusoids/square waves/etc. only exist as mathematical abstractions. A good starting point would be to get a feel for what, say, the square wave coming out of an analog synthesizer actually looks like - the noise floor, the distribution of harmonics, frequency jitter, under/overshoot, etc. I think you'll find that modern digital equipment generally can produce much, much closer approximations to these ideals than we ever see from analogue synthesizers. But more generally the answers to your questions depend on the specifics of the A/D/As in question. There are many different designs with different oversampling and interpolation stages, different noise shaping strategies, etc. So the question of where the quantization noise ends up doesn't have a general answer - it depends in on both the design of the A/D/A components, and on the specifics of the signal in question (these are nonlinear, time-variant operations we're talking about). Likewise, the effect of applying a half-sample delay depends heavily on how you do that. If you're talking about signals that are simply computed directly or delay in the analog domain, then you'll simply see a half-sample delay at the output. But if you're talking about sampling a signal and applying a half-sample delay, then it depends on what technique you use to implement that delay. Also, note that the function a*exp(b*x+c) is not, by itself, a well-defined test signal, since it has infinite energy. The sampling theorem does not apply to such signals in the first place. You would need to apply some kind of window/truncation to ensure finite energy and bandlimiting. More generally, these issues are all quite well studied and understood, and not particularly relevant to music-dsp as such. So I'm not sure what is the point of repeatedly bringing them up in this context. It sounds like what you want is more like a book or class on mixed-signal circuits. The main reference I use for that is Analog Integrated Circuit Design by Johns and Martin. It's a bit dated by now, so it's not going to cover the latest-greatest converter designs (not sure that any book really does), but the chapters on the noise modeling, theory of oversampled conversion, etc. are all still valid for learning the theoretical underpinnings of this stuff. E On Wed, Jun 3, 2015 at 1:47 PM, Theo Verelst theo...@theover.org wrote: Hi, Playing with analog and digital processing, I came to the conclusion I'd like to contemplate about certain digital signal processing considerations, I'm sure have been in the minds of pioneering people quite a while ago, concerning let's say how accurate theoretically and practically all kinds of basic DSP subjects really are. For instance, I care about what happens with a perfect sine wave getting either digitized or mathematically and with an accurate computer program put into a sequence of signal samples. When a close to perfect sample (in the sense of a list of signal samples) gets played over a Digital to Analog Converter, how perfect is the analog signal getting out of there? And if it isn't all perfect, where are the errors? As a very crude thinking example, suppose a square wave oscillator like in a synthesizer or an electronic circuit test generator is creating a near perfect square wave, and it is also digitized or an attempt is made in software to somehow turn the two voltages of the square wave into samples. Maybe a more reasonable idea is to take into account what a DAC will do with the signal represented in the samples that are taken as music, speech, a musical instrument's tones, or sound effects. For instance, what does the digital reconstruction window and the build in oversampling make of a exponential curve (like the part of an envelope could easily be) with it's given (usually FIR) filter length. In that context, you could wonder what happens if we shift a given exponential signal (or signal component) by half a sample ? Add to the consideration that a function a*exp(b*x+c) defines a unique function for each a,b and c. Anyone here think and/or work on these kinds of subjects, I'd like to hear. (I think it's an interesting subject, so I'm serious about it) T. Verelst -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Did anybody here think about signal integrity
Perfect sinusoids/square waves/etc. only exist as mathematical abstractions. A good starting point would be to get a feel for what, say, the square wave coming out of an analog synthesizer actually looks like - the noise floor, the distribution of harmonics, frequency jitter, under/overshoot, etc. I think you'll find that modern digital equipment generally can produce much, much closer approximations to these ideals than we ever see from analogue synthesizers. But more generally the answers to your questions depend on the specifics of the A/D/As in question. There are many different designs with different oversampling and interpolation stages, different noise shaping strategies, etc. So the question of where the quantization noise ends up doesn't have a general answer - it depends in on both the design of the A/D/A components, and on the specifics of the signal in question (these are nonlinear, time-variant operations we're talking about). Likewise, the effect of applying a half-sample delay depends heavily on how you do that. If you're talking about signals that are simply computed directly or delay in the analog domain, then you'll simply see a half-sample delay at the output. But if you're talking about sampling a signal and applying a half-sample delay, then it depends on what technique you use to implement that delay. Also, note that the function a*exp(b*x+c) is not, by itself, a well-defined test signal, since it has infinite energy. The sampling theorem does not apply to such signals in the first place. You would need to apply some kind of window/truncation to ensure finite energy and bandlimiting. More generally, these issues are all quite well studied and understood, and not particularly relevant to music-dsp as such. So I'm not sure what is the point of repeatedly bringing them up in this context. It sounds like what you want is more like a book or class on mixed-signal circuits. The main reference I use for that is Analog Integrated Circuit Design by Johns and Martin. It's a bit dated by now, so it's not going to cover the latest-greatest converter designs (not sure that any book really does), but the chapters on the noise modeling, theory of oversampled conversion, etc. are all still valid for learning the theoretical underpinnings of this stuff. E On Wed, Jun 3, 2015 at 1:47 PM, Theo Verelst theo...@theover.org wrote: Hi, Playing with analog and digital processing, I came to the conclusion I'd like to contemplate about certain digital signal processing considerations, I'm sure have been in the minds of pioneering people quite a while ago, concerning let's say how accurate theoretically and practically all kinds of basic DSP subjects really are. For instance, I care about what happens with a perfect sine wave getting either digitized or mathematically and with an accurate computer program put into a sequence of signal samples. When a close to perfect sample (in the sense of a list of signal samples) gets played over a Digital to Analog Converter, how perfect is the analog signal getting out of there? And if it isn't all perfect, where are the errors? As a very crude thinking example, suppose a square wave oscillator like in a synthesizer or an electronic circuit test generator is creating a near perfect square wave, and it is also digitized or an attempt is made in software to somehow turn the two voltages of the square wave into samples. Maybe a more reasonable idea is to take into account what a DAC will do with the signal represented in the samples that are taken as music, speech, a musical instrument's tones, or sound effects. For instance, what does the digital reconstruction window and the build in oversampling make of a exponential curve (like the part of an envelope could easily be) with it's given (usually FIR) filter length. In that context, you could wonder what happens if we shift a given exponential signal (or signal component) by half a sample ? Add to the consideration that a function a*exp(b*x+c) defines a unique function for each a,b and c. Anyone here think and/or work on these kinds of subjects, I'd like to hear. (I think it's an interesting subject, so I'm serious about it) T. Verelst -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
