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Blind Signal Separation and Blind Deconvolution
Published in Yu Hen Hu, Jenq-Neng Hwang, Handbook of Neural Network Signal Processing, 2018
where [q]m denotes the modulo-m or remainder operation on the integer q. A circulant matrix is completely specified by any one row or column, as the other rows or columns of the matrix are simply modulo-shifted versions of this row or column. For example, the first column of A is [a0a1…am−1]T, the second column of A is [am−1a0…am−2]T, and so on. The assumption of a circulant mixing matrix is completely artificial; practically no physical mixing system exhibits this structure. As we shall see, however, the circulant form of this BSS task yields a simple connection with the blind deconvolution task.
Subspace Algorithms
Published in Philipos C. Loizou, Speech Enhancement, 2013
For symmetric and Toeplitz matrices, different types of algorithms were proposed that exploit the Toeplitz structure [74–77]. For instance, one can make use of efficient methods to perform the matrix-vector multiplication required in the power method (Equation 8.230). By appropriately embedding a Toeplitz matrix in a circulant matrix C, we can use the FFT to efficiently perform the matrix-vector multiplication in less than O(K2) operations [78] (see Example 8.5). Let F denote the DFT matrix, and suppose that we are interested in performing C · x, where C is a circulant matrix. After making use of the fact that the DFT matrix F diagonalizes the circulant matrix C, that is, C = FHΔF, where Δ = diag(F · c) is a diagonal matrix containing the Fourier transform of the first column of C, we can express C · x as
Asymptotic behaviours of a class of threshold models for collective action in social networks
Published in International Journal of Control, 2018
Andrea Garulli, Antonio Giannitrapani
Matrix F in (54) is a circulant matrix, i.e. it has the form where , and fi = 0, for i ≠ 0, 1, n − 1. The eigenvalues and eigenvectors of a circulant matrix can be computed analytically (e.g. see Davis, 1979). Let , where . The eigenvalues of F are given by The eigenvectors of F are given by and form an orthonormal basis. A circulant matrix can always be diagonalised. Let U = [u0u1 … un − 1], then F = UΛU*, where Λ = diag(λ0,… , λn − 1) and U* is the conjugate transpose of U (Th. 3.2.1 in Davis, 1979). Observing that λ0 = 1, λk = λn − k, k = 1,… , h, and , the evolution of the thresholds θ(t) can be written as where the last equality comes from , since , k = 1,… , n − 1. The thesis (56) easily follows by noting that the entries of uk in (59) are such that , for l = 0,… , n − 1 and k = 1,… , h.