# Multiple ADCs and aliasing

Started by February 23, 2018
```On 1.3.18 02:43, Randy Yates 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 have practical experience from an interleaved system of
5 parallel ADC/DSP channels on a radar receiver I worked
with some 30 years ago. There were simply no units fast
enough for the received video in one channel.

--

-TV

```
```Niklas Holsti  <niklas.holsti@tidorum.invalid> wrote:

>On 18-03-01 06:28 , Steve Pope wrote:

>> 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.

(AOC .. ADC ... what's the difference?)

You have an infinite collection of samplers and select one sample
out of each.

>(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.)

It's a good exercise, yes, but not needed to prove anything.

Steve
```
```On Thursday, March 1, 2018 at 1:02:53 AM UTC-8, Niklas Holsti wrote:

[Snipped by Lyons]

> 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

Hi Niklas. For what it's worth, I agree with your statement.

[-Rick Lyons-]
```
```On Thursday, March 1, 2018 at 12:06:00 PM UTC-8, Steve Pope wrote:

[Snipped by Lyons]

>
> It's a good exercise, yes, but not needed to prove anything.
>
> Steve

No need to prove anything!!  Say that to a mathematician and then see the look of revulsion in his face.

[-Rick-]

```
```Richard (Rick) Lyons <r.lyons@ieee.org> wrote:

>On Thursday, March 1, 2018 at 12:06:00 PM UTC-8, Steve Pope wrote:

>> It's a good exercise, yes, but not needed to prove anything.

>No need to prove anything!!  Say that to a mathematician and then see
>the look of revulsion in his face.

Um, I am a mathematician!

Not that it's done anything but get me into trouble.

Steve
```
```spope384@gmail.com (Steve Pope) writes:

> Richard (Rick) Lyons <r.lyons@ieee.org> wrote:
>
>>On Thursday, March 1, 2018 at 12:06:00 PM UTC-8, Steve Pope wrote:
>
>>> It's a good exercise, yes, but not needed to prove anything.
>
>>No need to prove anything!!  Say that to a mathematician and then see
>>the look of revulsion in his face.
>
> Um, I am a mathematician!

PhD? I'm curious: by what reckoning you consider yourself a
mathematician?

I have an undergraduate degree in math, and I've had some graduate
coursework. I've studied real analysis, abstract algebra, advanced
linear algebra, complex analysis, number theory, etc.

I'm not sure that makes me a mathematician. Fortunately a PhD in math is
not required to see that if a model of a process is used to prove a
result, then if the process is changed the result can no longer be
concluded by the same model.

The model I am familiar with is the one in which the infinite impulse
train (commonly denoted p(t)) with period Ts is used to select the
samples. So Steve your remark about how the samples are selected
is really at the heart of my position.

If you select samples another way, I don't see how you can reach the
same conclusion by the standard p(t) model.

What may be the case is that another proof of the sampling theorem
exists which does not use this p(t) model, and that perhaps sampling
using N quantizers can be covered within this proof.
--
Randy Yates, Embedded Linux Developer
Garner Underground, Inc.
http://www.garnerundergroundinc.com
```
```Randy Yates  <randyy@garnerundergroundinc.com> wrote:

>spope384@gmail.com (Steve Pope) writes:

>> Um, I am a mathematician!

>PhD? I'm curious: by what reckoning you consider yourself a
>mathematician?

>I have an undergraduate degree in math, and I've had some graduate
>coursework. I've studied real analysis, abstract algebra, advanced
>linear algebra, complex analysis, number theory, etc.

I have a degree (B.S.) in mathematics.  I studied nearly enitrely algebra,
and only took the minimum in analysis.  Which is a a good thing, as I
also have a Ph.D. but that is in engineering, which finally allowed me
to exchange the order of integration willy-nilly.

