2012/01/23 by Suvrit Sra, Sra, Suvrit
Computer Science · Physics and Astronomy · #15A18 #15A26 #Digital Filter Design and Implementation #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1201.4651
openalex publication_date 2012/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a very recent paper "On eigenvalues and equivalent transformation of trigonometric matrices" (D. Zhang, Z. Lin, and Y. Liu, LAA 436, 71--78 (2012)), the authors motivated and discussed a trigonometric matrix that arises in the design of finite impulse response (FIR) digital filters. The eigenvalues of this matrix shed light on the FIR filter design, so obtaining them in closed form was investigated. Zhang et al. proved that their matrix had rank-4 and they conjectured closed form expressions for its eigenvalues, leaving a rigorous proof as an open problem. This paper studies trigonometric matrices significantly more general than theirs, deduces their rank, and derives closed-forms for their eigenvalues. As a corollary, it yields a short proof of the conjectures in the aforementioned paper.