[22906]

In the immortal words of Ice-T, "Don't hate the playa, hate the game"

 

[cjl]

 
castleofargh:

Let me start off by thanking you deeply for your criticism, without the likes of which scientific progress would not be possible. My response is as follows.

You are confusing "what the waveform looks like" with audio fidelity. No one has the right to say "what the waveform should look like," whether smooth, rough, staircased, sinusoid, or whatever. Visual representation in a graph is but one way of representing the real signal. The basis of the S-N theorem is that there exists something in reality called a "signal" which is independent of the way we choose to represent it (e.g. time domain voltage graph, FFT, sound pressure meter, what have you). My goal is simply to point out that the information contained in the signal is best preserved by a PCM encoding and R-2R conversion process, as sigma-delta conversion "smooths over" details that are present in the PCM encoded signal.

The waveform should look like the original analog waveform. The output of a DAC, by definition, is a continuous time signal. The input, by definition, is a discrete time signal. Discrete time signals are often represented, mathematically, by effectively a series of impulse functions, but that isn't because that's an accurate continuous time representation of the signal. Instead, it's a method of trying to show that on a discrete time signal, there is no value that the signal contains between those samples. As far as the discrete time signal is concerned, any continuous time signal that passes through each sample exactly is potentially a perfect representation of the signal, and no signal that perfectly passes through each data point is any more valid than any other that also meets that criterion.
 
Because this allows for a lot of possible signals, we need an additional criterion to figure out which waveform we want, and that's where Nyquist comes in. The S-N theorem states that for a signal that was originally bandlimited such that it contains no content above 0.5Fs, the sampled representation is unique. In other words, for a given discrete time signal sampled at sample rate Fs, there is only one possible waveform that both passes through every sample point perfectly and contains no content above a frequency of 0.5Fs. So, this means that the only correct reconstruction* will meet the following 2 criteria:
 
1) It must pass through each sampled point
2) It must contain no frequency content above 0.5Fs**
 
As such, your original statement that "No one has the right to say 'what the waveform should look like,' whether smooth, rough, staircased, sinusoid, or whatever." is incorrect. Since there is only one, unique, correct reconstruction, it is what the waveform should look like, and it will always look like a smooth curve connecting the samples, with no discontinuities or roughness. If the waveform passes through each sample point perfectly, it cannot, by definition, "smooth over details" that are present in the PCM signal, since it perfectly matches every single data point in the discrete time signal.
 
 
*This is assuming that we are dealing with a signal that was recorded correctly - in other words, that the original analog signal contained no frequencies above half the recording sample rate at the point where it was converted from continuous time to discrete time, and that any downsampling that was performed on the signal included a low pass filter at <0.5 times the new sample rate. If this isn't the case, it is not possible to reconstruct the original signal from the sample points, since if the signal isn't bandlimited, the analog reconstruction is nonunique.
 
**I find it fascinating that you are claiming to be defending the "real meaning" of the S-N theorem while at the same time pushing for reconstruction techniques that spectacularly fail this criterion, which is required for accurate signal reconstruction according to the theorem.

 

[22906]

cjl:
 
Thank you for your criticism. My response is as follows.
 

  Because this allows for a lot of possible signals, we need an additional criterion to figure out which waveform we want, and that's where Nyquist comes in. The S-N theorem states that for a signal that was originally bandlimited such that it contains no content above 0.5Fs, the sampled representation is unique. In other words, for a given discrete time signal sampled at sample rate Fs, there is only one possible waveform that both passes through every sample point perfectly and contains no content above a frequency of 0.5Fs. So, this means that the only correct reconstruction* will meet the following 2 criteria:

 
What the signal is and isn't depends on our definition. That every sampled signal should have one and only one "shape" as measured by a voltmeter at the DAC output, is an arbitrary standard, and one that, I am arguing, causes people to ignore the inferior resolution of sigma-delta designs, because no, they don't pass through every sample point perfectly. That is what a dynamic resolution test would show, if we could design one. Audio signals are band-limited by convention to 20 Khz, which most people agree is the limit of human hearing. That means any audio signal that is sampled at >40 Khz will contain all the information necessary to reconstruct the original AUDIO signal perfectly, without extra filtering. Smoothness, and other visual criteria that look nice on oscillloscopes, are not required for proper digital to analog conversion.

People are focusing their criticism on my lack of support for low-pass filtering, because the low-pass filter is the most important part of the sigma-delta design.
 

