Re: [music-dsp] Sampling theory "best" explanation
It is a most fascinating thread. The more one looks into it, the more one has to marvel that the process works at all. Richard Dobson On 07/09/2017 07:16, Nigel Redmon wrote: Somehow, combining the term "rat's ass" with a clear and concise explanation of your viewpoint makes it especially satisfying. ... ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theory "best" explanation
Somehow, combining the term "rat's ass" with a clear and concise explanation of your viewpoint makes it especially satisfying. > On Sep 7, 2017, at 11:57 AM, robert bristow-johnson > wrote: > > > > Original Message > Subject: Re: [music-dsp] Sampling theory "best" explanation > From: "Ethan Duni" > Date: Wed, September 6, 2017 4:49 pm > To: "robert bristow-johnson" > "A discussion list for music-related DSP" > -- > > > rbj wrote: > >>what do you mean be "non-ideal"? that it's not an ideal brick wall LPF? > > it's still LTI if it's some other filter **unless** you're meaning that > > the possible aliasing. > > > > Yes, that is exactly what I am talking about. LTI systems cannot produce > > aliasing. > > > > Without an ideal bandlimiting filter, resampling doesn't fulfill either > > definition of time invariance. Not the classic one in terms of sample > > shifts, and not the "common real time" one suggested for multirate cases. > > > > It's easy to demonstrate this by constructing a counterexample. Consider > > downsampling by 2, and an input signal that contains only a single sinusoid > > with frequency above half the (input) Nyquist rate, and at a frequency that > > the non-ideal bandlimiting filter fails to completely suppress. To be LTI, > > shifting the input by one sample should result in a half-sample shift in > > the output (i.e., bandlimited interpolation). But this doesn't happen, due > > to aliasing. This becomes obvious if you push the frequency of the input > > sinusoid close to the (input) Nyquist frequency - instead of a half-sample > > shift in the output, you get negation! > > > >>we draw the little arrows with different heights and we draw the impulses > > scaled with samples of negative value as arrows pointing down > > > > But that's just a graph of the discrete time sequence. > > well, even if the *information* necessary is the same, a graph of x[n] need > only be little dots, one per sample. or discrete lines (without arrowheads). > > but the use of the symbol of an arrow for an impulse is a symbol of something > difficult to graph for a continuous-time function (not to be confused with a > continuous function). if the impulse heights and directions (up or down) are > analog to the sample value magnitude and polarity, those graphing object > suffice to depict these *hypothetical* impulses in the continuous-time domain. > > > > > >>you could do SRC without linear interpolation (ZOH a.k.a. "drop-sample") > > but you would need a much larger table > >>(if i recall correctly, 1024 times larger, so it would be 512Kx > > oversampling) to get the same S/N. if you use 512x > >>oversampling and ZOH interpolation, you'll only get about 55 dB S/N for an > > arbitrary conversion ratio. > > > > Interesting stuff, it didn't occur to me that the SNR would be that low. > > How do you estimate SNR for a particular configuration (i.e., target > > resampling ratio, fixed upsampling factor, etc)? Is that for ideal 512x > > resampling, or does it include the effects of a particular filter design > > choice? > > this is what Duane Wise and i ( > https://www.researchgate.net/publication/266675823_Performance_of_Low-Order_Polynomial_Interpolators_in_the_Presence_of_Oversampled_Input > ) were trying to show and Olli Niemitalo (in his pink elephant paper > http://yehar.com/blog/wp-content/uploads/2009/08/deip.pdf ). > > so let's say that you're oversampling by a factor of R. if the sample rate > is 96 kHz and the audio is limited to 20 kHz, the oversampling ratio is 2.4 . > but now imagine it's *highly* oversampled (which we can get from polyphase > FIR resampling) like R=32 or R=512 or R=512K. > > when it's upsampled (like hypothetically stuffing 31 zeros or 511 zeros or > (512K-1) zeros into the stream and brick-wall low-pass filtering) then the > spectrum has energy at the baseband (from -Nyquist to +Nyquist of the > original sample rate, Fs) and is empty for 31 (or 511 or (512K-1)) image > widths of Nyquist, and the first non-zero image is at 32 or 512 or 512K x Fs. > > now if you're drop-sample or ZOH interpolating it's convolving the train of > weighted impulses with a rect() pulse function and in the frequency domain > you are multiplying by a sinc() function with zeros through every integer > times R x Fs except for the one at 0 x Fs (the baseband, where the sinc > multiplies by virtually 1) those reduce your image by a known amount. > multiplying the magnitude by sinc() is the same as multiplying the power > spectrum by sinc^2(). > > with linear interpolation, you're convolving with a triangular pulse, > multiplying the sample values by the triangular pulse function and in the > frequency domain you're multiplying by a sinc^2() function and in the power > spectrum you're multiplying by a sinc^4() function. > > now that sinc^2
