Multiple ADCs and aliasing

Started by Piotr Wyderski February 23, 2018
Randy Yates  <randyy@garnerundergroundinc.com> wrote:

>spope384@gmail.com (Steve Pope) writes:
>> Aliasing occurs when sampling. This signal is being sampled at 2 >> Ms/sec.
>That is an unproven assertion. The standard treatment of sampling theory >involves the use of a SINGLE quantizer sampling at Fs samples per second >with Fs/2 bandwidth.
According to whom?
>Creating an alternate sampling architecture which uses two quantizers >and claiming it is equivalent to sampling at twice the sample rate >requires a theoretical basis to establish its validity. Otherwise it's >just an assertion.
The theory is the following: Uniform sampling of x(t) at rate Fs is defined as creating the signal y(n) = f(n * t / Fs) for integers n. Full stop. Nowhere is it said you can't use one, two, thirteen, or perhaps a countably infinite number of samplers. Just that the signal y(n) must exist, in some format. Steve
spope384@gmail.com (Steve Pope) writes:

> Randy Yates <randyy@garnerundergroundinc.com> wrote: > >>spope384@gmail.com (Steve Pope) writes: > >>> Aliasing occurs when sampling. This signal is being sampled at 2 >>> Ms/sec. > >>That is an unproven assertion. The standard treatment of sampling theory >>involves the use of a SINGLE quantizer sampling at Fs samples per second >>with Fs/2 bandwidth. > > According to whom?
Oppenheim, Proakis/Manolakis, Lyons, etc. Btw, I don't think your summary of sampling their is accurate/complete. You are free to bridge the gap with intuition if you like. I've already proven this to myself analytically so I'm good. -- Randy Yates, Embedded Linux Developer Garner Underground, Inc. http://www.garnerundergroundinc.com
Steve Pope wrote:
> Randy Yates <randyy@garnerundergroundinc.com> wrote: > >> spope384@gmail.com (Steve Pope) writes: > >>> Aliasing occurs when sampling. This signal is being sampled at 2 >>> Ms/sec. > >> That is an unproven assertion. The standard treatment of sampling theory >> involves the use of a SINGLE quantizer sampling at Fs samples per second >> with Fs/2 bandwidth. > > According to whom? > >> Creating an alternate sampling architecture which uses two quantizers >> and claiming it is equivalent to sampling at twice the sample rate >> requires a theoretical basis to establish its validity. Otherwise it's >> just an assertion. > > The theory is the following: > > Uniform sampling of x(t) at rate Fs is defined as creating the signal > y(n) = f(n * t / Fs) for integers n. Full stop. > > Nowhere is it said you can't use one, two, thirteen, or perhaps a countably > infinite number of samplers. Just that the signal y(n) must exist, in some > format. > > Steve >
So imagine we have a 4MHz clock. Imagine we have a small distribution network/FSM ( CPLD/FPGA) that toggles between two "chip enable" outputs on every rising (falling?) edge of the 4 MHz clock. Tie each of the two "chip enable" to the "output enable" on the A/D chips. Then run that output to a FIFO that a CPU relieves in software. Wave hands at the duty cycle of the "output enables" for now ( should be tractable with well chosen crystals and a through understanding of the A/D chips ) Suppose also that the latency of the two A/D are the "same" to some limit. *In that case*, other than part-to-part variation, you'd be limited by the jitter of the highest frequency clock of the sampled streams. Aliasing of the bandlimit sort seems to be purely that - it'd move as expected by increasing the sample rate. You've created a single block-device that behaves as-if it were a single device. It's the unhappy spawn of a ... TDM MUX :) and an A/D converter... Now, given the ... agreeability of the transition from 1 to 2 such devices, we can imagine that we could extend the concept to 4,8,16... some limit of number of devices. We could easily find that at some limit, timing, signal or other implementation details would cause it to break down. But for perfect parts on the perfect board, no such limitations exist... indeed, you should be able to get away with a lot with the appropriate buffering. So we have the two halves of a "proof by induction"; P(2) AND P(n)->P(2^(n+1)) :) -- Les Cargill
Randy Yates  <randyy@garnerundergroundinc.com> wrote:

