# Non-uniform Sampling of White Gaussian Noise??

Started by February 25, 2005
```Hi there,

I have modelled a system that non-uniformly samples a
bandpass signal. Unfortunately I am not allowed to transform
to the equivalent bandpass. Now I have the problem that
I should also model the samples, when bandpass filtered
White Gaussian Noise is sampled. For uniform sampling,
the variance of the Gaussian Variable of each tap equals
N_0B. How can I evaluate the variance of the noise on each
sample for non-uniform sampling? What do I have to consider.

Hope to get some help.

Thanks
MIMO

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```
```mimo wrote:
> Hi there,
>
> I have modelled a system that non-uniformly samples a
> bandpass signal. Unfortunately I am not allowed to transform
> to the equivalent bandpass. Now I have the problem that
> I should also model the samples, when bandpass filtered
> White Gaussian Noise is sampled. For uniform sampling,
> the variance of the Gaussian Variable of each tap equals
> N_0B. How can I evaluate the variance of the noise on each
> sample for non-uniform sampling? What do I have to consider.
>
> Hope to get some help.

I don't understand all of the question, let alone the answer. If you
were allowed to do it, what would "transform [a bandpass signal] to the
equivalent bandpass" mean?

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```"mimo" <florian.troesch@gmx.ch> wrote in message
news:gf-dnVkt050Fk4LfRVn-uQ@giganews.com...
> .Now I have the problem that
> I should also model the samples, when bandpass filtered
> White Gaussian Noise is sampled. For uniform sampling,
> the variance of the Gaussian Variable of each tap equals
> N_0B. How can I evaluate the variance of the noise on each
> sample for non-uniform sampling?

The variance should still be N_0B for all the samples.  With uniform
sampling (at rate B samples per second), the noise samples are
independent Gaussian random variables.  With non-uniform sampling
the noise samples will be correlated in general , and this should be
accounted for in the model.

The correlation coefficient between two samples of "band-limited
white noise" (of bandwith B and center frequency f_c) that are
spaced t seconds apart is sinc(Bt)cos(2 pi f_c t). Thus, if t is a
multiple of 1/B, the correlation is 0 (the result for uniform sampling
mentioned above).  If the spacing between your non-uniform
happens to be samples a multiple of 1/f_c, then you will also get
zero correlation.  (Remember that uncorrelated (jointly) Gaussian
random variables are independent).

Hope this helps.

```
```Dilip V. Sarwate wrote:

...

> The correlation coefficient between two samples of "band-limited
> white noise" (of bandwith B and center frequency f_c) that are
> spaced t seconds apart is sinc(Bt)cos(2 pi f_c t). Thus, if t is a
> multiple of 1/B, the correlation is 0 (the result for uniform sampling
> mentioned above).  If the spacing between your non-uniform
> happens to be samples a multiple of 1/f_c, then you will also get
> zero correlation.  (Remember that uncorrelated (jointly) Gaussian
> random variables are independent).

If I interpret that correctly, noise samples at 1/B are uncorrelated,
but they become correlated to some extent at slightly greater spacing. I
suspect any formulation that predicts greater correlation with
increasing time (or greater optical coherence with increasing length).
Can you enlighten me, or at least lay my doubt to rest?

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```"Jerry Avins" <jya@ieee.org> wrote in message
news:AJqdnYiSROUcKL3fRVn-iQ@rcn.net...
> Dilip V. Sarwate wrote:
>
>   ...
>
>> The correlation coefficient between two samples of "band-limited
>> white noise" (of bandwith B and center frequency f_c) that are
>> spaced t seconds apart is sinc(Bt)cos(2 pi f_c t). Thus, if t is a
>> multiple of 1/B, the correlation is 0 (the result for uniform sampling
>> mentioned above).  If the spacing between your non-uniform
>> happens to be samples a multiple of 1/f_c, then you will also get
>> zero correlation.  (Remember that uncorrelated (jointly) Gaussian
>> random variables are independent).
>
> If I interpret that correctly, noise samples at 1/B are uncorrelated, but
> they become correlated to some extent at slightly greater spacing. I
> suspect any formulation that predicts greater correlation with increasing
> time (or greater optical coherence with increasing length). Can you
> enlighten me, or at least lay my doubt to rest?

I agree that it is intuitively suspect that the correlation increases at
spacing greater than 1/B, but if we want our model to have ideal
bandlimited white noise, then we have to put up with the fact that
the autocorrelation function of the bandlimited noise has support
(-infty, +infty).  Fortunately for intuition, for spacing greater than 1/B,
we are outside the main lobe of sinc(Bt) and thus the correlation
values are small compared to the correlation values for spacings
that are significantly smaller than 1/B.  Recall that the sinc function
has maximum magnitude less than 0.25 for arguments greater than 1...

Hope this helps.

```
```Dilip V. Sarwate wrote:

...

> I agree that it is intuitively suspect that the correlation increases at
> spacing greater than 1/B, but if we want our model to have ideal
> bandlimited white noise, then we have to put up with the fact that
> the autocorrelation function of the bandlimited noise has support
> (-infty, +infty).  Fortunately for intuition, for spacing greater than 1/B,
> we are outside the main lobe of sinc(Bt) and thus the correlation
> values are small compared to the correlation values for spacings
> that are significantly smaller than 1/B.  Recall that the sinc function
> has maximum magnitude less than 0.25 for arguments greater than 1...
>
> Hope this helps.

