Multiple ADCs and aliasing

rr <rr@somwhere.com> writes:

> 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

Nice! Thanks for the reference! 
-- 
Randy Yates, Embedded Linux Developer
Garner Underground, Inc.
http://www.garnerundergroundinc.com

On Friday, March 2, 2018 at 2:55:05 AM UTC-8, rr wrote:
> Hi Randy,
> 
> W dniu 27.02.2018 o 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

[Snipped by Lyons]

Hi Roman.  This has been an educational thread.

The paper you cited is interesting. On page 1810 of the paper the author wrote: "Since multiplexing of discrete-time signals is equivalent to addition,... ." That's not true, of course. We don't simply add the two discrete sequences' samples, we interleave the two sequences' samples. Perhaps the "addition" the author is referring to should be called "time-aligned addition" or "time-interleaved addition".

[-Rick-]

Hi Rick,

W dniu 02.03.2018 o 20:31, Richard (Rick) Lyons pisze:
(...)
> The paper you cited is interesting. On page 1810 of the paper the author wrote: "Since multiplexing of discrete-time signals is equivalent to addition,... ." That's not true, of course. We don't simply add the two discrete sequences' samples, we interleave the two sequences' samples. Perhaps the "addition" the author is referring to should be called "time-aligned addition" or "time-interleaved addition".
> 

hm, THIS IS simply addition of the two sample sequences, but as sampling 
moments of second one is shifted by Tck/2, they are upsampled by 2 in 
order to make this operation possible (at 2 times higher output frequency).
Well, if it is your point, "time-interleaved addition" is acceptable for 
me. ;-)

Thanks for vigilance,

Roman Rumian

In article <p7cegr$9a8$1@node1.news.atman.pl>, rr  <rr@somwhere.com> wrote:

> The paper you cited is interesting. On page 1810 of the paper the
> author wrote: "Since multiplexing of discrete-time signals is equivalent
> to addition,... ." That's not true, of course. We don't simply add the
> two discrete sequences' samples, we interleave the two sequences'
> samples. Perhaps the "addition" the author is referring to should be
> called "time-aligned addition" or "time-interleaved addition".

Actually it's true.  A discrete-time signal is non-zero only at
discrete points in time.  So if for example,

            X(t) is zero for t not in (0, 2, 4, 6...)
            Y(t) is zero for t not in (1, 3, 5, 7...)

Then Z(t) = X(t) + Y(t) where

       Z(t) = X(t) if t = 0, 2, 4, 6 ... , Y(t) otherwise.

So I would say addition and multiplexing are the same, so long as
you haven't mangled the time axis.

Steve

On 2018-03-02 22:19, Steve Pope wrote:
[...]
> Actually it's true.  A discrete-time signal is non-zero only at
> discrete points in time.  So if for example,

I would even say that a discrete-time signal is only
*defined* at discrete points in time.

bye,

-- 

piergiorgio

Piergiorgio Sartor  <piergiorgio.sartor.this.should.not.be.used@nexgo.REMOVETHIS.de> wrote:

>On 2018-03-02 22:19, Steve Pope wrote:

>> Actually it's true.  A discrete-time signal is non-zero only at
>> discrete points in time.  So if for example,

>I would even say that a discrete-time signal is only
>*defined* at discrete points in time.

Well, that certainly would shoot the idea of adding two 
differently-sampled signals together.

You're right of course.

Steve

On 2018-03-02 23:02, Steve Pope wrote:
> Piergiorgio Sartor  <piergiorgio.sartor.this.should.not.be.used@nexgo.REMOVETHIS.de> wrote:
> 
>> On 2018-03-02 22:19, Steve Pope wrote:
> 
>>> Actually it's true.  A discrete-time signal is non-zero only at
>>> discrete points in time.  So if for example,
> 
>> I would even say that a discrete-time signal is only
>> *defined* at discrete points in time.
> 
> Well, that certainly would shoot the idea of adding two 
> differently-sampled signals together.

Exactly, I think the original paper refers to "continuous
time representation of a discrete time signal".

In this case, the signal can be represented as a train
of Diracs, hence addition is possible anywhere.

My Prof. at the University was very picky about such things... :-)

> You're right of course.

