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Chapter 8:
Fast Hadamard Transforms for Arbitrary Orders

Hadamard matrices have received much attention in recent years, owing to their numerous known and promising applications. The FHT algorithm was developed for N = 2n, 12 · 2n, 4n. In this chapter, a general and efficient algorithm to compute 4t-point (t is an "arbitrary" integer) HTs is developed. The proposed scheme requires no zero padding of the input data to make it the size equal to 2n. The difficulty of the construction of the N ≡ 0 (mod 4)-point HT is related to the Hadamard problem, namely, we do not know if, for every integer n, there is or is not an orthogonal 4n × 4n matrix of plus and minus ones. The number of real operations is reduced from O(N2) to O(N log2 N). Comparative estimates revealing the efficiency of the proposed algorithms with respect to those known are given. In particular, it is demonstrated that, in typical applications, the proposed algorithm is significantly more efficient than the conventional WHT. Note that the general algorithm is more efficient for Hadamard matrices of orders ≥96 than the classical WHT, whose order is a power of 2. The algorithm renders a simple and symmetric structure. Note that there are many approaches and algorithms concerning HTs.

In this chapter, we present new algorithms for fast computation of HTs of any existing order. Additionally, using the structures of those matrices, we reduce the number of operations. The chapter is organized as follows. Section 8.1 presents three algorithms of Hadamard matrix construction. Sections 8.2 and 8.3 present the decomposition of the arbitrary Hadamard matrix by {(1, 1), (1, −1)} and by the {(1, 1, 1, 1), (1, 1, −1, −1), (1, −1, −1, 1), (1, −1, 1, −1)} vector system. Section 8.4 describes these decompositions based on N ≡ 0 (mod 4)-point FHT algorithms. Section 8.5 describes a multiply/add instruction-based FHT algorithm that primarily uses shifted operations. Section 8.6 presents the complexity of developed algorithms, as well as comparative estimates, revealing the efficiency of the proposed algorithms with respect to those known.

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