Jeff.Miller@samba.acs.unc.edu (Jeff Miller) (10/13/90)
Of course the first thing any treatment of digital audio and A/D converters geos into is the Nyquist frequency and the crucial need for effective input filtering to avoid aliasing. That I've got covered. But I don't recall any treatment of the need and mathematics of filtering the output of a D/A converter. I would imagine at CD or DAT frequencies it wouldn't be too critical as any harmonics would be ultrasonic (or would they? And I've always wondered: can two ultrasonic sounds or an ultrasonic and audible sound beat to create an audible tone?) but at lower frequencies, I would think that aproximating a sine wave with stepped squares would sound bad. If someone could tell me loosely what the requirements are I would appreciate it. I really must get a good handle on it as I am shooting for CD quality. If the requirements are as stringent as the input filtering requirements, I may have some trouble. Would mathematically interpolating a few output levels between each input sample help, and deos that technique have a name I should look out for? Thanks, -Jeff --
grayt@spock (Tom Gray) (10/15/90)
In article <1319@beguine.UUCP> Jeff.Miller@samba.acs.unc.edu (Jeff Miller) writes: >Of course the first thing any treatment of digital audio and A/D converters >geos into is the Nyquist frequency and the crucial need for effective input >filtering to avoid aliasing. That I've got covered. > >But I don't recall any treatment of the need and mathematics of filtering >the output of a D/A converter. I would imagine at CD or DAT frequencies >it wouldn't be too critical as any harmonics would be ultrasonic (or would > >If the requirements are as stringent as the input filtering requirements, >I may have some trouble. Would mathematically interpolating a few output >levels between each input sample help, and deos that technique have a name >I should look out for? > >Thanks, The answer to the above question is yes. Different types of sampling systems do have different requirements for filtering. One way to look at this is to regard the sampled signal as a Taylor series. Commonly only the current sample is used. This has been called a zero order sampler. It requires the sinx/x filter that we all know and love. If you start deriving the derivatives of the signal from various orders and summing these into the output sampe, you get higher order samplers. This is your idea of interpolation. it is really expanding the signal around the sampling time as a Taylor series. In the limit, if derivitives of all orders are supplied, you have ideal sampling and no filter is required. >
whit@milton.u.washington.edu (John Whitmore) (10/16/90)
In article <1319@beguine.UUCP> Jeff.Miller@samba.acs.unc.edu (Jeff Miller) writes: >Of course the first thing any treatment of digital audio and A/D converters >geos into is the Nyquist frequency and the crucial need for effective input >filtering to avoid aliasing. That I've got covered. > >But I don't recall any treatment of the need and mathematics of filtering >the output of a D/A converter. I would imagine at CD or DAT frequencies >it wouldn't be too critical as any harmonics would be ultrasonic (or would >they? Yes, they'd be ultrasonic, BUT that doesn't mean the sound reproducing equipment will ignore them. A classic problem arose when the early FM tuners were connected to tape recorders; the harmonics of the (19 kHz?) stereo pilot signal beat against the (circa 60 kHz) record bias oscillator to generate a horrendous howl (which wavered because the bias oscillator frequency wasn't terribly stable). >And I've always wondered: can two ultrasonic sounds or an ultrasonic >and audible sound beat to create an audible tone? Obviously, yes. > but at lower frequencies, >I would think that aproximating a sine wave with stepped squares would sound >bad. For this reason, many top-end CD players mask the low-amplitude 'warble' that a low-amplitude signal becomes when played back on a digital system (the problem here is that there's 96 dB signal/noise for LOUD signals, but a quiet passage, 80 dB down from the peak power in a given opus, will only have 16 dB signal/noise). Typical treatment is to mask the artifact with some pink noise (yes, there are noise generators in good CD players). > > Would mathematically interpolating a few output >levels between each input sample help, and deos that technique have a name >I should look out for? A generalized interpolation scheme is to consider each point in time to be a linear combination of the nearby sampled points; this is called a Finite Impulse Response filter (FIR). Such a scheme is a filter because it can be chosen (by twiddling the coefficients) to generate zero output amplitude of the first harmonic interval (which is called a '2X oversampling digital filter'), or of the first two harmonic intervals ('4X oversampling digital filter'). These so-called 'digital filters' are a hot topic; I don't know, offhand, what would be a good reference. John Whitmore
jeh@dcs.simpact.com (10/16/90)