  Because this allows for a lot of possible signals, we need an additional criterion to figure out which waveform we want, and that's where Nyquist comes in. The S-N theorem states that for a signal that was originally bandlimited such that it contains no content above 0.5Fs, the sampled representation is unique. In other words, for a given discrete time signal sampled at sample rate Fs, there is only one possible waveform that both passes through every sample point perfectly and contains no content above a frequency of 0.5Fs. So, this means that the only correct reconstruction* will meet the following 2 criteria:
 
1) It must pass through each sampled point
2) It must contain no frequency content above 0.5Fs**

Your second point is, I believe, incorrect. While the original sampled signal (what we define as the audio signal) should not contain frequency components above .5 Fs, the reconstructed signal will always contain aliasing, by the S-N theorem. But these aliases are not enemies, they are just unavoidable artifacts that do not affect the sound at all, because they are above the band-limit we have defined for our signal. In other words, the aliasing is not even part of the signal as we have defined it. If suddenly we decided that human hearing goes to 24 Khz, then yes, 44.1 Khz sampled audio would be introducing a lot of frequency noise, but until then, we are safe.
 
Thanks for not making science a personal issue.
 
- m3_arun

 

[Greenears]

FYI I am not an expert in R2R design but I believe the problem with it is that as you get more bits you need progressively more accuracy in the resistors. In modern CMOS semiconductor process below 0.18u I believe it is hard to get that accuracy, if not impossible.  Accurate resistors and resistor matching was always a problem in CMOS (hence the reason for switched C to make resistors out of capacitors).  That is probably the reason we don't offer any R2R - all the converters nowadays all are SAR or SD.
 
While this is an interesting discussion, my $0.02 there is no fundamental issue with SD that is not where any audible problems lie.  There is nothing magical about multi-segment SD it just happens to be the circuit design technique that best matches the current available technology process, if SAR or R2R was better they would use that.
 
Please also remember that Nyquist applies only to perfectly band-limited signals (zero energy above the cutoff) and is a continuous (analog) impulse sample at perfectly symmetric periodic intervals (commonly denoted by T).  Open any textbook to check my statement.  If all those conditions are met you get perfect reconstruction.  Since none of these three conditions are possible in real life due to the 2nd law of thermodynamics, there will always be some small compromise the issue is not with Nyquist theorem itself it is whether the compromises are audible (which usually they are not nowadays).
 
Just my $0.02.

 

[22906]

I think making the issue about reducing audible problems, rather than trying to reach perfect reconstruction, is counterproductive. And there is no objective standard for audibility, whereas we can objectively measure resolution, phase and frequency response. I dont think we should allow the semiconductor manufacturers to dictate which designs are better or not, and leave that to the tests. The problem is the tests are inadequate to describe the accuracy of DA conversion today, specifically the resolution. Once we develop a proper test for resolution, the resolution difference between R-2R and SD, which, you're right, is not that big, will become clear.
 
BTW SAR means successive approximation register, and refers to a particular kind of ADC (analog to digital converter) design. I believe these are the counterpart to R2R in achieving optimal resolution on the ADC side.

 

[stv014]

  Saying a problem is old and that you're focusing on it isn't a personal attack, but since you want to play petty, I'll just drop you.

 
Indeed, that seems to be the best decision, I do not think m3_arun will ever be convinced by any counter-arguments, and it is just one of those "you cannot prove I am wrong under the rules I created" debates that keep going in circles.

 

[22906]

   
Indeed, that seems to be the best decision, I do not think m3_arun will ever be convinced by any counter-arguments, and it is just one of those "you cannot prove I am wrong under the rules I created" debates that keep going in circles.

I welcome counter-arguments that are supported by evidence. Those are not my rules, those are the universal rules of argumentation, which you can find at the top of the forum page. Why would anyone give up their position unless proven wrong?

 

[Greenears]

I'm not "making the issue" anything. It is the 2nd law of thermodynamics that creates the problem, didn't invent that one sorry
 
People always quote Nyquist and focus on the sampling rate but neglect the critically important restrictions.  Nyquist knew about the restrictions and knew they were impossible in real life.  That doesn't make the theory not useful, but it is just something you have to know.  For instance just one example, zero energy beyond the cutoff is not possible, there is always cosmic background radiation with energy at all frequencies.  Some of that noise will bleed into the signal (aliasing).  The issue is whether it is audible (which fortunately it isn't). I didn't create it, it's just there.  Sorry.  Each restriction has a story like this.
 
