44kHz sampling freq. gives 1 sample per halfwave for 20kHz sine ?

[percy]

so the 'reconstruction filter' will ensure that you have a nice sine wave, correct ? But how exactly does it do that magic ?

..and hi to soongsc

 

[percy]

Bernhard, at what point in the digital<->analog conversion process is the signal in the pictures that you have attached in your first post ?

I'd be curious to see how a final 'reconsturcted' 10Khz or 20Khz sine wave actually looks like as compared to the original.

 

[Stocker]

Well that depends on how it's reconstructed, and how fast you can looking at it.

 

[Bernhard]

Bernhard, at what point in the digital<->analog conversion process is the signal in the pictures that you have attached in your first post ?

I'd be curious to see how a final 'reconsturcted' 10Khz or 20Khz sine wave actually looks like as compared to the original.

Both are taken directly from Vout of the DAC chip.

The final signal looks like a sinewave even if I look at it fast.

Maybe I can explain it my self:

Another theory (?) say that you can split any signal into a sinewave of first order plus sines of higher harmonic orders.

If you look at a square or triangle signal on a spectrum analyzer, you get the basic sine of same frequency as measured square or triangle plus lots of harmonics.

When the square is 20 kHz , the analyzer will show a line on 20kHz and lines for the higher harmonics.

Now if the square is fed through a filter that cuts everything above 20 kHz, all those harmonics are filtered out to a high degree and the poor analyzer can only show what is left of that good old square: The line on 20 kHz which represents the basic 20 kHz sinewave.
So only a sine is left over from the square or the triangle.
And it has the same amplitude and the same phase like the squarewave had.
So two samples that make a square are enough to reconstruct a sine with help of a filter.

Analyzers help you understand
Analyzers make you see what you hear
Analyzers keep you from listening to the music

Now what is with an unfiltered 10kHz signal ?
The shape is already more like a sine because of more samples, but if you filter over 20 kHz there is still left the second order harmonic at 20 kHz.
Is this reconstructed perfectly as well ?
Does the unfiltered signal not contain second order harmonic ?
Because the shape is already more like a sine because of more samples ?

 

[soongsc]

so the 'reconstruction filter' will ensure that you have a nice sine wave, correct ? But how exactly does it do that magic ?

..and hi to soongsc

Is that supposed to be some gesture to attact attention of a species of the opposit gender?

 

[jan.didden]

[snip]Now what is with an unfiltered 10kHz signal ?
The shape is already more like a sine because of more samples, but if you filter over 20 kHz there is still left the second order harmonic at 20 kHz.
Is this reconstructed perfectly as well ?
Does the unfiltered signal not contain second order harmonic ?
Because the shape is already more like a sine because of more samples ?

Bernhard,

I am not 100% sure here, but IIRC any harmonics in any signal fall above the 22 kHz half sample rate. So, your 10kHz signal will have harmonics, but not the second harmonic. So the filtering will still work as you described so well.

Jan Didden

 

[Werner]

Another theory (?) say that you can split any signal into a sinewave of first order plus sines of higher harmonic orders.

You're confusing two theorems.

One (Fourier series decomposition) says that you can represent any periodic signal as a discrete series of, harmonically-related, sinewaves (or whatever set of orthogonal functions you prefer).

The other (Fourier transform) says that any signal can be represented as a continuous spectrum of sinewaves.


So:

periodic signal <-> discrete spectrum (harmonics)

non-periodic signal <-> continuous spectrum


And while were at it, let's repeat Shannon/Nyquist in all of its detail:

any signal of which the (continuous) spectrum has been limited so that it does not contain anything above fs/2, when sampled at fs (using a stream of Dirac functions), shall be completely and unambiguously be reconstructed when the sampled stream is convolved with a Sinc function (boxcar low-pass at fs/2).


All very neat and very very perfect and very very very theoretical as these gentlemen don't tell you how to band-limit the source signal without screwing it up (and most ADCs indeed screw up*), and don't tell you how to implement Dirac and Sinc functions in this universe.
But OK, the latter two one can work around, but for the first one only sampling at way above 44.1k will really help.

