rokicki@polya.STANFORD.EDU (Tomas G. Rokicki) (05/19/88)
[ Sample *this*, you soundless hulk! ]
Why Sampling At 29K Is Usually Sufficient
First, I'm not sure I know what I'm talking about
here, as I'm not a musician and know very little about sound,
but let's pretend I do know what I'm talking about for a
second or two.
Some people have been bitching lately about having to
provide waveforms for each octave of a sound sample. I won't
go into issues here about how the relative frequency
distribution changes as a function of frequency for musical
instruments, other than to note that it does, so actually
sampling the different octaves will give you more realistic
sound, but rather I will explain how using separate samples for
the higher frequencies works better than undersampling higher
resolution samples, by stepping through them at a faster rate.
First, any infinite periodic waveform of period t with
n samples can be represented as a sum of n sine waves with
periods of t/1, t/2, t/3...t/n. The discrete fourier transform
can do (and undo) this transformation.
If you undersample such a waveform, aliasing will cause
some strange effects. Let us say our sampling rate that we are
undersampling at is m (samples per second.) According to Nyquist,
our sampling rate should be at least twice the frequency of the
highest component sine wave. Any components that have a
frequency greater than 2/m will have some portion of their energy
aliased or mirrored down to a lower frequency.
To demonstrate this, let us look at a 10KHz sine wave.
We'll sample it at 20KHz, exactly the Nyquist rate. If our
samples happen to fall exactly at the zero crossings of the sine
wave, we will have a zero amplitude output. If they fall on
the peaks, we will have a maximum amplitude output. Depending
on the random position of our first sample.
Now let's sample our 10KHz sine wave at 20.2KHz. We
are now slightly off our frequency, and we will see a 10KHz
tone modulated by a 200 Hz carrier; this will sound like two
narrowly separated frequencies beating against one another.
Ugly as sin.
Let's now sample a 16KHz sine wave at 20KHz. In this
case, we will get as output approximately a 4KHz tone at almost
the full amplitude as the original sine wave. (Draw it out on
graph paper to see what happens.)
So, by undersampling our highly accurate waveform, we
map the frequencies which are normally unheard (if they are
really there at all) into frequencies which can be heard and which
will greatly diminish the quality of the output.
How do we fix this? Easy. We generate sound samples at
the higher frequencies that do not contain the unheard higher frequency
components of the original sound, and sample this newer sample at
the correct rate. This is exactly the way it is recommended you do
it on the Amiga, and for good reason.
Someone recently complained that they had to go all the way
down to a 32 samples to get the high frequencies they needed. I claim
that that 32 sample sound, if properly created from the filtered
original, would sound superior to an undersampled larger sample size.
The final point is, because you know *each* sample will be
used, you can adjust the energy of any of the harmonics essentially
independently, and be assured that the output signal will have that
much energy. Undersampling does not give you this ability.
Any further comments?
Oh, so you want to know how to create the smaller samples from
the larger ones. I'll leave that to the experts out there.
