# Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model $$\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}$$ where $\mathbf{s}(w)$ is a known vector function, and $\mathbf{n}\sim \mathcal{CN}(0,\sigma^2\mathbf{I})$ is white Gaussian noise with variance $\sigma^2$.

1. Column vector $\mathbf{r}\in\mathbb{C}^N$ constitutes the noisy data.
2. Parameters $w_1,\ldots,w_Q$ are unknown and need to be estimated.
3. $\alpha_1,\ldots,\alpha_Q$ are zero-mean complex random Gaussian variables with unknown variance.
4. $Q$ is the number of parameters to be estimated and is known.

A very useful family of estimators rely on some properties of the correlation matrix which is $$\operatorname{E}\mathbf{r}\mathbf{r}^H = \mathbf{S}\left(\mathbf{w}\right) \mathbf{A} \mathbf{S}^H\left(\mathbf{w}\right) + \sigma^2\mathbf{I}$$ where $\mathbf{A}=\operatorname{E}\,[\alpha_1,\ldots,\alpha_Q]^T[\alpha_1,\ldots,\alpha_Q]^*$, $\mathbf{S}(\mathbf{w})=[\mathbf{s}(w_1),\ldots,\mathbf{s}(w_Q)]$, $(\cdot)^T$ is the transpose operator, $(\cdot)^*$ is the conjugate operator, and $(\cdot)^H$ is the transpose and conjugate operator. In practice, the correlation matrix is estimated using the sample covariance matrix defined as $$\hat{\mathbf{R}}=\frac{1}{K}\sum_{k=1}^{K}\mathbf{r}_k\mathbf{r}_k^H$$ where $\mathbf{r}_k$ are independent realizations of $\mathbf{r}$.

Denote $\lambda_1,\ldots,\lambda_N$ and $\mathbf{v}_1,\ldots,\mathbf{v}_N$ the eigenvalues and eigenvectors of $\hat{\mathbf{R}}$, respectively. The eigenvalues are ordered from largest to smallest (all are non-negative because $\hat{\mathbf{R}}$ can be shown to be a Gramian matrix). Then in such case it turns out that $$\begin{bmatrix} \mathbf{v}_1 \cdots \mathbf{v}_Q \end{bmatrix} \operatorname{diag}\left(\lambda_1-\sigma^2,\ldots,\lambda_Q-\sigma^2\right) \begin{bmatrix} \mathbf{v}_1 \cdots \mathbf{v}_Q \end{bmatrix}^H \rightarrow \mathbf{S}\left(\mathbf{w}\right) \mathbf{A} \mathbf{S}^H\left(\mathbf{w}\right)$$ in probability when $K\rightarrow +\infty$.

Without going into further unnecessary detail my question is

Is it known the exact/asymptotic/approximated distribution of $\begin{bmatrix} \mathbf{v}_1 \cdots \mathbf{v}_Q \end{bmatrix} \operatorname{diag}\left(\lambda_1,\ldots,\lambda_Q\right) \begin{bmatrix} \mathbf{v}_1 \cdots \mathbf{v}_Q \end{bmatrix}^H$?

As far as I know the sample covariance matrix has a Wishart distribution, but I do not know what is the distribution of such signal covariance matrix.

• Nice question. Note: the eigenvalues of a Hermitian matrix are real, not necy positive. They can, of course, be ordered. – JayInNyc Jan 8 '15 at 22:47
• Oops, my mistake @JayInNyc, I meant it's a Gramian matrix. – mermeladeK Jan 9 '15 at 0:15
• I've gotten that wrong, too. Yours is a very interesting question. Do you have a reference as to where to learn more about this family of estimators? – JayInNyc Jan 9 '15 at 1:00
• Sure. They are known usually as subspace methods. A very famous one is MUSIC. The typical use is in direction of arrival (DOA) estimation. Check these papers: part 1 part 2 – mermeladeK Jan 9 '15 at 1:17