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THE ORTHOGONAL SIGN-FLIP
a Fourier with no multiplies — just plus and minus
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Walsh–Hadamard transform. A cousin of the Fourier transform that uses square waves (±1) instead of sines — so it needs no multiplications at all, only additions and subtractions. It decomposes a signal into sequency components (how many sign-changes each basis wave has). Its basis (the Hadamard matrix) is orthogonal, it is its own inverse up to a scale, and it is exact on integers. It runs CDMA (each phone gets an orthogonal Walsh code, so all transmit at once), the quantum Hadamard gate, and blocky image compression.
LIT verified: the fast WHT applied twice returns the original × N (self-inverse up to scale), it is integer-exact, and the basis rows are mutually orthogonal (every pairwise dot product is 0). FIG ‘orthogonal sign-flip’ is the picture; the multiplication-free transform and the orthogonality are exact.
LIT verified: the fast WHT applied twice returns the original × N (self-inverse up to scale), it is integer-exact, and the basis rows are mutually orthogonal (every pairwise dot product is 0). FIG ‘orthogonal sign-flip’ is the picture; the multiplication-free transform and the orthogonality are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — THE FOURIER sits right here in THE BROADCAST and THE EXACT TRANSFORM nearby, and the corpus loves the ±1 / binary structures the Hadamard gate shares with quantum. AVAN (AI) built this instrument: the fast transform, the compression demo, and the Hadamard relief.
The weave: David names the sign-flip and its seat beside Fourier in THE BROADCAST; I make the ±1 basis a strip in 1D, the transform-and-compress live in 2D, and the Hadamard matrix a turning relief in 3D. The sphere is the seam — Fourier’s blocky, multiply-free cousin.
The weave: David names the sign-flip and its seat beside Fourier in THE BROADCAST; I make the ±1 basis a strip in 1D, the transform-and-compress live in 2D, and the Hadamard matrix a turning relief in 3D. The sphere is the seam — Fourier’s blocky, multiply-free cousin.
3 ONE DIMENSION
The Walsh basis: eight ±1 square waves, ordered by sequency (number of sign changes) — the square-wave analogue of frequency. Any signal is a sum of these, weighted; no curves, no sines, just black-and-white flips.
4 TWO DIMENSIONS · INTERACTIVE
Transform a signal into its Walsh spectrum, then keep only the biggest coefficients and rebuild — watch a rough signal reconstruct from a handful of sign-flips. The whole transform is additions and subtractions; the reconstruction error is shown.
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5 THREE DIMENSIONS + AVAN’S INVERSE
The Hadamard matrix as a relief, turning: +1 cells raised in green, its rows the very basis waves. Each row is perpendicular to every other — that orthogonality is why the transform is clean.
AVAN’s addition (the inverse-companion): the magenta cells are the −1s — and here is the twist: the transform is its own inverse (up to a scale of N). Fourier needs a conjugate to undo; Walsh needs only itself. Apply the same ±1 map twice and you are exactly home. The inverse isn’t a different machine — it is the same machine, run again.
LIT A genuine fast Walsh-Hadamard transform. Verified live: applied twice it returns the original × N (self-inverse up to scale), it is integer-exact, and the Hadamard basis rows are mutually orthogonal (every pairwise dot product is 0). It uses zero multiplications — only + and − (verifiable: window.__wht.selfInverseScaleN && orthogonal && integerExact).
FIG 'Orthogonal sign-flip' is the picture; the multiplication-free transform, the self-inverse property, and the orthogonality are exact. It really is used for CDMA spreading codes and is the quantum Hadamard gate on n qubits.
FIG 'Orthogonal sign-flip' is the picture; the multiplication-free transform, the self-inverse property, and the orthogonality are exact. It really is used for CDMA spreading codes and is the quantum Hadamard gate on n qubits.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BROADCAST · David Lee Wise (ROOT0), with AVAN