DOA estimation; Music (Multiple Signal Classification) algorithm

Question

Why do we say that signal subspace and noise subspace are orthogonal in e.g. MUSIC algorithm? precisely, Suppose we have 2 sources (signals) which are WSS, in WGN in a uniform linear array. After computing the covariance matrix of received data "Y=AS+N", and decomposing R (covariance matrix) into 2 subspaces, we say that these 2 subspaces are orthogonal. Why?

Answer 1

The receive-signals covariance matrix $\mathbf R$ is by its definition $$\DeclareMathOperator{\expect}{E}\newcommand{\E}[1]{\expect\left[#1\right]}\DeclareMathOperator{\diag}{diag}$$ $$\mathbf R= \E{\mathbf{YY}^*}=\mathbf R^*$$ hermitian.

We don't just find any decomposition of $\mathbf R$ into subspaces – we pick the Eigenvalue Decomposition, which gives us

$$\mathbf R= \mathbf U \mathbf \Lambda \mathbf U^* \text,$$

where

\begin{align} \mathbf\Lambda &= \diag(\lambda_1,\ldots,\lambda_N) &\text{diagonal matrix of Eigenvalues,}\\ \mathbf U &= ( \mathbf g_1,\ldots, \mathbf g_N)&\text{matrix: Eigenvectors as columns} \end{align}

We can without loss of generality have $\mathbf \Lambda$ sorted such that $\lambda_1,\ldots\lambda_M$ and thus $\mathbf g_1,\ldots \mathbf g_M$ correspond to transmit signals, and $\lambda_{M+1},\ldots,\lambda_N$ and $\mathbf g_{M+1},\ldots,\mathbf g_N$ come from to noise.

Assuming our receive array is well-formed and the signal is not incident from an angle that leads to ambiguities, $\lambda_i\ne\lambda_{l\ne i}$. Now, in the presence of uncorrelated noise, we know that

1. $\mathbf R$ has full rank, and
2. since $\mathbf R$ is hermitian, different Eigenvalues have orthogonal Eigenvectors.

So, if we have $M<N$ independent receive signal components, we get $M$ different, orthogonal vectors.

The $M+1,\ldots, N$th eigenvalues of $\mathbf R$ are the noise variance $\sigma^2\ne\lambda_i,\,\forall i\in\left\{1,\ldots,M\right\}$, we see that the space spanned by the signal-caused Eigenvector must be orthogonal to the noise subspace.