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Dynamic range question, Dynamic range of hearing vs. equipment
dhromed
post Jun 14 2011, 11:01
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QUOTE (Northpack @ Jun 14 2011, 09:15) *
Anyone interested in writing such an explanation?


Since I'm a little hazy on the concepts as well, here's what I've pieced together from crucial posts in this thread that seemed to gloss over a few details that I think were assumed to be self-evident, but are not (at least to me).

Here goes.

#
QUOTE (drewfx @ Jun 11 2011, 18:33) *
96 dB of dynamic range represents the ratio of the loudest sound to the quietest sound


# It is a ratio, and not a linear distance, because dB is a logarithmic scale, which in turn is because that's how our ears/brains perceive volume.

# This ratio is a number, obviously.

# This number can be expressed or approximated as a fraction of n:1

# To get a certain n, you need a certain amount of bits to put that number in. Self-evident to anyone familiar with computers (such as myself), but probably not to a sizeable section of curious listeners.

#
QUOTE (Notat @ Jun 12 2011, 19:10) *
During the development of the CD format, a 14-bit format (84 dB) was originally proposed. The 14-bit proposal was based on the capabilities of the ADCs and DACs available at the time and on calculations that you see in this thread of ideal dynamic range of real listening environments. The resolution was pushed to 16 bits at the insistence of Sony.


#
QUOTE (knutinh @ Jun 14 2011, 10:13) *
For CD (16 bits) this works out to be 65536:1 or 2^16 or about 96dB.


(reordered the above post to better sequence the train of concepts)

# Taken to an extreme: suppose you have 2-bit audio, the ratio would be tiny, and everything would drown in the noise.

And there you have it, I believe. If I made a mistake, please correct it. There's nothing worse than to learn you're actually fundamentally confused by objects that you use on a daily basis (i.e. foobar and my receiver).

This post has been edited by dhromed: Jun 14 2011, 11:05
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knutinh
post Jun 14 2011, 11:12
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QUOTE (Northpack @ Jun 14 2011, 12:00) *
QUOTE (knutinh @ Jun 14 2011, 09:13) *
Properly dithered digital behaves similar to analog systems where the noise floor is (best case) about 6*#bits below the maximum signal.

Sure, that's the way it is, you and I know that because we know the math behind digital audio. But it is not an explanation to those who don't know the math and are misguided by audiofoolish reasoning. If you want to enlighten people you have to explain things in a way they can follow instead of making assertions, how true they may be. That's why Plato always wrote in the form of dialogues...

For those who seek true insight, I would expect them to be able to search wikipedia or buy a book where the topic should be far greater explained than I am capable of. That involves doing actual calculations yourself.

For those who dont want true insight, simple assertions may be good enough. How to phrase those assertions, what level of detail/argumentation to include, what (if any) analogys to include, and how to be sure that the assertions are correct and relevant is an open question.

I think that given the knowledge that CD can (at best) do 96dB-ish SNR/DR or so, and that no analog media or acoustic listening-room/speaker-combo can do anything close to that, for pragmatic assertion-based discussions there is not much more to it. Of course, if you want to true insight, you could discuss the type of noise, the audibility of noise, how one noise source masks or dont mask another etc.

-k

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Arnold B. Kruege...
post Jun 16 2011, 13:05
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QUOTE (knutinh @ Jun 10 2011, 05:31) *
I seem to remember that people who record ambient sounds for movies etc (animals, weather, water) have a hard time finding spots on the earth that are not affected by passing planes, kids with a boomblaster etc. Acoustic pollution.


Once you avoid the human-related noises, you are still left with natural noises which are often quite loud. I'm talking about water, wind, animal calls, thunder, and leaves driven by wind.

It can be quite quiet out in nature. Along the Minnesota-Ontario border and in northern Ontario you can get 20-40 miles from the nearest road or other man-made noise source. The exhaust sound of a small gasolene generator can carry for over a dozen miles.
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dc2bluelight
post Jun 17 2011, 09:58
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Very interesting discussion. I'd like to add a couple of things.

