# Sampling a signal corrupted by AWGN

Started by November 13, 2008
```Let us assume that we have a bandlimited signal (with maximum frequency
f_n) corrupted by additive white Gaussian noise. Before we can sample this
signal, we pass the signal through an ideal anti-alias filter with cut-off
frequency f_c >=f_n to avoid noise aliasing. The output of the
anti-aliasing filter is fed into a matched filter matched to the symbol
rate, (1/T)>=2f_n (i used f_n here because we intend to keep useful signal
spectrum intact). As a prticular case we let I/T=10f_n and f_c=2f_n. My
question is what will be the impact of sampling on the white noise in this
case? Will it remain white? Will it not be the case that (bandlimited)
noise will get oversampled so that power spectral density of noise will no
more be flat over -pi and pi?
My understanding is that noise is white (theoretically) in discrete-time
domain if its PSD is flat over -pi to pi and hence over all frequencies in
the discrete-time domain. And the noise will be colored if it is not flat
over -pi to pi.

Secondly suppose we have signal-plus-noise (in discrete-time domain) such
that noise PSD is flat over -pi to pi whereas the spectrum of the signal is
non-zero over -pi/M<omega<pi/M where M is a positive integer. We upsample
signal-plus-noise by factor N. My question is what will be the impact of
upsampling on PSD of noise. Will it be magnitude and frequency scaled?
(Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald W.
Schafer)

RIMalhi

```
```On Nov 13, 6:12&#2013266080;pm, "RIMalhi" <m4ma...@yahoo.com> wrote:
> Let us assume that we have a bandlimited signal (with maximum frequency
> f_n) corrupted by additive white Gaussian noise. Before we can sample this
> signal, we pass the signal through an ideal anti-alias filter with cut-off
> frequency f_c >=f_n to avoid noise aliasing. The output of the
> anti-aliasing filter is fed into a matched filter matched to the symbol
> rate, (1/T)>=2f_n (i used f_n here because we intend to keep useful signal
> spectrum intact). As a prticular case we let I/T=10f_n and f_c=2f_n. My

I'm _assuming_ by symbol rate you mean sampling rate. Correct me if
that's not right. To get my bearings, we have

0 <= f_n <= f_c <= f_s

where f_s is the sampling frequency. With your numbers, normalized to
f_n = 1,

0 <= 1 <= 2  <= 10

So the Nyquist limit is at 5, therefore (- pi...+pi ) in discrete
domain corresponds to -5 to +5 in original continuous frequency
domain.

> question is what will be the impact of sampling on the white noise in this
> case? Will it remain white? Will it not be the case that (bandlimited)
> noise will get oversampled so that power spectral density of noise will no
> more be flat over -pi and pi?

It *was* flat until you filtered it . But then you filtered it. So you
now have (ideally) non-zero flat PSD from (-2 to 2). But you're
sampling with Nyquist mapped to (-5,5). So I'd guess that your PSD in
the discrete domain would be non-zero from (-2/5 pi ... +2/5 pi). That
doesn't sound like what you want to call "white" in the discrete
domain.

> My understanding is that noise is white (theoretically) in discrete-time
> domain if its PSD is flat over -pi to pi and hence over all frequencies in
> the discrete-time domain. And the noise will be colored if it is not flat
> over -pi to pi.

So it sounds like your filter coloured it, then.

>
> Secondly suppose we have signal-plus-noise (in discrete-time domain) such
> that noise PSD is flat over -pi to pi whereas the spectrum of the signal is
> non-zero over -pi/M<omega<pi/M where M is a positive integer. We upsample
> signal-plus-noise by factor N. My question is what will be the impact of
> upsampling on PSD of noise. Will it be magnitude and frequency scaled?
> (Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald W.
> Schafer)

It's like deja vu all over again.

You start with a signal that's "full" of noise, pregnant with entropy
for the entire omega spectrum. Then you stuff in some zeros
(modulation), then you filter (your sinc filter).  The filter is the
hint here. You have more samples now but you've also rescaled omega,
so the noise now looks like it lives only in (-pi/N to pi/N).