In article <1319@beguine.UUCP>, Jeff.Miller@samba.acs.unc.edu (Jeff Miller) writes: > [...] I don't recall any treatment of the need and mathematics of filtering > the output of a D/A converter. I would imagine at CD or DAT frequencies > it wouldn't be too critical as any harmonics would be ultrasonic (or would > they? No no no. The output filter (aka the "reconstruction filter") is an essential part of the process. Also be aware that the input to this filter is NOT a step waveform, but rather a series of pulses -- one pulse per sample. Ideally the pulses should be of zero width. > If the requirements are as stringent as the input filtering requirements, > I may have some trouble. yep. > Would mathematically interpolating a few output > levels between each input sample help, and deos that technique have a name > I should look out for? yes... I believe it's called oversampling with digital filtering. --- Jamie Hanrahan, Simpact Associates, San Diego CA Chair, VMSnet [DECUS uucp] and Internals Working Groups, DECUS VAX Systems SIG Internet: jeh@dcs.simpact.com, or if that fails, jeh@crash.cts.com Uucp: ...{crash,scubed,decwrl}!simpact!jeh
myers@hpfcdj.HP.COM (Bob Myers) (10/18/90)
>>And I've always wondered: can two ultrasonic sounds or an ultrasonic >>and audible sound beat to create an audible tone? > > Obviously, yes. Wait just a minute, John. As SOUND, the answer to this question is *no*. You will not, for example, hear a 1000 Hz tone when exposed to *sound sources* of, say, 20 kHz and 21 kHz. The ear doesn't work that way. (The phenonenon wherein one may tune a stringed instrument by listening for a "wavering" sound when one string approaches the frequency of the other is a case of interference producing a tremolo (varying intensity), not a vibrato (varying frequency).) (And I hope I got the musicalese right - it's been a while!) However, if such signals exist in an electronic circuit (as was the case in the instance you mention, with the ultrasonic pilot tone producing nasty results when mixed with 60 Hz), then these "beat frequencies" WILL very likely occur, as intermodulation occurs due to the non-linearities present in any amplifying device. Bob Myers KC0EW HP Graphics Tech. Div.| Opinions expressed here are not Ft. Collins, Colorado | those of my employer or any other myers@fc.hp.com | sentient life-form on this planet.
jbm@eos.arc.nasa.gov (Jeffrey Mulligan) (10/20/90)
myers@hpfcdj.HP.COM (Bob Myers) writes: >>>And I've always wondered: can two ultrasonic sounds or an ultrasonic >>>and audible sound beat to create an audible tone? >> >> Obviously, yes. >Wait just a minute, John. As SOUND, the answer to this question is *no*. >You will not, for example, hear a 1000 Hz tone when exposed to *sound sources* >of, say, 20 kHz and 21 kHz. The ear doesn't work that way. (The >phenonenon wherein one may tune a stringed instrument by listening for a >"wavering" sound when one string approaches the frequency of the other is >a case of interference producing a tremolo (varying intensity), not a vibrato >(varying frequency).) (And I hope I got the musicalese right - it's been a >while!) Excellent point! >However, if such signals exist in an electronic circuit (as was the case in >the instance you mention, with the ultrasonic pilot tone producing nasty >results when mixed with 60 Hz), then these "beat frequencies" WILL very >likely occur, as intermodulation occurs due to the non-linearities present >in any amplifying device. As Bob has pointed out, the key here is nonlinearities. Interestingly, the visual analog of this DOES work, i.e. the sum of two high frequency gratings (which cannot be resolved) can produce a visible beat. [To actually get these gratings on the retina, you have to use laser interferometry, otherwise the optical point spread function of the eye would blur them away]. This demonstrates not only the neural nonlinearity, but also the fact that these high frequencies ARE passed by the early stages of the visual process. In order to hear beats between ultrasonic tones, it would be necessary both to have a nonlinearity in the auditory system, AND a cochlea etc. that would pass the ultrasonic tones at least as far as the nonlinearity. I suspect that the ultrasonic tones don't make it as far as the nervous system for mechanical reasons. -- Jeff Mulligan (jbm@eos.arc.nasa.gov) NASA/Ames Research Ctr., Mail Stop 262-2, Moffett Field CA, 94035 (415) 604-3745
wolfgang@wsrcc.uucp (Wolfgang S. Rupprecht) (10/21/90)
>And I've always wondered: can two ultrasonic sounds or an ultrasonic >and audible sound beat to create an audible tone? An some people have pointed out, the whole trick to getting the beats is a non-linear addition of the two sine waves. The ear is fairly linear in that respect. You won't hear the beats. Electronics on the other hand are hard pressed to be called linear. You will definitely get sum and difference products. This is actually the basis for some bug-foilers. Just activate several powerful ultrasonic sources. Human ears won't hear the beats, but the electronics sure will! It usually best to have totally separate oscillators for this, since you don't want the sinusoids mixing in your electronics. -wolfgang -- Wolfgang Rupprecht uunet!{nancy,usaos,media!ka3ovk}!wsrcc!wolfgang Snail Mail Address: Box 6524, Alexandria, VA 22306-0524
myers@hpfcdj.HP.COM (Bob Myers) (10/23/90)
>As Bob has pointed out, the key here is nonlinearities. Interestingly, >the visual analog of this DOES work, i.e. the sum of two high frequency >gratings (which cannot be resolved) can produce a visible beat. Uh, Jeff, I'm still not certain I go along with this - isn't the above phenomenon the result of interference (two items of slightly different wavelengths - or in this case, spacing - resulting in constructive and destructive addition of the signals at a much longer, and hence visible, wavelength) rather than intermodulation (the actual multiplication of sinusoids of different frequencies)? Bob Myers KC0EW HP Graphics Tech. Div.| Opinions expressed here are not Ft. Collins, Colorado | those of my employer or any other myers@fc.hp.com | sentient life-form on this planet.
