Fast Walsh–Hadamard transform

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Divide-and-conquer algorithm to compute a Hadamard transform

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 $n=2^{m}$ would have a computational complexity of O( $n^{2}$ ). The FWHT_h requires only $n\log n$ additions or subtractions.

The FWHT_h is a divide-and-conquer algorithm that recursively breaks down a WHT of size $n$ into two smaller WHTs of size $n/2$ . This implementation follows the recursive definition of the $2^{m}\times 2^{m}$ Hadamard matrix $H_{m}$ :

H_{m}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}H_{m-1}&H_{m-1}\\H_{m-1}&-H_{m-1}\end{pmatrix}}.

The $1/{\sqrt {2}}$ 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 $H_{m}=A^{m}$ , where A is m-th root of $H_{m}$ .

Python example code

import math
def fwht(a) -> None:
    """In-place Fast Walsh–Hadamard Transform of array a."""
    assert math.log2(len(a)).is_integer(), "length of a is a power of 2"
    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
                y = a
                a = x + y
                a = x - y
        # normalize and increment
        a /= math.sqrt(2)
        h *= 2

References

Fino, B. J.; Algazi, V. R. (1976). "Unified Matrix Treatment of the Fast Walsh–Hadamard Transform". IEEE Transactions on Computers. 25 (11): 1142–1146. doi:10.1109/TC.1976.1674569. S2CID 13252360.
Yarlagadda and Hershey, "Hadamard Matrix Analysis and Synthesis", 1997 (Springer)

External links

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

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