A few other notes I should add: 1. Per the reasoning above, the PCM of a sine wave is going *tend to* to be proportional to the wave's physical amplitude (speaker displacement) *multiplied by* its frequency — again, in an ideal system without frequency-dependent effects (and those are inevitable).
2. Just because PCM encodes velocity doesn't mean there's some inherent delay or phase shift as suggested by others here. Those artifacts come from frequency-dependent effects. For example, if I have a circuit composed of two identical transducer coils wired to one another, and I whack the first one, the acceleration of its coil is going to create a voltage that drives an acceleration of the other coil. In an "ideal" physics scenario the second coil might copy the first's movement perfectly and instantaneously, but it doesn't because we have electrical resistance, friction, momentum, etc. – Evan Balster creator of imitone <http://imitone.com> On Sun, Oct 1, 2017 at 12:56 PM, Evan Balster <e...@imitone.com> wrote: > DSP programmer recently getting into transducer experimentation here. > Gonna walk through the dimensional analysis: > > > 1. Sound waves are a form of energy. Energy doesn't exchange directly > with other units like voltage, velocity, momentum, etc, because those units > are in different dimensions. > > 2. If the displacement distance is constant, a higher-frequency sound > wave means the air is traveling back an forth at higher velocity (directly > proportional to frequency). > > 3. Kinetic energy is proportional to velocity squared. > > 4. *Thus*, it takes four times as much energy to vibrate with the same > displacement at twice the frequency (because the velocity is doubled). > > 5. Power is the rate of energy flow in an electrical system. > > 6. In an ideal digital/analog conversion (with no frequency-dependent > effects) PCM values are directly proportional to voltage. > > 7. In an electrical system, power is (voltage x current). Current is > (voltage / impedance). Ideally impedance is fixed, so power is > proportional to voltage squared. > > 8. *Thus*, power is proportional to PCM squared, and energy is > proportional to PCM squared per second (mean-square). > > 9. Thus, because mean-square PCM is proportional to energy, and energy is > proportional to speaker displacement times frequency squared, PCM is *not* > proportional to displacement. In an ideal electrical system, it's > proportional to velocity. > > > Filter-like (frequency-dependent) effects can change things, because they > can mix a signal together with its integrals and derivatives. It's simple > to transform that velocity signal into a displacement signal or vice versa, > but in practice any filtering effects will result in a frequency-dependent > mixture of the two (and potentially other degrees of integration). > > – Evan Balster > creator of imitone <http://imitone.com> > > On Sun, Oct 1, 2017 at 11:31 AM, Renato Fabbri <renato.fab...@gmail.com> > wrote: > >> >> >> On Sun, Oct 1, 2017 at 1:23 PM, robert bristow-johnson < >> r...@audioimagination.com> wrote: >> >>> >>> >>> i know relatively little about transducers. but these are getting to be >>> pretty fundamental questions. >>> >>> Renato, we don't know exactly how much the digital value will translate >>> to a precise loudspeaker piston displacement because there is an amplifier >>> with a volume control in between the D/A converter and the loudspeakers. >>> >>> when we talk of gain and scaling of the digital numbers coming out of a >>> DSP process, we can only really discuss what is "full scale" and, perhaps >>> if you're a hardware geek, what the reference voltage is for the D/A >>> converter (or A/D on the input side) which will map to full scale. >>> usually, as integers, the A/D and D/A integer values are left-justified. >>> if your D/A word was a puny 16 bits, that means -32768 and +32767 are your >>> full scale values. how they get translated to loudspeaker piston >>> displacement depends on the analog hardware in between (and the volume >>> control) and of the loudspeaker. >>> >>> also about instantaneous pressure vs. displacement, it's really a >>> combination of the two and the two are related by some LTI acoustic >>> filtering function. >>> >> one is the derivative of the other I guess >> ( >> as in >> http://physics.bu.edu/~duffy/semester1/c20_disp_pressure.html >> ) >> >> PS. I am getting quite convinced that PCM samples by standard represent >> the displacement of the sonic wave. >> >>> for me, it's easiest to imagine that the instantaneous pressure >>> (actually the *difference* between instantaneous absolute pressure and the >>> ambient atmospheric pressure, the DC component, which is about 100 kPa) is >>> proportionally mapped to the signal sample in the DSP. but i never worry >>> about the constant of proportionality because it simply is what it after i >>> set the volume control to a desired setting. >>> >>> that's my pedantic spin on it. >>> >>> r b-j >>> >>> >>> >>> >>> ---------------------------- Original Message >>> ---------------------------- >>> Subject: Re: [music-dsp] PCM audio amplitudes represent pressure or >>> displacement? >>> From: "Renato Fabbri" <renato.fab...