# Beating Nyquist?

Started by July 25, 2007
```Friends,

just now I stumbled upon this webpage:

http://www.edi.lv/dasp-web/

Specifically, in this chapter

http://www.edi.lv/dasp-web/sec-5.htm

they state that they can sample a 1.2GHz signal using a pseudo-random
sampling instants with an average rate of 80MHz (in the last line of
section "5.2 Aliasing, how to avoid it").

I know that for nonuniform sampling, a generalization of the Sampling
Theorem has been proved [1], which states that a bandlimited signal
maybe reconstructed from nonuniformly spaced samples if the "average"
sampling rate is higher than twice the bandwidth of the signal.

This doesn't immediately contradict the claim above - it just says
that if the average sampling rate exceeds a certain limit, it can be
shown that the samples are sufficient for reconstruction. It might
well be that if the average rate is below the limit, reconstruction is
still possible. However, the claim still seems like magic to me,
specifically in the light that the sampled signals underlie no
restrictions (apart from the 1.2GHz bandwidth).

Regards,
Andor

[1] F. J. Beutler, "Error-free recovery of signals from irregularly
spaced
samples," SIAM Rev., vol. 8, pp. 328-335, July 1966.

```
```On 25 Jul, 11:24, Andor <andor.bari...@gmail.com> wrote:
> Friends,
>
> just now I stumbled upon this webpage:
>
> http://www.edi.lv/dasp-web/
>
> Specifically, in this chapter
>
> http://www.edi.lv/dasp-web/sec-5.htm
>
> they state that they can sample a 1.2GHz signal using a pseudo-random
> sampling instants with an average rate of 80MHz (in the last line of
> section "5.2 Aliasing, how to avoid it").
>
> I know that for nonuniform sampling, a generalization of the Sampling
> Theorem has been proved [1], which states that a bandlimited signal
> maybe reconstructed from nonuniformly spaced samples if the "average"
> sampling rate is higher than twice the bandwidth of the signal.
>
> This doesn't immediately contradict the claim above - it just says
> that if the average sampling rate exceeds a certain limit, it can be
> shown that the samples are sufficient for reconstruction. It might
> well be that if the average rate is below the limit, reconstruction is
> still possible. However, the claim still seems like magic to me,
> specifically in the light that the sampled signals underlie no
> restrictions (apart from the 1.2GHz bandwidth).
>
>
> Regards,
> Andor
>
> [1] F. J. Beutler, "Error-free recovery of signals from irregularly
> spaced
> samples," SIAM Rev., vol. 8, pp. 328-335, July 1966.

I don't like "random" algorithms which work "on average."
Maybe the claim that random sampling works "on average"
can be formally justified; I don't have the competence
to comment on such a *formal* claim either way.

>From a practical perspective, assume you have a signal
x(t) which comprises one transient of the type of
signal I prefer to call an "energy signal", i.e.

+inf
integral |x(t)|^2 dt < inf                           [1]
-inf

AS you know, one of the consequences of [1] is that
the magnitude of x(t) is "significantly larger than 0,
|x(t)| > eps" only inside some finite domain, say,
a < t < b.

This means that you *can* get the "random" samplingh to
work, beating Nyquist, provided most of the sampling
points are "spent" inside the domain [a,b]. On the
other hand, you are screwed if the converse happens
to be the case, that no sampling instances are inside
the domain [a,b]. This is a perfectly valid case, as
the properties of the random sampling scheme is
defined on average, not per instance.

Basically, by employing a randomized sampling scheme
you exchange a global scheme with well-defined,
well understood, guaranteed properties, for a
random scheme which might be better "on average"
but where every the worst case scenario is that
you loose the properties of the signal you want.

It's a matter of economy and damage control: Is it
acceptable that the properties of a particular
instance of a sampled signal can not be guaranteed?
application require deterministic, if poor,
performance characteristics?

Rune

```
```On Jul 25, 5:24 am, Andor <andor.bari...@gmail.com> wrote:
> Friends,
>
> just now I stumbled upon this webpage:
>
> http://www.edi.lv/dasp-web/
>
> Specifically, in this chapter
>
> http://www.edi.lv/dasp-web/sec-5.htm
>
> they state that they can sample a 1.2GHz signal using a pseudo-random
> sampling instants with an average rate of 80MHz (in the last line of
> section "5.2 Aliasing, how to avoid it").

I agree with Rune's comments too, but the thing that kind of sticks
out to me is that you're throwing away most of the basic tools of DSP
(like linear filtering), as they have the implicit assumption that the
data is periodically sampled. I've never studied nonuniform sampling
techniques, so there may be a way around this, but it seems like you
would have to exert a lot more effort to account for the times at
which each sample was taken.

Jason

