Correlation between sample rate and audible frequency?

[dc655321]

Now extrapolate that so that so that exhibit A is made from a note just within hearing - say 18Khz and a combination of 18Khz and 24Khz would be exhibit B. Now will they sound the same?

If one's hearing range cuts off above 18 kHz?
Yes, B would sound the same as A.

 

[dc655321]

So a byproduct of higher sample rates is the capacity to reconstruct higher audible frequencies?

Yes. Sampling Theorem says that sampling a bandwidth limited signal at frequency Fs permits capturing all frequencies up to at most Fs/2.

Is this of secondary importance to data resolution achieved through higher sample rates?

Maybe I'm wrong, but it seems you've yet to tease out a clear differentiation in your mind between amplitude resolution (bits per sample) and bandwidth (frequency)?

An analogy is a video camera which records a more accurate picture/captures more information when shooting at higher frames per second. In this analogy color bit depth is similar to higher bit depth per sample in an audio example = greater resolution of the reconstructed waveform.

Higher frames-per-second in your analogy has nothing to do with greater color range.
Greater color range translates to more bits-per-pixel (amplitude resolution), while higher frame rate translates to smoother motion capture.
A greater density of pixels would result in the capacity to capture higher spatial frequencies ("details").

For all intents and purposes 48khz 24bit conversion may be a suitable for audio applications though I'm not certain the 20khz human hearing limit is relevant to why.

Audio sampling rates and bit-depths correspond to limits of human sensory perception.

Our ears have a dynamic range (loud to quiet) of ~100dB. At 6 bits per dB, that indicates around 16-17 bits of amplitude resolution is required.
The highest frequency (pitch) that our ears can hear is around 20 kHz. Since we have to sample at twice the highest frequency to capture that, we arrive at around 40kHz sampling rates.

No magic required...

 

[milezone]

dc:

I'm not familiar with Sampling Theorem but appreciate that information.

I have some understanding of amplitude resolution and bandwidth. That said in what you seem to be suggesting -- a relevance between human hearing limitations/bandwidth and sample rate -- I'm having trouble understanding how bandwidth is relevant to sample rate, as isn't the function of sample rate applicable to the time domain as the speed at which information is extracted from a waveform, and that information is extracted in samples the size of which is defined by bit depth?

Perhaps unclear -- what I intended to articulate: FPS = Samplerate; Color Bit Depth = Sample Bit Depth or amplitude resolution.

How does the 6bits per db apply to DSD conversion where sample size = 1bit? Someone mentioned earlier that DSD has a dynamic range of 6db -- is it 6db per bit? This being the case, would the effect be a significantly more compressed audio signal?

Appreciate the explanation.

 

[BDWoody]

dc:

I'm not familiar with Sampling Theorem but appreciate that information.

Maybe I'm wrong, but it seems you've yet to tease out a clear differentiation in your mind between amplitude resolution (bits per sample) and bandwidth (frequency)?


Higher frames-per-second in your analogy has nothing to do with greater color range.
Greater color range translates to more bits-per-pixel (amplitude resolution), while higher frame rate translates to smoother motion capture.
A greater density of pixels would result in the capacity to capture higher spatial frequencies ("details"

I think looking more into sampling theory will answer this question. You need two samples at a given frequency to recreate that wave with zero loss of information. It doesn't connect the dots of the samples, it uses the samples to reconstruct the waveform. Beyond those two samples per frequency, you are not adding more fidelity. Higher sample rate means you can sample a higher frequency, not sample the rest more accurately.

 

[Soniclife]

I'm not familiar with Sampling Theorem but appreciate that information.

It's key to have a grasp of this, it's the bedrock that digital works on. It's a beautiful but unintuitive solution.

 

[scott wurcer]

If one's hearing range cuts off above 18 kHz?
Yes, B would sound the same as A.

It should be noted that it is virtually impossible to actually do these experiments. You would not be able to find a tweeter that would not produce the 6kHz IM tone probably perceptually louder to most people than either 18k or 24k.

 

[amirm]

How does the 6bits per db apply to DSD conversion where sample size = 1bit?

It fully applies. With no post processing, you only get 6 dB of dynamic range with single bit ADC/DAC!

A technique called noise shaping is used to push out the incredible amount of noise in audible band, into ultrasonic range. Here is a quick sample from my music library:

The hump is all the pushed out noise. Otherwise it would be in audible band.

 

[dc655321]

It should be noted that it is virtually impossible to actually do these experiments. You would not be able to find a tweeter that would not produce the 6kHz IM tone probably perceptually louder to most people than either 18k or 24k.

Yeah, I was assuming an ideal playback channel.
Even on my IEMs I hear a distorted tone at a frequency much less than either of the 17kHz or 21kHz tones I mixed.

