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 18.104.22.168: 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.