2009/10/26 by Øyvind Ryan, Ø. Ryan, Mérouane Debbah +1 · 14 citations
Computer Science · Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Circular convolution #Computer science #Convolution (computer science) #Convolution theorem #Eigenvalues and eigenvectors #Fourier analysis #Fourier transform #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Multiplicative function #Pure mathematics #Random Matrices and Applications #Toeplitz matrix #Vandermonde matrix #cs.IT #math.IT
paper · pdf · doi:10.1109/tit.2011.2145990
published in IEEE Transactions on Information Theory 57(7), 4647-4659 (Institute of Electrical and Electronics Engineers) · Submitted to IEEE Transactions on Information Theory. 16 pages, 1 figure
arxiv created 2009/10/26 · openalex publication_date 2011/06/21 · openalex created_date 2016/06/24 · arxiv updated 2016/11/15 · openalex updated_date 2026/08/05
Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution, and include additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices, and cases where two independent Vandermonde matrices are involved. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. The convolutions are divided into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of the presented convolutions are provided and discussed. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Connections with related matrices, such as Toeplitz and Hankel matrices, are also discussed.