--
/-- Tomas Rokicki /// Box 2081 Stanford, CA 94309
/ o Radical Eye Software /// (415) 326-5312
\ / | . . . or I \\\/// Gig 'em, Aggies! (TAMU EE '85)
V | won't get dressed \XX/ Bay area Amiga Developer's GroupEphil@eos.UUCP (Phil Stone) (05/19/88)
In article <2845@polya.STANFORD.EDU> rokicki@polya.STANFORD.EDU (Tomas G. Rokicki) writes: > > Why Sampling At 29K Is Usually Sufficient > >.....Lecture on sampling theory omitted..... > > If you undersample such a waveform, aliasing will cause >some strange effects. Let us say our sampling rate that we are >undersampling at is m (samples per second.) According to Nyquist, >our sampling rate should be at least twice the frequency of the >highest component sine wave. Any components that have a >frequency greater than 2/m will have some portion of their energy >aliased or mirrored down to a lower frequency. > >.....7 paragraphs on the evils of undersampling omitted..... Obviously, one should not try to play frequencies higher than the Nyquist frequency; this is what you so laboriously illustrate. Your examples are not very realistic, however. Let's take a simple sine-wave synthesis instead (not the most realistic scenario either, but so simple as to be an example of what an audio driver *should* be able to handle). Compute a 256-point sine wave and put it in memory (32-byte sine waves don't cut it in my book - even a tin ear can hear the interpolation noise in fixed, jagged steps that big). With a fixed sample-playback increment and a maximum rate of 29 KHz, the highest frequency you can generate is 29000/256 = 113 Hz! - just about TWO OCTAVES below A440(!) With this same sine wave and a variable (intergral.fractional) sampling increment, one could generate a maximum frequency of 14 KHz! Now, I realize that more complex waveforms will alias at this frequency, but at more reasonable, musical frequencies, say 1 KHz, fairly complex waveforms can be easily generated with no Nyquist aliasing, and without the need to spin-off shorter tables just to get higher frequencies. My point is, it is the combination of a relatively low sampling rate *and* a simple increment method that makes the sound driver so inconvenient to use. Fix one or the other if you want a high quality and flexible sound device (I'd pick the increment as the easier to fix, though I don't appreciate the subtleties of the Amiga's DMA). Also, I'm all for multi-sampling, if what you are trying to do is *re-create* a musical instrument. Not everyone is using the Amiga as a sampler, though. I enjoy playing with its synthesis capabilities, and wouldn't mind if one waveform of say, 256 bytes, could be used over five octaves - not a tall order. I used to be able to do *that* with a 6502. Your postscript tosses off a reference to *making* the other octaves of sound - this is called interpolation, a process which has earned many Ph.D's over the last ten years. Try interpolating 4 octaves from one sample *live* sometime (can you say "Cray killer?") > Any further comments? Yes. Why is it that articles that start out with "I really don't know much about this but..." always turn out to be the longest ones? :-) Phil Stone (I still love the Amiga sound driver) phil@eos.arc.nasa.gov phil@eos.UUCP { uunet, hplabs, hao, ihnp4, decwrl, allegra, tektronix } ames!eos!phil
rokicki@polya.STANFORD.EDU (Tomas G. Rokicki) (05/19/88)
> Obviously, one should not try to play frequencies higher than the Nyquist > frequency; this is what you so laboriously illustrate. Your examples are > not very realistic, however. Let's take a simple sine-wave synthesis > instead (not the most realistic scenario either, but so simple as to be > an example of what an audio driver *should* be able to handle). But if you try to play back a waveform that *includes* high frequencies by undersampling, those high frequencies come back and haunt you. That is the point I am trying to make. > Compute a 256-point sine wave and put it in memory (32-byte sine waves > don't cut it in my book - even a tin ear can hear the interpolation > noise in fixed, jagged steps that big). As chuck said, a square wave at 14KHz should be filtered to a very nice little sine wave; that is what the filter is for. A square wave at frequency f looks something like sin(ft+a_1) + 1/3 sin(3ft+a_2) + 1/5 sin(5ft+a_3) . . . where the a_i's are phase shifts. The filter should attenuate the third and fifth etc. harmonics almost entirely, leaving you with a very nice sine wave. > With a fixed sample-playback > increment and a maximum rate of 29 KHz, the highest frequency you can > generate is 29000/256 = 113 Hz! - just about TWO OCTAVES below A440(!) Correct, assuming you stick with that 256 sample table. > With this same sine wave and a variable (intergral.fractional) sampling > increment, one could generate a maximum frequency of 14 KHz! Yes, but because of the integral.fractional nature of your incrementing, you are guaranteed to introduce some amount of garbage into your sound; all your peaks won't peak at the same point, etc. Although this effect probably won't become really bothersome until you are up to about 1/8 of the maximum frequency or so. > My point is, it is the combination of a relatively low sampling rate > *and* a simple increment method that makes the sound driver so > inconvenient to use. Fix one or the other if you want a high quality > and flexible sound device (I'd pick the increment as the easier to fix, > though I don't appreciate the subtleties of the Amiga's DMA). But the sampling rate is variable; this is what makes it a win. I really don't understand what is so hard about having multiple tables; if you do octave tables, you need only twice the storage of the original table (1+1/2+1/4...