The amplitude scaling idea is really tripping you up. Look at it this
way: Imagine your original signal was DC:

1,1,1,1,1,...

Now zero-stuff (N=4)

1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0...

If you filter the stuffed signal with an ideal *unity-gain* filter,

0.25, 0.25, 0.25, 0.25, .....

This is where the amplitude is lost. You either keep track of it in
your head as a loss (i.e a fudge factor of 0.25) , or you redefine
your upsampling filter to have a gain of 4 buried in it somewhere to
make the upsampling unity gain as far as signal amplitude goes.  It's
all in our heads anyway :-)

>
>
> RIMalhi

Maybe :-)

- Kenn
```
```On Thu, 13 Nov 2008 17:12:57 -0600, RIMalhi wrote:

> Let us assume that we have a bandlimited signal (with maximum frequency
> f_n) corrupted by additive white Gaussian noise. Before we can sample
> this signal, we pass the signal through an ideal anti-alias filter with
> cut-off frequency f_c >=f_n to avoid noise aliasing. The output of the
> anti-aliasing filter is fed into a matched filter matched to the symbol
> rate, (1/T)>=2f_n (i used f_n here because we intend to keep useful
> signal spectrum intact). As a prticular case we let I/T=10f_n and
> f_c=2f_n. My question is what will be the impact of sampling on the
> white noise in this case? Will it remain white? Will it not be the case
> that (bandlimited) noise will get oversampled so that power spectral
> density of noise will no more be flat over -pi and pi?

At the point that you are sampling it is no longer white -- it is
colored, because you have filtered it.  If you have filtered it to have a
bandwidth strictly less than the Nyquist rate, then it'll have the same
spectrum in the sampled-time domain.

You could construct a filter that rolls off symmetrically around the
Nyquist rate; were you to do this then the resulting sampled-time noise
would be white, even though sampled-time "white" means something
different from continuous-time "white".

> My understanding is that noise is white (theoretically) in discrete-time
> domain if its PSD is flat over -pi to pi and hence over all frequencies
> in the discrete-time domain. And the noise will be colored if it is not
> flat over -pi to pi.

Correct.
>
> Secondly suppose we have signal-plus-noise (in discrete-time domain)
> such that noise PSD is flat over -pi to pi whereas the spectrum of the
> signal is non-zero over -pi/M<omega<pi/M where M is a positive integer.
> We upsample signal-plus-noise by factor N. My question is what will be
> the impact of upsampling on PSD of noise. Will it be magnitude and
> frequency scaled? (Ref: Discrete-time signal processing by Alan V.
> Oppenheim, Ronald W. Schafer)
>
That depends on how you upsample.  If you upsample by keeping all the
original samples and filling in the spaces with N-1 long strings of ones,
then your resulting noise will be white, although it will no longer be
stationary.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" gives you just what it says.
See details at http://www.wescottdesign.com/actfes/actfes.html
```
```Kenn Wrote

>On Nov 13, 6:12=A0pm, "RIMalhi" <m4ma...@yahoo.com> wrote:
>> Let us assume that we have a bandlimited signal (with maximum
frequency
>> f_n) corrupted by additive white Gaussian noise. Before we can sample
thi=
>s
>> signal, we pass the signal through an ideal anti-alias filter with
cut-of=
>f
>> frequency f_c >=3Df_n to avoid noise aliasing. The output of the
>> anti-aliasing filter is fed into a matched filter matched to the
symbol
>> rate, (1/T)>=3D2f_n (i used f_n here because we intend to keep useful
sig=
>nal
>> spectrum intact). As a prticular case we let I/T=3D10f_n and
f_c=3D2f_n. =
>My
>
>I'm _assuming_ by symbol rate you mean sampling rate. Correct me if
>that's not right. To get my bearings, we have
>
>0 <= f_n <= f_c <= f_s
>
>where f_s is the sampling frequency. With your numbers, normalized to
>f_n =1,
>
>0 <= 1 <= 2  <= 10
>
>So the Nyquist limit is at 5, therefore (- pi...+pi ) in discrete
>domain corresponds to -5 to +5 in original continuous frequency
>domain.