@gmail.com> >>> Date: Sun, October 1, 2017 12:58 am >>> To: "A discussion list for music-related DSP" < >>> music-dsp@music.columbia.edu> >>> ------------------------------------------------------------ >>> -------------- >>> >>> > tx for the infos and thoughts. >>> > >>> > I am getting the same difficulty when I >>> > look at texts in the subject. >>> > >>> > Making the question very objective, >>> > is anyone able to state very briefly if >>> > LPCM samples 1000, 5, 500, 70 >>> > will be converted by a loudspeaker >>> > into displacements proportional to such >>> > values or to pressure (differentials)? >>> > >>> > I mean, if the samples are constant, >>> > say 6000, the speaker surface is displaced >>> > an amount proportional to 6000 and >>> > stays there until the samples end. >>> > That is displacement, right? >>> > >>> > Apologies if I am trying to hard to >>> > keep things too simple, but >>> > I believe that there might be >>> > a simple answer. >>> > (Although probably not if we >>> > think also about what is input >>> > by mics.) >>> > >>> > >>> > >>> > On Sat, Sep 30, 2017 at 6:40 PM, Nigel Redmon <earle...@earlevel.com> >>> wrote: >>> > >>> >> > I'm pretty sure an instantaneous [audio] voltage (or a number in a >>> >> > stream of PCM values) represents a pressure differential, and not >>> >> > displacement. >>> >> >>> >> This is one of those point-of-view things…I used the “usually” caveat >>> for >>> >> that reason… >>> >> >>> >> You strike a drum, you displace the head, create a pressure wave >>> because >>> >> the room has air, the pressure wave displaces the conducer capsule >>> membrane >>> >> resulting in a change of capacitance, which is converted to a change >>> in >>> >> voltage, it’s amplifier and used to displace a speaker, which creates >>> a >>> >> pressure wave, which displaces your ear drum… >>> >> >>> >> So, whether the electrical signal is of the displacement of the >>> capsule >>> >> membrane, or the pressure differential over time hitting it…ultimately >>> >> we’re normally recording sound as it exists in the air, so you could >>> >> rightly say the electrical signal is an analog of the pressure >>> changes over >>> >> time—or you could look at it on the electro-mechanical level and say >>> we use >>> >> the pressure to displace and element and record that displacement. >>> >> >>> >> I guess how firmly you stick to one or the other depends on >>> conventions >>> >> you're used to. As an engineer, I see it as representing the >>> displacement. >>> >> The reason I view it that way is because I’m intimately aware of the >>> masses >>> >> involved with dynamic or condenser mics, and their shortcomings. So, I >>> >> think of it as the mic diaphragm trying its best to approximate the >>> changes >>> >> in pressure, and we convert that displacement approximation to an >>> >> electrical signal. >>> >> >>> >> It’s probably easier to view the flip side, the speaker—so many >>> reasons >>> >> for a bad approximation; you need a big surface to move a lot of air, >>> >> particularly for low frequencies, but a big surface has enough mass >>> that it >>> >> sucks for quicker changes so we split up the audio band; all the >>> while the >>> >> surfaces are flexing and the cabinet and surface attachments are >>> messing >>> >> with the attempt, and you wonder how the heck we manage to put >>> something >>> >> out that’s listenable ;-) Anyway, that’s why I view it as >>> displacement; >>> >> we’re trying like heck to make the displacement true, and the pressure >>> >> changes follow (for speakers—the other way with a mic). It may be a >>> >> different story with “plasma” approaches, I’m talking about our usual >>> >> practical transducers. >>> >> >>> >> >>> >> > On Sep 30, 2017, at 1:37 PM, Ben Bradley <ben.pi.brad...@gmail.com> >>> >> wrote: >>> >> > >>> >> > I'm pretty sure an instantaneous [audio] voltage (or a number in a >>> >> > stream of PCM values) represents a pressure differential, and not >>> >> > displacement. A loudspeaker driver in air (within its rated >>> response) >>> >> > is constrained by the air, above its resonant frequency (at the low >>> >> > end of its frequency range - for a woofer, this would be near the >>> >> > resonant frequency of a ported bass cabinet). Below its resonant >>> >> > frequency the output is a position proportional to voltage or >>> current, >>> >> > but the coupling efficiency to the air goes down with frequency, so >>> >> > this isn't a good operating range. A woofer with a 1Hz input is >>> going >>> >> > to have the same displacement as with a 0.1Hz input at the same >>> >> > voltage, because it doesn't have good coupling to the air at such >>> low >>> >> > frequencies. >>> >> > >>> >> > A speaker in air (operating within its intended frequency range) is >>> >> > like an oar in water. You can move it back and forth very far of >>> >> > you're doing it at a slow enough rate (low enough frequency). If you >>> >> > do it at a higher frequency, it takes more force to move it back and >>> >> > forth the same distance. If you use the same force, you end up >>> moving >>> >> > back and forth a smaller distance due to the "strong coupling" of >>> the >>> >> > oar to the water. This is how a speaker cone sees the air, and shows >>> >> > how cone displacement goes down as frequency goes up, even though >>> the >>> >> > acoustic energy is the same. The voltage is proportional to >>> >> > [differential] pressure, and not (as one might easily believe, and >>> >> > probably some books say!) displacement. >>> >> > >>> >> > Regarding phase, as displacement is the integral of pressure, >>> >> > displacement goes down with an increase in frequency, and there's a >>> >> > phase shift between pressure and displacement. I vaguely recall that >>> >> > the integral of a cosine is a sine, so there's a 90 degree (or pi/2, >>> >> > not pi/4 - you're perhaps thinking of 1/4 of a complete wave) phase >>> >> > shift between these two. But a dynamic microphone does the exact >>> >> > inverse of a speaker, so the sound-to-sound conversion actually >>> works >>> >> > out without a phase shift. Even presuming a condenser mic does this >>> >> > phase shift (I can't quite visualize how or whether it does >>> offhand), >>> >> > human ears are almost completely insensitive to phase shift vs. >>> >> > frequency, so in practice it doesn't matter. >>> >> > >>> >> > >>> >> > On Sat, Sep 30, 2017 at 3:39 PM, Stefan Sullivan >>> >> > <stefan.sulli...@gmail.com> wrote: >>> >> >> >>> >> >> so there might be a phase >>> >> >> offset between the recorded >>> >> >> and the reproduced sound. >>> >> >> >>> >> >> >>> >> >> Ah, I think I might be understanding your question more >>> intuitively. Is >>> >> your >>> >> >> question about positive voltages from microphones being >>> represented as >>> >> one >>> >> >> direction of displacement, whereas the positive voltages from >>> speakers >>> >> being >>> >> >> represented as the opposite displacement? To be honest I'm not sure >>> >> what the >>> >> >> convention is here, but there must be an industry-wide convention >>> or >>> >> even >>> >> >> one speaker manufacturer to the next might be phase incoherent? I >>> >> actually >>> >> >> don't know the answer here, but maybe somebody else on the list >>> does? >>> >> >> >>> >> >> It is worth pointing out that Nigel is right about phase being >>> frequency >>> >> >> dependent. Even the mechanical system has dynamic components that >>> have a >>> >> >> frequency response, which means their phase response could be >>> nonlinear, >>> >> >> which transducer engineers would either need to compensate for with >>> >> other >>> >> >> reactive mechanical components, or with the electrical components, >>> or >>> >> DSP. >>> >> >> >>> >> >> Interestingly, the acoustical and mechanical systems of >>> transducers can >>> >> be >>> >> >> modeled as electrical circuit complements themselves. I assume >>> that all >>> >> >> speaker/microphone manufacturers model their systems this way, but >>> again >>> >> >> it's not actually my industry so I can't speak to what actually >>> happens. >>> >> >> Marshall Leach has a really good book on the subject: >>> >> >> https://he.kendallhunt.com/product/introduction- >>> >> electroacoustics-and-audio-amplifier-design >>> >> >> >>> >> >> Stefan >>> >> >> >>> >> >> >>> >> >> _______________________________________________ >>> >> >> dupswapdrop: music-dsp mailing list >>> >> >> music-dsp@music.columbia.edu >>> >> >> https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> > _______________________________________________ >>> >> > dupswapdrop: music-dsp mailing list >>> >> > music-dsp@music.columbia.edu >>> >> > https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> > >>> >> >>> >> _______________________________________________ >>> >> dupswapdrop: music-dsp mailing list >>> >> music-dsp@music.columbia.edu >>> >> https://lists.columbia.edu/mailman/listinfo/music-dsp >>> >> >>> > >>> > >>> > >>> > -- >>> > Renato Fabbri >>> > GNU/Linux User #479299 >>> > labmacambira.sourceforge.net >>> > _______________________________________________ >>> > 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 >>> >> >> >> >> -- >> Renato Fabbri >> GNU/Linux User #479299 >> labmacambira.sourceforge.net >> >> _______________________________________________ >> dupswapdrop: music-dsp mailing list >> music-dsp@music.columbia.edu >> https://lists.columbia.edu/mailman/listinfo/music-dsp >> > >
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