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Chapter 11:
Extended Hadamard Matrices

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 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 = HH = 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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