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[Mark_W]

Nika Aldrich wrote on Fri, 30 April 2004 17:11

Make sense? Nika. * With due respect to George and his upsampled EQ plugin, there are arguments for why this should not be necessary if certain implementations are used.

Nika, yes that makes sense, thank you. But you raised another interesting issue regarding differing implementations of EQ and the possible need for upsampling in some cases but not others. Would you be willing to share some more details on that topic as well?

[danlavry]

Oh man, So may questions, and I have spent too much time on the web... I certainly can not answer everything. But that one deserves it:
I can see all sorts of reasons to up sample to even very extremely high rate than come down (decimate).

One example that comes to mind is that certain types of digital signal processing is very difficult to perform when you are near Nyquist.
Say you have a 44.1KHz system and wish to process (EQ, Reverb and so on) an 20KHz, free of problems. NyQuest is near by at 22.05KHz. So you up sample by X2 (or use 88.2KHz data if already available), and your Nyquist is now at  44.1KHz. Your 20KHz is in a good place now.

Another example: You wish to do a non linear operation, such as a tube sound simulator, fast attack compressor, limiter and so on. Non linear processes will introduce a lot of energy in various places. Well that is fine. A non linear analog, such as a tube, may yield all sorts of sum and differences of frequencies. Even the case of a simple pure tone will. Well, what sounds good in analog may not sound good in digital. Or at least may be very different in digital. Why? In one word, aliasing. There energy produced due to non linearity in analog gear may cover a lot of frequency range, and some of it you don’t hear, because it is just too high for the ear (be it 30KHz or a MHz). But With digital, if you are using say a 44.1KHz converter, all that high frequency stuff re-appears at the audible range.  
What if you went to say 88.2KHz sampling? Well, there are 2 frequency regions:
L – the one you hear (say 0-22Khz or so) and H- that you do not hear, 22-48KHz.  If anyone wants to move the number 22 up or down be my guest, the concept is still here.
Of course, if one would up-sample by say 64 or 1024… very little gets aliased. And of course one can design the gear (the math) in such a way that you do not need to go that far. Say I want to approximate a non linearity with y=a0+a1*X+a2*X^2, that is a 2nd order curve. I start with a 44.1KHz system (22.05 is the maximum possible tone).  The math tells me that no matter how complex the input is,  the highest energy produced will be 22.05KHz *2= 44.1KHz.
Nothing will alias if I use 88.2KHz data. But If I wished to process say a 10th order curve, I may have tones 220.5KHz, and thus need a 441KHz data. That is a lot of up-sampling… As always, do not just get to filter things to solve aliasing problems.
I can go on, but hope that what said shows that I am not against working at high rates, when it is needed. We up sample to high rates for DA’s, for processing, for AD input circuits (modulators). There is a whole technology that deal with noise shaping, a tradeoff between few bits at high rates and many bits at low rate. I am all for it. And all of it is about LOCALIZED processes, getting from signal A to signal B the best way, whatever it takes. Non of it conflicts what I said about ridicules it is to use 192KHz for conversion or data format.  

Best Regards
Dan Lavry  

[Nika Aldrich]

Mark_W wrote on Fri, 30 April 2004 18:28

Nika, yes that makes sense, thank you. But you raised another interesting issue regarding differing implementations of EQ and the possible need for upsampling in some cases but not others. Would you be willing to share some more details on that topic as well?

It has to do with what happens when the band in the EQ gets close to Nyquist.  A low-Q EQ with a center frequency of, say, 20KHz will "bump" into the 22.05KHz Nyquist limit and the behavior of that EQ gets funky up there.  It is said to be "cramped" against the brick-wall of Nyquist.  There are a couple of solutions to the problem that I know of - we can oversample the EQ so that we can let the EQ operate properly, uncramped, even with high center frequencies, or we can use math that circumvents the problem through a "decramping" algorithm.  The Massenburg EQ can have a center frequency of 26KHz as I recall.  The Oxford derivation of this can actually handle this with only a 44.1KS/s sample rate.  George got around it by upsampling first to 88.2KS/s and then applying the EQ when he did his own signature plugin version.

Nika.

[danlavry]

[quote title=Nika Aldrich wrote on Fri, 30 April 2004 19:53]

Mark_W wrote on Fri, 30 April 2004 18:28

"Nika, yes that makes sense, thank you. But you raised another interesting issue regarding differing implementations of EQ and the possible need for upsampling in some cases but not others. It has to do with what happens when the band in the EQ gets close to Nyquist.  A low-Q EQ with a center frequency of, say, 20KHz will "bump" into the 22.05KHz Nyquist limit and the behavior of that EQ gets funky up there.  It is said to be "cramped" against the brick-wall of Nyquist.... Nika.

"

This answer is way beyond over simplification. And there are different reasons for FIR and IIR.

Dan Lavry

[Nika Aldrich]

danlavry wrote on Fri, 30 April 2004 23:59

This answer is way beyond over simplification. And there are different reasons for FIR and IIR. Dan Lavry

...granted.

Nika.

