Understanding Sample Rate

[Ralf11]

I would ignore Hans

[buonassi]

26 minutes ago, Don Hills said:

 

This is a commonly held belief, but it is incorrect.

The time resolution of a 16 bit, 44.1khz PCM channel is not limited to the 22.7µs time difference between samples. The actual minimum time resolution is equivalent to 1/(2pi * quantization levels * sample rate). For 16/44.1, that is 1/(2pi * 65536 * 44100), which is about 55 picoseconds. To put that in perspective, light travels less than an inch in that time.  
 
Shannon and Nyquist showed that as long as you keep all components of the input signal below half the sampling frequency, you can reconstruct the original signal perfectly - not just in terms of amplitude, but in terms of temporal relationships too. They only addressed sampling, and assumed infinite resolution in amplitude. With a digital signal the precision is limited by the number of amplitude steps, leading to the above formula.

 

If you want to see a real-world example, watch the following video:

 

 

The whole video is worth watching, but the most relevant part starts at about 21:50. It shows the edge of a square wave being smoothly moved in time between 2 sample points.

 

 I'll be sure and watch it all and report back later.  On the surface, I'm having trouble grasping that time resolution could be partially a function of amplitude and not purely based on sample rate.  One is about "loudness" of a sample (not time), the other is more obviously about the rate (in time).  But perhaps the video will help me grasp that better.  And my noob brain just needs to go a little deeper! 

 

 

[Don Hills]

In theory, the time resolution is infinite, but this assumes infinite amplitude resolution. In practice, the smallest time resolution is that timing change required for at least one sample to change by 1 LSB.

[buonassi]

ok, I now understand why transients are still preserved (regardless of the sample rate) despite them being sampled discreetly, and that you CANT lose the "edge" of a transient between samples. Excellent video and pretty darn easy to follow to boot!

 

Thanks @Don Hills .  Good stuff.  

 

@Ralf11 - I think I'll point Hans to this video and see if he can comment on it.  I have enjoyed watching him over the past year or so, but have to award credence to the EE I just watched, who I'd think is better accredited to explain this topic.  Though TBH, I don't know Han's background so maybe that isn't fair just yet. 

 

I'd like to pick your guys' brains on the subject of reconstruction filter design and the effects of signal delay with a linear phase filter vs a minimum phase filter sometime.  I'm convinced I hear the arrival of lower frequencies before I should (or in relation to the high frequencies) when a linear phase filter is employed.  To my ears, a minimum phase just feels tighter and better timed.   This could be another argument for higher sampling rates = the slower rolloff filtering.

[jhwalker]

45 minutes ago, buonassi said:

 

 I'll be sure and watch it all and report back later.  On the surface, I'm having trouble grasping that time resolution could be partially a function of amplitude and not purely based on sample rate.  One is about "loudness" of a sample (not time), the other is more obviously about the rate (in time).  But perhaps the video will help me grasp that better.  And my noob brain just needs to go a little deeper! 

 

 

 

Think again about the nature of a wave - frequency is the horizontal axis (the higher the frequency, the steeper the slope), amplitude is the vertical axis (likewise).  The wave is sampled 44,000 / second, and both X and Y change each time.  So, as others have pointed out, you're not limited to accuracy along the horizontal axis, but the horizontal TIMES the vertical axis (i.e., sampling rate times amplitude / bit depth) TIMES the formula for the curve, which allows an extremely fine resolution.

[esldude]

To expand upon Don's reply, imagine sampling a 1 khz sine wave that started exactly at the first sample. The next sample will be some amplitude above zero.  Now imagine instead the wave starts some small time after the first sample. The next sample will be some amplitude above zero, but by a smaller amount. The accuracy by which you can place the beginning of that wave is poor if your step between bits is large.  If you have more bits, in other words smaller step sizes, you can more accurately place they beginning point after the first sample. If you can picture that, it becomes clear smaller step sizes improves time resolution.  All without changing sample rate.

 

With large step sizes you could slide the beginning of the wave forward and backwards in time without registering a change in sample value. Small step values reduce the time period before the sample value changes by the LSB.  Note the range is far smaller than time between samples. 

 

In other words I'm saying the same thing as Don and jhwalker.

[Don Hills]

In theory then, it's an argument for 24 bit sampling.

In practice, 55 picoseconds ought to be "close enough for rock'n'roll".

As a thought exercise, what is the accepted minimum time resolution for human hearing? From that, we can work out how many bits are required to encode it.

[buonassi]

6 minutes ago, esldude said:

To expand upon Don's reply, imagine sampling a 1 khz sine wave that started exactly at the first sample. The next sample will be some amplitude above zero.  Now imagine instead the wave starts some small time after the first sample. The next sample will be some amplitude above zero, but by a smaller amount. The accuracy by which you can place the beginning of that wave is poor if your step between bits is large.  If you have more bits, in other words smaller step sizes, you can more accurately place they beginning point after the first sample. If you can picture that, it becomes clear smaller step sizes improves time resolution.  All without changing sample rate.

yes, it's making sense.... wasn't very apparent at first, but sketching that out in my mind I get it.  This is also presumably why there's an increase in 24/44.1 material becoming available.  NOT ONLY because we need the extra dynamic range for say classical music, but because it has better resolution in time as well?