Re: [music-dsp] Sampling theory "best" explanation
Original Message Subject: Re: [music-dsp] Sampling theory "best" explanation From: "Ethan Duni" Date: Wed, September 6, 2017 4:49 pm To: "robert bristow-johnson" "A discussion list for music-related DSP" -- > rbj wrote: >>what do you mean be "non-ideal"? that it's not an ideal brick wall LPF? > it's still LTI if it's some other filter **unless** you're meaning that > the possible aliasing. > > Yes, that is exactly what I am talking about. LTI systems cannot produce > aliasing. > > Without an ideal bandlimiting filter, resampling doesn't fulfill either > definition of time invariance. Not the classic one in terms of sample > shifts, and not the "common real time" one suggested for multirate cases. > > It's easy to demonstrate this by constructing a counterexample. Consider > downsampling by 2, and an input signal that contains only a single sinusoid > with frequency above half the (input) Nyquist rate, and at a frequency that > the non-ideal bandlimiting filter fails to completely suppress. To be LTI, > shifting the input by one sample should result in a half-sample shift in > the output (i.e., bandlimited interpolation). But this doesn't happen, due > to aliasing. This becomes obvious if you push the frequency of the input > sinusoid close to the (input) Nyquist frequency - instead of a half-sample > shift in the output, you get negation! > >>we draw the little arrows with different heights and we draw the impulses > scaled with samples of negative value as arrows pointing down > > But that's just a graph of the discrete time sequence. well, even if the *information* necessary is the same, a graph of x[n] need only be little dots, one per sample. �or discrete lines (without arrowheads). but the use of the symbol of an arrow for an impulse is a symbol of something difficult to graph for a continuous-time function (not to be confused with a continuous function). �if the impulse heights and directions (up or down) are analog to the sample value magnitude and polarity, those graphing object suffice to depict these *hypothetical* impulses in the continuous-time domain. > >>you could do SRC without linear interpolation (ZOH a.k.a. "drop-sample") > but you would need a much larger table >>(if i recall correctly, 1024 times larger, so it would be 512Kx > oversampling) to get the same S/N. if you use 512x >>oversampling and ZOH interpolation, you'll only get about 55 dB S/N for an > arbitrary conversion ratio. > > Interesting stuff, it didn't occur to me that the SNR would be that low. > How do you estimate SNR for a particular configuration (i.e., target > resampling ratio, fixed upsampling factor, etc)? Is that for ideal 512x > resampling, or does it include the effects of a particular filter design > choice? this is what Duane Wise and i ( https://www.researchgate.net/publication/266675823_Performance_of_Low-Order_Polynomial_Interpolators_in_the_Presence_of_Oversampled_Input ) were trying to show and Olli Niemitalo (in his pink elephant paper�http://yehar.com/blog/wp-content/uploads/2009/08/deip.pdf ). so let's say that you're oversampling by a factor of R. �if the sample rate is 96 kHz and the audio is limited to 20 kHz, the oversampling ratio is 2.4 . but now imagine it's *highly* oversampled (which we can get from polyphase FIR resampling) like R=32 or R=512 or R=512K. when it's upsampled (like hypothetically stuffing 31 zeros or 511 zeros or (512K-1) zeros into the stream and brick-wall low-pass filtering) then the spectrum has energy at the baseband (from -Nyquist to +Nyquist of the original sample rate, Fs) and is empty for 31 (or 511 or (512K-1)) image widths of Nyquist, and the first non-zero image is at 32 or 512 or 512K x Fs. now if you're drop-sample or ZOH interpolating it's convolving the train of weighted impulses with a rect() pulse function and in the frequency domain you are multiplying by a sinc() function with zeros through every integer times R x Fs except for the one at 0 x Fs (the baseband, where the sinc multiplies by virtually 1) �those reduce your image by a known amount. �multiplying the magnitude by sinc() is the same as multiplying the power spectrum by sinc^2(). with linear interpolation, you're convolving with a triangular pulse, multiplying the sample values by the triangular pulse function and in the frequency domain you're multiplying by a sinc^2() function and in the power spectrum you're multiplying by a sinc^4() function. now that sinc^2 or sinc^4 functions really puts a hole in those images, reducing the area in those power spectrum images greatly. now when we resample at an arbitrary rate, we can expect in worse case that *all* of those images get folded back into the baseband. with 3rd-order B-spline (which i don't recommend) it's sinc^4 in the frequency domain and sinc
