# Fast Walsh–Hadamard transform

Jump to: navigation, search
The fast Walsh–Hadamard transform applied to a vector of length 8
Example for the input vector (1,0,1,0,0,1,1,0)

In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHTh) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT would have a computational complexity of O($N^2$). The FWHTh requires only $N \log N$ additions or subtractions.

The FWHTh 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 $2N \times 2N$ Hadamard matrix $H_N$:

$H_N = \frac{1}{\sqrt 2} \begin{pmatrix} H_{N-1} & H_{N-1} \\ H_{N-1} & -H_{N-1} \end{pmatrix}.$

The $1/\sqrt2$ normalization factors for each stage may be grouped together or even omitted.

The Sequency ordered, also known as Walsh ordered, fast Walsh–Hadamard transform, FWHTw, is obtained by computing the FWHTh as above, and then rearranging the outputs.

## 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.