[jhwalker]

8 minutes ago, Don Hills said:

In theory then, it's an argument for 24 bit sampling.

In practice, 55 picoseconds ought to be "close enough for rock'n'roll".

As a thought exercise, what is the accepted minimum time resolution for human hearing? From that, we can work out how many bits are required to encode it.

 

Exactly - if I'm doing my math correctly (always a shaky proposition for me!), 48/24 sampling would be 279x more sensitive in time discrimination than 44/16 - down to an incredible 198 femtoseconds! 

[Ralf11]

here is some grist for the mill:

 

https://www.sciencedirect.com/science/article/pii/0028393270900254

 

http://www.physiology.org/doi/full/10.1152/jn.1997.78.5.2438#F12

 

 

[buonassi]

So, agreeing now that bit depth/word length increases time resolution......would any of you agree there is merit in a higher sample rate insomuch as you can use the final octave for a slower rolloff reconstruction filter?  IE, an 88.2 file would have a very gentle sloped rolloff between 22K and 44K?

 

I'm thinking this is why @mansr (whom I've been following) has 24/96 as his avatar?

 

And also why there is such overt disdain for MQA here at CA - especially from those who know a lot about digital sampling?  Especially if all you need is 24/96 PCM! 

 

I believe the bigger picture is starting to come into focus for me here!  

[esldude]

23 minutes ago, Don Hills said:

In theory then, it's an argument for 24 bit sampling.

In practice, 55 picoseconds ought to be "close enough for rock'n'roll".

As a thought exercise, what is the accepted minimum time resolution for human hearing? From that, we can work out how many bits are required to encode it.

Well the research showing 5 micro seconds is shaky.  10 microseconds is detected between ears.  Either way at 44.1 1 bit is more than enough just for timing. According to the formula. Haha!

[audiventory]

1 hour ago, esldude said:

To expand upon Don's reply, imagine sampling a 1 khz sine wave that started exactly at the first sample. The next sample will be some amplitude above zero.  Now imagine instead the wave starts some small time after the first sample. The next sample will be some amplitude above zero, but by a smaller amount. The accuracy by which you can place the beginning of that wave is poor if your step between bits is large.  If you have more bits, in other words smaller step sizes, you can more accurately place they beginning point after the first sample. If you can picture that, it becomes clear smaller step sizes improves time resolution.  All without changing sample rate.

 

With large step sizes you could slide the beginning of the wave forward and backwards in time without registering a change in sample value. Small step values reduce the time period before the sample value changes by the LSB.  Note the range is far smaller than time between samples.

 

 

Digital to analog conversion is like to binding of sample values (its tops) by a "steel-edge ruler". The "ruler as curve". It is interpolation. It made analog filter.

Bit depth cause curve form error. We can observe it as noise at spectrum of digital signal. And analog one too.

 

By Naquist theorem sample rate must be 2 times more, than restored curve frequency (if the curve is sine).

 

But accuracy of restoring depend on sample number that bend the "ruler". In ideal need infinite sample number. And analog filter must be "infinite brickwall".

Thus sample rate increasing is easier way to more accurate restoring of original sine at given frequency. Bit depth too.

But it is no need to worry more than accuracy of spectrum. Because it show any distortions more clear than oscillogram (in proper settings for considered case). Sometimes need to consider spectrum development in time (time-frequency diagram).
 

[Don Hills]

49 minutes ago, esldude said:

Well the research showing 5 micro seconds is shaky.  10 microseconds is detected between ears.  Either way at 44.1 1 bit is more than enough just for timing. According to the formula. Haha!

 

Well, let's take the worst case and add an order of magnitude to be sure. Say 500 nano seconds.

 

1/(2pi * quantization levels * sample rate).

1/(2 pi * 8 * 44100) = 450 nano seconds if I've done the sum correctly.

That's 3 bits.

So even 3 bit encoding is more than enough as far as timing is concerned. You'd have trouble hearing a 3 bit (of 16) encoded signal at all, let alone discern a timing difference. And if you boosted it to normal listening levels it'd be too noisy to be called "hi fi". 

 

Can we agree that time resolution, even at 16/44.1, is a non issue?

[beerandmusic]

 

4 hours ago, Spacehound said:

Good for you. I try not to have any beliefs at all.

 

I already knew what i was up against....and I can probably guess with pretty good accuracy others with similar views.

[esldude]

5 minutes ago, Don Hills said:

 

Well, let's take the worst case and add an order of magnitude to be sure. Say 500 nano seconds.

 

1/(2pi * quantization levels * sample rate).

1/(2 pi * 8 * 44100) = 450 nano seconds if I've done the sum correctly.

That's 3 bits.

So even 3 bit encoding is more than enough as far as timing is concerned. You'd have trouble hearing a 3 bit (of 16) encoded signal at all, let alone discern a timing difference. And if you boosted it to normal listening levels it'd be too noisy to be called "hi fi". 

