>> Kenn Wrote
>>> On Nov 13, 6:12=A0pm, "RIMalhi" <m4ma...@yahoo.com> wrote:
>>>> Let us assume that we have a bandlimited signal (with maximum
>>>> f_n) corrupted by additive white Gaussian noise. Before we can
>>>> signal, we pass the signal through an ideal anti-alias filter with
>>>> 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
>>>> rate, (1/T)>=3D2f_n (i used f_n here because we intend to keep
>>>> spectrum intact). As a prticular case we let I/T=3D10f_n and
>> f_c=3D2f_n. =
>>> 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
>>>> question is what will be the impact of sampling on the white noise
>>>> case? Will it remain white? Will it not be the case that
>>>> noise will get oversampled so that power spectral density of noise
>>>> more be flat over -pi and pi?
>>> It *was* flat until you filtered it . But then you filtered it. So
>>> 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).
>>> doesn't sound like what you want to call "white" in the discrete
>>>> My understanding is that noise is white (theoretically) in
>>>> domain if its PSD is flat over -pi to pi and hence over all
>>>> the discrete-time domain. And the noise will be colored if it is not
>>>> over -pi to pi.
>>> So it sounds like your filter coloured it, then.
>>>> Secondly suppose we have signal-plus-noise (in discrete-time domain)
>>>> that noise PSD is flat over -pi to pi whereas the spectrum of the
>> signal =
>>>> non-zero over -pi/M<omega<pi/M where M is a positive integer. We
>>>> signal-plus-noise by factor N. My question is what will be the
>>>> upsampling on PSD of noise. Will it be magnitude and frequency
>>>> (Ref: Discrete-time signal processing by Alan V. Oppenheim, Ronald
>>> 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:
>>> Now zero-stuff (N=3D4)
>>> 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
>> its impact on the spectrum. We know that contraction in time domain
>> 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
>> 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
>> when we stuff time domain signal corrupted by noise with zeros, we
>> observe contraction of the spectrum of the signal and noise (noise is
>> additive and is independent of the signal!). Before upsampling, the
>> had flat PSD=N_0 over -pi to pi. If the suggestion that noise remains
>> 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
>> 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 ..."
>Engineering is the art of making what you want from things you can get.
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
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?