>
>> question is what will be the impact of sampling on the white noise in
thi=
>s
>> case? Will it remain white? Will it not be the case that (bandlimited)
>> noise will get oversampled so that power spectral density of noise will
n=
>o
>> more be flat over -pi and pi?
>
>It *was* flat until you filtered it . But then you filtered it. So you
>now have (ideally) non-zero flat PSD from (-2 to 2). But you're
>sampling with Nyquist mapped to (-5,5). So I'd guess that your PSD in
>the discrete domain would be non-zero from (-2/5 pi ... +2/5 pi). That
>doesn't sound like what you want to call "white" in the discrete
>domain.
>
>> My understanding is that noise is white (theoretically) in
discrete-time
>> domain if its PSD is flat over -pi to pi and hence over all frequencies
i=
>n
>> the discrete-time domain. And the noise will be colored if it is not
flat
>> over -pi to pi.
>
>So it sounds like your filter coloured it, then.
>
>>
>> Secondly suppose we have signal-plus-noise (in discrete-time domain)
such
>> that noise PSD is flat over -pi to pi whereas the spectrum of the
signal =
>is
>> non-zero over -pi/M<omega<pi/M where M is a positive integer. We
upsample
>> signal-plus-noise by factor N. My question is what will be the impact
of
>> upsampling on PSD of noise. Will it be magnitude and frequency scaled?
>> (Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald W.
>> Schafer)
>
>It's like deja vu all over again.
>
>You start with a signal that's "full" of noise, pregnant with entropy
>for the entire omega spectrum. Then you stuff in some zeros
>(modulation), then you filter (your sinc filter).  The filter is the
>hint here. You have more samples now but you've also rescaled omega,
>so the noise now looks like it lives only in (-pi/N to pi/N).
>
>The amplitude scaling idea is really tripping you up. Look at it this
>way: Imagine your original signal was DC:
>
>1,1,1,1,1,...
>
>Now zero-stuff (N=3D4)
>
>1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0...
>
>If you filter the stuffed signal with an ideal *unity-gain* filter,
>
>0.25, 0.25, 0.25, 0.25, .....
>
>This is where the amplitude is lost. You either keep track of it in
>your head as a loss (i.e a fudge factor of 0.25) , or you redefine
>your upsampling filter to have a gain of 4 buried in it somewhere to
>make the upsampling unity gain as far as signal amplitude goes.  It's
>all in our heads anyway :-)

Hi Kenn,
We have flat spectrum in the frequency domain. But we do not stuff any
zeros in the flat spectrum. We stff zeros in time domain and are seeking
its impact on the spectrum. We know that contraction in time domain results
in expansion in frequency domain and vice versa. So when we upsample a
signal in the time domain, we are in fact expanding it. Therefore, in
frequency domain we should observe an equal contraction in the spectrum.
What confuses me is this: Some people (e.g., look at url
http://sipc.eecs.berkeley.edu/ee123/ee123handoutPSD.pdf) suggest that
after upsampling (or downsampling) white noise remains white. By theory,
when we stuff time domain signal corrupted by noise with zeros, we should
observe contraction of the spectrum of the signal and noise (noise is
additive and is independent of the signal!). Before upsampling, the noise
had flat PSD=N_0 over -pi to pi. If the suggestion that noise remains white
after upsampling is TRUE, then there must not be any change in both
magnitude of PSD of noise and the frequency. Why? Because for noise to be
white, its PSD must remain flat over -pi to pi which requires that
Upsampling must not cause any contraction in the spectrum of noise.

And if that suggestion is NOT true, white noise should be colored after
upsampling!

So my question is whether the noise remains white or becomes colored after
Upsampling?

And regarding Tim's explanation:

>
> Secondly suppose we have signal-plus-noise (in discrete-time domain)
> such that noise PSD is flat over -pi to pi whereas the spectrum of the
> signal is non-zero over -pi/M<omega<pi/M where M is a positive integer.