Re: [music-dsp] Sampling theory "best" explanation
Okay, no big deal. It's easy to come off the wrong way in complicated, fast moving email threads. Ethan D On Wed, Sep 6, 2017 at 6:37 PM, Nigel Redmon wrote: > Ethan, I wasn't taking a swipe at you, by any stretch. In fact, I wasn't > even addressing your ADC comment. It was actually about things like the > idea of making DACs with impulses. As I noted, we don't because there are > ways that are easier and accomplish the same goal, but it is feasible. I've > had people say in the past to me it's absurd, and I've assured them that a > reasonable and practical approximation of it would indeed produce a > reasonable approximation of a decent DAC. That's a pretty relative > statement because the quality depends on how hard you want try, but I > subsequently saw that Julius Smith make the same assertion on his website. > > Sorry you misinterpreted it. > > On Sep 7, 2017, at 5:34 AM, Ethan Duni wrote: > > Nigel Redmon wrote: > >As an electrical engineer, we find great humor when people say we can't > do impulses. > > I'm the electrical engineer who pointed out that impulses don't exist and > are not found in actual ADCs. If you have some issue with anything I've > posted, I'll thank you to address it to me directly and respectfully. > > Taking oblique swipes at fellow list members, impugning their standing as > engineers, etc. is poisonous to the list community. > > >What constitutes an impulse depends on the context—nano seconds, > milliseconds... > > If it has non-zero pulse width, it isn't an impulse in the relevant sense: > multiplying by such a function would not model the sampling process. You > would need to introduce additional operations to describe how this finite > region of non-zero signal around each sample time is translated into a > unique sample value. > > >For ADC, we effectively measure an instantaneous voltage and store it as > an impulse. > > I don't know of any ADC design that stores voltages as "impulse" signals, > even approximately. The measured voltage is represented through modulation > schemes such as PDM, PWM, PCM, etc. > > Impulse trains are a convenient pedagogical model for understanding > aliasing, reconstruction filters, etc., but there is a considerable gap > between that model and what actually goes on in a real ADC. > > >If you can make a downsampler that has no audible aliasing (and you > can), I think the process has to be called linear, even if you can make a > poor quality one that isn't. > > I'm not sure how you got onto linearity, but the subject is > time-invariance. > > I have no objection to calling resamplers "approximately time-invariant" > or "asymptotically time-invariant" or somesuch, in the sense that you can > get as close to time-invariant behavior as you like by throwing resources > at the bandlimiting filter. This is qualitatively different from other > archetypical examples of time-variant systems (modulation, envelopes, etc.) > where explicitly time-variant behavior is the goal, even in the ideal case. > Moreover, I agree that this distinction is important and worth > highlighting. > > However, there needs to be *some* qualifier - the bare statement > "(re)sampling is LTI" is incorrect and misleading. It obscures that fact > that addressing the aliasing caused by the system's time-variance is the > principle concern in the design of resamplers. The fact that a given design > does a good job is great and all - but that only happens because the > designer recognizes that the system is time-invariant, and dedicates > resources to mitigating the impact of aliasing. > > >If you get too picky and call something non-linear, when for practical > decision-making purposes it clearly is, it seem you've defeated the purpose. > > If you insist on labelling all resamplers as "time-invariant," without any > further qualification, then it will mess up practical decision making. > There will be no reason to consider the effects of aliasing - LTI