Re: Sampling a signal corrupted by AWGN

Kenn Wrote

On Nov 13, 6:12=A0pm, "RIMalhi" <m4ma...@xxxxxxxxx> 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,
you'll get (in steady-state)

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,






.

Relevant Pages

  • Re: Sampling a signal corrupted by AWGN
    ... f_n) corrupted by additive white Gaussian noise. ... we pass the signal through an ideal anti-alias filter with cut-off ... It *was* flat until you filtered it. ... Secondly suppose we have signal-plus-noise such ...
    (comp.dsp)
  • Re: Sampling a signal corrupted by AWGN
    ... anti-aliasing filter is fed into a matched filter matched to the ... noise will get oversampled so that power spectral density of noise will ... make the upsampling unity gain as far as signal amplitude goes. ... We have flat spectrum in the frequency domain. ...
    (comp.dsp)
  • Re: Sampling a signal corrupted by AWGN
    ... f_n) corrupted by additive white Gaussian noise. ... anti-aliasing filter is fed into a matched filter matched to the symbol ... were you to do this then the resulting sampled-time noise ... Secondly suppose we have signal-plus-noise ...
    (comp.dsp)
  • Re: help with navtex/medium wave receiver sensitivity and decoding
    ... the system is too high to pick up anything but the strongest navtex ... above the background noise, and navtex signals are much lower power. ... Commercial stuff will filter, mix up to maybe a 60Mc IF to lose image ...
    (sci.electronics.design)
  • Re: Line Noise Interference Question
    ... X-10 module with the heavy transformer inside a linear power supply. ... They may filter their output to prevent noise from reaching the mains. ... >> The filters may also filter 120kHz thus draining the X-10 signals. ...
    (comp.home.automation)