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  "body": "\n## Summary\n\nThe Walsh functions are functions on $[0, 1)$ that take only the values $1$ and $-1$.\nThey are the finite products of [[Rademacher functions|51GRFQ4WY6]], and together they form a complete orthonormal system of $L^2([0, 1))$.\nThey are a digital analogue of the trigonometric system, and they lead to the [[Walsh–Hadamard transform|K784BA2WHV]].\n\n## Explanations\n\n### L1 human\n\nThink of a square wave that jumps between $+1$ and $-1$.\nThe Rademacher functions are square waves that flip sign faster and faster, one for each binary digit.\nA Walsh function multiplies some of these square waves together, chosen by the binary digits of an index $n$.\nFor example, $w_3 = r_1 r_2$ equals $1$ on $[0, \\tfrac14)$, $-1$ on $[\\tfrac14, \\tfrac34)$, and $1$ on $[\\tfrac34, 1)$.\nEvery square-integrable function on $[0, 1)$ can be expanded in Walsh functions, much as in a Fourier series.\n",
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