Haar: where the book changes
The basis that localizes, sister of Walsh
Walk the Haar coefficients by scale and position and see in which stretches of the book the changes concentrate.
The Walsh transform decomposes a signal according to which frequency (which subset of lines) it lives in. Haar's asks something else: where it changes. Its matrix is built by recursion: one constant row (the mean) and, for each scale and position, a wave that is +1 on the first half of its stretch and −1 on the second. Unlike Walsh, its rows have different norms (2 raised to the scale), which is why Parseval is written with those norms.
Applied to the King Wen sequence (position → hexagram number, the same signal as Walsh's and as the Fourier of the ring), the coefficients mark in which stretches of the book the irregularities gather.
rows = scales (finer at the bottom) · green = positive coefficient, blue = negative · intensity = magnitude · box = the largest
| scale | stretch of the book | coefficient | energy c²/norm² |
|---|---|---|---|
| 2^2 | v [48, 64) (Lín … Qián) | +198 | 2450 |
| 2^2 | v [0, 16) (Kūn … Dùn) | −180 | 2025 |
| 2^4 | v [16, 20) (Shī … Huàn) | −77 | 1482 |
| 2^5 | v [54, 56) (Duì … Lǚ) | +48 | 1152 |
| 2^4 | v [48, 52) (Lín … Zhōng Fú) | −61 | 930 |
| 2^5 | v [62, 64) (Guài … Qián) | +42 | 882 |
The two largest live in the scale of the blocks (width 16): the stretches [0, 16) and [48, 64), the ends of the book by upper trigram. The irregularities of King Wen are not fine: they are in its block structure.
The same King Wen signal, read in three orthogonal geometries of 64 points: Walsh is the cube (in which frequency), the Fourier of the ring is the circle (in which cyclic harmonic) and Haar is the localisation (where it changes). Three different questions about the same sequence.
The Haar basis is the other classical orthogonal basis of 64 points, sister of Walsh: its 64 rows are orthogonal (the Gram matrix is diagonal) though of different norms, the reconstruction is exact and Parseval holds with those norms.
verificar_haarApplied to the King Wen sequence, the Haar coefficients localise where the book changes; the largest live in the scale of the trigram blocks (the stretches 0 to 16 and 48 to 64) and are frozen as fixed values. It is the same signal as Walsh's and as the Fourier of the ring, on the line 0..63.
verificar_haarThe types of claim and the full bibliography (APA) are in Foundations.