> We upsample signal-plus-noise by factor N. My question is what will be
> the impact of upsampling on PSD of noise. Will it be magnitude and
> frequency scaled? (Ref: Discrete-time signal processing by Alan V.
> Oppenheim, Ronald W. Schafer)
>
That depends on how you upsample.  If you upsample by keeping all the
original samples and filling in the spaces with N-1 long strings of ones,

then your resulting noise will be white, although it will no longer be
stationary.

Yes Tim,
we keep original samples and stuff N-1 zeros between two consecutive
samples. You said that noise will no more be stationary which is something
confusing me. Could you please explain a bit?

Thanks,

```
```RIMalhi wrote:
> Kenn Wrote
>
>> On Nov 13, 6:12=A0pm, "RIMalhi" <m4ma...@yahoo.com> wrote:
>>> Let us assume that we have a bandlimited signal (with maximum
> frequency
>>> f_n) corrupted by additive white Gaussian noise. Before we can sample
> thi=
>> s
>>> signal, we pass the signal through an ideal anti-alias filter with
> cut-of=
>> f
>>> frequency f_c >=3Df_n to avoid noise aliasing. The output of the
>>> anti-aliasing filter is fed into a matched filter matched to the
> symbol
>>> rate, (1/T)>=3D2f_n (i used f_n here because we intend to keep useful
> sig=
>> nal
>>> spectrum intact). As a prticular case we let I/T=3D10f_n and
> f_c=3D2f_n. =
>> My
>>
>> I'm _assuming_ by symbol rate you mean sampling rate. Correct me if
>> that's not right. To get my bearings, we have
>>
>> 0 <= f_n <= f_c <= f_s
>>
>> where f_s is the sampling frequency. With your numbers, normalized to
>> f_n =1,
>>
>> 0 <= 1 <= 2  <= 10
>>
>> So the Nyquist limit is at 5, therefore (- pi...+pi ) in discrete
>> domain corresponds to -5 to +5 in original continuous frequency
>> domain.
>>
>>> question is what will be the impact of sampling on the white noise in
> thi=
>> s
>>> case? Will it remain white? Will it not be the case that (bandlimited)
>>> noise will get oversampled so that power spectral density of noise will
> n=
>> o
>>> more be flat over -pi and pi?
>> It *was* flat until you filtered it . But then you filtered it. So you
>> now have (ideally) non-zero flat PSD from (-2 to 2). But you're
>> sampling with Nyquist mapped to (-5,5). So I'd guess that your PSD in
>> the discrete domain would be non-zero from (-2/5 pi ... +2/5 pi). That
>> doesn't sound like what you want to call "white" in the discrete
>> domain.
>>
>>> My understanding is that noise is white (theoretically) in
> discrete-time
>>> domain if its PSD is flat over -pi to pi and hence over all frequencies
> i=
>> n
>>> the discrete-time domain. And the noise will be colored if it is not
> flat
>>> over -pi to pi.
>> So it sounds like your filter coloured it, then.
>>
>>> Secondly suppose we have signal-plus-noise (in discrete-time domain)
> such
>>> that noise PSD is flat over -pi to pi whereas the spectrum of the
> signal =
>> is
>>> non-zero over -pi/M<omega<pi/M where M is a positive integer. We
> upsample
>>> signal-plus-noise by factor N. My question is what will be the impact
> of
>>> upsampling on PSD of noise. Will it be magnitude and frequency scaled?
>>> (Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald W.
>>> Schafer)
>> It's like deja vu all over again.
>>
>> You start with a signal that's "full" of noise, pregnant with entropy
>> for the entire omega spectrum. Then you stuff in some zeros
>> (modulation), then you filter (your sinc filter).  The filter is the
>> hint here. You have more samples now but you've also rescaled omega,
>> so the noise now looks like it lives only in (-pi/N to pi/N).
>>
>> The amplitude scaling idea is really tripping you up. Look at it this
>> way: Imagine your original signal was DC:
>>
>> 1,1,1,1,1,...
>>
>> Now zero-stuff (N=3D4)
>>
>> 1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0...
>>
>> If you filter the stuffed signal with an ideal *unity-gain* filter,
>>
>> 0.25, 0.25, 0.25, 0.25, .....