myers@hpfcdj.HP.COM (Bob Myers) (10/23/90)
>An some people have pointed out, the whole trick to getting the beats >is a non-linear addition of the two sine waves. Still "no." Non-linear ADDITION (whatever that means) is not the source of true intermodulation. (By "non-linear addition", I can only assume that you mean the two input signals are multiplied by different values before being summed. But this is not required to produce *interference*, which is the phenomenon which gives audible "beats" and visible interference patterns - interference happens when adding any two signals of different phase or frequency.) Non-linear *multiplication* (i.e., amplification) is the means through which electronic equipment can produce the sum and difference frequencies of two inputs. Here's what happens: Ideally, an amplifier would produce and output which is simply a "bigger copy" of the input; this would equate to a linear function as follows (in terms of input and output voltages): Eout = A Ein where A is the amplification factor; if the output is a "ten times bigger" version of the input, A = 10. As this is of the same form as the equation for a line, such amplification is said to be "linear," and the gain does not depend on the input amplitude. However, NO electronic device can provide perfectly linear behavior over all possible inputs; at some point, at least, the output will no longer be properly described by such a simple equation. If there is any non-linearity in the output function, then there must be at least some "higher-order" terms in this function. We might imagine that a simple case would be at least: Eout = AEin + BEin^2 Here, the "linear" portion of the action (the amplification that we really want) is described by the first term, and the non-linearity is modelled as a "squared" term. Here's where the intermodulation comes from, for if the input is anything other than a simple sinusoid, there will be at least two sinusoidal components. In the simplest case, there would be exactly two: Ein = Xsin(w1t) + Ysin(w2t) where w1 and w2 are indicating different frequencies. It should be clear that if THIS value of "Ein" is squared (by the non-linearity of the amplifier), then the output will include some terms which are the product of these two sinusoids (plus the multiplying factor B). Trig identities will show that the multiple of two sine waves are cosines at the sum and difference frequencies, and so we have the "intermodulation" components. (This has some practical application; for example, a diode is about the simplest totally-nonlinear device one can imagine, and forms the basis - along with a resistive summing network - for a simple AM modulator or mixer.) Bob Myers KC0EW HP Graphics Tech. Div.| Opinions expressed here are not Ft. Collins, Colorado | those of my employer or any other myers@fc.hp.com | sentient life-form on this planet.
rosentha@sierra.STANFORD.EDU (rosentha) (10/26/90)
Can a 20 and 21khz tone combine to give an audible difference frequency? Yes they can, if they are loud enough. The ear has some nonlinearity and should mix the two signals down to a kiloherz.