~=2). > I enjoy playing with its synthesis capabilities, and > wouldn't mind if one waveform of say, 256 bytes, could be used over five > octaves - not a tall order. I used to be able to do *that* with a 6502. Ever feed the 6502-generated signal into a spectrum analyzer? > Your postscript tosses off a reference to *making* the other octaves of > sound - this is called interpolation, a process which has earned many Ph.D's > over the last ten years. Try interpolating 4 octaves from one sample *live* > sometime (can you say "Cray killer?") But it only needs be done once for each sample . . . even a full FFT, filter, and reverse FFT doesn't take long for a 256 byte sample. Not real time, true, but for synthesis? Faster than you can figure parameters for the sample. > > Any further comments? > > Yes. Why is it that articles that start out with "I really don't know > much about this but..." always turn out to be the longest ones? :-) Well, I'm trying to learn myself, so I presented what I think I understand to be the case, and I hope to be corrected where I am wrong. > Phil Stone (I still love the Amiga sound driver) -- /-- Tomas Rokicki /// Box 2081 Stanford, CA 94309 / o Radical Eye Software /// (415) 326-5312 \ / | . . . or I \\\/// Gig 'em, Aggies! (TAMU EE '85) V | won't get dressed \XX/ Bay area Amiga Developer's GroupE
cmcmanis%pepper@Sun.COM (Chuck McManis) (05/19/88)
In article <734@eos.UUCP> phil@eos.UUCP (Phil Stone) writes: >Compute a 256-point sine wave and put it in memory (32-byte sine waves >don't cut it in my book - even a tin ear can hear the interpolation >noise in fixed, jagged steps that big). Phil you still don't get it do you? If take a 32 sample sine wave and run it through a spectrum analyzer, you will find that it has a big fundamental at the frequency of interest, and a bunch of higher order frequencies that are caused by the 'interpolation' noise. The key feature is that *all* of these frequencies are above the 'nyquist' frequency. The Amigas low pass filters start cutting of frequencies above 7Khz and pretty much eliminate everything above 14Khz. That means that even your golden ear may have difficulty in hearing the differences. (You will always get .4% distortion because that is the monotonic difference between to 'points' in space.) Where's my DFT when I need it. So to determine when a 32 sample waveform is 'good' enough, run it through a DFT, and look at the first frequency component above the fundamental. Is it below your filter point? If so you need to lengthen your sample, if not then your sample *cannot* be made any better so why waste the bytes? The constants in the equation are the low pass filter, the sample frequency, and the DAC resolution. --Chuck McManis uucp: {anywhere}!sun!cmcmanis BIX: cmcmanis ARPAnet: cmcmanis@sun.com These opinions are my own and no one elses, but you knew that didn't you.
phil@eos.UUCP (Phil Stone) (05/19/88)
In article <53788@sun.uucp> cmcmanis@sun.UUCP (Chuck McManis) writes: <In article <734@eos.UUCP> phil@eos.UUCP (Phil Stone) writes: <<Compute a 256-point sine wave and put it in memory (32-byte sine waves <<don't cut it in my book - even a tin ear can hear the interpolation <<noise in fixed, jagged steps that big). < <Phil you still don't get it do you? If take a 32 sample sine wave <and run it through a spectrum analyzer, you will find that it has <a big fundamental at the frequency of interest, and a bunch of <higher order frequencies that are caused by the 'interpolation' noise. <The key feature is that *all* of these frequencies are above the <'nyquist' frequency. The Amigas low pass filters start cutting of <frequencies above 7Khz and pretty much eliminate everything above <14Khz.....etc. <--Chuck McManis Wow - now the 7Khz cutoff of the filter on the Amiga sound channels is being touted as an advantage! The reason the cutoff is so low (and therefore has inspired hacks and now a 500/2000 option to bypass it in order to get any decent high frequency response) is precisely because of the low sampling rate. Saying that the low sampling rate is not a problem because the absurdly low cutoff frequency of the filter "fixes" it, is like equating not swimming with solving the problem of shark attacks. Damn, I'm not saying scrap the Amiga sound driver, just suggesting some areas where it could be improved. I am amazed at how many people (not all of whom even program the sound device) have reacted with "That's not a problem!" I've taken the original discusion to email, as I sense that its welcome is about to wear out here. Please feel free to write. >uucp: {anywhere}!sun!cmcmanis BIX: cmcmanis ARPAnet: cmcmanis@sun.com >These opinions are my own and no one elses, but you knew that didn't you. Phil Stone phil@eos.arc.nasa.gov phil@eos.UUCP { uunet, hplabs, hao, ihnp4, decwrl, allegra, tektronix } ames!eos!phil I'm sneaking these opinions out without my employer's knowledge.