>>
>> This is where the amplitude is lost. You either keep track of it in
>> your head as a loss (i.e a fudge factor of 0.25) , or you redefine
>> your upsampling filter to have a gain of 4 buried in it somewhere to
>> make the upsampling unity gain as far as signal amplitude goes.  It's
>> all in our heads anyway :-)
>
>
>
> Hi Kenn,
> We have flat spectrum in the frequency domain. But we do not stuff any
> zeros in the flat spectrum. We stff zeros in time domain and are seeking
> its impact on the spectrum. We know that contraction in time domain results
> in expansion in frequency domain and vice versa. So when we upsample a
> signal in the time domain, we are in fact expanding it. Therefore, in
> frequency domain we should observe an equal contraction in the spectrum.
> What confuses me is this: Some people (e.g., look at url
> http://sipc.eecs.berkeley.edu/ee123/ee123handoutPSD.pdf) suggest that
> after upsampling (or downsampling) white noise remains white. By theory,
> when we stuff time domain signal corrupted by noise with zeros, we should
> observe contraction of the spectrum of the signal and noise (noise is
> additive and is independent of the signal!). Before upsampling, the noise
> had flat PSD=N_0 over -pi to pi. If the suggestion that noise remains white
> after upsampling is TRUE, then there must not be any change in both
> magnitude of PSD of noise and the frequency. Why? Because for noise to be
> white, its PSD must remain flat over -pi to pi which requires that
> Upsampling must not cause any contraction in the spectrum of noise.

Think about what the range -pi < w < +pi means before upsampling and
what it means after. I suspect that you are confusing yourself with
equations and trying to correct that with logic.

Complete this thought: "The frequencies pi and -pi are normalized to the
sample rate. When you alter the sample rate ..."

...

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;
```
```>RIMalhi wrote:
>> Kenn Wrote
>>
>>> On Nov 13, 6:12=A0pm, "RIMalhi" <m4ma...@yahoo.com> wrote:
>>>> Let us assume that we have a bandlimited signal (with maximum
>> frequency
>>>> f_n) corrupted by additive white Gaussian noise. Before we can
sample
>> thi=
>>> s
>>>> signal, we pass the signal through an ideal anti-alias filter with
>> cut-of=
>>> f
>>>> frequency f_c >=3Df_n to avoid noise aliasing. The output of the
>>>> anti-aliasing filter is fed into a matched filter matched to the
>> symbol
>>>> rate, (1/T)>=3D2f_n (i used f_n here because we intend to keep
useful
>> sig=
>>> nal
>>>> spectrum intact). As a prticular case we let I/T=3D10f_n and
>> f_c=3D2f_n. =
>>> My
>>>
>>> I'm _assuming_ by symbol rate you mean sampling rate. Correct me if
>>> that's not right. To get my bearings, we have
>>>
>>> 0 <= f_n <= f_c <= f_s
>>>
>>> where f_s is the sampling frequency. With your numbers, normalized to
>>> f_n =1,
>>>
>>> 0 <= 1 <= 2  <= 10
>>>
>>> So the Nyquist limit is at 5, therefore (- pi...+pi ) in discrete
>>> domain corresponds to -5 to +5 in original continuous frequency
>>> domain.
>>>
>>>> question is what will be the impact of sampling on the white noise
in
>> thi=
>>> s
>>>> case? Will it remain white? Will it not be the case that
(bandlimited)
>>>> noise will get oversampled so that power spectral density of noise
will
>> n=
>>> o
>>>> more be flat over -pi and pi?
>>> It *was* flat until you filtered it . But then you filtered it. So
you
>>> now have (ideally) non-zero flat PSD from (-2 to 2). But you're
>>> sampling with Nyquist mapped to (-5,5). So I'd guess that your PSD in
>>> the discrete domain would be non-zero from (-2/5 pi ... +2/5 pi).
That
>>> doesn't sound like what you want to call "white" in the discrete
>>> domain.
