Difference in following random proccess

Question

Let just say I understand the first process that is white noise,

Process 1: If $x(n)$ is a Gaussian random variable and a process is formed from a sequence of $x(n)$ and all random variables are uncorrelated then this process is white noise.

Process 2: Now if $x(n) = \alpha $ where now $\alpha$ is Gaussian random variable then each process in ensamble is equal to constant for all $n$

But I am not able to get what difference does it makes when we form Process 2 it seems same to me. Can any one explain the difference? Also how the ensamble became constant in the second case?

Answer 1

In the first case, $x(0)$ and $x(1)$ and $x(2)$ etc are all uncorrelated. (And since they're gaussian, they're independent). This means that, to get a realization of the first random process, you have to take $\infty$ independent samples from a gaussian distribution, and assign each of these $\infty$ samples to a $x(n)$.

The second process is much simpler. You just take one sample from a gaussian distribution (and call it $\alpha$), and then assign all of the $x(n)$ to this $\alpha$. In this way, $\alpha$ may be random, but you will get $x(0)=x(1)=x(2)=\cdots=\alpha$.

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I'll post some plots, as you asked. I used the following code in MATLAB:

N = 3;
T = 10;
t = 0:T;
x1 = randn(N,T+1);
x2 = repmat(randn(N,1), 1, T+1);

figure
subplot(3,1,1); stem(t, x1(1,:));
subplot(3,1,2); stem(t, x1(2,:));
subplot(3,1,3); stem(t, x1(3,:));

figure
subplot(3,1,1); stem(t, x2(1,:));
subplot(3,1,2); stem(t, x2(2,:));
subplot(3,1,3); stem(t, x2(3,:));

In this first figure, I plot three independent realizations of Random Process 1:

And in this figure, three independent realizations of Random Process 2. Note that in this case, all the $x$ signals are constant, but the actual level changes from one realization to another: