draw a signal · decompose it into square waves
Walsh functions are the digital world's answer to sine waves. Where Fourier analysis
decomposes signals into smooth sinusoids, Walsh analysis decomposes them into sharp square waves
that only take values of +1 and -1.
The Hadamard matrix shown above is the key — each row is one Walsh basis function. The matrix is orthogonal: every row is perpendicular to every other row. To decompose your signal, we take the dot product of your signal with each row, giving a coefficient for each basis function.
Toggle components on and off to see how they combine. The first component (W0) is always the average — the DC offset. Higher-order components capture increasingly fine detail, oscillating more rapidly. This is exactly how JPEG compression works in concept — discard the high-frequency components, keep the important low-frequency ones.
Walsh functions underpin CDMA (how your phone shares a channel with millions of others), quantum error correction, and digital signal processing. They're everywhere in computing, yet almost nobody has seen them.