>spope384@gmail.com (Steve Pope) writes:
>> Randy Yates <randyy@garnerundergroundinc.com> wrote:
>>>spope384@gmail.com (Steve Pope) writes:
>>>> Aliasing occurs when sampling. This signal is being sampled at 2 >>>> Ms/sec.
>>>That is an unproven assertion. The standard treatment of sampling theory >>>involves the use of a SINGLE quantizer sampling at Fs samples per second >>>with Fs/2 bandwidth.
>> According to whom?
>Oppenheim, Proakis/Manolakis, Lyons, etc.
How off that I would use these references for my entire career and never run across this tidbit.
>Btw, I don't think your summary of sampling their is accurate/complete.
It's accurate, and complete enough for our purposes here. Steve
On Monday, February 26, 2018 at 2:57:08 AM UTC-5, Randy Yates wrote:
> > I guess the phallacy in my logic is that, once aliasing is introduced, > it is impossible to remove. That has been driven into my mind for so > long it took some effort to thwart.
Randy, another example of recovering from aliasing is in Radar systems. If you transmit a linearly frequency modulated pulse, in an arbitrary case you need to sample at a rate to accommodate the pulse bandwidth. Another scheme is to use deramp on receive - basically you mix the RF signal with a chirp signal having the opposite frequency chirp. The bandwidth of the resulting signal depends on the range swath/extent of your signal. For small range extents you can have a smaller sampling rate. So the above 2 schemes are the typical approach. It has been shown that you can also do the sampling at the lower rate and then do the deramp in the digital domain. So you are effectively subsampling the signal and then compensating for the aliasing with the digital deramping. So you are able to compensate for aliasing in certain cases - if you have some additional information. The Nyquist / Shannon theorem is the most general case, when you don't have any other information. Cheers, David
On 27.2.18 21:56, Randy Yates wrote:
> Tauno Voipio <tauno.voipio@notused.fi.invalid> writes: > >> On 27.2.18 17:48, Randy Yates wrote: >>> rr <rr@somwhere.com> writes: >>> >>>> Hi Randy, >>>> >>>> W dniu 27.02.2018 o&nbsp;03:47, Randy Yates pisze: >>>>> rr <rr@somwhere.com> writes: >>>> (...) >>>>>> No, Piotr question is about TWO 1 Msps ADCs converting 1 MHz bandwith >>>>>> signal in two phase, interleave mode: >>>>>> "... >>>>>> I have two 1MHz ADCs and would like to interleave them >>>>>> by a half of the sampling cycle in order to get a single >>>>>> ADC with effective sampling frequency of 2MHz ... " >>>>> >>>>> I am clear on Piotr's question. The point is, if you consider one of the >>>>> two ADCs (either one) independent of the other, it will (in general) >>>>> experience aliasing. This was actually part of Piotr's original >>>>> question, and Piotr is correct on this point. >>>> >>>> yes, of course, if we sample 1 MHz bandwidth analog signal at 1 Msps >>>> rate, the sample stream will contain aliasing, but dependency has no >>>> meaning, because of too low sampling FREQUENCY, and single ADC is >>>> independent (making them dependent has no sense). Regular time >>>> intervals between samples counts only. Each of two sample streams from >>>> dependent ADCs experience aliasing, but not together. >>> >>> Roman, >>> >>> In my opinion you are overlooking the most relevent portion of >>> Piotr's question, which was also the question I had in my mind. >>> >>> All my life I've had it drilled it into my head, unconditionally, that >>> once aliasing occurs, it is irreversible. Until now I don't ever >>> remember anyone teaching me, "Aliasing can be reversed in certain >>> cases." >>> >>> Contributing (in my opinion) to the confusion in this thread is that >>> some of the answers (including yours, and including the Analog Devices >>> article that was cited) mix theoretical with practical considerations. >>> Again, I think Piotr's first and foremost question was, "How can >>> aliasing in an ADC ever be undone?" The answer to that question is >>> purely theoretical and has absolutely nothing to do with practical >>> considerations. >>> >>> Further contributing (in my opinion) to the confusion in this thread is >>> that nowhere (until my answer describing the modulation model of >>> sampling) was a clear theoretical explanation provided. >>> >>> I'm not saying practical considerations aren't important. I'm just >>> saying I don't think they answer Piotr's most fundamental question, and >>> I know they didn't answer mine. >>> >>> In any case, I appreciate the discussion. I learned something here and >>> that is very valuable. >> >> >> If the ADC samplings are properly staggered and the resulting >> number stream interleaved, how does the result differ from one >> sampled at twice the original rate? > > It does not differ. The question in my mind was, why didn't the aliasing > in the individual converters screw things up? Until you can prove > analytically how this aliasing is "undone", you're waving your hands, no > matter how intuitive it may seem from other points-of-view. Intuition is > not proof! If it is true, it should be provable. > > Analogy: Did Shannon/Nyquist merely assert that the input bandwidth to a > single, real (time) quantizer running at a rate of Fs quantizations per > second must be less than Fs / 2? Or did they prove why it must be so?!? > > All the responses I've seen so far have been faith-based.
Ther is no faith needed: The main problem with sparse sampling is that you do not know what the signal does between the samples. Shannon and Nyquist formulated the limit mathematically. If the signal is band-limited according to their theories, it simply cannot wiggle out of way between the samples. Another way of looking to it comes from the lower sideband of the sampling frequency toward the arithmetic center, where it collides with the baseband signal and spoils the show. If you limit the input so that the two cannot meet, you're OK, and that means input below fs/2. -- -TV
If I sample 0.9 MHz at 2 MHz, it meets Nyquist and there is no aliasing.

