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To my knowledge, white Gaussian noise (WGN) is defined as a process with a correlation function:

$$ R[k]=\sigma^2 \delta[k] $$

and whose symbols are distributed according to $N(0,\sigma^2)$.

Naturally, when simulating such noise, the mean won't be zero for a specific realization but it would be close. However, if I try to force a zero mean (simply subtract the mean), I then get the undesirable characteristic in the frequency domain. I get that my noise process does not have a DC term (instead of having a constant spectrum).

Can I simulate WGN with zero mean and flat spectrum? Is the problem numeric, or is it my interpretation of the results which is flawed?

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It is not clear if you're thinking of continuous-time or discrete-time noise. It is impossible to simulate CT white Gaussian noise; it only exists as equations on paper and as an abstraction in our minds.

Usually this is not an actual problem, though. What one normally does is to consider a band-limited analog system with WGN input, followed by an ADC sampling at some given rate. We can simulate this discrete-time WGN.

As you say, a given realization will not have zero DC and will not have a flat spectrum. This, again, is not a problem. If you heat a resistor, band-limit the noise, and sample it, you'll find that no given set of samples has exactly zero mean, nor a flat spectrum.

Zero mean and a flat spectrum are consequences of the theory, but you must interpret it correctly: in the limit, if you take an infinite number of samples, theory says that the mean will tend to zero and the spectrum will tend to flatness. That' all.

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  • $\begingroup$ Good answer except I question the 2nd sentence: my thinking was that we can indeed simulate SAMPLES of a mythical continuous time WGN process and it would be mathematically accurate (without the requirement that it be bandlimited). What would end up being a CT WGN with PSD =0 would all alias to a DT WGN with finite non-zero PSD. When you get a chance Mbaz, can you review my answer here and let me know there if I am missing something? dsp.stackexchange.com/a/87654/21048 $\endgroup$ May 18, 2023 at 15:03
  • $\begingroup$ Thanks for the answer. Indeed, I meant DT band-limited WGN. That being said, my question remains. Obviously, the problem of a finite would also exist in an experimental setup and not only in simulation. However, this brings another question, what does this imply to estimators which usually assume zero-mean noise? If I follow you correctly, it means that in order to get unbiased estimates (experimentally) we would have to take infinite an amount of samples. $\endgroup$
    – Yair M
    May 18, 2023 at 18:17
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    $\begingroup$ Another comment. Perhaps would like to add to your answer the fact that due to the issues faced by characterizing noise using a specific realization, methods such as Welch's method, which does a cascaded estimation with two stages of averaging, tend to yield spectral estimates closer to theory. $\endgroup$
    – Yair M
    May 18, 2023 at 18:25
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An additional point that may add additional intuition to MBaz's answer is that the for any of the results given in the frequency domain, each of the frequency domain samples from any simulation or measurement represent a power spectral density; each sample depicts the power over a given bandwidth (as the resolution bandwidth for the simulation or measurement). For example, the resolution bandwidth (also called "equivalent noise bandwidth" since it is the same power we would measure with a brick-wall filter of that bandwidth given a WGN process) of the FFT is $f_s/N$ where $f_s$ is the sampling rate and $N$ is the total number of samples (this is the case with no further windowing applied).

That said, there is no power at DC which has infinitely narrow bandwidth (Similarly for WGN there is no power at any single frequency...DC is not special here), but there is power over the frequency range given by the resolution bandwidth that includes DC. The DC sample in frequency for any measurement or simulation result will include this total power. So consistent with what MBaz has already stated, as the time duration approaches infinity, the resolution bandwidth approaches zero, and therefore as we approach infinity we can approach measuring what the true power is at DC.

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  • $\begingroup$ Those are very good points! $\endgroup$
    – MBaz
    May 18, 2023 at 15:19
  • $\begingroup$ Thanks, this answer really sheds light on the subject :) I enjoyed your line of thought. $\endgroup$
    – Yair M
    May 18, 2023 at 18:21

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