doug-merritt@cup.portal.com (05/20/88)
A number of questions have been raised about sampled sound by several people, and the following posting is long because it answers all of the ones that weren't answered previously, as well as correcting some errors in statements by previous posters. If you don't care about high quality sound generation, don't read it. Since Tom got a mild flame for saying he was not an expert, I guess I should say that I *am* an expert, as far as the following goes. Trust me. 1/2 :-) Tom Rokicki wrote: > Now let's sample our 10KHz sine wave at 20.2KHz. We >are now slightly off our frequency, and we will see a 10KHz >tone modulated by a 200 Hz carrier; this will sound like two >narrowly separated frequencies beating against one another. >Ugly as sin. The beat frequencies are the sum and difference of the sampling frequency and the highest frequency component in the sample. There will always be a beat frequency, the question is what to do with it. If you have the lowpass filter turned on (always on the the 1000, btw), then you aim to put the beat frequency above the range of the lowpass filter (see pg 156, Fig 5-7 of the Hardware Reference Manual). Otherwise: There are two ways to remove aliasing of the sampling frequency. One is to remove it via an add-on hardware filter (a 200hz notch filter, for instance). The other is to transform the energy of the beat frequency into random noise, which is far less annoying to the ear, and distributes the energy across the entire spectrum, so that any given component of the noise will be very low energy. This is actually the *preferred* method (in absence of a low pass filter), according to the best minds in sampling theory (not me; but I've been to talks on the subject). It's better than a notch filter because you don't lose the original signal at the notch frequency. The way that you transform the alias into noise is to randomly vary the sampling frequency (actually ideally it should follow a Poisson distribution, but random works pretty well). In other words, instead of sampling exactly every N microseconds, you sample every N+random() microseconds, where 0 <= random() <= N. Thus you still get an alias/beat, but its frequency varies randomly on each sample. (BTW the equivalent method for graphics is to sample on a randomly disturbed grid, and Pixar uses this method for improving ray tracing. The human eye does the same thing: outside the fovea, the rods are placed by a Poisson distribution, to eliminate aliasing.) As far as I can tell from a quick glance at the Hardware Reference Manual, this is perfectly feasible if you use Direct (Non-DMA) output, with a maximum resolution of 280 nanoseconds. See page 161, The Audio State Machine. As far as DMA is concerned, intuitively you would think you're stuck with aliasing due to a fixed period, but you *might* be able to pull the same trick if you period-modulate one channel with random data in another. See pg 151, Table 5-5. Tom again: > Oh, so you want to know how to create the smaller samples from >the larger ones. I'll leave that to the experts out there. It's easy...you just do a (weighted) average down. If you go from 256 to 32 samples then you just average each 8 adjacent samples (weight of 1 since 256 = 8*32) in the source into one sample in the original. If you wanted to go from 256 to say 49 samples, then you'd average each 5.224 samples (5.224 = 256/49) together. In other words, the first destination sample equals the sum of the first five source samples, plus 0.224 times the sixth source sample, divided by 5.224. This leaves you with 0.776 worth of the sixth sample leftover, so you for the second destination sample you take that, plus the next 4 samples, plus 0.448 (= 5.224-4-0.776) times the fifth sample, all divided by 5.224. And continue. The reason this works is that averaging is the time domain equivalent of a frequency domain low pass filter. Now as to Phil's posting: Phil Stone then posts to critique Tom's perfectly good article, in particular about its length and unrealistic examples. Just for the record, Phil's article was almost as long as Tom's (66 vs 84 lines), and used even less realistic examples, because he concentrated on what he wanted to see added to the system, where everyone else has been talking mostly about what *is*. People who live in glass houses... Phil also writes: >Compute a 256-point sine wave and put it in memory (32-byte sine waves >don't cut it in my book - even a tin ear can hear the interpolation >noise in fixed, jagged steps that big). With a fixed sample-playback >increment and a maximum rate of 29 KHz, the highest frequency you can >generate is 29000/256 = 113 Hz! - just about TWO OCTAVES below A440(!) That's why Tom (or was it Chuck?) talked about 32 byte samples...they were being realistic about the current hardware, you