Thanks,

bye,

-- 

piergiorgio

On Friday, March 2, 2018 at 1:08:49 PM UTC-8, rr wrote:
> Hi Rick,
> 
> W dniu 02.03.2018 o 20:31, Richard (Rick) Lyons pisze:
> (...)
> > The paper you cited is interesting. On page 1810 of the paper the author wrote: "Since multiplexing of discrete-time signals is equivalent to addition,... ." That's not true, of course. We don't simply add the two discrete sequences' samples, we interleave the two sequences' samples. Perhaps the "addition" the author is referring to should be called "time-aligned addition" or "time-interleaved addition".
> > 
> 
> hm, THIS IS simply addition of the two sample sequences, but as sampling 
> moments of second one is shifted by Tck/2, they are upsampled by 2 in 
> order to make this operation possible (at 2 times higher output frequency).
> Well, if it is your point, "time-interleaved addition" is acceptable for 
> me. ;-)
> 
> Thanks for vigilance,
> 
> Roman Rumian

Hey Roman, It just "hit me". Isn't it true that the author's Eq. (19) only applies to periodic signals?  If that's true then his spectral example in Figure 6 would not be correct for real-world, unpredictable, information-carrying signals.  Am I missing something here?

[-Rick-]

On Friday, March 2, 2018 at 1:19:22 PM UTC-8, Steve Pope wrote:
> In article <p7cegr$9a8$1@node1.news.atman.pl>, rr  <rr@somwhere.com> wrote:
> 
> > The paper you cited is interesting. On page 1810 of the paper the
> > author wrote: "Since multiplexing of discrete-time signals is equivalent
> > to addition,... ." That's not true, of course. We don't simply add the
> > two discrete sequences' samples, we interleave the two sequences'
> > samples. Perhaps the "addition" the author is referring to should be
> > called "time-aligned addition" or "time-interleaved addition".
> 
> Actually it's true.  A discrete-time signal is non-zero only at
> discrete points in time.  So if for example,
> 
>             X(t) is zero for t not in (0, 2, 4, 6...)
>             Y(t) is zero for t not in (1, 3, 5, 7...)
> 
> Then Z(t) = X(t) + Y(t) where
> 
>        Z(t) = X(t) if t = 0, 2, 4, 6 ... , Y(t) otherwise.
> 
> So I would say addition and multiplexing are the same, so long as
> you haven't mangled the time axis.
> 
> Steve

Hi Steve.  I claim that multiplexing two sequences is not the same as adding two two sequences.  Multiplexing your above X(t) and Y(t) sequences yields

  Seq# 1 --> 0, 1, 2, 3, 4, 5, ... .

Adding your X(t) and Y(t) sequences produces:

  Seq# 2 --> 1, 5, 9, 13, 17, ... .

Seq# 1 and Seq# 2 are not equal to each other.

If you e-mailed your X(t) and Y(t) sequences to a DSP-practitioner friend and asked him to "add" the sequences and send the resulting "summed" sequence to you, he'd e-mail my above Seq# 2 to you.

[-Rick-]

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

>On Friday, March 2, 2018 at 1:19:22 PM UTC-8, Steve Pope wrote:

>> Actually it's true.  A discrete-time signal is non-zero only at
>> discrete points in time.  So if for example,

>>             X(t) is zero for t not in (0, 2, 4, 6...)
>>             Y(t) is zero for t not in (1, 3, 5, 7...)

>> Then Z(t) = X(t) + Y(t) where

>>        Z(t) = X(t) if t = 0, 2, 4, 6 ... , Y(t) otherwise.

>> So I would say addition and multiplexing are the same, so long as
>> you haven't mangled the time axis.

>Hi Steve.  I claim that multiplexing two sequences is not the same as
>adding two two sequences.  Multiplexing your above X(t) and Y(t)
>sequences yields
>
>  Seq# 1 --> 0, 1, 2, 3, 4, 5, ... .
>
>Adding your X(t) and Y(t) sequences produces:
>
>  Seq# 2 --> 1, 5, 9, 13, 17, ... .

You've completely lost me.  I claim that you get a sequence that
is equal to X(t) for t = 0, 2, 4, 6 ... and equal to Y(t) for 
t = 1, 3, 5, 7.

But, as has been pointed out, I departed from the definition of
"discrete time signal" in order to get to this.

S.