(*Because many designers think that what is nice as an anti-imaging reconstruction filter is also nice as an anti-aliasing filter ...)

 

[soongsc]

Um, no. You have very good phase resolution. In fact if you have a full height (i.e. +32,767 to -32,768) sine wave at 20kHz you have a phase resolution of well under 1/100th of a degree. At lower amplitudes it drops proportionally. Only at an amplitude of the LSB do you lose most of your phase resolution. Think about the amount of information in the stream.

More thinking on the subject, it seems that the theory is perfect if you have one freqhency. But if you have music, which is a a complex frequency content, How would this work? I think if you record and playback the same music contetn multiple times, you will find that the wave forms would be totally different. Has anyone here tried? Even with the best equipment you can find?

This would be clear verification and validation of a theory.

 

[gmarsh]

More thinking on the subject, it seems that the theory is perfect if you have one freqhency. But if you have music, which is a a complex frequency content, How would this work? I think if you record and playback the same music contetn multiple times, you will find that the wave forms would be totally different. Has anyone here tried? Even with the best equipment you can find?

This would be clear verification and validation of a theory.

Superposition!

Music has "complex frequency content" in that it consists of hundreds of different simultaneous tones. Sure they're not continuous tones, but it's all predictable - when you go from one tone to the next, you're effectively "AM modulating" a tone down to zero, and "modulating" another one up. And this creates extra tones, but they're tones nontheless. There's nothing in music that doesn't consist of tones. Hell, your ear works by detecting tones.

Assuming that the ADC, discrete time processing, DAC, etc... are all linear systems - eg, the entire system is a linear system, then superposition applies. Basically it means that if a system can pass any two single tones accurately, then it can accurately pass both tones simultaneously.

So assuming that your recording/playback system is capable of reproducing every frequency which exists in the music you're listening to, then your entire music signal will make it through OK.

 

[soongsc]

I don't question what the ADC can to, it is how often sampe should takes place. If you have two 20K signals superimposed at different times just 1/10th of a wave length apart and only one cycle of these, How can it mathematically be reconstructed since the two samples can also be a samples of a different combination of modulation? I would have to be proven that two samples can only be the results of one unique modulation combination for perfect reconstruction to be true. Is there any reference information that proves this?

 

[gmarsh]

I don't question what the ADC can to, it is how often sampe should takes place. If you have two 20K signals superimposed at different times just 1/10th of a wave length apart and only one cycle of these, How can it mathematically be reconstructed since the two samples can also be a samples of a different combination of modulation? I would have to be proven that two samples can only be the results of one unique modulation combination for perfect reconstruction to be true. Is there any reference information that proves this?

The signal you described has infinite frequency content. It won't pass through a 22.05KHz *analog* low pass filter, and it certainly can't be represented in 44.1KHz discrete time. Single cycles of sine waves (abruptly starting and stopping) have infinite frequency content; effectively you're modulating your 20KHz tone with a 10KHz "on-off" square wave, yielding a square wave spectrum (10KHz, 30KHz, 50KHz... to infinity) shifted by 20KHz.

These signals don't exist in nature anyway. I've covered this in a few of the 'first cycle distortion' threads that have popped up.

 

[Bernhard]

complex signals

On the scope, music looks like rocky mountains.

 

[jan.didden]

I don't question what the ADC can to, it is how often sampe should takes place. If you have two 20K signals superimposed at different times just 1/10th of a wave length apart and only one cycle of these, How can it mathematically be reconstructed since the two samples can also be a samples of a different combination of modulation? I would have to be proven that two samples can only be the results of one unique modulation combination for perfect reconstruction to be true. Is there any reference information that proves this?

The combination wave you describe is a combination of two 20kHz waves, one shifted in phase. If you add those, what have you got? A 20kHz wave. Try it. The problem you think to see isn't there.