see. Besides, what you're talking about has to do with accuracy of reproduction of the higher frequency harmonics need to synthesize, say, a musical instrument's timbre, not "the highest frequency you can generate". The highest frequency component is strictly a function of sample rate. What you're talking about is the highest frequency *fundamental*. And 113hz doesn't give much range, now does it? Nonetheless I'd have to agree that it would be nice if you could sample faster so as to raise the frequency of the highest achievable fundamental, while still accurately reproducing timbre. But note that you *have* the following feature: >With this same sine wave and a variable (intergral.fractional) sampling >increment, one could generate a maximum frequency of 14 KHz! My guess is that you can do this already, as I discussed above. >Your postscript tosses off a reference to *making* the other octaves of >sound - this is called interpolation, a process which has earned many Ph.D's >over the last ten years. Try interpolating 4 octaves from one sample *live* >sometime (can you say "Cray killer?") Gross exaggeration. The papers on the topic of interpolating down from larger to smaller waveforms occurred a lot more than 10 years ago, and a 68000 with a math chip is perfectly capable of keeping up with the demands. A Cray could do a full orchestra in real time. Then Tom posts again: >Yes, but because of the integral.fractional nature of your incrementing, >you are guaranteed to introduce some amount of garbage into your >sound; all your peaks won't peak at the same point, etc. Although >this effect probably won't become really bothersome until you are up >to about 1/8 of the maximum frequency or so. Quite accurate, but there will ALWAYS be some garbage of SOME sort introduced. The integral.fractional notion does a better job of minimizing it than if you just truncate!!! The main problem introduced by that method is that you can no longer have a fixed length sample that is exactly one period long, which nominally forces you to use a non-DMA method. Except that 1) Phil was making a wish list for something new, and you could conceivably add hardware features to restart the sample at the appropriate point beyond the beginning, and 2) my suggestion about using a DMA channel for period modulation could probably fix this, too, if that in fact works. >But it only needs be done once for each sample . . . even a full FFT, >filter, and reverse FFT doesn't take long for a 256 byte sample. Not >real time, true, but for synthesis? Faster than you can figure parameters >for the sample. I guess everyone is assuming you need to do a full FFT, that's why they have problems with speed. But the type of filtering required doesn't need a full FFT; it's just a question of averaging. >Well, I'm trying to learn myself, so I presented what I think I >understand to be the case, and I hope to be corrected where I am >wrong. I hope Phil and Chuck feels the same way, or I'm gonna get flamed! :-) And finally, Chuck McManis writes: > The Amigas low pass filters start cutting of >frequencies above 7Khz and pretty much eliminate everything above >14Khz. That should be 5Khz and 7Khz, respectively. See pg 154-157, Aliasing Distortion. >That means that even your golden ear may have difficulty >in hearing the differences. (You will always get .4% distortion >because that is the monotonic difference between to 'points' in >space.) Exactly. Except that again, this is where the trick of adding a random jitter to the sampling frequency wins, because it distributes the distortion throughout the audio range, rather than consistently creating the same error on every tone. Wow, made it all the way to the end, did you? Must be real interested in the topic! Doug --- Doug Merritt ucbvax!sun.com!cup.portal.com!doug-merritt or ucbvax!eris!doug (doug@eris.berkeley.edu) or ucbvax!unisoft!certes!doug
augi@cbmvax.UUCP (Joe Augenbraun) (05/24/88)
In article <5637@cup.portal.com> doug-merritt@cup.portal.com writes: > A number of questions have been raised about sampled sound by several > people, and the following posting is long because it answers all of > the ones that weren't answered previously, as well as correcting some > errors in statements by previous posters. If you don't care about > high quality sound generation, don't read it. > > Since Tom got a mild flame for saying he was not an expert, I guess I > should say that I *am* an expert, as far as the following goes. > Trust me. 1/2 :-) > I've had a couple of questions that I've been meaning to ask an expert. :-) The top waveform (let's say) is an 18 KHz signal and the lines underneath it are the 44 khz sampling signal used by a CD player: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | | | | | | | Resulting in the following samples: . . | | | . | | | | | | | | | | | | | | . | | | | | | | Which a normal DA converter would probably recunstruct into: . . |. . .