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If we filter out ideal white noise using an ideal LPF of cutoff frequency 10 KHz and then sample it at 30 KHz , is the resulting discrete signal statistically independent? I would like to know the statistical behaviour of the output signal.

I was attending an on-line test. My answer was told wrong.Below is the snapshot from the on-line test.

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I would also like to know what would be the result if sampling frequency is below Nyquist frequency?

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No, it wouldn't be "statistically independent" or uncorrelated, as I would prefer to say. If you sampled it with 20 kHz then it would be. –  Jazzmaniac Dec 23 '13 at 12:35
Keeping in mind that statistical independence and uncorrelated-ness are two different things, with the former implying the latter but not vise-versa. But indeed, in DSP contexts the latter is usually what we're interested in. –  stootoon Dec 23 '13 at 14:38
I have made an edit (an addition) in the question. –  dexterdev Dec 23 '13 at 15:02
@Jazzmaniac : Below Nyquist frequency it will be correlated and above Nyquist rate the discrete will be uncorrelated right? When will it be statistically independent and orhtogonal? –  dexterdev Dec 24 '13 at 11:55
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2 Answers

up vote 5 down vote accepted

The output of your filter is what is sometimes called band-limited white noise. In your particular case, the autocorrelation function of the output noise is a sinc function whose zeroes are every $100$ microseconds, that is, samples taken at the rate of $10^4$ samples/second are uncorrelated. Your samples at $3\times 10^4$ samples per second are closer together and thus are correlated. In fact, the correlation coefficient between successive samples is $\displaystyle \frac{\sin(\pi/3)}{\pi/3} = \frac{3\sqrt{3}}{2\pi} \approx 0.827$

Note added after question was edited: Samples taken $0.03$ milliseconds apart are at a frequency of $33.33\ldots$ kHz, not $30$ kHz as you say in the part of the question that you typed in yourself. Regardless, the answer (B) is incorrect and the reasoning given in support of answer (B) is bogus. What is being sampled is not white noise but filtered white noise, and Answer A is correct.

For (B) to be the correct answer, the sampling rate must be a sub-multiple of $10$ kilosamples per second, that is the samples must be spaced $100$ microseconds apart or an integer multiple of $100$ microseconds apart. Sampling every $0.3$ milliseconds, that is, every $300$ microseconds meets this criterion; sampling every $0.03$ milliseconds does not. Now sampling at $10$ kilosamples/second makes the samples uncorrelated (answer (C)) and to get from this to the stronger result that the samples are statistically independent (answer (B)), we need further assumptions about the noise. The standard assumption is that the noise is Gaussian.

In summary, (B) would be the correct answer if the sampling were done every $0.3$ milliseconds (since the problem statement already includes the assertion that the noise is Gaussian) and (C) would be the correct answer if it did not say that the noise is Gaussian since we could not the make the specialization from uncorrelated to independent. But when the sampling is done every $0.03$ milliseconds, then (A) is the correct answer.

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If you have any suggestion of references for me to see the explanation for this topic elaborately, please kindly share. I can grasp this but it will take time. –  dexterdev Dec 23 '13 at 15:36
@dexterdev As I said in answer to a different question, More than what you probably want to know about white Gaussian noise can be found in the Appendix of this lecture note of mine. –  Dilip Sarwate Dec 23 '13 at 20:12
Sir, If I am not troubling you, can you just show me by math that how as sampling frequencies vary the properties like correlation, uncorrelation , statistical independence etc are attained. Also how you arrived at the correlation coefficient of 0.827 ie sinc(pi/3)...How pi/3 comes? Sir I know you are a busy person and I don't want to disturb you. When you are free please show the math behind this. I didn't understand the sub-multiple theory etc. –  dexterdev Dec 24 '13 at 6:05
The correlation coefficient of two noise samples (separated in time by $t$ seconds) from bandlimited white noise is $\operatorname{sinc}(t/T)= \frac{\sin(\pi (t/T))}{\pi (t/T)}$ where $T=1/W$ is the minimum separation that gives zero correlation and $W$ is the bandwidth of the noise. In your case, $W=10^4, T=10^{-4}, t=(3\times 10^{4})^{-1} = (1/3)\times 10^{-4}$ since you claim a sampling rate of $30$ kHz (the problem says the sampling interval is $3\times 10^{-4}$). The correlation for your sampling interval is $\sin(\pi/3)/(\pi/3)$; for the problem, it is different but $\neq 0$ –  Dilip Sarwate Dec 24 '13 at 14:55
Can I get a link/reference of the derivation this correlation coefficient. –  dexterdev Dec 25 '13 at 6:22
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If by 'resulting discrete signal statistically independent' you mean whether the samples will be independent, i.e. whether $P(x[1],x[2],\dots,x[N]) = \prod_{i=1}^N P(x[i])$, the answer is in general no, because the filtering will induce correlations between nearby elements: the LPF operation means that values of successive elements can't change arbitrarily quickly, hence not independent.

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