```
```cincydsp@gmail.com wrote in news:1185365183.924874.258340

> On Jul 25, 5:24 am, Andor <andor.bari...@gmail.com> wrote:
>> Friends,
>>
>> just now I stumbled upon this webpage:
>>
>> http://www.edi.lv/dasp-web/
>>
>> Specifically, in this chapter
>>
>> http://www.edi.lv/dasp-web/sec-5.htm
>>
>> they state that they can sample a 1.2GHz signal using a pseudo-random
>> sampling instants with an average rate of 80MHz (in the last line of
>> section "5.2 Aliasing, how to avoid it").
>
> I agree with Rune's comments too, but the thing that kind of sticks
> out to me is that you're throwing away most of the basic tools of DSP
> (like linear filtering), as they have the implicit assumption that the
> data is periodically sampled. I've never studied nonuniform sampling
> techniques, so there may be a way around this, but it seems like you
> would have to exert a lot more effort to account for the times at
> which each sample was taken.
>
> Jason
>
>

If the signal can be reconstructed, you can always resample, so you throw
away nothing (but we can argue about the "if")

--
Scott
```
```On Jul 25, 5:24 am, Andor <andor.bari...@gmail.com> wrote:
> Friends,
>
> just now I stumbled upon this webpage:
>
> http://www.edi.lv/dasp-web/
>
> Specifically, in this chapter
>
> http://www.edi.lv/dasp-web/sec-5.htm
>
> they state that they can sample a 1.2GHz signal using a pseudo-random
> sampling instants with an average rate of 80MHz (in the last line of
> section "5.2 Aliasing, how to avoid it").
>
> I know that for nonuniform sampling, a generalization of the Sampling
> Theorem has been proved [1], which states that a bandlimited signal
> maybe reconstructed from nonuniformly spaced samples if the "average"
> sampling rate is higher than twice the bandwidth of the signal.
>
> This doesn't immediately contradict the claim above - it just says
> that if the average sampling rate exceeds a certain limit, it can be
> shown that the samples are sufficient for reconstruction. It might
> well be that if the average rate is below the limit, reconstruction is
> still possible. However, the claim still seems like magic to me,
> specifically in the light that the sampled signals underlie no
> restrictions (apart from the 1.2GHz bandwidth).
>
>
> Regards,
> Andor
>
> [1] F. J. Beutler, "Error-free recovery of signals from irregularly
> spaced
> samples," SIAM Rev., vol. 8, pp. 328-335, July 1966.

It says actually..
"... for fully digital analysis  of RF signals in Time, Frequency and
Modulation Domains in the frequency range from dc up to 1.2 GHz...."

So the RF carrier frequenies can be DC to 1.2GHz but it does not say
anything about the instantaneous bandwidth of those signals....   I
suspect this is simple sub-sampling...

Mark

```
```Rune Allnor wrote:
> I don't like "random" algorithms which work "on average."
> Maybe the claim that random sampling works "on average"
> can be formally justified; I don't have the competence
> to comment on such a *formal* claim either way.

Aren't algorithms that work statistically the very essence of signal
processing? :-\

Steve
```
```On Wed, 25 Jul 2007 06:27:47 -0700, Mark wrote:

> On Jul 25, 5:24 am, Andor <andor.bari...@gmail.com> wrote:
>> Friends,
>>
>> just now I stumbled upon this webpage:
>>
>> http://www.edi.lv/dasp-web/
>>
>> Specifically, in this chapter
>>
>> http://www.edi.lv/dasp-web/sec-5.htm
>>
>> they state that they can sample a 1.2GHz signal using a pseudo-random
>> sampling instants with an average rate of 80MHz (in the last line of
>> section "5.2 Aliasing, how to avoid it").
>>
>> I know that for nonuniform sampling, a generalization of the Sampling
>> Theorem has been proved [1], which states that a bandlimited signal
>> maybe reconstructed from nonuniformly spaced samples if the "average"
>> sampling rate is higher than twice the bandwidth of the signal.
>>
>> This doesn't immediately contradict the claim above - it just says
>> that if the average sampling rate exceeds a certain limit, it can be
>> shown that the samples are sufficient for reconstruction. It might
>> well be that if the average rate is below the limit, reconstruction is
>> still possible. However, the claim still seems like magic to me,
>> specifically in the light that the sampled signals underlie no
>> restrictions (apart from the 1.2GHz bandwidth).
>>
>>
>> Regards,
>> Andor
>>
>> [1] F. J. Beutler, "Error-free recovery of signals from irregularly
>> spaced
>> samples," SIAM Rev., vol. 8, pp. 328-335, July 1966.
>
> It says actually..
> "... for fully digital analysis  of RF signals in Time, Frequency and
> Modulation Domains in the frequency range from dc up to 1.2 GHz...."
>
> So the RF carrier frequenies can be DC to 1.2GHz but it does not say
> anything about the instantaneous bandwidth of those signals....   I
> suspect this is simple sub-sampling...
>
> Mark

If that is, indeed, what they mean, this could be a dandy way to take a
signal at a single frequency and smear it over a spectrum so that it gets
lost in the noise.  It wouldn't really be _anti_ aliasing, but it may
still be a good thing.

--
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
```
```On 25 Jul, 15:57, Steve Underwood <ste...@dis.org> wrote:
> Rune Allnor wrote:
> > I don't like "random" algorithms which work "on average."
> > Maybe the claim that random sampling works "on average"
> > can be formally justified; I don't have the competence
> > to comment on such a *formal* claim either way.
>
> Aren't algorithms that work statistically the very essence of signal
> processing? :-\

No. They play a large part -- adaptive DSP and model-based
DSP based on parameter estimation, come to mind -- but all
suffer from the same weakness: They only work to whatever
extent the statistical model fit reality. Or, alternatively,
to the extent the signal fits the model.

As long as there is a certain degree of match between signal
and statistsical model, OK, the algorithms work. The problems
occur when the mismatch becomes large and go undetected.
Are these instances acceptable "on average" or does the
application require *guaranteed* worst-case behaviour?

Rune