| . | . . . . | . . | . | . . | | . | . | . . | | . . | . | . . | | . . | . | | | | | | | I know that in theory a sampled signal should be recontructed using some kind of funny exponential, but I assume that this is real hard to do in real time. So what did I do wrong here? Is the distortion on the waveform just the aliased sampling frequency, and its not easily recognizable because I am drawing in the time domain? Or is this an artifact of not reconstructing using the exponential? (I just used CD parameters to give me reasonable numbers. I am not interested in CD issues here, just the theoretical reason of what causes the distortion that I drew.) joe -- Joe Augenbraun ucp: {uunet|ihnp4|rutgers}!cbmvax!augi System Engineering arpa: cbmvax!augi@uunet.uu.net Commodore Business Machines Phone: 215-431-9332
doug-merritt@cup.portal.com (05/25/88)
Joe Augenbraun writes: >I've had a couple of questions that I've been meaning to ask an expert. :-) Thanks for the smiley face; I was only kidding about being an expert! >The top waveform (let's say) is an 18 KHz signal and the lines underneath >it are the 44 khz sampling signal used by a CD player: >[ waveforms deleted ] The junk that you're seeing in your example is due to 26Khz and 62Khz aliasing. With a sampling rate of A of a sine of frequency B, you'll always get two aliased components of frequencies A-B (here 26Khz) and A+B (62Khz). This isn't really a problem, since even the 26Khz frequency is beyond the range of human hearing. You could look at it as though your ear is imposing a low pass filter on the incoming signal, with a cutoff around 20Khz. If you filter your example waveform through a 20Khz lowpass filter, you'd get a nice clean 18Khz signal out of it. As I said in the other posting, the other way (besides a high sampling rate) to get rid of unpleasant aliasing is to randomize your sampling frequency, which distributes the aliasing into low level white noise. >I know that in theory a sampled signal should be recontructed using some kind >of funny exponential, but I assume that this is real hard to do in real time. The "funny exponential" you're referring to is simply one in the domain of complex numbers, and happens to be equivalent to sines and cosines. The mathematics of sampling theory, Fourier transforms, etc, can be phrased either in terms of trigonometry (sines and cosines) or of complex analysis (complex-domain exponentials). It's exactly the same thing either way. See Euler's theorem in an algebra book to see why. Doug -- Doug Merritt ucbvax!sun.com!cup.portal.com!doug-merritt or ucbvax!eris!doug (doug@eris.berkeley.edu) or ucbvax!unisoft!certes!doug
phil@eos.UUCP (Phil Stone) (05/25/88)
In article <5637@cup.portal.com> doug-merritt@cup.portal.com writes: ....Doug Merritt explains some interesting ways of avoiding sampling aliasing using random fluctuations in sampling frequency.... >Now as to Phil's posting: > >Phil Stone then posts to critique Tom's perfectly good article, >in particular about its length and unrealistic examples. Just for >the record, Phil's article was almost as long as Tom's (66 vs 84 lines), >and used even less realistic examples, because he concentrated on >what he wanted to see added to the system, where everyone else has >been talking mostly about what *is*. People who live in glass houses... Never thought suggestions for improvement were unwelcome here, Doug. I made an effort to point out that I regularly program and perform live using the Amiga sound interface, and had encountered some practical drawbacks. Alot of the postings on this subject have been very educational treatises on sampling theory that seem to come from very informed people, but none seemed to have the perspective of actually writing software for the sound device. I apologized to Tom via Email (and now publicly) for flaming his well-written article, but it came from a sense of people approaching the issue from a purely theoretical point of view. How many people who have said "29 KHz is not a problem" actually write code for the sound device (never mind try to use it live!) Your suggestions on direct (non-DMA) access of the sound device are, once again, wonderful from a theoretical point of view, but would drain so much CPU time as to make the Amiga difficult to use for other tasks in a live context. Allow me to summarize my position: I *love* the Amiga sound device, and find no parallel to it in music: low-price, flexible programmability, built-in to a superior microcomputer system with all that implies (peripheral access, excellent software, etc). But to say that it has *no problems*, even if you back this up with a ream of sampling theory, is pure defensiveism. Maybe we can work some of these problems out before the next generation of machines goes into production. P.S. I promise to try to think a little more before following up in the future. Sorry, Tom. Phil Stone phil@eos.arc.nasa.gov phil@eos.UUCP { uunet, hplabs, hao, ihnp4, decwrl, allegra, tektronix } ames!eos!phil
