Nyquist Frequency (was Re: Looperlative LP1 - sample rate)

["jondrums" <jondrums@hotmail.com>]

>     Even though higher frequencies are indeed possible with a 192K 
> sampling rate, it also means
> that each waveform has more samples representing it, thus higher 
>fidelity. 
> I never did understand
> how the Nyquist theorum could claim fidelity in the upper end by only 
> having two samples per
> waveform.  Seems to me like you wouldn't be able to distinguish a 
>sawtooth 
> from a sine from a
> square wave if you only sampled at twice the highest frequency.

Stephen -  It is my understanding that the Nyquist theorum actually 
specifies the theoretical maximum frequency that can be represented with a 
digital signal.  It says nothing of the quality of that representation. 
There's a key issue that makes the nyquist theorum all important to the 
A/D 
conversion process - frequencies higher than the Nyquist freq. will 
manifest 
themselves in the resultant digital signal as much lower frequency noise. 
That noise is basically garbage and is nearly unpredictable.

Furthermore, frequencies that are close to but still less than the Nyquist 
frequency will be represented, however the amplitude of the representation 
will vary over time based on the _difference_ in frequencies.  How much 
the 
output varies in amplitude is a more complex equasion - important thing is 
that frequencies close to the Nyquist freq. (even though they are still 
less 
than) will be VERY distorted but will still be represented "perfectly 
in-tune".  Example: 48KHz digital A/D converter could capture a 23KHz pure 
sine wave, and output a 23KHz sine wave whose amplitude varies at 1Khz - 
you 
would hear that 1Khz signal for sure!

So this is a long story to explain the real meat of the issue - LOW PASS 
FILTERING.  Any digital audio capture system must have low pass filtering 
before the A/D converter.  Ideally you would like to make sure that 
absolutely no signals whose frequency is greater than the Nyquist 
frequency 
can be input to the A/D converter - because those will just cause useless 
garbage noise out the other end.  You also want to roll off frequencies 
that 
are close to the Nyquist frequency, because though could create ugly 
garbage 
noise too even though they will be represented somewhat.  These filters 
are 
called anti-aliasing filters and every digital audio system has them.

The thing about filters is that even the most expensive filters don't cut 
off sharply at a certain frequency - instead they "roll off" slowly 
starting 
at 0Hz and gradually roll off more and more as the frequency goes up. 
People refer to the "cut-off" frequency of a filter, but be aware that 
this 
is just the frequency at which the sound is sufficiently attenuated so 
that 
its much quiter than the original signal.   This means that a simple 
filter 
with cutoff frequency of 20KHz still passes 22KHz signals they're just 
real 
quiet.  It also means that this filter is rolling off your 18KHz signals 
too.   The more money a manufacturer spend on the filter design and 
implimentation, the "steeper" it is, but none can be infinately steep.

So you've got to set your anti-aliasing filter well below the nyquist 
frequency to be sure that nothing above the nyquist freq. of significant 
volume can get through to the A/D converter.  If you don't have a lot of 
money to spend on anti-aliasing filters (and most music equipment falls 
into 
this catagory) you use an off-the-shelf filter solution which doesn't have 
very steep filtering (and therefore has a cutoff freq. well below the 
nyquist freq.).  This cheap filter will roll off frequencies well below 
the 
cutoff freq. too.    This could be what most people are hearing when they 
say 44.1KHz digital doesn't sound good.

Ok, tired yet???   Well how about this one here - Time to learn about 
anti-imaging filters.  This is just about the same thing as an 
anti-aliasing 
filter, but its on the other side of things - the outputs.   Every device 
that uses digital audio has to have one.   Typically it will be a similar 
or 
identical low pass filter to the anti-aliasing filter on the input.  This 
further "rolls-off" some of the high end of your signal (even stuff below 
the Nyquist freq. - same as above).

As a side note - the anti-imaging filter is the reason why you can buy a 
really high end CD player and it can actually sound much better than a 
crappy one even though both are using the exact same digital content. 
(There are other reasons like upsampling though too) Same with sound 
cards, 
mixing desks, ect.   That's why a lot of people use SPDIF and do the final 
conversion only once on a high quality audio system.

All of these things lead me to believe that even the most high quality 
digital audio system can only do a very good job of representing 
frequencies 
less than about 1/3 of the sampling frequency.  This means that I consider 
44.1KHz digital audio systems can represent up to 14.7KHz very well.  
That's 
probably fine for most of the public, but there are many people who can 
hear 
this.   I believe that 48KHz is a pretty significant improvement with the 
ability to represent frequencies up to 16KHz quite well.   This is 
probably 
right around the actual threshold of most human beings and above this 
starts 
to become somewhat esoteric and specialized.  (keep in mind though that 
DVD 
supports 192KHz - so its not THAT esoteric -smile-).

Jon

If you got this far you don't need this link, but here is a decent 
explaination of some of this with diagrams:
http://www.cems.uwe.ac.uk/~lrlang/multimedia/audio2.pdf
upsampling on high end cd players:
http://www.mlssa.com/pdf/Upsampling-theory-rev-2.pdf