Archive for Mai, 2010

Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 35 exercise 2.14

Author: Panagiotis

28 Mai

In [1, p. 35 exercise 2.14] we are asked to consider the solution to the set of linear equations

$\mathbf{R}\mathbf{x}=\mathbf{-r}$

(1)

where $\mathbf{R}$ is a $n \times n$ hermitian matrix given by:

$\mathbf{R}=P_1\mathbf{e_1}\mathbf{e_1}^H+P_2\mathbf{e_2}\mathbf{e_2}^H$

(2)

and $\mathbf{r}$ is a complex $n \times 1$ vector given by

$\mathbf{r}=P_1 e^{j2\pi f_1}\mathbf{e_1}+P_2 e^{j2\pi f_2}\mathbf{e_2}$

(3)

The complex vectors are defined in [1, p.22, (2.27)]. Furthermore we are asked to assumed that $f_1=k/n$ , $f_2=l/n$ for $k,l$ are distinct integers in the range $[-n/2,n/2-1]$ for $n$ even and $[-(n-1)/2,(n-1)/2]$ for $n$ odd. $\mathbf{x}$ is defined to be a $n \times 1$ vector. It is requested to show that $\mathbf{R}$ is a singular matrix (assuming $n>2$ ) and that there are infinite number of solutions. A further task is to find the general solution and also the minimum norm solution of the set of linear equations. The hint provided by the exercise is to note that $\mathbf{e}_1/\sqrt{n},\mathbf{e}_2/\sqrt{n}$ are eigenvectors of $\mathbf{R}$ with nonzero eigenvalues and then to assume a solution of the form

$\mathbf{x}=\xi_1 \mathbf{e}_1 + \xi_2 \mathbf{e}_2 + \sum\limits_{i=3}^{n}\xi_i \mathbf{e}_i$