 

[Fluffy]

This video might help:

So a byproduct of higher sample rates is the capacity to reconstruct higher audible frequencies? Is this of secondary importance to data resolution achieved through higher sample rates?

In the time domain, "data resolution" is the same as maximum allowed frequency. Your time resolution is defined by how small each division of a second you can achieve. If you can produce a 10 hertz signal, that means you can divide the second by 10, and have a complete wave cycle in each division. For doing that, you need a sample rate of at least 20 hertz, in order to represent at least two points in each cycle. Of course, with a 20hz sample rate you can also represent signals with a lower frequency (or definition) than 10, but not higher.

Since human hearing (generally) cannot perceive air pressure changes that have a higher time resolution than 20 thousand times per second, we say that the maximum frequency we can hear is 20khz. In order to achieve a perfect representation of a signal that contains 20k cycles per second or lower, we need to sample it at a rate of at least 40k times per second – 40khz. The basis for this is the Nyquist–Shannon sampling theorem.
So that's basically why all you really need to achieve full reproducibility of any audible sound is a sample rate of 40khz. CDs use a 44.1khz for technical concerns, like preventing aliasing because of imperfect out of band frequency filters.

The bit depth is an entirely different property, and it determines the amplitude resolution of the signal.

This is all true for PCM. DSD works differently, and I don't see any real value in it.

 

[rodtor]

Ethan Winer has a good discussion of the relevant sampling theory in his The Audio Expert (2nd ed., 2018). See chapter 8, and esp. pp. 246-7 for a discussion of what greater bit depth does and doesn't do. I'm finding this very helpful.

 

[dc655321]

My apologies @milezone, I made a very misleading typo: I wrote "6 bits per dB", but it should be "6dB per bit".
Freaky Friday on a Tuesday...?

I'm having trouble understanding how bandwidth is relevant to sample rate, as isn't the function of sample rate applicable to the time domain as the speed at which information is extracted from a waveform, and that information is extracted in samples the size of which is defined by bit depth?

Sampling a band-limited waveform at certain rate determines the maximum frequency that any waveform reconstructed from those samples may contain.

If I sample at 48kHz, I capture information from 0-24kHz. Sample at 66kHz? I can capture frequencies from 0-33kHz.

More examples:
Given an analog waveform that we know has frequencies from 0-10kHz, if we then sample it at > 20kHz, we can be assured that we can perfectly capture all frequencies of the analog waveform in our sampled data.

However, if our analog waveform has a bandwidth of 0-16kHz and we sample at 20kHz, our samples will not be able to perfectly reconstruct the original waveform. Why the failure? We did not sample at a rate greater than or equal to twice the highest frequency content of our signal, which the math says we must.

Now, the bit-depth of each sample is a measure of how precisely we can represent the amplitude of the waveform.
Or looking at it another way, the more bits we use to represent a sample, the smaller the error in amplitude when we reconstruct it.
16-bits may not sound like much with today's 64-bit computers, but it is 1/65636 = 0.000015 (unipolar), or 15 parts-per-million.
24-bits is 0.000000060, or 0.06 ppm, covering a dynamic range of 144dB. This is well beyond human capabilities, or COTS electronics for that matter.

 

[rodtor]

@Fluffy and @dc655321 , I found your explanations particularly helpful for understanding the different implications of sample rates and bit depth. This leaves me with a question about what people have in mind when they advocate on behalf of hi res recordings. I realize that the definition of this term sometimes is contested, but I am writing here of music recorded at higher than a 16 bit depth and a 44.1 k sample rate. If the latter reaches 22 k hertz, then presumably this is at or beyond the limits of human hearing, which is around 20 k hertz. So there would be no possible advantage in listening to a recording with a higher sample rate. Is that correct? If so, do the claims on behalf of hi res instead have to do with its higher bit depth, which gives it higher amplitude? (I understand the latter to refer to the extent of variation in an audio signal from its average, and to be measured by dynamic range, which refers to the distance in dbs between the highest and quietest sounds.) 16 bit recordings can achieve a DR of 96, while 24 bits can reach 144. If human hearing is limited to around 120 db of DR, this would presumably be where hi res could prove its audible superiority to the red book CD. Is this, in fact, what is being claimed? Apologies for the lengthy question.

 

[Fluffy]

The question if humans can perceive ultra-sonic sounds (higher than the theoretical 20khz limit) is a very debated topic. There have been studies to try to prove or disprove that, and I'm not really an expert on that subject. Personally I never saw a convincing absolute proof that ultra-sonics are meaningful in music reproduction. And testing on myself, I found that I hear no difference between regular CD quality and hi-res quality. Moreover, I can't reliably detect frequencies higher than 17khz, so I don't see why I would benefit from hi-res anyway. You can try and blind test yourself to see it you can distinguish "redbook" from hi-res. Most people can't as far as I'm aware.