The 96dB FS to Noise ratio of a 16 bit system is theoretical. It's almost never exactly 96dB, though. Dithering shaves off 3dB or so, then the so called "noise shaping" D/A converters put 8dB or so back in, if that's even real. But FS is a maximum peak figure, we measure noise using an RMS meter with (hopefully) a weighting filter in front of it, either A or ITU-R468, either of which will tend to push the reading downward making 96dB look more like 103 or so.

Rooms, on the other hand, generally have their noise figures expressed as NC (Noise Criterion) figures. NC is a set of curves, and to qualify for a given NC curve, each octave band of measured noise must fall below that standardized curve. So, the actual measurement of room noise is done flat, but the result has to fit a curve that allows lots more energy at the low end. It's not an A weighting curve, so it's not directly related to a SNR figure of a digital system. And, though I can't site a reference, it's been shown recently that the "average" living room hovers around NC-25. By the way, you can't actually measure below an NC-20 with a 1/4" measurement mic. Too much self noise., you need a 1" mic with a quiet pre to do it.

Also, when we say we listen to our music at 80dB SPL, again, that's an average level, and might be measured with an average meter (probably) or an RMS meter. Depending on how a CD is mastered, the average levels might be as much as 15dB - 20dB below FS, or as little as 8dB below FS (hope not, but they crunch 'em pretty hard these days). It's not unreasonable to expect to listen at 80dB SPL (average, unweighted) and still get a peak or two at 95 - 100dB SPL. Depending on your SPL meter ballistics, you may or may not see that peak or read it properly, though.

It would still seem that 16 bits is more than enough to handle a 95dB SPL peak in an NC-20 room.

There is an advantage to recording at 24 bits. For one, if you're recording live, you've got (theoretically!) 48dB more room to make a level error in. If you're mixing, or processing, you've got more resolution in which to adjust gain, process, eq, etc. But there's a really big rub: Almost without exception, 24 bit sound cards NEVER produce 144dB of dynamic range! Just check the specs of almost any "affordable" sound device,oh heck, even the unaffordable ones. In fact, I've found exactly one A/D that is commercially available that has a real 24 bit's worth of dynamic range: http://www.stagetec.com They get 153dB A, and don't even bother with a mic preamp! Everything else is WAY less.

Which is the point. Why would to stretch your system to handle 24 bits when even the so-called 24 bit material has a dynamic range of more like 18 - 20 bits?

So I'm saying 16 bits is enough for "release" material, perhaps a little shy for capture. And I'm reminded that ratio of 0dB SPL to the Threshold of Pain is about 22+ bits.
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pdq
post Jun 17 2011, 13:21
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Very nice first post. Thanks. beer.gif
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Arnold B. Kruege...
post Jun 17 2011, 14:02
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QUOTE (pdq @ Jun 17 2011, 08:21) *
Very nice first post. Thanks. beer.gif


First serious omission - a real world discussion of noise levels in the place where the recording is made. For example my live recording kit actually has close to 100 dB dynamic range under ideal conditions. I don't think I've ever made a live recording with better than about 67 dB dyanmc range. The balance of the noise is usually due to HVAC and humans being in the room.

Second serious omission - discusison of what kind of dynamic range we can actually perceive.

As JJ has said from time to time, the ear has about 30 dB instantaneous dynamic range that slides up and down. The sliding is is subject to a number of timing and memory conditions.

I think most people have been around some loud sound that caused a temporary threshold shift that was readily perceptible. IOW, they went to a loud concert or were in a noisy industrial environment and noticed that they were having a hard time hearing soft sounds for hours or even days later. That reperesnts a memory effect that vastly reduces the effective dynamic range of the human ear.
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Arnold B. Kruege...
post Jun 17 2011, 14:20
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QUOTE (dc2bluelight @ Jun 17 2011, 04:58) *
In fact, I've found exactly one A/D that is commercially available that has a real 24 bit's worth of dynamic range: http://www.stagetec.com They get 153dB A, and don't even bother with a mic preamp! Everything else is WAY less.


The means that they use to obtain the 153 dB dynamic range has been around for a long time.

They basically run a bunch of converters in parallel. Each converter has an input attenuator with a different amount of attenuation. They then pick an output from a converter that is neither clipping nor close to the bottom of its range, and scale it digitally based on that converter's input attenuation.

