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

The N N trigonometric matrix P (!) whose entries are P (!)(i;j) = 1 (i + j 2) cos(i j)! 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 (!) 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 +N 2 and the two negative eigenvalues to N 2 , where = (1 2= p 3)=8. Furthermore, an equivalent transformation diagonalizing P (!) is described.