[nick_charles]

This is pretty interesting, I do not have sufficient higher maths to determine if it is correct. If it is correct, perhaps we have a mathematician in the house , then it demonstrates that a DAC should be capable of a perfect reconstruction of any bandwdith limited continuous signal regardless of its complexity if sampled correctly
 
 
Maths from St. Andrews
 
The next question then is are modern DACs capable of this feat ? or how close do they get ?

 

[arnyk]

  This is pretty interesting, I do not have sufficient higher maths to determine if it is correct. If it is correct, perhaps we have a mathematician in the house , then it demonstrates that a DAC should be capable of a perfect reconstruction of any bandwdith limited continuous signal regardless of its complexity if sampled correctly
 
 
Maths from St. Andrews
 
The next question then is are modern DACs capable of this feat ? or how close do they get ?

 
A common benchmark for a good modern DAC is flat response up to from 95 to 99% the Nyquist frequency. The lower number still covers 20 KHz for 44.1 KHz sampling.

 

[mikeaj]

The mathematics are covered in a DSP course and are clear and unambiguous.
 
This is the result with that perfect sinc (brickwall) filter. Keep in mind that the sinc filter has infinite support and is non-causal. The correct output now depends on values potentially infinitely in the future. Not great for realtime output. Real DACs working on actual data (also on the ADC side) need to be an approximation. Actual filter designs are approximations, both for the input and the output. And most actual DACs for audio are delta-sigma types with other kinds of approximations and noise shaping going on.
 
"How close" should be primarily evaluated by proper listening tests, maybe like the A/D/A loop like in Meyer/Moran.

 

[arnyk]

  The mathematics are covered in a DSP course and are clear and unambiguous.
 
This is the result with that perfect sinc (brickwall) filter. Keep in mind that the sinc filter has infinite support and is non-causal. The correct output now depends on values potentially infinitely in the future. Not great for realtime output. Real DACs working on actual data (also on the ADC side) need to be an approximation. Actual filter designs are approximations, both for the input and the output. And most actual DACs for audio are delta-sigma types with other kinds of approximations and noise shaping going on.
 
"How close" should be primarily evaluated by proper listening tests, maybe like the A/D/A loop like in Meyer/Moran.

 
Sinc filters are far from ideal for audio because by definition they cause potentially audible losses above 10 KHz in 44 KHz sampled system. Aperture correction is required to product ideal flat response.
 
DACs used in music reproduction always operate in the past, so there is no problem with anticipating future or even present events. The digital filters used in audio include digital delays in order to obtain the correct timiing.

 

[dazzerfong]

On one hand, it's great that my uni has taught me enough to understand what's going on there. On the other, oh God, I was hoping to take a break from this stuff after so many tests with it...
 
That being said, what you said is true: in theory, provided your sampling rate is at least twice your intended frequency, yes, any signal with frequencies below the Nyquist frequency (the frequency half of your sampling rate, or Nyquist rate) is capable of being reconstructed perfectly. When they mean perfectly, they usually include a phase (delay). When I say it's being capable of being reconstructed theory, that just means that the sampling part of it doesn't contribute to the inaccuracy: that doesn't mean all DAC's will provide a perfect output.
 
To put it in simple terms, if Nyquist frequency <= half of Nyquist rate, then all is good.

 

[cjl]

  On one hand, it's great that my uni has taught me enough to understand what's going on there. On the other, oh God, I was hoping to take a break from this stuff after so many tests with it...
 
That being said, what you said is true: in theory, provided your sampling rate is at least twice your intended frequency, yes, any signal with frequencies below the Nyquist frequency (the frequency half of your sampling rate, or Nyquist rate) is capable of being reconstructed perfectly. When they mean perfectly, they usually include a phase (delay). When I say it's being capable of being reconstructed theory, that just means that the sampling part of it doesn't contribute to the inaccuracy: that doesn't mean all DAC's will provide a perfect output.

One more caveat here:
 
Nyquist-shannon sampling theorem states that provided that:
 
1) Your sampling rate is greater than twice the maximum intended frequency that you want to capture
2) Your signal does not contain any content at > 0.5Fs (your signal must be perfectly bandlimited)
 
You can then create a perfect reconstruction of the original signal.
 
If you don't filter out ultrasonics before you perform the sampling/ADC process, you will not get the correct result out.

 

[superjawes]

One more caveat here:

Nyquist-shannon sampling theorem states that provided that:

1) Your sampling rate is greater than twice the maximum intended frequency that you want to capture
2) Your signal does not contain any content at > 0.5Fs (your signal must be perfectly bandlimited)

You can then create a perfect reconstruction of the original signal.

If you don't filter out ultrasonics before you perform the sampling/ADC process, you will not get the correct result out.

Indeed. If anyone wants to read more on this, it's called Aliasing.