Jan Didden

 

[gmarsh]

The combination wave you describe is a combination of two 20kHz waves, one shifted in phase. If you add those, what have you got? A 20kHz wave. Try it. The problem you think to see isn't there.

Jan Didden

I think he's describing a transient like the attachment's red curve...

 

[jan.didden]

I think he's describing a transient like the attachment's red curve...

Hi,

Remember: anti-alias filters ARE PART OF THE SYSTEM, both before the ADC as well as after the DAC. The waveform you have drawn has components far above 20kHz, is illegal in this system and will never occur. Sorry, this is a very strict definition that those engineers set up, to make the system work. You've got to stick to the rules.

Jan Didden

 

[gmarsh]

Hi,

Remember: anti-alias filters ARE PART OF THE SYSTEM, both before the ADC as well as after the DAC. The waveform you have drawn has components far above 20kHz, is illegal in this system and will never occur. Sorry, this is a very strict definition that those engineers set up, to make the system work. You've got to stick to the rules.

Jan Didden

Exactly what i'm saying.

 

[soongsc]

I think he's describing a transient like the attachment's red curve...

So notice the ends of the red curve. In reality, music signals like this can possibly occur, but the ends would not be reproduced. So the reproduction is not perfect. All music signals are transient signals.

Hi,

Remember: anti-alias filters ARE PART OF THE SYSTEM, both before the ADC as well as after the DAC. The waveform you have drawn has components far above 20kHz, is illegal in this system and will never occur. Sorry, this is a very strict definition that those engineers set up, to make the system work. You've got to stick to the rules.

Jan Didden

So the engineers define this not such that perfect reproduction can be made, but maybe due to cost limitations and assumption that the mass population won't hear the difference.

I guess unless the majority people demand better quality, we're stuck with this for now. I wonder if DVD audio or SACD is better?

 

[SY]

Re: complex signals

On the scope, music looks like rocky mountains.

It's useful to look at the spectrum of real music. Very 1/f-like.

 

[jan.didden]

So notice the ends of the red curve. In reality, music signals like this can possibly occur, but the ends would not be reproduced. So the reproduction is not perfect. All music signals are transient signals.[snip]

Music signals like this CANNOT occur. What we call transients in music are sudden attacks of a tone TO OUR EARS, but compared with the electronic 'instanteneous' start of a wave as in the red graph, music starts slowly and leisurely, even as 'attack'.
Much what SY is saying above.

This is also the tragedy behind Graham Maynard's 6-part article in EW on 'first cycle distortion'. The waveform that is the basis for his point is one that cannot occur in music.

Jan Didden

Edit: with the possible exception of electronically generated music. Possibly that can have attacks that are unknown in 'natural' music generated by instruments. But I am no expert on that.

 

[gmarsh]

So notice the ends of the red curve. In reality, music signals like this can possibly occur, but the ends would not be reproduced. So the reproduction is not perfect. All music signals are transient signals.

We've shifted gears in this thread. It can be shown that a 44.1KHz sampled discrete time system, theoretically, can perfectly reproduce any musical signal that only contains frequency content up to nyquist, assuming 'ideal' antialiasing/reconstruction filters.

But the latest signal you've described has content well beyond 20KHz... it has infinte frequency content. So it can't be perfectly reproduced by any bandlimited system, and certainly can't be recorded on a CD.

But a few things... first of all, instantaneously-starting sine waves can't occur in reality, and they certainly can't be recorded. The instrument producing the sound (guitar string, drum head, etc) would have to start moving *instantaneously*, requiring infinite acceleration and infinite force. The air surrounding the instrument would have to do the exact same thing, the recording microphone's diaphram will have to instantaneously move, etc. The whole world is a low pass filter.

And your ears are low pass filters too. Your eardrum has mass, and the basilar (sp?) membrane in your ears can only resonate to frequency content up to ~20KHz.

So if your signal has content at 30/50/70 KHz but only the 10 and 20KHz components make it through the recording system, and your ear can only hear the 10KHz/20KHz components anyway, was the signal actually distorted?