White noise and Colored noise

First of all I would like to say that this was an article I wrote for friends at my institute, when we were discussing about the topic white noise and its power spectrum etc. Since I am an engineer by training, I think I was able to pass on my intuition to my "theoretical" friends. :) If you find any mistake please do comment below.

Consider a collodial particle in a fluid. The corresponding langevin equation is as below:
$m \ddot{x} = -\gamma \dot{x} + \eta(t)$

'm' is mass of the particle.
' $\ddot{x}$ ' (x-doubledot)is the second derivative of particle position with respect to time a.k.a acceleration.
' $\dot{x}$ ' (x-dot)is the first derivative of particle position with respect to time a.k.a particle velocity.
' $\gamma$ ' is the damping coefficient.
' $\eta(t)$ ' is the noise signal.

So mass times acceleration becomes force. The force which is acting on the particle is written as a sum of a viscous force proportional to the particle's velocity and noise term ' $\eta(t)$ '.

We will focus on the noise term ' $\eta(t)$ ' which is assumed to be a normal distribution of fashion $\mathcal{N}(0,\sigma^2 \cdot t)$ . You will notice that this distribution is not stationary, because the variance changes as time progresses. But once you fix the time and look at different realizations of this "signal" you will get a stationary normal distributed probability density function.

The autocorrelation function of $\eta(t)$ is written as:
$<\eta(t) \eta(t+\tau)>=k\delta(\tau)$

where 'k' is some constant. So now where is "whiteness" coming from? You will easily recognize that the fourier transform of this "time-domain" autocorrelation function i.e the scaled delta function will give you a flat "frequency-domain" power spectral density(PSD) function. The flat PSD says that each infinitesimal band contains equal power content contributed by each frequency component. Something analogous to white color. But keep in mind that there is no ideal white noise in our real world. So in a colored noise you will see a PSD something like in Figure 3, which has different power contributions from different frequencies.

The fourier transform of autocorrelation is power spectral density.
$\mathscr{F}$ (Autocorrelation)= Power spectral density
$\mathscr{F}$ ( $\delta(t)$ )=1

One more point which I need to emphasis is that the ideal white noise is a mathematical construct. And practically all noises you generate(whether using a computer or not) will have some kind of correlation instead of delta correlation and so practically you end up generating colored noise.

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bornprince (52) 8 years ago

My friend, I am not an engineer but can tell by reading this post that you have done a great work. Thumb up!

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dexterdev (63) 8 years ago (edited)

Thank you @bornprince :) It was painful to incorporate math equations though. I think steemit will bring latex formatting for equations.

Always welcome my friend! I think so.

dexterdev (63) 8 years ago

By the way I just wanted to say that why white noise is called "white". You may know know that white light contains all colors (or all possible visible color frequencies). So calling white noise white is because this noise signal is a superposition of signals with all (0 to infinity) frequencies. Colored noise will have a band of frequencies only. :) Thank you