For example, let say we have two medium-priced converters with 110 dB dynamic range. We connect one to the source with no attenuation, and put a 30 dB attenuator in line with the other converter. We then monitor the status of the two converters. In general we use the output of the unattenuated converter, but when it gets within 3 dB of clipping, we switch over the the converter with the 30 db attuator. On the digital side we multiply the digital output of the attenuated converter by 30 dB in the digital domain. When we switch over to the output of the unattenuated converter we simply use its digital output as is.

The net result is a system with about 140 dB dynamic range made out of out of two $5 ADC chips, and a few other relaitvely inexpensive parts. We do have to pay attention to the recovery time of the unattenuated converter, because it may take a few milliseconds to recover from clipping when the input signal gets too high and it clips.

The other approach to extending the dynamic range of conveters is to simply run them in parallel and obtain a 3 dB noise reduction every time we double the number of converters. I've never seen more than 8 convertors used this way.

Neither of these approaches reduce nonlinear distortion. They just address noise.
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dc2bluelight
post Jun 17 2011, 16:40
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Wow. You guys are tough here.
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dc2bluelight
post Jun 18 2011, 00:53
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QUOTE (Arnold B. Krueger @ Jun 17 2011, 08:02) *
First serious omission - a real world discussion of noise levels in the place where the recording is made. For example my live recording kit actually has close to 100 dB dynamic range under ideal conditions. I don't think I've ever made a live recording with better than about 67 dB dyanmc range. The balance of the noise is usually due to HVAC and humans being in the room.

Let me apologize for my "serious omission". What you say is certainly true of live recording, but recording studios are built to NC-15 and below, including HVAC, and NC-10 and NC-5 is not unachievable. Keep in mind, you won't be able to correlate an unweighted dynamic range measurement with a room noise measurement based on NC curves, though. If you take into account positioning a mic close to a source (inverse square law), you can indeed record a dynamic range in excess of 100dB, even in an NC-20 room. No, it's not typical, but it's done quite often.
QUOTE (Arnold B. Krueger @ Jun 17 2011, 08:02) *
Second serious omission - discusison of what kind of dynamic range we can actually perceive.

As JJ has said from time to time, the ear has about 30 dB instantaneous dynamic range that slides up and down. The sliding is is subject to a number of timing and memory conditions.

I think most people have been around some loud sound that caused a temporary threshold shift that was readily perceptible. IOW, they went to a loud concert or were in a noisy industrial environment and noticed that they were having a hard time hearing soft sounds for hours or even days later. That reperesnts a memory effect that vastly reduces the effective dynamic range of the human ear.


Again, if I've made another "serious omission", I apologize. As to the "discusison" (sic), however, the instantaneous dynamic range you refer to has a time element. If the music in question has a volume envelope that makes level transitions slower than the sliding dynamic range window you describe, it is possible to perceive a dynamic range to the limits of an individual's hearing.

I thought I was agreeing with you re: 16 bit being more than adequate. My point on it being a little light for capture is, as I'm sure you're very aware, live situations often present unexpected levels. Would you rather "waste" a few bits, store some more data and capture the entire event without issue, or go ahead and clip the A/D every so often? HDD storage is cheap, 24 bit A/D, even if they have a noise floor of a 20 bit A/D, are affordable to everyone. Why not use them, then once you're finished in post, release in a well-mastered, controlled, real, unclipped, perfectly adequate 16 bit format?

I thought the Stagetec thing was a cool idea, and unique in the market. If not, I guess I've had my head under a rock, which is completely possible. But I don't see anyone else doing the multiple (cheap??) A/D trick in a product at any price. And if it doesn't address nonlinear distortion (I assume you refer to the distortion products caused by linear quantization), ok, but we're discussing dynamic range here, right?

I'm not sure why in a discussion of dynamic range, the idea that 16 bits is "overkill" because of environmental noise in recording and reproduction is so pertinent. You know, when you listen to music in a car at 65mph, the available dynamic range is less than 18dB, and in a truck, more like 10. So when we know we're recording for the car, should we use a 3 or 4 bit converter because the other 12 bits are overkill? We already have 16 bits in common use, for better or worse. It's not going to change to anything else quickly for the main stream consumer. And the direction it's going is more bits at a higher rate, as wasteful and ineffective as that is. We can't reproduce 16 bits, true. But why not record with more bit depth so we can work with some elbow room in post? Seems the recording industry from music to location film sound thinks it's a good idea.

