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11.1 Generalized Hadamard Matrices The generalized Hadamard matrices were introduced by Butson in 1962. Generalized Hadamard matrices arise naturally in the study of error-correcting codes, orthogonal arrays, and affine designs (see Refs. 2-4). In general, generalized Hadamard matrices are used in digital signal/image processing in the form of the fast transform by Walsh, Fourier, and Vilenkin-Chrestenson-Kronecker systems. The survey of generalized Hadamard matrix construction can be found in Refs. 2 and 5-12. 11.1.1 Introduction and statement of problems Definition 11.1.1.1: A square matrix H(p, N) of order N with elements of the p'th root of unity is called a generalized Hadamard matrix if HH∗ = H∗H = NIN, where H∗ is the conjugate-transpose matrix of H. Remarks: The generalized Hadamard matrices contain the following: • A Sylvester-Hadamard matrix if p = 2, N = 2n. • A real Hadamard matrix if p = 2, N = 4t. • A complex Hadamard matrix if p = 4, N = 2t. • A Fourier matrix if p = N, N = N. Note: Vilenkin-Kronecker systems are generalized Hadamard H(p, p) and H(p, pn) matrices, respectively. |
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