I am trying to work through and understand the theory behind the methods in this paper. I have been staring at this sentence on pg 6 for several hours now, and just can't seem to work it out:
Noise and signal are assumed to be uncorrelated and therefore $Rx − σ^2 I = A(A^T)$
How do they get from point A to point B here? I have no idea how
$$corr(s,η) = 0 => Rx − σ^2 I = A(A^T).$$
What I have worked out is that $Rx = xx^T$ and $σ^2 = E[η^2]$...yeah, I haven't gotten that far. I thought maybe I could derive this conclusion by starting with $x = As + η$, and continuing as such:
$$x(x^T) = (As + η)(As + η)^T$$
$$x(x^T) = (As + η)((s^T)(A^T) + η^T)$$
$$x(x^T) = As(s^T)(A^T) + As(η^T) + η(s^T)(A^T) + η(η^T)$$
I thought that perhaps I could simplify by using the assumption that s and η are uncorrelated to conclude $s(η^T) = 0$ and $η(s^T) = 0$, thereby getting
$$x(x^T) = As(s^T)(A^T) + η(η^T)$$
And then using the assumption that the sources are independent, conclude
$$s(s^T) = I $$
Thus getting
$$x(x^T) = A(A^T) + η(η^T)$$
I am confused on what it means for the noise to be isotropic and how this fits into the equation. Can I somehow use that assumption to simplify $η(η^T)$ to $σ^2 I$??
Help would be much appreciated, thank you!
Ok, guys, I figured it out.
We start with $$x_i = A s_i + η_i$$
We are given that $R_x$ is the covariance matrix of $x_i$, or more specifically, the variance-covariance matrix of $x_i$. So
$$R_x = vc(x) = vc(A s_i + η_i)$$
From this, we can expand to
$$vc(A s_i + η_i)$$ $$= vc(As_i) + cov(As_i, η_i) + cov(η_i, A s_i) + vc(η_i)$$ $$= vc(As_i) + A*cov(s_i, η_i) + [A*cov(s_i, η_i)]^T + vc(η_i)$$
We are given that the signal is uncorrelated, thus $$cov(s_i, η_i) = 0$$
And the equation can be simplified to $$vc(As_i) + vc(η_i)$$
At this point, let's take it one piece at a time. $$vc(As_i) = A vc(s_i) A^T$$ $$vc(s_i) = cov(s,s) = E[ss^T]-E[s]E[s^T]$$
I'm new to stack exchange, so I'm going to avoid showing what this looks like in matrix form. If you want to, try it out, but you should end up with $vc(s_i) = I$ based on the assumptions that (1) the sources are independent, and (2) the sources have unit variance.
Thus we can conclude that $$vc(As_i) = AA^T$$
Similarly, $vc(η_i)$ can be simplified to $σ^2 I$ because (1) each noise component is independent, and (2) since $η_i$ is isotropic, the variance of each η in η_i is equivalent.
And that's how you end up with $$R_x = AA^T + σ^2 I$$