Ok, I hear ammunition being loaded... I'm ducking back into the fox hole and putting my helmet on...





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Woodinville
post Jun 18 2011, 04:32
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Just so it's clear, the 30dB number applies for instantaneous masking inside of one ERB/Critical Band.

Over all frequency, you can have an instantaneous dynamic range of at least 70dB.


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dc2bluelight
post Jun 18 2011, 07:17
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Yes, that clears it up. Critical band masking would be a special case of dynamic range limiting in hearing.

I did look look around for something about a temporary 30dB dynamic range limit in hearing, nothing found of course. I thought it might be a reference to temporal masking, in which the masking effect exists for at least 50ms after the masker terminates, and can extend to 200ms (post-masking), with pre-masking also working up to 20ms before the masker. But since the masking effect in time is a curve, the 30dB figure didn't make sense, which is why I mentioned the time factor.

Thanks for the clarification.
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knutinh
post Jun 18 2011, 09:31
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QUOTE (dc2bluelight @ Jun 18 2011, 01:53) *
I'm not sure why in a discussion of dynamic range, the idea that 16 bits is "overkill" because of environmental noise in recording and reproduction is so pertinent. You know, when you listen to music in a car at 65mph, the available dynamic range is less than 18dB, and in a truck, more like 10. So when we know we're recording for the car, should we use a 3 or 4 bit converter because the other 12 bits are overkill? We already have 16 bits in common use, for better or worse. It's not going to change to anything else quickly for the main stream consumer. And the direction it's going is more bits at a higher rate, as wasteful and ineffective as that is. We can't reproduce 16 bits, true. But why not record with more bit depth so we can work with some elbow room in post? Seems the recording industry from music to location film sound thinks it's a good idea.

I think that this forum in many ways is an "anti-Stereophile-mag". Both camps claim to be interested in music and good sound, but other than that, whenever some audiophool claims to be able to hear staircaising in lowly 24-bit recordings, the gut-feeling of any hydrogenaudio-member is to strike him hard in the head with science (or rather her perception of what is science).

The debate should not be read as "using (e.g.) 7-bit recording in studio would be a smart choice that we recommend for all recordists", but perhaps rather "given the constraints that your particular end-to-end chain is presenting, 7 bits would be marginally enough, and anything in excess of that would only buy you margin for user-error, calculation error etc". The point is that if this marginal number is considerably smaller than regular available technology (16bits, 20bits, whatever), then one would be well adviced to spend money and research time on other parts of the business rather than drooling over 150dB SNR ADC/DACs.

I would think that when 100dB of recording SNR is really worth it, you typically would not need 40dB of headroom above that to allow for error in setting input gain? In cases where I have had problems setting input levels sort of right, it has invariable been live recordings where sound quality was so-so anyways. The gear most priced by my studio friends is typically old analog gear and tube microphones that was not designed to have extremely wide dynamic range, but rather other qualities. This indicates to me that most studio people (even really good ones) really are not looking for 120dB SNR, even though they may say otherwise. Perhaps there are exceptions in the classical/acoustic camp.

-k

This post has been edited by knutinh: Jun 18 2011, 09:37
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Arnold B. Kruege...
post Jun 18 2011, 18:58
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QUOTE (dc2bluelight @ Jun 17 2011, 19:53) *
I thought the Stagetec thing was a cool idea, and unique in the market. If not, I guess I've had my head under a rock, which is completely possible. But I don't see anyone else doing the multiple (cheap??) A/D trick in a product at any price.


I think that most of the market addresses needs that are widely perceived. The argument that you should do it just because you can do it doesn't sell a lot of equipment.

Right now converters priced for installation in middle-of-the-road production equipment have about 110 dB dynamic range. Converters in production equipment for the low end market are about 10 dB worse. Even low end consumer gear has converters that spec out above 80-85 dB.

