White noise vs. delta pulse and Ultraviolet catastrofe

Everybody explains that white noise has all frequencies equally strong. But, this immediately means that

1. Ultraviolet catastrofe inevitably happens if power > 0 stays constant at any frequency and, what is similarly unacceptable,
2. White noise is identical to single Dirac impulse since delta pulse is a constant in the Fourier basis). Note that constant is the opposite to the notion of noise.

Is average over time is implied, like you obtain uniform distribution, or what?

Answers say that white noise does not mean that signal spectrum is limited. It means that that power spectrum is flat, which also means that there is no correlation between samples so that it is correlation matrix, which is delta function (and I do not know what is its fourier transform is, which has the infinite spectrum).

I see a several problems with that. At first, the power spectrum is simple a square of the spectrum. If you say that you prevent the Utraviolet catastrophe by cutting all frequencies above some threshold $f_{max},$ you cannot have a flat spectrum anymore.

Secondly, I understand that you can have a mean and variance of a uniform distribution which has value $v$ in $[a,b]$ and 0 outside. But what is a mean and variance of a perfectly flat power spectrum? Ok, mean might be zero if you admit negative frequencies. But you say that variance is $\sigma^2$. How is that?

Lastly, I have determined that rapid changes are less likely in the the limited spectrum signal, which means that samples tend to each other, which means that they are correlated. Ok, might be you say that they are correlated but pink nose does not say if they are positively or negatively correlated, so samples are not correlated in case of pink noise. Ok, this is great. But we have just concluded that pink noise is white (or can be white). Is it right?

I also see Wikipedia saying that white noise can be Gaussian, which means that the samples are normally distributed. This means that they will tend to each other, like in the pink noise.

• Quick answers: white noise is a random process, and therefore its power spectral density is flat. This is different from saying that a particular signal (which might be a single realization of a white noise process) has a constant spectrum. Jul 5 '13 at 13:39
• Secondly, there is no true white noise source that provides a flat power spectrum across all frequencies from zero to infinity. All noise sources are bandlimited in some way. The white noise model is very useful, however, because there are many situations where a noise process is flat over some particular bandwidth of interest. In this case, assuming that the input noise source is white can be a valid method of simplifying the problem analysis. Jul 5 '13 at 13:43
• Thanks, but delta-function is a constant function and its power spectral density is flat. I do not understand how you achieve flat spectral density for random functions.
– Val
Jul 5 '13 at 13:47
• @Val Flat in the sense that if you integrate the spectrum over some arbitrary section, the result will be proportional to the length of the section, i.e. the density is constant. Jul 5 '13 at 14:40
• @Val I recommend that you follow the suggestion "More than what you probably want to know about white Gaussian noise can be found in the Appendix of this lecture note of mine." that I gave in an answer there. You are trying to use a model in a situation where it does not apply. Ohm's Law says that a gazillion volts applied to a $1\ \Omega$ resistor makes a gazillion amps flow through the resistor but in fact, the result is a flash of light and a puff of smoke. Jul 6 '13 at 11:32

The difference between both signals is the phase of the spectral components. The phase of the noise signal is completely random. The phase of the dirac is zero (in case of an dirac at t=0).

Take for example a noise signal, transform it to the frequency domain, change phase to zero, transform back, and you will obtain an approximate dirac.

Sample Matlab code:

x = rand(1000,1)-0.5; %random signal
y = real(ifft(abs(fft(x)))); %abs() removes the phase
plot([x y]);
legend('x','y');

Edit: trying to clear things up

In this example x is a realization of a band limited white noise, as can be seen by plotting the magnitude of its spectrum via plot(abs(x)). The next step is not calculating its autocorrelation function, that would be ifft(fft(x).*conj(fft(x))) and result in a band limited approximativ dirac function. Instead it takes the magnitude, which is effektively setting the phase of the complex spectrum to zero and transforms that signal back into time domain, and the result is also a band limited appromximative dirac function.

The goal of this example is answering this part of the question:

I cannot find any intelligible explanation what makes white noise different from both dirac and resolves the frequency limit.

Noting that the theoretical signals white noise and dirac impulse are not bandlimited, therefore they do not exist in real world. I am just giving a descriptive example using real world approximizations, what features have to be altered to convert a noise signal with a flat amplitude spektrum into an impulse signal.

• Ok, this resolves the paradox. Though, I do not understand the magic of phase. Can we produce an arbinary function by just shifting the phases so that arbitrary time-domain function has the flat spectrum?
– Val
Jul 5 '13 at 18:25
• the spectrum has a magnitude and a phase, so changing phase changes the spectrum. If you mean amplitude spectrum or power spectrum, which is what is normaly meant by saying "spectrum", than the answer is no. Changing the phase does not change the amplitude or power spectrum. Jul 5 '13 at 19:07
• The dc part is removed with the "-0.5" term. The absolute value is taken in frequency domain. Look at en.wikipedia.org/wiki/… to see that this is correct. I think you are confusing time domain and frequency domain here. Jul 5 '13 at 19:19
• This answer is incorrect. The white noise whose autocorrelation function is an impulse (a.k.a. delta function) is a mythical continuous-time process. For the discrete-time process generated by the MATLAB code fragment, white noise means a sequence of independent identically distributed zero-mean finite variance random variables so that the discrete-time autocorrelation function $R[m] = E[X[n]X[n-m]$ equals $\sigma^2\cdot u[n]$ where $\sigma^2$ is the variance of all the random variables and $u[n]$ is the unit pulse sequence: $$u[n]=\begin{cases}1, & n = 0,\\0, & n \neq 0.\end{cases}$$ Jul 5 '13 at 19:39
• I am not stating, that if ifft(abs(fft(x))) is a dirac, x is a white noise process! I am stating, that if x is a white noise process, then ifft(abs(fft(x))) is a dirac. But with a different scaling than the dirac resulting from the AKF of x, of course. Anyways, i provided a working example, that gives the correct result. I can't see why it should be wrong then. Jul 7 '13 at 5:56

As Jason was saying in the comments, the power spectral density of white noise is flat. This is equivalent to saying that the autocorrelation of white noise is a delta dirac function (i.e. that there is no correlation, positive or negative, between one noise sample and another), not that the noise itself is a delta dirac function.

• I have some misunderstandings regarding the power spectrum and correlations. I think that you cannot have limited spectrum altogether with flat power density and 0 correlations. Please see my updated question.
– Val
Dec 12 '13 at 16:27