```
```"Mark" <makolber@yahoo.com> wrote in message
> It says actually..
> "... for fully digital analysis  of RF signals in Time, Frequency and
> Modulation Domains in the frequency range from dc up to 1.2 GHz...."
>
> So the RF carrier frequenies can be DC to 1.2GHz but it does not say
> anything about the instantaneous bandwidth of those signals....   I
> suspect this is simple sub-sampling...
>
> Mark
>

Yes, it does say that but I'm persuaded to believe that he meant the
bandwidth.

Without studying it too hard it looks to me like the proposed approach tends
to spread the spectrum of the aliases - generating what looks more like
noise.  Thus creating a tradeoff between having aliases and noise.  Seems
like it would depends on lot on the nature of the signal for it to work.
Periodic waveforms might work fine while more random waveforms might not -
because how do you differentiate between "signal" and "noise" in that latter
case?

I'm not endorsing the claim, just pondering it.

Knowing the sample times precisely only positions the samples for
reconstruction.  Pure sinc reconstruction (or lowpass filtering of
precisely-placed samples) works fine for the fundamental components.  But,
for higher frequency components that would alias, the dithering might spread
their spectrum, would't it?

Here's another way of looking at it - a thought experiment:
Assume a really high sample rate that is equivalent to the temporal
resolution of the samples to be taken "randomly".  Surely there is a time
grid that must underlie the method - for example, on which to place the
reconstructing waveforms (e.g. sincs).
This creates a sample rate that is (by definition) "high enough" for the
actual bandwidth and Nyquist.

Sample at this rate.  You know what the spectrum will look like if you know
the signal being sampled.

Then, decimate but randomly.  That is, decimate (but not regularly) so that
the average sample rate is what you wanted in the first place.  Note that
this is a nonlinear operation so familiar methods of analysis don't work.

Now, use a test signal that is above the "new" fs/2.
It seems to me that the randomized sample points will modulate the heck out
of a higher-frequency sinusoid - turning its samples into what looks like
random noise - because of the rather drastic phase hops between samples.

Thus, no "alias" tonals in the result - but, more noise.

If you switch the input to a test signal that is below the "new" fs/2 then
there will be no modulation (spectral spreading) because of the careful
temporal registration of the samples and the reconstructing sincs.  Well,
that's an arm-waving description and I'm sure others can do a better, more
thoughtful, job of explaining why this might be.

I think that's what the author is driving at without being very clear about
it.
So, no free lunch, and no cheating Nyquist, just a method to trade tonal
aliases into added noise with a nonlinear step in the "contructed" process
above.

*****

OK - I thought about it some more...
The "nonlinear" step above can be replaced with a linear step.
Create the desired sample points in time.  By definition they align on the
fine temporal grid.  Note that the gross view of these samples looks like
they are regularly spaced.  But they aren't of course - on purpose.

Because the temporal location will be dithered at decimation, using a new
set of sample points, there is a spectrum of this lower-frequency unit
sample train that's not the typical picket fence repeating at fs.  Rather,
there's a broader clump at fs - the width determined by how much temporal
deviation is built in between the samples.

Now, multiply the high-frequency signal samples by the lower-frequency
dithered unit samples.  This linear operation zeros out all the unwanted
samples from the higher fs sampled set and results in a set of signal
samples with dithered sample times.
At the same time, this convolves their spectra.

I don't know.... there's something about the reconstruction that should work
for the signals below fs/2 just fine.  Since the reconstruction steps are
linear, it should be easy to figure out.

Fred

```
```Rune Allnor wrote:
> On 25 Jul, 15:57, Steve Underwood <ste...@dis.org> wrote:
>> Rune Allnor wrote:
>>> I don't like "random" algorithms which work "on average."
>>> Maybe the claim that random sampling works "on average"
>>> can be formally justified; I don't have the competence
>>> to comment on such a *formal* claim either way.
>> Aren't algorithms that work statistically the very essence of signal
>> processing? :-\
>
> No. They play a large part -- adaptive DSP and model-based
> DSP based on parameter estimation, come to mind -- but all
> suffer from the same weakness: They only work to whatever
> extent the statistical model fit reality. Or, alternatively,
> to the extent the signal fits the model.
>
> As long as there is a certain degree of match between signal
> and statistsical model, OK, the algorithms work. The problems
> occur when the mismatch becomes large and go undetected.
> Are these instances acceptable "on average" or does the
> application require *guaranteed* worst-case behaviour?

Pretty much anything in comms is statistical.
Pretty much anything in sensing is statistical.

I thought you'd worked in acoustic sensing. That's a very statistical area.

Steve
```