If I sample 0.9 MHz at 1 MHz, I get aliasing at 100 kHz.

If I INTERLEAVE a second ADC also sampling at 1 MHz the second one also
has aliasing at 100 kHz. 

Now if you combine the two interleaved ADCs,  such that the effective sampling is 2
MHz ___and the interleaving is perfect___ the two 100 kHz alias will cancel.  This
illustrates the critical nature of the phasing of the interleaving of the 2 ADCs. 
If they are not exactly 180 deg apart, the 100 kHz aliass  will not completely
cancel.

Likewise if you consider a single ADC sampling at 2 MHz but the 2 MHz clock has 1
MHz jitter  the same 100 kHz alias will appear.

mark

Tauno Voipio <tauno.voipio@notused.fi.invalid> writes:

> On 27.2.18 21:56, Randy Yates wrote: >> All the responses I've seen so far have been faith-based. > > > Ther is no faith needed:
I disagree. -- Randy Yates, Embedded Linux Developer Garner Underground, Inc. http://www.garnerundergroundinc.com
Randy Yates  <randyy@garnerundergroundinc.com> wrote:

>Tauno Voipio <tauno.voipio@notused.fi.invalid> writes:
>> On 27.2.18 21:56, Randy Yates wrote:
>>> All the responses I've seen so far have been faith-based.
>> Ther is no faith needed:
>I disagree.
I just had a metaphysical thought: Randy does not accept the axiom of choice! The AOC says that you can look at any non-empty set, and select a member of that set. Most scientists use this all the time, but Randy is among the resistance. Steve
On 18-03-01 06:28 , Steve Pope wrote:
> Randy Yates <randyy@garnerundergroundinc.com> wrote: > >> Tauno Voipio <tauno.voipio@notused.fi.invalid> writes: > >>> On 27.2.18 21:56, Randy Yates wrote: > >>>> All the responses I've seen so far have been faith-based. > >>> Ther is no faith needed: > >> I disagree. > > I just had a metaphysical thought: > > Randy does not accept the axiom of choice! > > The AOC says that you can look at any non-empty set, and select > a member of that set.
Ah... not quite, for the axiom of choice, one has a (possibly) infinite collection of non-empty sets, and selects one member from each set in the collection. https://en.wikipedia.org/wiki/Axiom_of_choice (Getting back to the point, I tend to agree with Tauno that no faith is needed to understand why the two-ADC method works. However, I can also understand the interest in showing mathematically how the aliasing is removed when the two ADC sample streams, with aliasing, are combined into one stream, without aliasing.) -- Niklas Holsti Tidorum Ltd niklas holsti tidorum fi . @ .