stever@videovax.Tek.COM (Steven E. Rice) (05/25/88)
In article <3854@cbmvax.UUCP>, Joe Augenbraun (augi@cbmvax.UUCP) asked how a digital system would reconstruct the original signal when the sampling rate was not very much higher than that required to give two samples per cycle of the signal. He included some diagrams of an 18 kHz signal sampled at 44 kHz. He conjectured that the D/A converter would recreate the signal as follows: > > . . > |. . .| . > | . . . . | . . | . > | . . | | . | . > | . . | | . . | . > | . . | | . . | . > | | | | | | | (ref) ----------------------------------------------------------------------- Ignoring overshoots, ringing, and so forth (and assuming negligible risetimes) the reconstructed output would actually look like this: .----------. .----------. | | .----------. | | | | .----------' | | | | | | | | | | | | | | | | |__________| |----------| |_ | (ref) ----------------------------------------------------------------------- This then goes to a "reconstruction filter" (which is just a low-pass filter), to remove the high-frequency harmonics, leaving a faithful reproduction of the original signal. (There is more to it. Email if you are interested in the gory details.) Steve Rice ----------------------------------------------------------------------------- * Every knee shall bow, and every tongue confess that Jesus Christ is Lord! * new: stever@videovax.tv.Tek.com [phone (503) 627-1320] old: {decvax | hplabs | ihnp4 | uw-beaver}!tektronix!videovax!stever
doug-merritt@cup.portal.com (05/26/88)
Phil Stone wrote: >....Doug Merritt explains some interesting ways of avoiding sampling >aliasing using random fluctuations in sampling frequency.... [...] >Your suggestions on direct (non-DMA) access of the sound device are, >once again, wonderful from a theoretical point of view, but would drain >so much CPU time as to make the Amiga difficult to use for other tasks Totally agree. Do you know whether my idea about doing the same technique via DMA (period modulation of another channel) would work? >But to say that it [ the Amiga audio driver ] has *no problems*, even if >you back this up with a ream of sampling theory, is pure defensiveism. Once again, totally agree. Good point. Sorry to have implied otherwise. >Never thought suggestions for improvement were unwelcome here, Doug. Oops! Sorry! I take it back, by all means, please do continue making suggestions for improvements, they're more than welcome! I meant only to rebut your criticism of Tom, not to flame you! Actually I'd say that your ideas about fractional period support sound very useful. As I think you implied, they may require actual hardware support in a future system rather than just a driver change. And while we're asking for 4K by 4K by 24 screens :-) why not ask for a 44Khz sampling rate, too? Only half kidding. Some day. Doug -- Doug Merritt ucbvax!sun.com!cup.portal.com!doug-merritt or ucbvax!eris!doug (doug@eris.berkeley.edu) or ucbvax!unisoft!certes!doug
phil@eos.UUCP (Phil Stone) (05/28/88)
Doug Merritt wrote: >Phil Stone wrote: >>....Doug Merritt explains some interesting ways of avoiding sampling >>aliasing using random fluctuations in sampling frequency.... [...] >>Your suggestions on direct (non-DMA) access of the sound device are, >>once again, wonderful from a theoretical point of view, but would drain >>so much CPU time as to make the Amiga difficult to use for other tasks > >Totally agree. Do you know whether my idea about doing the same technique >via DMA (period modulation of another channel) would work? > I thought it depended on *random* fluctuations in sampling frequency - wouldn't this just introduce a higher-order periodicity? BTW - sampler hardware would also have to do this on the input end wouldn't it? If it *could* be worked out, it might be a great way of doing hi-fi sampling. >And while we're asking for 4K by 4K by 24 screens :-) why not ask for a >44Khz sampling rate, too? Only half kidding. Some day. > Doug Merritt ucbvax!sun.com!cup.portal.com!doug-merritt Yeah! Presents some interesting possibilities for direct downloading from compact disks. C-A has taken the first step in making a quality, programmable sound device built in to a great computer (Mac sound? Get serious). It would be great to see them develop this idea further in future products. Can anybody speak for C-A re: their interest in improving the sound device? Phil
augi@cbmvax.UUCP (Joe Augenbraun) (05/28/88)
First of all, for those who have pointed out that my D/A convertor is
predicting the future by drawing the nice diagonal lines to the next
sample, you are right, I was not thinking clearly.
