White noise vs. delta pulse and Ultraviolet catastrofe

Question

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?

Why answers are not satisfactory

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.

Answer 1

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.

Answer 2

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.