systems > cannot produce aliasing - when making practical system design decisions. > You only end up with approximately-LTI behavior because you recognize at > the outset that the system is *not* LTI, and make appropriate design > decisions to limit the impact of aliasing. So this is putting the cart > before the horse. > > The appropriate way to deal with this is not to get hung up on the label > "LTI" (or any specialized variations thereof), but to simply quote the > actual performance of the system (SNR, spurious-free dynamic range, etc.). > In that way, everything is clear to the designers and clients: the system > is fundamentally non-LTI, and deviation from LTI behavior is bounded by the > performance figures. Then the client can look at that, and make an > informed, practical decision about whether they need to worry about > aliasing in their specific context. If not, they are free to say to > themselves "close enough to LTI for me!" If so, they can dig into the > non-LTI behavior and figure out how to deal with it. Ins
Re: [music-dsp] Sampling theory "best" explanation
Of course I mean that we store a representation of an impulse. I've said many times that the sample values "represent" impulses. > On Sep 7, 2017, at 5:34 AM, Ethan Duni wrote: > > >For ADC, we effectively measure an instantaneous voltage and store it as an > >impulse. > > I don't know of any ADC design that stores voltages as "impulse" signals, > even approximately. The measured voltage is represented through modulation > schemes such as PDM, PWM, PCM, etc. ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theory "best" explanation
Ethan, I wasn't taking a swipe at you, by any stretch. In fact, I wasn't even addressing your ADC comment. It was actually about things like the idea of making DACs with impulses. As I noted, we don't because there are ways that are easier and accomplish the same goal, but it is feasible. I've had people say in the past to me it's absurd, and I've assured them that a reasonable and practical approximation of it would indeed produce a reasonable approximation of a decent DAC. That's a pretty relative statement because the quality depends on how hard you want try, but I subsequently saw that Julius Smith make the same assertion on his website. Sorry you misinterpreted it. > On Sep 7, 2017, at 5:34 AM, Ethan Duni wrote: > > Nigel Redmon wrote: > >As an electrical engineer, we find great humor when people say we can't do > >impulses. > > I'm the electrical engineer who pointed out that impulses don't exist and are > not found in actual ADCs. If you have some issue with anything I've posted, > I'll thank you to address it to me directly and respectfully. > > Taking oblique swipes at fellow list members, impugning their standing as > engineers, etc. is poisonous to the list community. > > >What constitutes an impulse depends on the context—nano seconds, > >milliseconds... > > If it has non-zero pulse width, it isn't an impulse in the relevant sense: > multiplying by such a function would not model the sampling process. You > would need to introduce additional operations to describe how this finite > region of non-zero signal around each sample time is translated into a unique > sample value. > > >For ADC, we effectively measure an instantaneous voltage and store it as an > >impulse. > > I don't know of any ADC design that stores voltages as "impulse" signals, > even approximately. The measured voltage is represented through modulation > schemes such as PDM, PWM, PCM, etc. > > Impulse trains are a convenient pedagogical model for understanding aliasing, > reconstruction filters, etc., but there is a considerable gap between that > model and what actually goes on in a real ADC. > > >If you can make a downsampler that has no audible aliasing (and you can), I > >think the process has to be called linear, even if you can make a poor > >quality one that isn't. > > I'm not sure how you got onto linearity, but the subject is time-invariance. > > I have no objection to calling resamplers "approximately time-invariant" or > "asymptotically time-invariant" or somesuch, in the sense that you can get as > close to time-invariant behavior as you like by throwing resources at the > bandlimiting filter. This is qualitatively different from other archetypical > examples of time-variant systems (modulation, envelopes, etc.) where > explicitly time-variant behavior is the goal, even in the ideal case. > Moreover, I agree that this distinction is important and worth highlighting. > > However, there needs to be *some* qualifier - the bare statement > "(re)sampling is LTI" is incorrect and misleading. It obscures that fact that > addressing the aliasing caused by the system's time-variance is the principle > concern in the design of resamplers. The fact that a given design does a good > job is great and all - but that only happens because the designer recognizes > that the system is time-invariant, and dedicates resources to mitigating the > impact of aliasing. > > >If you get too picky and call something non-linear, when for practical > >decision-making purposes it clearly is, it seem you've defeated the purpose. > > If you insist on labelling all resamplers as "time-invariant," without any > further qualification, then it will mess up practical decision making. There > will be no reason to consider the effects of aliasing - LTI systems cannot > produce aliasing - when making practical system design decisions. You only > end up with approximately-LTI behavior because you recognize at the outset > that the system is *not* LTI, and make appropriate design decisions to limit > the impact of aliasing. So this is putting the cart before the horse. > > The appropriate way to deal with this is not to get hung up on the label > "LTI" (or any specialized variations thereof), but to simply quote the actual > performance of the system (SNR, spurious-free dynamic range, etc.). In that > way, everything is clear to the designers and clients: the system is > fundamentally non-LTI, and deviation from LTI behavior is bounded by the > performance figures. Then the client can look at that, and make an informed, > practical decision about whether they need to worry about aliasing in their > specific context. If not, they are free to say to themselves "close enough to > LTI for me!" If so, they can dig into the non-LTI behavior and figure out how > to deal with it. Insisting that everyone mislabel time-variant systems as LTI > short-circuits that whole process and so undermi
Re: [music-dsp] Sampling theory "best" explanation
Nigel Redmon wrote: >As an electrical engineer, we find great humor when people say we can't do impulses. I'm the electrical engineer who pointed out that impulses don't exist and are not found in actual ADCs. If you have some issue with anything I've posted, I'll thank you to address it to me directly and respectfully. Taking oblique swipes at fellow list members, impugning their standing as engineers, etc. is poisonous to the list community. >What constitutes an impulse depends on the context—nano seconds, milliseconds... If it has non-zero pulse width, it isn't an impulse in the relevant sense: multiplying by such a function would not model the sampling process. You would need to introduce additional operations to describe how this finite region of non-zero signal around each sample time is translated into a unique sample value. >For ADC, we effectively measure an instantaneous voltage and store it as an impulse. I don't know of any ADC design that stores voltages as "impulse" signals, even approximately. The measured voltage is represented through modulation schemes such as PDM, PWM, PCM, etc. Impulse trains are a convenient pedagogical model for understanding aliasing, reconstruction filters, etc., but there is a considerable gap between that model and what actually goes on in a real ADC. >If you can make a downsampler that has no audible aliasing (and you can), I think the process has to be called linear, even if you can make a poor quality one that isn't. I'm not sure how you got onto linearity, but the subject is time-invariance. I have no objection to calling resamplers "approximately time-invariant" or "asymptotically time-invariant" or somesuch, in the sense that you can get as close to time-invariant behavior as you like by throwing resources at the bandlimiting filter. This is qualitatively different from other archetypical examples of time-variant systems (modulation, envelopes, etc.) where explicitly time-variant behavior is the goal, even in the ideal case. Moreover, I agree that this distinction is important and worth highlighting. However, there needs to be *some* qualifier - the bare statement "(re)sampling is LTI" is incorrect and misleading. It obscures that fact that addressing the aliasing caused by the system's time-variance is the principle concern in the design of resamplers. The fact that a given design does a good job is great and all - but that only happens because the designer recognizes that the system is time-invariant, and dedicates resources to mitigating the impact of aliasing. >If you get too picky and call something non-linear, when for practical decision-making purposes it clearly is, it seem you've defeated the purpose. If you insist on labelling all resamplers as "time-invariant," without any further