jbm@eos.arc.nasa.gov (Jeffrey Mulligan) (10/26/90)
I wrote: >>As Bob has pointed out, the key here is nonlinearities. Interestingly, >>the visual analog of this DOES work, i.e. the sum of two high frequency >>gratings (which cannot be resolved) can produce a visible beat. myers@hpfcdj.HP.COM (Bob Myers) writes: >Uh, Jeff, I'm still not certain I go along with this - isn't the above >phenomenon the result of interference (two items of slightly different >wavelengths - or in this case, spacing - resulting in constructive and >destructive addition of the signals at a much longer, and hence visible, >wavelength) rather than intermodulation (the actual multiplication of >sinusoids of different frequencies)? You are right up to a point; what you describe is the formation of a single interference grating. In practice this is done using two plane waves of the same frequency, but different orientations in space. Two focussed beams enter through different points in the pupil, with the spacing of the entrance pupils controlling the spatial frequency on the retina. The upper limit on spatial frequency produced is determined by how far apart the beams can be spaced; sometimes drugs may be used to dilate the pupil. In the experiments which demonstrated the nonlinearity, two pairs of beams produced two interference gratings. The beams were pulsed so that only one pair was on at a time. Thus there is no interference between the beams making up each of the two gratings. The pulse rate was high ( > 1KHz ), so the gratings were superimposed as far as the visual system is concerned. This is pretty much state-of-the-art research which has been done at the University of Rochester. The results about the nonlinearity are still unpublished, although they have been presented at several scientific meetings. There are some papers out of the same lab in which high-frequency laser interference gratings generated visible low-frequency aliases with the photoreceptor sampling array. -- Jeff Mulligan (jbm@eos.arc.nasa.gov) NASA/Ames Research Ctr., Mail Stop 262-2, Moffett Field CA, 94035 (415) 604-3745
roy@phri.nyu.edu (Roy Smith) (10/26/90)
rosentha@sierra.STANFORD.EDU (rosentha) writes: > Can a 20 and 21khz tone combine to give an audible difference frequency? > Yes they can, if they are loud enough. The ear has some nonlinearity and > should mix the two signals down to a kiloherz. I missed the beginning of this, but rosentha's statement doesn't make sense. You don't need any nonlinearities to mix two signals to get their sum and differences, all you need is a (linear) adder. What you need a nonlinearity for is to get detection without a mixer. -- Roy Smith, Public Health Research Institute 455 First Avenue, New York, NY 10016 roy@alanine.phri.nyu.edu -OR- {att,cmcl2,rutgers,hombre}!phri!roy "Arcane? Did you say arcane? It wouldn't be Unix if it wasn't arcane!"
mrj@cs.su.oz (Mark James) (10/26/90)
In article <17660121@hpfcdj.HP.COM> myers@hpfcdj.HP.COM (Bob Myers) writes: >Wait just a minute, John. As SOUND, the answer to this question is *no*. >You will not, for example, hear a 1000 Hz tone when exposed to *sound sources* >of, say, 20 kHz and 21 kHz. The ear doesn't work that way. The auditory system is able to convert an amplitute modulated tone to a pitch percept at the modulating frequency. e.g the combination of a 1.0, 1.1 and 1.2 kHz tone (a 1.1kHz tone amplitude modulated at 100Hz) has a perceived pitch of 100Hz. This is not caused by non-linearity in the cochlear. This ability falls off somewhat as the component frequencies increase. Mark
gja@mullian.ee.mu.oz.au (Grenville Armitage) (10/26/90)
In article <17660121@hpfcdj.HP.COM> myers@hpfcdj.HP.COM (Bob Myers) writes: >Wait just a minute, John. As SOUND, the answer to this question is *no*. >You will not, for example, hear a 1000 Hz tone when exposed to *sound sources* >of, say, 20 kHz and 21 kHz. The ear doesn't work that way. Hmmm. Try telling that to guitar players who tune their instruments by ear. At least one technique I know of (and use) involves listening for the 'beat' notes as you tune each higher string to a harmonic of the previous string. And this works at quite low volumes too..... gja
jbm@eos.arc.nasa.gov (Jeffrey Mulligan) (10/27/90)
gja@mullian.ee.mu.oz.au (Grenville Armitage) writes: >In article <17660121@hpfcdj.HP.COM> myers@hpfcdj.HP.COM (Bob Myers) writes: >>Wait just a minute, John. As SOUND, the answer to this question is *no*. >>You will not, for example, hear a 1000 Hz tone when exposed to *sound sources* >>of, say, 20 kHz and 21 kHz. The ear doesn't work that way. >Hmmm. Try telling that to guitar players who >tune their instruments by ear. At least one technique >I know of (and use) involves listening for the 'beat' notes >as you tune each higher string to a harmonic of the previous >string. There seems to be a lot of confusion about this. Sure a guitarist can hear the beat; but he can because he can hear the carrier, not because the beat itself might be at an audible frequency. There has to be a nonlinearity before there will be any spectral energy at the beat frequency. Another poster mentioned the psychological phenomenon of "periodicity pitch", where a complex of tones having frequencies 1000, 1100, 1200 will have a subjective pitch of 100 Hz. This is fine, but again the tones in the complex have to get into the auditory system first. Ultrasonic tones don't. -- Jeff Mulligan (jbm@eos.arc.nasa.gov) NASA/Ames Research Ctr., Mail Stop 262-2, Moffett Field CA, 94035 (415) 604-3745