I also agree that you need a 22khz low pass filter, and that it will
make things better. And I have gone through all of the math proving that
in an ideal system that this is adequate (although it was several years
ago when I was an undergrad).
The thing that I don't quite see is how in the real world type system
that I drew that a low pass filter would actually output the original
signal. Generally a low pass filter will take the 'edge' off of corners
in the time domain, and there are places in my reconstructed signal
that never reach zero, but were zero in the original signal. Those parts
will Fourrier transform into a low frequency, and the low pass won't
touch them. I think if you would actually low-pass my sampled signal
(or actually the correct sampled signal that someone posted just a
couple messages ago) you would find that the amplitude and phase of the
output will waver. Does anyone have a program that would let you actually
do this?
joe
--
Joe Augenbraun ucp: {uunet|ihnp4|rutgers}!cbmvax!augi
System Engineering arpa: cbmvax!augi@uunet.uu.net
Commodore Business Machines Phone: 215-431-9332 doug-merritt@cup.portal.com (05/30/88)
I wrote: > Do you know whether my idea about doing the same technique >via DMA (period modulation of another channel) would work? Phil Stone wrote: >I thought it depended on *random* fluctuations in sampling frequency - >wouldn't this just introduce a higher-order periodicity? I'm not sure. Ordinarily you have problems with a sample that is not an integral multiple of its constituents, because the discontinuity where it wraps from end to start introduces a high frequency click. I was thinking it might not be perceptible in the case of a sample filled with random values, especially one that was simply period-modulating the actual output. You could certainly detect it via autocorrelation, though, so likely your ear could, too (for some waveforms, anyway). Intuitively I would expect that generating a random sample that is windowed as usual (rising from a zero amplitude envelope at the beginning, and falling to zero again at the end) would give perfectly adequate results. That's the usual trick with getting finite samples to behave the way that you expect the theoretical model of an infinite Fourier series to work. Mainly my question is whether or not the Amiga audio subsystem works the way I'm inferring it does...I.e. is it possible to even try this idea using DMA? It *looks* like it is. >BTW - sampler hardware would also have to do this on the input end wouldn't it? >If it *could* be worked out, it might be a great way of doing hi-fi sampling. Yes, you do want it on input as well. As I recall from the driver available for the Applied Visions Future Sound digitizer, this is quite possible via a software mod. I wouldn't know about other sound digitizers. Doug Doug Merritt ucbvax!sun.com!cup.portal.com!doug-merritt or ucbvax!eris!doug (doug@eris.berkeley.edu) or ucbvax!unisoft!certes!doug
cheung@vu-vlsi.Villanova.EDU (Wilson Cheung) (06/01/88)
In article <5872@cup.portal.com>, doug-merritt@cup.portal.com writes: > >it are the 44 khz sampling signal used by a CD player: > >[ waveforms deleted ] > aliasing. With a sampling rate of A of a sine of frequency B, you'll > always get two aliased components of frequencies A-B (here 26Khz) and > A+B (62Khz). that aliasing is occuring here since 44 Khz is more than double 18 Khz. What he is looking at is the result of the sampling process, repeated spectra of the original signal. To obtain the original signal you must pick out one of these repeated spectra. This is done by using a low pass filter that cuts off around 22 Khz (half the sampling rate). Wilson Cheung