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In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT_h) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order would have a computational complexity of O(). The FWHT_h requires only additions or subtractions.

The FWHT_h is a divide-and-conquer algorithm that recursively breaks down a WHT of size into two smaller WHTs of size . ^[1] This implementation follows the recursive definition of the Hadamard matrix :

The normalization factors for each stage may be grouped together or even omitted.

The sequency-ordered, also known as Walsh-ordered, fast Walsh–Hadamard transform, FWHT_w, is obtained by computing the FWHT_h as above, and then rearranging the outputs.

A simple fast nonrecursive implementation of the Walsh–Hadamard transform follows from decomposition of the Hadamard transform matrix as , where A is m-th root of . ^[2]

Python example code

def fwht(a) -> None:
    """In-place Fast Walsh–Hadamard Transform of array a."""
    h = 1
    while h < len(a):
        # perform FWHT
        for i in range(0, len(a), h * 2):
            for j in range(i, i + h):
                x = a[j]
                y = a[j + h]
                a[j] = x + y
                a[j + h] = x - y
        # normalize and increment
        a /= math.sqrt(2)
        h *= 2

See also

• Fast Fourier transform

References

External links

• Charles Constantine Gumas, A century old, the fast Hadamard transform proves useful in digital communications