It gives me hope that for real band-limited noise (as opposed to
brick-wall filtered noise), intuition is not overturned. I won't attempt
the analysis, though.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```Jerry Avins wrote:
> Dilip V. Sarwate wrote:
>
>   ...
>
>> I agree that it is intuitively suspect that the correlation increases at
>> spacing greater than 1/B, but if we want our model to have ideal
>> bandlimited white noise, then we have to put up with the fact that
>> the autocorrelation function of the bandlimited noise has support
>> (-infty, +infty).  Fortunately for intuition, for spacing greater than
>> 1/B,
>> we are outside the main lobe of sinc(Bt) and thus the correlation
>> values are small compared to the correlation values for spacings
>> that are significantly smaller than 1/B.  Recall that the sinc function
>> has maximum magnitude less than 0.25 for arguments greater than 1...
>>
>> Hope this helps.
>
>
> It gives me hope that for real band-limited noise (as opposed to
> brick-wall filtered noise), intuition is not overturned. I won't attempt
> the analysis, though.
>
> Jerry

No, it is observable in real life with real filters.  The correlation
drops off with time, but there are certain intervals where the
correlation is zero.  Look in Burdic's book on Underwater Acoustics if
you want to see a derivation.
```
```>Jerry Avins wrote:
>> Dilip V. Sarwate wrote:
>>
>>   ...
>>
>>> I agree that it is intuitively suspect that the correlation increases
at
>>> spacing greater than 1/B, but if we want our model to have ideal
>>> bandlimited white noise, then we have to put up with the fact that
>>> the autocorrelation function of the bandlimited noise has support
>>> (-infty, +infty).  Fortunately for intuition, for spacing greater than

>>> 1/B,
>>> we are outside the main lobe of sinc(Bt) and thus the correlation
>>> values are small compared to the correlation values for spacings
>>> that are significantly smaller than 1/B.  Recall that the sinc
function
>>> has maximum magnitude less than 0.25 for arguments greater than 1...
>>>
>>> Hope this helps.
>>
>>
>> It gives me hope that for real band-limited noise (as opposed to
>> brick-wall filtered noise), intuition is not overturned. I won't
attempt
>> the analysis, though.
>>
>> Jerry
>
>No, it is observable in real life with real filters.  The correlation
>drops off with time, but there are certain intervals where the
>correlation is zero.  Look in Burdic's book on Underwater Acoustics if
>you want to see a derivation.
>

Hi,
So, if I am right a can sample a bandpass filtered white Gaussian
noise with "carrier frequency" f_c = mB (m=integer) at time
instances:
t = p/2f_c (inphase) and
to achieve a set of i.i.d samples.

Cheers

This message was sent using the Comp.DSP web interface on
www.DSPRelated.com
```
```>>Jerry Avins wrote:
>>> Dilip V. Sarwate wrote:
>>>
>>>   ...
>>>
>>>> I agree that it is intuitively suspect that the correlation
increases
>at
>>>> spacing greater than 1/B, but if we want our model to have ideal
>>>> bandlimited white noise, then we have to put up with the fact that
>>>> the autocorrelation function of the bandlimited noise has support
>>>> (-infty, +infty).  Fortunately for intuition, for spacing greater
than
>
>>>> 1/B,
>>>> we are outside the main lobe of sinc(Bt) and thus the correlation
>>>> values are small compared to the correlation values for spacings
>>>> that are significantly smaller than 1/B.  Recall that the sinc
>function
>>>> has maximum magnitude less than 0.25 for arguments greater than 1...
>>>>
>>>> Hope this helps.
>>>
>>>
>>> It gives me hope that for real band-limited noise (as opposed to
>>> brick-wall filtered noise), intuition is not overturned. I won't
>attempt
>>> the analysis, though.
>>>
>>> Jerry
>>
>>No, it is observable in real life with real filters.  The correlation
>>drops off with time, but there are certain intervals where the
>>correlation is zero.  Look in Burdic's book on Underwater Acoustics if
>>you want to see a derivation.
>>
>
>
>Hi,
>So, if I am right a can sample a bandpass filtered white Gaussian
>noise with "carrier frequency" f_c = mB (m=integer) at time
>instances:
> t = p/2f_c (inphase) and
>to achieve a set of i.i.d samples.
>
>Cheers
>
>
>
>This message was sent using the Comp.DSP web interface on
>www.DSPRelated.com
>

Hi again,

just do be sure:
I do quadrature sampling with an interpolant sinc(Bt)cos(2 pi f_c t)
and B the bandwidth of the signal. I sample twice with 1/B, i.e.
f(p/B) [inphase] and f(p/B+k) [quadrature] with k=1/f_c and
p=...,-1,0,1,2...... . Am I right, if a say that I get uncorrelated
samples
only if the sampled signal f is half integer position, i.e. f_c =
(m+1)/2B
with m any integer?

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```