 

Can we agree that time resolution, even at 16/44.1, is a non issue?

No argument from me. With 3 bits you could have dither. 

[beerandmusic]

1 hour ago, audiventory said:

 

 

But accuracy of restoring depend on sample number that bend the "ruler". In ideal need infinite sample number. And analog filter must be "infinite brickwall".

Thus sample rate increasing is easier way to more accurate restoring of original sine at given frequency. Bit depth too.

But it is no need to worry more than accuracy of spectrum. Because it show any distortions more clear than oscillogram (in proper settings for considered case). Sometimes need to consider spectrum development in time (time-frequency diagram).
 

 

of anyone to post on subject, i agree mostly with you in my thinking or belief.

 

i can likely agree with this, and thus "good enough"

more accuracy obtainable with higher sample rate, but "good enough" where "good enough" is subjective...i have conceded that long ago.

 

i also think distortions you speak of are likely more prominent in production, than in live. 

 

I think you have a good grasp of my belief, and that you have more truth on subject than anyone here.

[audiventory]

28 minutes ago, beerandmusic said:

i also think distortions you speak of are likely more prominent in production, than in live. 

 

Distortions is very useful thing in development of audio tools. It allow to aim and exactly control results of work.

Though binding of figures and subjective perception is not simple work.

[beerandmusic]

31 minutes ago, audiventory said:

 

Distortions is very useful thing in development of audio tools. It allow to aim and exactly control results of work.

Though binding of figures and subjective perception is not simple work.

i "think" this coincides with my statement that with greater sample rate comes greater accuracy to the point where engineering cannot process without error?

 

in live, we have music and ears that are not so prone.

[Superdad]

A one week old thread with over 1,000 replies? What the heck?!  

[Don Hills]

 

[firedog]

7 hours ago, tmtomh said:

 

Also, one more important point: You started this thread asking if we could all agree on a techterms.com definition as a starting point for sample rate discussion, and you provided that definition in your opening post.

 

As you can see by the part I've highlighted in bold and in red in the above quoted post of yours (you have to expand it by clicking "read more"), the definition you started us off with, and which you proposed was correct, says exactly what you have recently denied claiming: that higher sample rates can provide "smoother sound."

 

This is another example of why a refusal to be accountable for your own prior statements (and in this case, a statement by someone else that you explicitly aligned yourself with) makes rational discussion impossible. Accountability is essential for a productive discussion, and with all due respect, my view is that you have no one but yourself to blame if the thread goes sideways as a result of your lack of accountability for your own prior words and views.

Stop wasting your time. Whatever you write he will somehow turn it around and come back to his original position. 

[firedog]

5 hours ago, bobbmd said:

@Spacehound but he authored my freshman general chemistry text at St Lawrence University in 1965=i still have it somewhere-just about the same time he uttered that nonsense about Vit C and the human body doesn't store VitC you just pee excess out-Laureates don't know everything

goodnite time for Homeland...

bobbmd

Which has absolutely nothing to do with the topic at hand. In any way, shape, or fashion. What one individual scientist thinks (even a Nobel Winner) means nothing without evidence to back it up and testing of the hypothesis. That's the difference between science and belief.  A Nobel winner can believe anything he wants. His beliefs aren't science. 

[Spacehound]

3 hours ago, buonassi said:

ok, I now understand why transients are still preserved (regardless of the sample rate) despite them being sampled discreetly, and that you CANT lose the "edge" of a transient between samples. Excellent video and pretty darn easy to follow to boot!

 

Thanks @Don Hills .  Good stuff.  

 

@Ralf11 - I think I'll point Hans to this video and see if he can comment on it.  I have enjoyed watching him over the past year or so, but have to award credence to the EE I just watched, who I'd think is better accredited to explain this topic.  Though TBH, I don't know Han's background so maybe that isn't fair just yet. 

 

I'd like to pick your guys' brains on the subject of reconstruction filter design and the effects of signal delay with a linear phase filter vs a minimum phase filter sometime.  I'm convinced I hear the arrival of lower frequencies before I should (or in relation to the high frequencies) when a linear phase filter is employed.  To my ears, a minimum phase just feels tighter and better timed.   This could be another argument for higher sampling rates = the slower rolloff filtering.

The Xiph  video is what you might think of as the "World Standard" And it's not only demonstrated, it's mathematically proven.

 

Anyone who disagrees with it is wrong. It's as simple as that.

[Spacehound]

15 minutes ago, firedog said:

Which has absolutely nothing to do with the topic at hand. In any way, shape, or fashion. What one individual scientist thinks (even a Nobel Winner) means nothing without evidence to back it up and testing of the hypothesis. That's the difference between science and belief.  A Nobel winner can believe anything he wants. His beliefs aren't science. 

Right or wrong, it's a demonstration that you should never 'argue by example' as opposing ones can always be found.

 

But many continue  to do it,  including (I suspect) the majority of  CA members present on 'controversial'  threads, which often actually begin with an example.