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

• Main Authors: Lin, Zhiping, Liu, Yiying., Molteni, Giuseppe., Zhang, Dongye.
• Other Authors: School of Electrical and Electronic Engineering
• Format: Article
• Language: English
• Published: 2013
• Subjects: DRNTU::Engineering::Electrical and electronic engineering
• Online Access: https://hdl.handle.net/10356/85257; http://hdl.handle.net/10220/17655
• Institution: Nanyang Technological University

The N × N trigonometric matrix P(ω) whose entries are P(ω)(i, j) =1/2 (i+j−2) cos(i−j)ω 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 ω/π is an integer, while it has two positive and two negative eigenvalues when ω/π is not an integer. We also show that for ω/π 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 α± = (1 ± 2/√3)/8. Furthermore, an equivalent transformation diagonalizing P(ω) is described.