>>>
>>>> My understanding is that noise is white (theoretically) in
>> discrete-time
>>>> domain if its PSD is flat over -pi to pi and hence over all
frequencies
>> i=
>>> n
>>>> the discrete-time domain. And the noise will be colored if it is not
>> flat
>>>> over -pi to pi.
>>> So it sounds like your filter coloured it, then.
>>>
>>>> Secondly suppose we have signal-plus-noise (in discrete-time domain)
>> such
>>>> that noise PSD is flat over -pi to pi whereas the spectrum of the
>> signal =
>>> is
>>>> non-zero over -pi/M<omega<pi/M where M is a positive integer. We
>> upsample
>>>> signal-plus-noise by factor N. My question is what will be the
impact
>> of
>>>> upsampling on PSD of noise. Will it be magnitude and frequency
scaled?
>>>> (Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald
W.
>>>> Schafer)
>>> It's like deja vu all over again.
>>>
>>> You start with a signal that's "full" of noise, pregnant with entropy
>>> for the entire omega spectrum. Then you stuff in some zeros
>>> (modulation), then you filter (your sinc filter).  The filter is the
>>> hint here. You have more samples now but you've also rescaled omega,
>>> so the noise now looks like it lives only in (-pi/N to pi/N).
>>>
>>> The amplitude scaling idea is really tripping you up. Look at it this
>>> way: Imagine your original signal was DC:
>>>
>>> 1,1,1,1,1,...
>>>
>>> Now zero-stuff (N=3D4)
>>>
>>> 1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0...
>>>
>>> If you filter the stuffed signal with an ideal *unity-gain* filter,
>>>
>>> 0.25, 0.25, 0.25, 0.25, .....
>>>
>>> This is where the amplitude is lost. You either keep track of it in
>>> your head as a loss (i.e a fudge factor of 0.25) , or you redefine
>>> your upsampling filter to have a gain of 4 buried in it somewhere to
>>> make the upsampling unity gain as far as signal amplitude goes.  It's
>>> all in our heads anyway :-)
>>
>>
>>
>> Hi Kenn,
>> We have flat spectrum in the frequency domain. But we do not stuff any
>> zeros in the flat spectrum. We stff zeros in time domain and are
seeking
>> its impact on the spectrum. We know that contraction in time domain
results
>> in expansion in frequency domain and vice versa. So when we upsample a
>> signal in the time domain, we are in fact expanding it. Therefore, in
>> frequency domain we should observe an equal contraction in the
spectrum.
>> What confuses me is this: Some people (e.g., look at url
>> http://sipc.eecs.berkeley.edu/ee123/ee123handoutPSD.pdf) suggest that
>> after upsampling (or downsampling) white noise remains white. By
theory,
>> when we stuff time domain signal corrupted by noise with zeros, we
should
>> observe contraction of the spectrum of the signal and noise (noise is
>> additive and is independent of the signal!). Before upsampling, the
noise
>> had flat PSD=N_0 over -pi to pi. If the suggestion that noise remains
white
>> after upsampling is TRUE, then there must not be any change in both
>> magnitude of PSD of noise and the frequency. Why? Because for noise to
be
>> white, its PSD must remain flat over -pi to pi which requires that
>> Upsampling must not cause any contraction in the spectrum of noise.
>
>Think about what the range -pi < w < +pi means before upsampling and
>what it means after. I suspect that you are confusing yourself with
>equations and trying to correct that with logic.
>
>Complete this thought: "The frequencies pi and -pi are normalized to the

>sample rate. When you alter the sample rate ..."
>
>   ...
>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;&#65533;
>

Hi Jerry,
Thank you very much making me clear about the confusion that i described.
-pi < w < +pi is the fundamental spectrum of the sampled sgnal where pi is
equal to half the sampling frequency. When we upsample or downsample the
sequence, we also change pi < w < +pi.

There is one more thing that i wish to ask. At the receiver end in a
communication system, before we can sample we have to limit the bandwidth
of the received signal. Suppose we have the following model of received
signal in continuous-time domain

y(t)=c(t)s(t)+n(t);

whete c(t) and s(t) are bandlimited processes and n(t) is AWGN.
