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