Eigenvalues and Equivalent Transformation of A Trigonometric Matrix Associated with Filter Design

The $N\times N$ trigonometric matrix $P(\omega)$ whose entries are $P(\omega)(i,j)=\frac{1}{2}(i + j - 2)\cos(i-j)\omega$ appears in connection with the design of finite impulse response (FIR) digital filters with real coefficients. We prove several results about its eigenvalues; in particular, assuming $N\geq 4$ we prove that $P(\omega)$ has one positive and one negative eigenvalue when $\frac{\omega}{\pi}$ is an integer, while it has two positive and two negative eigenvalues when $\frac{\omega}{\pi}$ is not an integer. We also show that for $\frac{\omega}{\pi}$ not being an integer and a sufficiently large $N$, the two positive eigenvalues converge to $\alpha_+N^2$ and the two negative eigenvalues to $\alpha_-N^2$, where $\alpha_\pm = (1\pm 2/\sqrt{3})/8$. Furthermore, an equivalent transformation diagonalizing $P(\omega)$ is described.