qualification, then it will mess up practical decision making. There will be no reason to consider the effects of aliasing - LTI systems cannot produce aliasing - when making practical system design decisions. You only end up with approximately-LTI behavior because you recognize at the outset that the system is *not* LTI, and make appropriate design decisions to limit the impact of aliasing. So this is putting the cart before the horse. The appropriate way to deal with this is not to get hung up on the label "LTI" (or any specialized variations thereof), but to simply quote the actual performance of the system (SNR, spurious-free dynamic range, etc.). In that way, everything is clear to the designers and clients: the system is fundamentally non-LTI, and deviation from LTI behavior is bounded by the performance figures. Then the client can look at that, and make an informed, practical decision about whether they need to worry about aliasing in their specific context. If not, they are free to say to themselves "close enough to LTI for me!" If so, they can dig into the non-LTI behavior and figure out how to deal with it. Insisting that everyone mislabel time-variant systems as LTI short-circuits that whole process and so undermines practical decision-making. Ethan D On Tue, Sep 5, 2017 at 1:05 AM, Nigel Redmon wrote: > As an electrical engineer, we find great humor when people say we can't do > impulses. What constitutes an impulse depends on the context—nano seconds, > milliseconds... > > For ADC, we effectively measure an instantaneous voltage and store it as > an impulse. Arguing that we don't really do that...well, Amazon didn't > really ship that Chinese garlic press to me, because they really relayed an > order to some warehouse, the shipper did some crazy thing like send it in > the wrong direction to a hub, to be more efficient...and it was on my > doorstep when I checked the mail. What's the diff... > > Well, that's the most important detail (ADC), because that defined what > we're dealing with when we do "music-dsp". But as far as DAC not using > impulses, it's only because the shortcut is trivial. Like I said, audio > sample rates are slow, not
[music-dsp] Open Position: Chair of the School of Music @ Georgia Tech
Dear list, apologies for cross-posting. Please see below for a position as Chair of the School of Music here at Georgia Tech. Best, Alexander The Georgia Institute of Technology invites nominations and applications for the position of Chair of the School of Music in the College of Design in Atlanta, Georgia. The School offers a B.S. in Music Technology, a Master of Music Technology, and a Ph.D., as well as a Music Minor open to all students at Georgia Tech. The School is positioned to be the premier School of Music Technology in the country. In addition to its three degree programs in music technology the School also provides musical instruction to over 900 students per semester from all six colleges of Georgia Tech through the Georgia Tech Marching Band, two orchestras and numerous ensembles as well as a strong set of vocal groups. The School is a unique blend of music technology and traditional music, including performance and composition. The Chair will have the opportunity to build on the success of the previous two decades to create a School of Music that is well tuned to the opportunities of the 21 st Century. The new Chair will have the opportunity to provide overall leadership and vision for the development of a comprehensive music technology program at both the undergraduate and graduate levels. The Chair manages the academic, fiscal, and personnel matters and links the School of Music to the strategic objectives of the College and Institute. Candidate must have a terminal degree in music with a distinguished record of scholarly achievement qualifying for a tenurable position. The candidate must maintain an active commitment to the profession and to the promotion of excellence in teaching and research, and must have the ability to engage with community and corporate leaders and work effectively with faculty, students, and administrators. The salary will be competitive with qualifications and experience. Appointment is anticipated on or before July 1, 2018. Review of applications will begin November 1, 2017 but will continue until the position is filled. Interested individuals should email the following materials to the search committee chair (mailto:musicchairsea...