Eigenvalues and equivalent transformation of a trigonometric matrix associated with filter design

The N × N trigonometric matrix P(ω) whose entries are 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⩾4 we prove that P(ω) has one positive and one negative eigenvalue when is an integer, while it has two positive and two negative eigenvalues when is not an integer. We also show that for not being an integer and a sufficiently large N, the two positive eigenvalues converge to α+N2 and the two negative eigenvalues to α-N2, where . Furthermore, an equivalent transformation diagonalizing P(ω) is described.