'Mathmatical trickery'
I'm going to copy and paste my comments from another thread... maybe you'll get something from it, maybe you won't. Enjoy!
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Let's use an example to think about this.
You have a white noise generator and you're recording it's output through your a/d converter's input.
What is the full spectral content of white noise? White noise has equal spectral power density in any band.
What is the highest frequency band that is of relevance when taking into account the capacity of human hearing. For most of us, that would be 20khz.
So with your white noise there is a combination of all of these randomized sine waves playing all at once. The smallest variation in this chaotic wave that we're interested in capturing is the wave of a 20khz signal, right? Any wave shorter than that of a 20khz wave would be higher in frequency than 20khz and thus outside of the range of most human ears.
Square or triangle waves at 20khz are just regular multiples of the frequency of the sine wave. So if we can capture the 20khz signal accurately due to the frequency rate that our converter samples the sound, then we can easily capture longer, lower frequency sine waves with less frequent peaks and troughs.
So you see, any complex combination of frequencies between 20hz and 20khz can be captured perfectly with 44.1k.
The difference is in the converter design.. getting the analog components to best carry out this task without error or unintended variation.
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noolness says....
"However, not all signals can be represented accurately by a set of sine waves below 20kHz. In particular, sawtooth waves (also known as triangular waves) and square waves can only be represented by an infinite set of sine waves with frequencies extending all the way to infinity. When you capture either sawtooth waves or sine waves digitally, you don't get back a perfect waveform when you play it back due to filtering."
So technically you can only accurately recreated a sawtooth or square wave digitally by having an infinite sample rate. norman_nomad:
This is true! A 20khz square wave captured at 44.1k will not capture the harmonic content created much further past the 20khz fundamental.
The question you have to ask yourself is: Do you care and does it matter?
For most of us, the answer is: No, it doesn't.
You (at least I can't) hear past 20khz. So any harmonic content created above this range due to a square or triangle wave, whether captured by your converter or not, does not and should matter to us because we wouldn't be able to hear it anyways.
An example to think about
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If a 1khz square wave is generated by an imaginary synth with infinite bandwidth played by a speaker of infinite bandwidth we would hear the 1khz fundamental along with all of the increasingly diminishing harmonics up to 20khz after which harmonics would exist, but we would not be able to hear them due to the limitations of our physical hearing.
For that same reason, a converter only needs to capture spectral content up to the threshold of human capacity, past which point there is a dubious return in utility.
This is why a 44.1khz sampling rate is considered to be sufficient - it covers the entire range of human hearing.
There are some that will argue that harmonics past the point of human hearing have some effect on the sounds we can hear. They may be right. I haven't done enough research to know if there is any merit to this or not... but before you tread down this path, consider the bandwidth limitations of your microphones and your speakers/headphones. Many of these devices are not capable of capturing or playing back much usable content past 25khz...
Also consider this: Any interaction that happens with an acoustic instrument due to harmonics past 20khz is happening in the air during the recording and thus you, in essence, record those interaction as they manifest in the 20khz and under frequency range (in other words, in the range you actually hear). You won't loose the effects of the ultra high frequencies by excluding them from the digital capture - they've already done their work in the air during the recording and their effect will be maintained on playback. Make sense?
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Idiophonic says...
I still can't help but feel that the amplitude issue exists at the lower end of sample rates because, as you indicated, the chance that your 2 samples hit the peaks is small. It seems that even with 2.2 samples (only fractionally more!) you are still in danger of under-representing the amplitude of the upper frequencies...4.4 samples would have a better chance of being more accurate in this respect, yes? Norman_nomad:
Hey Idiophonic,
The samples do not need to be at the peak of the sample to represent the frequency accurately.
From the link that noolness posted:
"How is this possible? How does the DAC know how to "plot" the signal in between sample points? ...
How can removing frequencies above Nyquist restore the correct peak levels (which are higher than any individual sample)? This is hard to explain without resorting to mathematics, but essentially Fourier Theory postulates that any complex continuously varying signal can be represented by a set of sine waves of varying frequencies and amplitudes. When you sum all these sine waves, you get the original signal.
In the case of Figure 4, if you draw a set of straight lines between sample points, you are plotting a signal composed of many sine waves summed together. These sine waves "modify" the original sine wave and lower the actual peaks of the waveforms. When these extraneous sine waves are removed, the peaks of the sine wave between sample points are restored.
If you still find this difficult to swallow, try thinking of it in a different way. The reason a perfect sine wave is output even though the samples did not capture the peaks is that a 19997Hz sine wave is the only possible plot that you can draw that passes through all the sample points but does not contain frequencies higher than Nyquist. This can actually be proven mathematically, but I will spare you the calculations."
So to sum it up:
If you sample a twice the rate of the highest frequency you wish to capture you will able to capture that frequency accurately and reconstruct it without anomalies using a theoretically perfect analog circuit. Because there is no such thing as a perfect analog circuit, each converter will handle this task with more or less grace. The samples don't need to catch the peaks of the frequency to know how to redraw them because there is only ONE POSSIBLE PLOT THAT CAN BE DRAWN after filtering.