@t-square.gatech.edu), scanned in order into a single PDF: 1. Cover letter describing your interest in the position, academic goals and leadership style; 2. Curriculum Vitae 3. Name, address (including email), and telephone number of five academic/professional references. Information on Georgia Tech, the College of Design, and the School of Music is best accessed from our website http://www.design.gatech.edu. The Georgia Institute of Technology is one of the nation's premier research universities. Ranked seventh among U.S. News & World Report's top public universities, Georgia Tech's more than 25,000 students are enrolled in its Colleges of Design, Computing, Engineering, Sciences, Liberal Arts, and Business. Tech is among the nation's top producers of women and African-American engineers. The Institute offers research opportunities to both undergraduate and graduate students and is home to more than 100 interdisciplinary research units plus the Georgia Tech Research Institute. The Georgia Institute of Technology is an Equal Education/Employment Opportunity Institution -- Alexander Lerch Assistant Professor, GT Center for Music Technology www.gtcmt.gatech.edu www.AudioContentAnalysis.org www.musicinformatics.gatech.edu ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
Re: [music-dsp] Sampling theory "best" explanation
rbj wrote: >what do you mean be "non-ideal"? that it's not an ideal brick wall LPF? it's still LTI if it's some other filter **unless** you're meaning that the possible aliasing. Yes, that is exactly what I am talking about. LTI systems cannot produce aliasing. Without an ideal bandlimiting filter, resampling doesn't fulfill either definition of time invariance. Not the classic one in terms of sample shifts, and not the "common real time" one suggested for multirate cases. It's easy to demonstrate this by constructing a counterexample. Consider downsampling by 2, and an input signal that contains only a single sinusoid with frequency above half the (input) Nyquist rate, and at a frequency that the non-ideal bandlimiting filter fails to completely suppress. To be LTI, shifting the input by one sample should result in a half-sample shift in the output (i.e., bandlimited interpolation). But this doesn't happen, due to aliasing. This becomes obvious if you push the frequency of the input sinusoid close to the (input) Nyquist frequency - instead of a half-sample shift in the output, you get negation! >we draw the little arrows with different heights and we draw the impulses scaled with samples of negative value as arrows pointing down But that's just a graph of the discrete time sequence. >you could do SRC without linear interpolation (ZOH a.k.a. "drop-sample") but you would need a much larger table >(if i recall correctly, 1024 times larger, so it would be 512Kx oversampling) to get the same S/N. if you use 512x >oversampling and ZOH interpolation, you'll only get about 55 dB S/N for an arbitrary conversion ratio. Interesting stuff, it didn't occur to me that the SNR would be that low. How do you estimate SNR for a particular configuration (i.e., target resampling ratio, fixed upsampling factor, etc)? Is that for ideal 512x resampling, or does it include the effects of a particular filter design choice? Ethan D On Tue, Sep 5, 2017 at 9:44 AM, robert bristow-johnson < r...@audioimagination.com> wrote: > > > Original Message > Subject: Re: [music-dsp] Sampling theory "best" explanation > From: "Ethan Duni" > Date: Tue, September 5, 2017 1:07 am > To: "A discussion list for music-related DSP" < > music-dsp@music.columbia.edu> > -- > > > rbj wrote: > > > >>1. resampling is LTI **if**, for the TI portion, one appropriately scales > > time. > > > > Have we established that this holds for non-ideal resampling? It doesn't > > seem like it does, in general. > > what do you mean be "non-ideal"? that it's not an ideal brick wall LPF? > it's still LTI if it's some other filter **unless** you're meaning that > the possible aliasing. > > > > If not, then the phrase "resampling is LTI" - without some kind of > "ideal" > > qualifier - seems misleading. If it's LTI then what are all these aliases > > doing in my outputs? > > > >>no one *really* zero-stuffs samples into the stream > > > > Nobody does it *explicitly* > > people using an IIR filter for reconstruction might be putting in the > zeros explicitly. > > > but it seems misleading to say we don't > > *really* do it. We employ optimizations to handle this part implicitly, > but > > the starting point for that is exactly to *really* stuff zeroes into the > > stream. This is true in the same sense that the FFT *really* computes the > > DFT. > > > > Contrast that with pedagogical abstractions like the impulse train model > of > > sampling. Nobody has ever *really* sampled a signal this way, because > > impulses do not exist in reality. > > it's the only direct way i can think of to