Again I'm not an expert but it seems to me that the SAR are better for higher sample rate but lower ENOB like 10-14 bit range.  I don't think it is anything fundamental about SD vs SAR it is all just tradeoffs.  SAR result in good power/area for communications like Wifi etc.  I don't know where you got your information that the tests are inadequate I've never heard any of our designers say that.  There are a whole range of tests that they throw at these things, and remember all these converters are used in real world application if the Wifi or Fiber didn't work we would know about it.
 
I also don't think semi manufacturers are dictating anything.  There is a reality about what fabs are available.  It's not economic to spend billions on a fab just tuned for audio components, and there is no need.  They use what is available and work the designs to match the best features of the technology and avoid the worst. That's just my opinion I think they've done an excellent job and in some ways we are in audio nirvana at least from the perspective of ADC and DAC chips available.

 

[22906]

Well I am making the issue about measurable resolution, not about audibility. The ambiguous standard of audibility is what allows chip manufacturers to justify sigma-delta to naive consumers. Sigma-delta is motivated, like you said, by the foundries' economic concerns, and not by high-fidelity. But for the consumer, high fidelity at low cost is most important.

 

[cjl]

   
What the signal is and isn't depends on our definition. That every sampled signal should have one and only one "shape" as measured by a voltmeter at the DAC output, is an arbitrary standard, and one that, I am arguing, causes people to ignore the inferior resolution of sigma-delta designs, because no, they don't pass through every sample point perfectly. That is what a dynamic resolution test would show, if we could design one. Audio signals are band-limited by convention to 20 Khz, which most people agree is the limit of human hearing. That means any audio signal that is sampled at >40 Khz will contain all the information necessary to reconstruct the original AUDIO signal perfectly, without extra filtering. Smoothness, and other visual criteria that look nice on oscillloscopes, are not required for proper digital to analog conversion.
 
People are focusing their criticism on my lack of support for low-pass filtering, because the low-pass filter is the most important part of the sigma-delta design.
 
Your second point is, I believe, incorrect. While the original sampled signal (what we define as the audio signal) should not contain frequency components above .5 Fs, the reconstructed signal will always contain aliasing, by the S-N theorem. But these aliases are not enemies, they are just unavoidable artifacts that do not affect the sound at all, because they are above the band-limit we have defined for our signal. In other words, the aliasing is not even part of the signal as we have defined it. If suddenly we decided that human hearing goes to 24 Khz, then yes, 44.1 Khz sampled audio would be introducing a lot of frequency noise, but until then, we are safe.
 
Thanks for not making science a personal issue.
 
- m3_arun

The reconstructed signal need not contain aliasing, and aliases are typically a result of the original signal having content above 0.5Fs that shows up as <0.5Fs content in the reconstruction. You seem to have the definition backwards here, since you seem to think that it implies that the reconstructed signal will have "aliases" above 0.5Fs. Also, every sampled, bandlimited signal should have one and only one shape. That's the whole point of the Nyquist theorem in the first place - that a signal with content up to 0.5Fs can be uniquely represented by a set of sample points at sample rate Fs.
 
Oh, and smoothness is absolutely a requirement. If a signal isn't smooth at a timescale near the sampling period, it contains substantial content at frequencies >0.5Fs, which means that something is very, very wrong with your reconstruction of the signal from the sampling points.

 

[22906]

According to http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem it is mathematically impossible to reconstruct a continuous time signal from a discrete one without aliasing.
 
If what you mean by smoothness is that the output shouldn't contain frequencies above .5 Fs, maybe that's your preference, but not mine. I don't care if ultrasonic frequencies get to my ears, as long as the signal is at the proper resolution, and phase-coherent with the original.

 

[cjl]

Where, exactly, in that article, does it state that it is mathematically impossible to reconstruct a continuous time signal from a discrete one without aliasing? Can you give me a direct quote?

 

[22906]

I'll let you find it yourself; hint: it's under the heading "Aliasing"

 

[cjl]

  I'll let you find it yourself; hint: it's under the heading "Aliasing"

I have a masters degree that included a significant focus on control theory. My day job is as a servo (control system) engineer. I'm quite familiar with sampled systems, thank you very much.
 
Now, direct quote please. I'll be a little more direct about my reasoning this time: I'm pretty certain you're misinterpreting something in that article, and I want you to quote the area you think states that so I can explain what it's actually stating. Nothing under that heading states that aliasing is unavoidable in reconstruction.

 

[22906]

Ok, I will revise my statement; DA conversion by an R2R DAC will create aliases. And since I prefer the resolving ability of R2R designs vs sigma-delta, I am willing to accept unfiltered aliases in ultrasonic frequencies.