As for higher bit depth, it comes down to dynamic range. Indeed 16 bit gives a possible dynamic range of 96 db, but in the real world you won't encounter any situation that requires even that much. Quiet listening rooms have background noise that is at least 30 dB SPL, and you are unlikely to listen to music in SPL exceeding 110 db. So you can see that a dynamic range of 80-85 db will cover almost all scenarios. Most recorded music doesn't have even that much dynamic range. And as for the 144 db of 24 bit, no dac or amp in the world has that big of a dynamic range, so it's pointless there also.

In the end, the ability to faithfully reproduce the audio signal depends on the weakest link in the chain. Sometimes it's the room, sometimes it's the gear, sometimes it's our own ears. Striving to maximize each individual component or feature is pointless if there is a bottleneck somewhere that nulls out it's superiority. Audiophiles commonly refuse to acknowledge that the bottleneck is themselves.

 

[scott wurcer]

And as for the 144 db of 24 bit, no dac or amp in the world has that big of a dynamic range, so it's pointless there also.

Now, now LIGO does at least a 100dB past that. .

 

[dc655321]

@Fluffy and @dc655321 , I found your explanations particularly helpful for understanding the different implications of sample rates and bit depth. This leaves me with a question about what people have in mind when they advocate on behalf of hi res recordings. I realize that the definition of this term sometimes is contested, but I am writing here of music recorded at higher than a 16 bit depth and a 44.1 k sample rate. If the latter reaches 22 k hertz, then presumably this is at or beyond the limits of human hearing, which is around 20 k hertz. So there would be no possible advantage in listening to a recording with a higher sample rate. Is that correct? If so, do the claims on behalf of hi res instead have to do with its higher bit depth, which gives it higher amplitude? (I understand the latter to refer to the extent of variation in an audio signal from its average, and to be measured by dynamic range, which refers to the distance in dbs between the highest and quietest sounds.) 16 bit recordings can achieve a DR of 96, while 24 bits can reach 144. If human hearing is limited to around 120 db of DR, this would presumably be where hi res could prove its audible superiority to the red book CD. Is this, in fact, what is being claimed? Apologies for the lengthy question.

If human hearing extends up to 20kHz, then technically, yes all audible frequencies can be captured by sampling at rates twice or above twice that (> 40kHz).

However, there is a practical reason to sample at rates somewhat greater than 2x the highest frequency content. That has to do allowing for a range of frequencies to be affected by the anti-aliasing filter that must accompany a band-width limited sampling system. That's maybe a topic for another day, though...

Higher bit depth extends the noise-floor: it allows for quieter quiet bits, not louder loud bits.
I suppose if the noise-floor of a recording had issues (was modulated by the signal, for example), then pushing that issue further down into inaudibility could be perceived as a good thing.

IMO, having more than 16 bits can do no more harm than take up slightly more storage/bandwidth.

 

[Fluffy]

Now, now LIGO does at least a 100dB past that. .

LIGO is the best!
But I think they focus on sensitivity, not dynamic range. I believe there is a very narrow amplitude range within they expect to be able to detect gravitational waves. You actually sent me on hunt to find an exact value, but I couldn't really track one down . I did see that they made detections with an indicated SNR of 24, so it's not very high.

 

[rodtor]

Thanks to both @Fluffy and @dc655321 for your very informative reponses. I can see that this is a complex issue.

 

[pkane]

Now, now LIGO does at least a 100dB past that. .

But did you try LIGO in a blind comparison with a Topping DAC ?

 

[Julf]

As I understand, audio signal from source to speaker is a single waveform comprised of infinite overtones in an analog state.

An analog signal is always bandwidth limited, so it does not contain an infinite number of overtones.

 

[ElNino]

It's important IMHO not to characterize Shannon's sampling theory in a misleading way. Shannon's sampling theory tells us that we can perfectly reconstruct an input signal that has been bandlimited by a perfect sinc filter by sampling at 2x the bandwidth. (Alternatively, sampling itself can viewed as bandlimiting by a perfect sinc filter; same thing.) The sinc function is symmetric in the time domain.

For real signals, which are not symmetric in time, this means that you cannot perfectly reconstruct signals up to fs/2. If you want to preserve frequency response and phase up to fs/2, you get pre-ringing in the time domain that was not present in the original signal upon reconstruction. If you want to avoid this, you have to sacrifice either frequency response or phase response. The sampling theorem is not a free lunch.