To get sampled version of y(t), we first have to use anti-alias filter to
limit the bandwidth of the input signal to avoid aliasing. Firstly the
queston is what should be the bandwidth of the anti-aliasing filter
ideally? An intuitive answer to this question is that the filter bandwidth
should be equal the maximum frequency in the useful signal. In our case
that signal is u(t)=c(t)s(t). As we know multiplication in the time-domain
implies convolution in the frequency domain, the maximum frequency present
in the useful signal is equal to the SUM of the maximum frequencies in
individual spectra of c(t) and s(t). Let that frequency be f_N. If the
sampling frequency is P times the Nyquist rate corresponding to that
frequency i.e., fs=P 2 f_N, the noise will no more be white in the
discrete-time domain (it will be white if we sample at Nyquist rate
corresponding to f_N). Then why do we make an assumption of white noise in
the discrete-time domain? Is it just a matter of mathematical convenience?
Do the state-of-the-art receivers assume the noise to be white?

If we have the following sequence (recall that u[n]=s[n]c[n])

... c[-3] u[-2] u[-1] c u u c u u....

and u(t) has maximum fequency f_N and c(t) has maximum frequency f_c. My
question is what will be the maximum frequency of this sequence?

Regards,
RIMalhi

```
```RIMalhi wrote:

...

> Hi Jerry,
> Thank you very much making me clear about the confusion that i described.
> -pi < w < +pi is the fundamental spectrum of the sampled sgnal where pi is
> equal to half the sampling frequency. When we upsample or downsample the
> sequence, we also change pi < w < +pi.

Indeed! some things become much simpler when we stand back and look at
the larger picture.

> There is one more thing that i wish to ask. At the receiver end in a
> communication system, before we can sample we have to limit the bandwidth
> of the received signal. Suppose we have the following model of received
> signal in continuous-time domain
>
> y(t)=c(t)s(t)+n(t);
>
> whete c(t) and s(t) are bandlimited processes and n(t) is AWGN.

The noise is also bandlimited. We call it white because it it behaves
white within the band of interest. (All noise is bandlimited. If it
weren't, what would its upper frequency be?)

> To get sampled version of y(t), we first have to use anti-alias filter to
> limit the bandwidth of the input signal to avoid aliasing. Firstly the
> queston is what should be the bandwidth of the anti-aliasing filter
> ideally? An intuitive answer to this question is that the filter bandwidth
> should be equal the maximum frequency in the useful signal. In our case
> that signal is u(t)=c(t)s(t). As we know multiplication in the time-domain
> implies convolution in the frequency domain, the maximum frequency present
> in the useful signal is equal to the SUM of the maximum frequencies in
> individual spectra of c(t) and s(t). Let that frequency be f_N. If the
> sampling frequency is P times the Nyquist rate corresponding to that
> frequency i.e., fs=P 2 f_N, the noise will no more be white in the
> discrete-time domain (it will be white if we sample at Nyquist rate
> corresponding to f_N). Then why do we make an assumption of white noise in
> the discrete-time domain? Is it just a matter of mathematical convenience?
> Do the state-of-the-art receivers assume the noise to be white?
>
> If we have the following sequence (recall that u[n]=s[n]c[n])
>
> ... c[-3] u[-2] u[-1] c u u c u u....
>
> and u(t) has maximum fequency f_N and c(t) has maximum frequency f_c. My
> question is what will be the maximum frequency of this sequence?

the signal through tuned circuits. There is not usually an additional
anti-alias filter in front of the sampler, and even if there were, the
received noise would be filtered by it. By the time the signal is handed
off to the sampler, noise and signal have the same bandwidth.
incomplete and therefore misleading. Don't be misled!

Jerry
--
Engineering is the art of making what you want from things you can get.
```
```>> Hi Jerry,
>> Thank you very much making me clear about the confusion that i
described.
>> -pi < w < +pi is the fundamental spectrum of the sampled sgnal where pi
is
>> equal to half the sampling frequency. When we upsample or downsample
the
>> sequence, we also change pi < w < +pi.
>
>Indeed! some things become much simpler when we stand back and look at
>the larger picture.