[Richard]

Dan, first thank you.
You say " What is true for say pixels and video or computer screen, is not true for sampling limited bandwidth signal. Making an analogy here is wrong! You need 2 points to draw a straight line. No need for more. You need 3 point for a circle. Well, the bandwidth restriction ends up with: you need only to exceed twice the highest frequency you deal with, thus 88.2KHz accommodates 44.1KH of audio."

I feel that not only the "points" matter but the journey between them. I agree that today's higher sampling rates are not all that useful, in fact mostly just expensive in terms of time wasting. Digital by nature being 'samples' lacks detail.
Faster (more frequent0 sampling should improve this detail but in fact, after a point, just adds to the difficulty of management and manipulation. I feel that the development of greater 'honest' bit depth in the digital mixing domain will
provide a much needed improvement to the 'final sound'.
Richard

[Loco]

Richard wrote on Sat, 01 May 2004 12:45

I feel that not only the "points" matter but the journey between them.

Sorry Richard. I didn't got your last name....

The journey between points are given by the Hi-cut filters on the DAC. For another analogy, the filters are the sandpaper that smooth the staircase effect of sampling. How good is the sandpaper? It depends on the designer of the DAC.

[danlavry]

Richard wrote on Sat, 01 May 2004 17:45

Dan, first I feel that not only the "points" matter but the journey between them. Richard

Richard. You did not read my paper, or you did not understand it. I stated that use of common sense to understand Nyquist may fail you. Everbodys common sense would figure out, wrongly, that if you want to "plot that curve" you are best to just draw the "whole thing", one uninteruptable line. My son could figure it when he was 6 years old, and the guy accross the street that paint buildings.
But it took a bigger mind, a Phd from Yell, a leader at Bell Labs reasrech at thier golden age, to find out that as long as you take just enough points, you are able to reconstruct not just the points, but the exactly THE WAY YOU GET THERE BETWEEN THE POINTS.
It would not work for ANY wave. It ALWAYS works for band limited wave. My paper “Sampling Theory” explains it at www.lavryengineering.com under support.
Sorry if it is not easy to understand.

Br
Dan Lavry

[Nika Aldrich]

Richard wrote on Sat, 01 May 2004 17:45

You need 2 points to draw a straight line. No need for more. You need 3 point for a circle.

Richard,

This point that Dan made is very important and one that I would recommend reviewing again.  The first part is obvious - in order to represent all information needed to re-draw a straight line you only need two points.  And the next part is more important.  All you need to represent a circle well enough that you can completely accurately redraw it is 3 points.  In other words, take a circle of any size.  Mark off three points on it and tell me the coordinates of those three points and I can re-draw your circle with 100% accuracy.  There is absolutely no need for any more data in order to re-draw your circle.

Now sampling theory is similar to this.  Nyquist told us that with a certain amount and type of information we can accurately record just enough essential information in order to re-draw the original waveform with complete accuracy.  Any more data than that is not any more helpful to the process.  That is the essence of sampling theory.

Now I know that you are wondering how we possibly re-draw the waveform with only the information given, but that work is left to certain types of filters in the D/A conversion process.  And so long as those filters follow the rules that Nyquist implied the waveform will be 100% accurately reconstructed.

Make sense?

Nika.

[davidc]

Nika Aldrich wrote on Sat, 01 May 2004 20:48

so long as those filters follow the rules that Nyquist implied the waveform will be 100% accurately reconstructed.

Yet there are lots of different filters, and they all sound different, so are you saying that they are not following the rules, or is there room for variation in accuracy?

Best Regards

DC

[Richard]

Richard,

This point that Dan made is very important and one that I would recommend reviewing again.  The first part is obvious - in order to represent all information needed to re-draw a straight line you only need two points.  And the next part is more important.  All you need to represent a circle well enough that you can completely accurately redraw it is 3 points.  In other words, take a circle of any size.  Mark off three points on it and tell me the coordinates of those three points and I can re-draw your circle with 100% accuracy.  There is absolutely no need for any more data in order to re-draw your circle.

Now sampling theory is similar to this.  Nyquist told us that with a certain amount and type of information we can accurately record just enough essential information in order to re-draw the original waveform with complete accuracy.  Any more data than that is not any more helpful to the process.  That is the essence of sampling theory.

Now I know that you are wondering how we possibly re-draw the waveform with only the information given, but that work is left to certain types of filters in the D/A conversion process.  And so long as those filters follow the rules that Nyquist implied the waveform will be 100% accurately reconstructed.

Make sense?

Nika.[/quote]

Thank you Nika.
Question: Where is the information that states that it is a circle? Those same coordinates present the option for many other results. Progress is often hindered by what is considered to be fact. In this case the time between samples, at any rate, is treated as two points destined to be represented as a fictitious line whatever their original shape might have been.
I would suggest that Nyquist's "complete accuracy" might be replaced with "sufficient illusion"
Richard Dodd

[Nika Aldrich]

Richard wrote on Sat, 01 May 2004 23:27

Thank you Nika. Question: Where is the information that states that it is a circle? Those same coordinates present the option for many other results.

