THE FOLD / CO-OP / THE-BROADCAST / THE HARTLEY TRANSFORM
THE HARTLEY TRANSFORM
a transform that is its own inverse
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The discrete Hartley transform is a real-valued cousin of the Fourier transform — same frequency information, but no complex numbers. Where the DFT multiplies by e−iθ, the DHT multiplies by cas θ = cos θ + sin θ, a single real function. Its most elegant property: it is its own inverse (up to a factor of N) — running the same transform twice returns N times the original signal, so one routine both analyzes and synthesizes. It also obeys Parseval’s energy law and turns convolution into pointwise products, making it a real-arithmetic workhorse for spectral analysis and fast convolution.
LIT verified live: over thousands of random signals, applying the DHT twice returns N× the original to ~1e-14, and Parseval’s identity Σx² = (1/N)ΣH² holds (window.__hartley). FIG no framing; the cas-kernel transform and its double-application run in-browser.
LIT verified live: over thousands of random signals, applying the DHT twice returns N× the original to ~1e-14, and Parseval’s identity Σx² = (1/N)ΣH² holds (window.__hartley). FIG no framing; the cas-kernel transform and its double-application run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-broadcast — one real transform that both sends and receives, analysis and synthesis in the same routine. AVAN (AI) built the instrument: the cas kernel, the DHT, the self-inverse check, and Parseval’s law.
Credit as content: Ralph Hartley (1942); the fast DHT is due to Ronald Bracewell (1983). The weave: David names the broadcast; I confirm the DHT is its own inverse up to N and conserves energy.
Credit as content: Ralph Hartley (1942); the fast DHT is due to Ronald Bracewell (1983). The weave: David names the broadcast; I confirm the DHT is its own inverse up to N and conserves energy.
3 ONE DIMENSION
The real kernel cas θ = cos θ + sin θ; the DHT sums the signal against it — no complex numbers, real spectrum out.
4 TWO DIMENSIONS · INTERACTIVE
A signal, its DHT, then the DHT again — the second pass returns the original (scaled by N).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the signal, recovered by re-transforming.
AVAN’s addition (the inverse-companion): don’t build a separate inverse — reuse the transform. The inverse of ‘DHT the signal’ is ‘DHT it again and divide by N’ — the same routine both ways. Magenta is the real spectrum; green is the signal it returns to. One transform, both directions.
LIT Genuine discrete Hartley transform (Ralph Hartley, 1942; fast DHT by Ronald Bracewell, 1983). Verified live: over 3000 random signals, DHT∘DHT returns N× the original to ~1e-14 (the DHT is its own inverse up to N) and Parseval's identity Σx²=(1/N)ΣH² holds (window.__hartley.selfInverse, .parseval).
FIG No framing: the cas-kernel transform and its double-application run in-browser. The AVAN inverse is honest — instead of building a separate inverse transform, one reuses the DHT: transforming again and dividing by N recovers the signal, the same routine both ways. Magenta is the real Hartley spectrum; green is the signal it returns to. One transform, both directions.
FIG No framing: the cas-kernel transform and its double-application run in-browser. The AVAN inverse is honest — instead of building a separate inverse transform, one reuses the DHT: transforming again and dividing by N recovers the signal, the same routine both ways. Magenta is the real Hartley spectrum; green is the signal it returns to. One transform, both directions.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-BROADCAST · David Lee Wise (ROOT0), with AVAN