# Standard deviation of the spectrum of white noise

I have a temporal signal which looks like $f(t) = t\eta(t)$, where $\eta(t)$ is a white noise with the mean $\eta_m$ and STD $\sigma$. I want to calculate the corresponding spectrum, $F(\omega)$, take its magnitude, $|F(\omega)|$, and finally compute the STD of $|F(\omega)|$ in terms of the given parameters. I have searched some literatures but I couldn't find anything which helps me directly tackle the problem. So, anyone know how to do this, or know which resources you would refer me to?

The following pictures display $f(t)$ and $|F(\omega)|$.

The one below is the histogram of $|F(\omega)|$ excluding some values around the central peak.

I want to derive a mathematical analysis which can compute the standard deviation of the above histogram.

• What do you mean by the standard deviation of the spectrum? Also: you can't calculate the spectrum of a random noise signal, only its power spectral density. You can, however, calculate the spectrum of a realization of the noise. Can you explain which you are trying to do? – MBaz Dec 2 '15 at 19:32
• I should say the STD of the PDF of $|F(\omega)|$. I definitely do not understand some terms you are saying because my background is not in signal processing. – nougako Dec 2 '15 at 19:58
• Feel free to ask new questions (or look around the site) regarding background questions. Back to the topic: there is no such thing as the spectrum of white noise. – MBaz Dec 2 '15 at 20:02
• It doesn't have one -- the power spectral density is not probabilistic. – MBaz Dec 2 '15 at 20:28
• nougako, not trying to pick on you, but you should be more careful differentiating between the notions of continuous-time signals (normally depicted "$x(t)$") and discrete-time signals (normally depicted "$x[n]$"). your problem statement show a lot of integrals and other continuous-time mathematical expressions, but once you're in MATLAB, you're discrete-time. – robert bristow-johnson Dec 2 '15 at 21:35

There are a few misconceptions here and confusion in what you've plotted versus what you've asked. This is an attempt to clarify things.

1. ${\tt CORRECTED}$ : OK, so your noise $\eta(t)$ is non-zero mean, which is why the $t\eta(t)$ term increases.

2. As robert says, "white noise" is a useful construct in continuous time. Only "bandlimited white noise" exists in discrete time.

3. Your question's title Standard deviation of the spectrum of white noise needs interpretation to make any sense.

• The power spectral density of bandlimited white noise is known, and is constant. If the variance of the noise is $\sigma^2$ then the value of the power spectral density is $\sigma^2$ for all $\omega$.
• This means that the power spectral density does not have a standard deviation.
• It's possible to take the DFT of one realization of the bandlimited white noise. The DFT of bandlimited white noise is... bandlimited white noise.
• One interpretation of your question is then: what is the variance of one realization of the DFT of bandlimited white noise? Does that get you the answer you need?
• As I said, $\eta(t)$ in my $t\eta(t)$ has nonzero mean. Well, I think the 4th point is really what I should have mentioned, but it's not purely white noise, it has been modulated by a linear function. – nougako Dec 2 '15 at 21:58
• OK, thanks for pointing that out, I've corrected point 1 here. We get that it's modulated by a linear function. What we don't understand is what you're trying to measure? Is it the power spectral density? Or is it the statistics of the DFT of one realization of your signal? – Peter K. Dec 2 '15 at 22:07

true white noise has infinite power (because the power spectrum with height $\frac{\eta}{2}$ has infinite area) which means infinite variance $\sigma^2$ which means infinite standard deviation. white noise is a conceptual instrument.

to do any analysis of a system with white noise in it, you must somehow determine the (single-sided) bandwidth of the system, $B$, and then the power or variance is $\eta B$.

• Well, let's then say that $f(t)$ is windowed with width $B$. I want to calculate the DFT of $f(t)$, which is denoted as $F(\omega)$ and then compute the STD of the probability density function of $F(\omega)$. By the way, $f(t)$ is a product between a linear function and a white noise, it is not purely white noise. – nougako Dec 2 '15 at 19:56
• where is this unlimited bandwidth $f(t)$ coming from that you're windowing? and how can you DFT it without first sampling it? and how can you sample it without first limiting the bandwidth to below half the sample rate? – robert bristow-johnson Dec 2 '15 at 21:00