Correct.  But if I give you three points on an X-Y axis and tell you that they represent a circle - that each point is on the circle - you fully agree that there is only one possible shape that you can draw that would fulfill these criteria, and your result will be the exact same shape that I started with.

What Nyquist said was the same.  He said that for any waveform on an X-Y axis where X equals time, and wherein no frequency content exists within the waveform above a given frequency (N) that the waveform can be completely represented with measurements of its amplitude, evenly spaced, at a frequency of greater than 2N.

It is the same as the circle.  Nyquist told us that two very valuable pieces of information are required for this to work.  First is the sample data and second is the knowledge that these sample points represent a waveform that contains no frequency content greater than N.  

With only that information we can fill in the rest of the waveform.  Just like with two crucial pieces of data you can fill in the rest of the circle - one being that the shape is a circle, and the other being the three data points.

Quote:

In this case the time between samples, at any rate, is treated as two points destined to be represented as a fictitious line whatever their original shape might have been.

This is absolutely fallacious.  The two points are NOT destined to be represented as a ficitious line - that is a complete violation of Nyquist, Shannon, and the bevvy of other mathematicians who have proved them right.  

Let's put it this way - mathematical law tells us that, given a set of points equally spaced along an X-Y axis in equal units of time (X) there is only one waveform that can pass through them that does not contain any frequency content above half the sample rate.  Recognizing this, the connection between the two dots is NOT a "fictitious line" because if it is anything other than accurate it inherently contains frequency content above N, and that is not allowed.  Therefore, if we limit the frequency response of the system to N (using filters) there is only one waveform that can pass through those sample points, and it, ipso facto, will not contain any "fictitious lines."

Quote:

I would suggest that Nyquist's "complete accuracy" might be replaced with "sufficient illusion"

After very intense mathematical analysis, Nyquist proposed his theory in 1929, and it is the theory that I have described above.  It took 20 years for another mathematician to prove him correct - that a waveform CAN be represented completely with only the two pieces of data I described above.  Claude E Shannon took years studying Fourier, LaPlace, and the other theories of the day before mathematically proving Nyquist correct in 1948 in a paper entitled "Communication in the Presence of Noise."  In that paper, Shannon very clearly spells out the entire mathematical proof (about 20 pages) of this concept.  Shannon and Nyquist both worked at AT&T research (and possibly Bell Labs?) and Shannon has authored several mathematical proofs - his writings fill a 1000 page book.  Since then the entire telecommunications industry has been based on this premise now called either the Nyquist Theorem or the Shannon Theorem ...

... and now, in 2004, you are saying that they were wrong?  That the original waveform CAN'T be represented with only those sample points with complete accuracy?

Please - disprove Shannon's proof.  Mr. Lavry and I will be interested in looking at the math.

Nika.

[Innominandum]

Fantastic thread! You've dispelled some myths from my mind.

"A bandlimited signal can be reconstructed exactly from its samples if the bandwidth is less than Nyquist frequency. Otherwise, aliasing occurs: high frequencies alias, appearing to be a lower frequency."

What is aliasing?

[sdevino]

Aliasing occurs when frequencies greater than the nyquist frequency (sample freq/2) are presented to the sampler. Band limiting eliminates aliasing.

One thing for all you wanna be nyquist experts to understand is that the reconstruction process is more analog than you might think.  The definition of a perfect reconstruction filter can be described using  a waveform instead of a filter.

The waveform represents a perfectly linear impulse response sampled at the target sample rate. An impulse response is a kind of resonant frequency for the system. Its like a perfectly tuned tuning fork.

The impulse response represents that passive resonance of the circuit with no out of band frequencies. Mathematically this works in a very similar way to the newer "room" reverbs that use convolution to apply the charactoristics of a room to your audio. The impulse response also contains an equal amount of energy at all frequencies in the passband.

If you convolve sampled data with the perfect impulse response you will get 100% accurate waveform reconstruction (minus a predictable loss). This is why a 3 sample waveform that is somewhat complex can be reconstructed. there is additional data in impulse response waveform.

All circuits resonate and we can use this resonance to help manipulate audio when we want.

If you want to have some fun playing with the idea  of filters and convolution, take any digitized audio you have on your DAW, zoom in to the sample level. Set your grid to 1 sample, select 1 sample or 2 or 3 or 4 samples and apply an Audio suite EQ and watch how changing the EQ settings change the way the various sample sizes react. i.e the same EQ setting will do something different to 2 samples than it will to 3 samples etc.

[natpub]

[quote title=Loco wrote on Sat, 01 May 2004 12:39]

Richard wrote on Sat, 01 May 2004 12:45

I feel that not only the "points" matter but the journey between them.

Sorry Richard. I didn't got your last name....

quote]

 http://recforums.prosoundweb.com/index.php/mv/msg/273/2213/9 05/?SQ=e5519fce30db51a82faf28c4d5add42a#msg_2213

http://mixonline.com/ar/audio_richard_dodd/


[edit: oops, I didn't see that he had added his last name on his last post--oh well, I will leave the mix link, is interesting]