There are converters that do 120 dB and up that are priced right for equipment in the $500-1K range, but so far they are only showing up in high end audio products that aren't needs-based, anyway. I think that most practitioners know that it isn't a lot of work getting systems and productions working subjectively noise free with 110 dB or worse converters. Therefore there isn't a lot of market for anything better.


QUOTE
And if it doesn't address nonlinear distortion (I assume you refer to the distortion products caused by linear quantization), ok, but we're discussing dynamic range here, right?


By the many standards spurious responses due to nonlinear distoriton (By which I include the things that are commonly called IM and THD) count the same towards dynamic range the same as the noise floor.

IOW there is a school of thought that counts THD+N towards dynamic range, and a less-critical school of thought that only counts noise.

QUOTE
I'm not sure why in a discussion of dynamic range, the idea that 16 bits is "overkill" because of environmental noise in recording and reproduction is so pertinent.


At this point you head off into a discussion of playback contexts, but I think that the recording context is more important since it affects the playback no matter where you play back.

The general thrust of this all heads into the same place, which is clearly a place that you want to be, which is that 16 bits is really all we need for most applications.
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thesurfingalien
post Jun 18 2011, 21:04
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@All,

I am ashamed to tell that, although the discussion has been very informative, I hve not been able to distill an answer for my question from it.

Sticking to CD specs, the 16-bit signed integers (-32768 to 32767; 65536) in my mind represent the steps to map the analogue waveform to. Since CD specs also state a 96.### dB dynamic range, it seems logical to me that the same 65536 steps also have a relation to the dB scale.

I am sure that, if we would map a waveform into 4 bits and played that waveform back at 96 dB, everybody would hear it as a rather coarse sound, and if we would lover the sound to 15 dB (theoretically, not taking into account background noise ) that same coarseness would be significantly lower. At some point however, if we increase the number of bits in which the waveform is coded in, the coarseness disappears, and the sound gets smooth enough.

There must be research done by Philips / Sony that would prove the 16 bits coding / mapping and the 96 dB were good enough for CD playback.


Peter

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knutinh
post Jun 18 2011, 21:40
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QUOTE (thesurfingalien @ Jun 18 2011, 22:04) *
Sticking to CD specs, the 16-bit signed integers (-32768 to 32767; 65536) in my mind represent the steps to map the analogue waveform to. Since CD specs also state a 96.### dB dynamic range, it seems logical to me that the same 65536 steps also have a relation to the dB scale.

The CD-specs probably does not state some dynamic range. It depends on a lot of things.

The relationship between 65536, 96 and the dB-scale is a direct one, and I believe that it has been presented in this thread:
2^16 = 65536
20*log10(2^16) ~= 96dB.

Briefly: Add one more bit of (true) resolution, and the AD or DA will have 6dB more of SNR, everything else equal.
QUOTE
There must be research done by Philips / Sony that would prove the 16 bits coding / mapping and the 96 dB were good enough for CD playback.

The fact that 16 bits was economically feasible and sounded better than existing media would have been enough.

-k
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lvqcl
post Jun 18 2011, 21:45
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QUOTE (thesurfingalien @ Jun 19 2011, 00:04) *
Since CD specs also state a 96.### dB dynamic range, it seems logical to me that the same 65536 steps also have a relation to the dB scale.

Yes. 20 * lg (65536) ≈ 96.3295986.

QUOTE (thesurfingalien @ Jun 19 2011, 00:04) *
everybody would hear it as a rather coarse sound

Not coarse, but noisy.
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Wombat
post Jun 19 2011, 00:55
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QUOTE (Woodinville @ Jun 18 2011, 04:32) *
Just so it's clear, the 30dB number applies for instantaneous masking inside of one ERB/Critical Band.

Over all frequency, you can have an instantaneous dynamic range of at least 70dB.


Very interesting number here besides the nice discussion in this thread. That may correlate to times of a day i can hear the low noise of my Aquarium pump working into my living and listening room. At nights it gets more obvious and of cause i donīt need loud music then. On a typical day i never recognize it. When i listen music loud and switch off i donīt hear it at all!
So besides all that talk about the need of more bits for bigger dynamics that little phenomenon gets ignored totaly and may smaller the needed number again smile.gif
To sad i canīt find a study about that...