>
>> There is one more thing that i wish to ask. At the receiver end in a
>> communication system, before we can sample we have to limit the
bandwidth
>> of the received signal. Suppose we have the following model of
>> signal in continuous-time domain
>>
>> y(t)=c(t)s(t)+n(t);
>>
>> whete c(t) and s(t) are bandlimited processes and n(t) is AWGN.
>
>The noise is also bandlimited. We call it white because it it behaves
>white within the band of interest. (All noise is bandlimited. If it
>weren't, what would its upper frequency be?)
>
>> To get sampled version of y(t), we first have to use anti-alias filter
to
>> limit the bandwidth of the input signal to avoid aliasing. Firstly the
>> queston is what should be the bandwidth of the anti-aliasing filter
>> ideally? An intuitive answer to this question is that the filter
bandwidth
>> should be equal the maximum frequency in the useful signal. In our
case
>> that signal is u(t)=c(t)s(t). As we know multiplication in the
time-domain
>> implies convolution in the frequency domain, the maximum frequency
present
>> in the useful signal is equal to the SUM of the maximum frequencies in
>> individual spectra of c(t) and s(t). Let that frequency be f_N. If the
>> sampling frequency is P times the Nyquist rate corresponding to that
>> frequency i.e., fs=P 2 f_N, the noise will no more be white in the
>> discrete-time domain (it will be white if we sample at Nyquist rate
>> corresponding to f_N). Then why do we make an assumption of white noise
in
>> the discrete-time domain? Is it just a matter of mathematical
convenience?
>> Do the state-of-the-art receivers assume the noise to be white?
>>
>> If we have the following sequence (recall that u[n]=s[n]c[n])
>>
>> ... c[-3] u[-2] u[-1] c u u c u u....
>>
>> and u(t) has maximum fequency f_N and c(t) has maximum frequency f_c.
My
>> question is what will be the maximum frequency of this sequence?
>
>the signal through tuned circuits. There is not usually an additional
>anti-alias filter in front of the sampler, and even if there were, the
>received noise would be filtered by it. By the time the signal is handed

>off to the sampler, noise and signal have the same bandwidth.
>incomplete and therefore misleading. Don't be misled!
>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>

Hi Jerry,
frequency in the above sequence? I think it can be considered as two
multiplexed streams. But the question is how to determine spectrum of the
composite stream?

RIMalhi
```
```RIMalhi wrote:

...

> Hi Jerry,
> frequency in the above sequence? I think it can be considered as two
> multiplexed streams. But the question is how to determine spectrum of the
> composite stream?

come through the same filters and therefore have the same bandwidth. The
receiver does generate some internal noise, but that is only significant
in the input stage and most of the filtering happens after that.

Jerry
--
Engineering is the art of making what you want from things you can get.
```
```>RIMalhi wrote:
>
>
>   ...
>
>> Hi Jerry,
>> frequency in the above sequence? I think it can be considered as two
>> multiplexed streams. But the question is how to determine spectrum of
the
>> composite stream?
>
>come through the same filters and therefore have the same bandwidth. The

>receiver does generate some internal noise, but that is only significant

>in the input stage and most of the filtering happens after that.
>
>Jerry
>--
>Engineering is the art of making what you want from things you can get.
>

Hi Jerry,
Thanks for your reply. The bandwidth of the filters will be dicatated by
the spectrum of the useful signal. My question was related to the bandwidth
of the useful signal. I  restate my question here. Let us forget any
filters or any receiver. I have an i.i.d. random process s(t). We make a
new process s'(t) such that we have the following realizations of s'(t)

....s1 s2 s3 1 s4 s5 s6 1 s7 s8 s9 1....

where s1,s2... are realization from random process s(t). Now if random
process   s(t) has maximum frequency f_c.

My question is what will be the maximum frequency of
s'(t)? Can we say something at least qualitatively that the maximum
frequency in the spectrum of s'(t) is equal to or greater than s(t)?

The above sequence is simply s(t) with 1's multiplexed into it (we can
consider that way because s(t) is i.i.d.).

```