demonstrate that we are > discarding all of the information between samples, yet keeping the > information at the sampling instances. it's what dirac impulses are for the > "sampling" or "sifting" property (but the math guys are unhappy if we don't > immediately surround that with an integral, they don't like naked dirac > impulse functions). > > > > > >>7. and i disagree with the statement: "The other big pedagogical problem > > with impulse train representation is that it can't be graphed in a > >useful > > way." graphing functions is an abstract representation to begin with, so > > we can use these abstract vertical arrows to represent >impulses. > > > > That is my statement, so I'll clarify: you can graph an impulse train > with > > a particular period. But how do you graph the product of the impulse > train > > with a continuous-time function (i.e., the sampling operation)? Draw a > > graph of a generic impulse train, with the scaling of each impulse > written > > out next to it? That's not useful. > > and that's not how we do it, of course. we draw the little arrows with > different heights and we draw the impulses scaled with samples of negative > value as arrows pointing down. just as it might look if you had nascent > deltas of *fixed* widt
Re: [music-dsp] Sampling theory "best" explanation
�pretty much agree. �i actually try to be anal only about the pragmatic things, but once in a while i will get into an argument with someone about an arcane detail. �Linearity (in the context of something having a quantizer) is one of these topics. �Time-Invariance (in the context of a sample-rate-converter) is another. �and, most often for me, it's the nature of the dirac impulse "function". �i am happy to treat it like a function as we normally do in an EE class. �but a pure math person who learns about "distributions" might pick a fight. �so now i try to be clear about it. �i also get in arguments with EE authors about that pesky T factor that they don't put in the correct place in their Sampling Theorem. �i think that it's inexcusable because it leads to confusion. �especially when describing the Zero-Order Hold. �we still have to fix "helpful edits" to the ZOH page on Wikipedia because some people just don't understand that scaling issue which comes up because EE textbooks fuck it up. �i wish they never did. �(just as i wish that MATLAB would allow changing the origin of their arrays so that the FFT doesn't put DC into X(1)). �there's a lotta wrong conventions in our world.bestest,r b-j Original Message Subject: Re: [music-dsp] Sampling theory "best" explanation From: "Nigel Redmon" Date: Wed, September 6, 2017 2:31 am To: r...@audioimagination.com music-dsp@music.columbia.edu -- > Ooo, I like that, better than being vague... > > I was implying that what constitutes a impulse depends on the context, but I > like your idea. > > Btw, interesting that when the LTI topic with downsampling came up years ago, > several people shot down the TI part, and this time the discussion has been > around L. > > However, if you take L too literally, even a fixed point butterworth lowpass > fails to be "linear". I think we have to limit ourselves to practicality on a > mailing list called "music-dsp". If you can make a downsampler that has no > audible aliasing (and you can), I think the process has to be called linear, even if you can make a poor quality one that isn't. > > Linear and Tim Invariant are classifications, and we use them to help make > decisions about how we might use a process. No? If you get too picky and call > something non-linear, when for practical decision-making purposes it clearly > is, it seem you've defeated the purpose. > >> On Sep 5, 2017, at 11:57 PM, robert bristow-johnson >> wrote: >> >> >> >> Original Message >> Subject: Re: [music-dsp] Sampling theory "best" explanation >> From: "Nigel Redmon" >> Date: Tue, September 5, 2017 4:05 am >> To: music-dsp@music.columbia.edu >> -- >> >> > As an electrical engineer, we find great humor when people say we can't do >> > impulses. What constitutes an impulse depends on the context—nano >> > seconds, milliseconds... >> >> >> how 'bout a Planck Time. i will define *my* rbj-dirac-impulse as a nascent >> impulse that has area of 1 and a width of 1 Planck time. Is that close >> enough? and the math guys cannot deny it's a real "function". >> >> >> -- >> >> r b-j r...@audioimagination.com >> >> "Imagination is more important than knowledge." >> >> ___ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp > -- � r b-j � � � � � � � � �r...@audioimagination.com � "Imagination is more important than knowledge." ___ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