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Notat
post Jun 19 2011, 01:21
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QUOTE (dc2bluelight @ Jun 17 2011, 17:53) *
But why not record with more bit depth so we can work with some elbow room in post? Seems the recording industry from music to location film sound thinks it's a good idea.

This is the thinking among mainstream audio engineers. No one uses 16-bit ADCs for professional recording. Arnold is being argumentative. He does that.

To my knowledge, the Stagetec converter is a novel idea. Like many such good ideas, one can only discount it as obvious once one has been introduced and understands it.
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Arnold B. Kruege...
post Jun 19 2011, 11:44
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QUOTE (thesurfingalien @ Jun 18 2011, 16:04) *
I am sure that, if we would map a waveform into 4 bits and played that waveform back at 96 dB, everybody would hear it as a rather coarse sound, and if we would lover the sound to 15 dB (theoretically,


There would be no coarse sound because dither must be applied for everything to be kosher.

Dither inherently randomizes the coarse steps into background noise. It is not a mask but an actual randomzing of the data. Since threre are only 4 steps, the background nnoise level would be very high, but it would be noise, not any sort of coarsness related to the step size or the sampling frequency,
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drewfx
post Jun 19 2011, 18:56
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QUOTE (thesurfingalien @ Jun 18 2011, 15:04) *
I am sure that, if we would map a waveform into 4 bits and played that waveform back at 96 dB, everybody would hear it as a rather coarse sound, and if we would lover the sound to 15 dB (theoretically, not taking into account background noise ) that same coarseness would be significantly lower. At some point however, if we increase the number of bits in which the waveform is coded in, the coarseness disappears, and the sound gets smooth enough.

There must be research done by Philips / Sony that would prove the 16 bits coding / mapping and the 96 dB were good enough for CD playback.


Peter


If you play back 4bit audio at 96dB SPL, that "96dB SPL" has little to do with the dynamic range. You are amplifying both the peaks and and the noise floor (i.e. quantization error).

The ratio of the loudest possible peak to the noise floor is always about 6dB per bit no matter how you set the playback volume (i.e. dB SPL). Adding bits lowers the level of the quantization error relative to the loudest possible signal.

So if you consider how loud (in terms of dB SPL) you might want the highest peaks in your audio to play back at, and compare it to how loud (in terms of dB SPL) the quietest sound you can hear above the ambient noise in the listening environment, that tells you how much dynamic range you need in your recorded media, which tells you how many bits you need.

If you amplify anything enough (regardless of its bit depth), you will always reach a point where you can hear the quantization. However, if the amount of amplification this requires results in the loud parts being so loud that they cause permanent hearing damage, most of us would say we've got more than enough dynamic range.
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Soap
post Jun 19 2011, 20:01
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I belive the problem thesurfingalien is having is in understanding why 4 bits = "noisy" and not "dramatic stair-stepping volume fluctuations", something I don't see clearly addressed yet.

While he is correct that 4 bits means every sample is at one of only 16 volume levels, I believe thesurfingalien thinks this means the volume of the track will coarsely jump up and down.

I think I understand why thesurfingalien is incorrect, but I'll give the experts a fair chance to answer it before I risk bastardizing the answer.


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Woodinville
post Jun 20 2011, 02:43
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QUOTE (Wombat @ Jun 18 2011, 16:55) *
QUOTE (Woodinville @ Jun 18 2011, 04:32) *
Just so it's clear, the 30dB number applies for instantaneous masking inside of one ERB/Critical Band.

Over all frequency, you can have an instantaneous dynamic range of at least 70dB.


Very interesting number here besides the nice discussion in this thread. That may correlate to times of a day i can hear the low noise of my Aquarium pump working into my living and listening room. At nights it gets more obvious and of cause i donīt need loud music then. On a typical day i never recognize it. When i listen music loud and switch off i donīt hear it at all!
So besides all that talk about the need of more bits for bigger dynamics that little phenomenon gets ignored totaly and may smaller the needed number again smile.gif
To sad i canīt find a study about that...