It's fascinating though that your memories of signal processing
instructional texts as so vastly different from my own.

Steve
```
```spope384@gmail.com (Steve Pope) writes:

> Randy Yates  <randyy@garnerundergroundinc.com> wrote:
>
>>spope384@gmail.com (Steve Pope) writes:
>
>>> Um, I am a mathematician!
>
>>PhD? I'm curious: by what reckoning you consider yourself a
>>mathematician?
>
>>I have an undergraduate degree in math, and I've had some graduate
>>coursework. I've studied real analysis, abstract algebra, advanced
>>linear algebra, complex analysis, number theory, etc.
>
> I have a degree (B.S.) in mathematics.  I studied nearly enitrely algebra,
> and only took the minimum in analysis.  Which is a a good thing, as I
> also have a Ph.D. but that is in engineering, which finally allowed me
> to exchange the order of integration willy-nilly.

We had "Baby Rudin" and even that was tough for me.

> It's fascinating though that your memories of signal processing
> instructional texts as so vastly different from my own.

Oppenheim et al. [1] discuss infinite-impulse train sampling in section
8.1.1. Mitra [2] does the same in in section 5.2.1. Proakis/Manolakis
[3] took a different approach so I was incorrect to include them. Did
you not remember this approach in [1] and [2]? I especially love
Oppenheim's text.

[1] @BOOK{signalsandsystems,
title = "{Signals and Systems}",
author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}",
publisher = "Prentice Hall",
year = "1983"}

[2] @BOOK{mitra,
title = "{Digital Signal Processing: A Computer-Based Approach}",
author = "Sanjit~K.~Mitra",
publisher = "McGraw-Hill",
edition = "second",
year = "2001"}

[3] @BOOK{proakisdsp4,
title = "{Digital Signal Processing: Principles, Algorithms, and Applications}",
author = "John~G.~Proakis and Dimitris~G.~Manolakis",
publisher = "Prentice Hall",
edition = "fourth",
year = "2007"}

--
Randy Yates, DSP/Embedded Firmware Developer
Digital Signal Labs
http://www.digitalsignallabs.com
```
```Randy Yates  <yates@digitalsignallabs.com> wrote:

>Oppenheim et al. [1] discuss infinite-impulse train sampling in section
>8.1.1. Mitra [2] does the same in in section 5.2.1. Proakis/Manolakis
>[3] took a different approach so I was incorrect to include them. Did
>you not remember this approach in [1] and [2]? I especially love
>Oppenheim's text.

I simply don't recall Oppenheim, or the others, singling out any
architectures or implementations as verboten.  I mean, they discuss
sampling algorithms.  Not the detail of a hardware design. Although
sometimes, a signal processing text will stray into discussing
architectures, but these are -- one hopes meant as examples, and not
as excluding other architectures.

But, I do not have any of these in my personal library, which is
rather weighted towards algebra and coding, so I will stipulate
to your reading that in such a referecne, a two-sampler approach to
producing a sample stream is NFG.

Steve
```
```Hi Randy,

W dniu 27.02.2018 o&nbsp;21:01, Randy Yates pisze:
> rr <rr@somwhere.com> writes:
>> [...]
>> Yes, that is true, but we have no aliasing in combined, interleaved
>> sample stream. :-)
>
> That is an assertion, not a proof. For all the time you've spent
> discussing this, you could have easily come up with the proof. It took
> me about a half an hour.

It is already done. Look at page 5(1810) of this paper:
http://www.seas.ucla.edu/brweb/papers/Journals/BRAug13.pdf

"... Shown in Fig. 6, these spectra exhibit heavy aliasing, which
is eventually undone by the back-end multiplexer. Since multiplexing of
discrete-time signals is equivalent to addition, the spectral copies
around &plusmn;Fck cancel each other in Y1(f)+Y2(f), thus yielding the original
X(f). Such cancellation is reminiscent of image rejection in RF
receivers, pointing to the precise matching required of the channels.
..."

Regards,

Roman Rumian

```