There are quite a few studies about TTS (Temporary Threshold Shift). Look for something under that phrase, perhaps.

By the way, I'm not sure how this relates to the aquarium pump, though. If it's low frequncy hum, it won't be masked by high frequencies.

But a bit of low frequency noise will bury it.

Airplanes flying overhead provide a startling amount of low frequency energy. During the post 9-11 flight ban, the silence in our part of New Jersey (USA) was amazing. Then the flights started again, rumble 24/7.


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DigitalMan
post Jun 20 2011, 02:59
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@Soap - agree we've not given thesurfingalien an answer they can digest (plus we got off topic which doesn't help.....) I'll take a shot based on your observations...

@thesurfingalien - Remember that as we increase the number of bits of resolution, all we're really doing is reducing the size of the error in representing (measuring) the waveform at each sample. In your 4-bit example, with only 16 distinct amplitude levels (2^4 = 16), you can picture that a few things will be true:

1) We don't have very many measurement levels, so we'll have to pick the level that is closest to the waveform during that sample and accept that there will be an error. This error will lead to effectively adding noise when we play back the sample.
2) The error will be different for every sample - some times we'll get it exactly right with our lowly 4-bit system but usually we'll be off as much as 1/2 of level (or, halfway between levels; any more than that and we should pick the next level). The amount of the error should be random across the range of zero to 1/2 of a measurement level for each sample in a well designed analog to digital conversion system (using dither, etc.), so the noise added during playback will also be random, leading to a white noise addition to the playback (white noise is the sound of pure randomness).
3) The ratio of how loud the white noise is due to the errors vs. the music signal (that ratio being equal to the dynamic range) will depend on how many measurement levels we have - basically the errors get smaller the more bits we have to accurately measure the waveform which leads to a lower level of noise vs. the signal and an improved dynamic range.

Also remember that in audio recording, the dynamic range of the recording is related to the ratio of the largest signal to the noise floor. The playback dynamic range is a totally separate discussion. The level recorded on a CD does not correspond to a volume level in playback due to variations in how loud you're playing it, the listening environment, etc. So these ratios define the dynamic range of the recording system (in this case its the CD). There is also the idea of whether the playback system can reproduce a 96dB or greater dynamic range using the same ideas - noise floor vs. loudest signal. It is possible that even with a 24 bit recording system we would have a tough time finding a playback system that could reproduce the 24 bit dynamic range for the listener (that is ~144dB, after all).

Does that help or just muddy the water more?


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Soap
post Jun 20 2011, 03:45
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What I was going to say (please correct me where wrong) is that a waveform represented by 4 bits (16 distinct possible "volumes") doesn't stairstep because stair steps would be square waves and square waves = lots of high frequency components and said HF components get cut off in the post DAC filter ("smoothing" the waveform) but the reproduced waveform is still not the original one, and the difference between the 4 bit waveform and the original one (the error) is really when it comes down to it nothing but additional energy in different (unintended) frequencies, which is what noise is.

Again correct me where I'm wrong, but if there wasn't a filter post DAC the 4 bit waveform would be stairstepped (but of course any real-world speaker would act as a low-pass filter effectively "smoothing" said signal somewhat because to perfectly reproduce a square wave the speaker cone would need to instantly move from peak to trough and no object with a non-zero mass can do such).

/ end rant


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WernerO
post Jun 20 2011, 11:33
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QUOTE (thesurfingalien @ Jun 18 2011, 21:04) *
I am sure that, if we would map a waveform into 4 bits and played that waveform back at 96 dB, everybody would hear it as a rather coarse sound, and if we would lover the sound to 15 dB (theoretically, not taking into account background noise ) that same coarseness would be significantly lower.


You can give it a try yourself.

These are music excerpts reduced to an equivalent of 4 bits (or slightly more, IIRC).


Undithered:

http://www.audiochrome.net/clips/Venice_4b_nodither.mp3

Dithered:

http://www.audiochrome.net/clips/Venice_4b_dither.mp3

Noise-shaped:

http://www.audiochrome.net/clips/Venice_4b_noiseshapeE.mp3


And here are solo drums at 5 bit equivalent.

http://www.audiochrome.net/clips/drums_dither.html
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