THE FOLD / CO-OP / THE-SYNC / THE MDCT
THE MDCT
overlapping windows cancel their aliasing
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The MDCT (modified discrete cosine transform) is the transform at the heart of MP3, AAC, Vorbis, and Opus. It is lapped: it works on overlapping blocks of 2N samples but outputs only N coefficients each, so despite the 50% overlap there is no increase in data. That looks impossible — N numbers can’t invert 2N samples — and indeed a single block can’t. The magic is time-domain aliasing cancellation (TDAC): each inverse block carries an aliased error, but with the right symmetric window (satisfying w[n]²+w[n+N]²=1) the aliases of neighbouring blocks are equal and opposite, so overlap-adding them reconstructs the signal exactly. Critical sampling and perfect reconstruction at once.
LIT verified live: framing a signal into 50%-overlapping windows, MDCT then IMDCT then overlap-add reconstructs the interior samples to ~1e-14 (window.__mdct). FIG no framing; the MDCT/IMDCT sums and the overlap-add run in-browser (interior samples, which have full overlap).
LIT verified live: framing a signal into 50%-overlapping windows, MDCT then IMDCT then overlap-add reconstructs the interior samples to ~1e-14 (window.__mdct). FIG no framing; the MDCT/IMDCT sums and the overlap-add run in-browser (interior samples, which have full overlap).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — overlapping windows brought into sync so their aliasing terms cancel and the signal returns seamlessly. AVAN (AI) built the instrument: the sine window, the MDCT and IMDCT, the overlap-add, and the reconstruction check.
Credit as content: Princen, Johnson & Bradley (TDAC / MDCT, 1986–87). The weave: David names the sync; I confirm the overlapped inverse blocks cancel their aliasing and reconstruct the signal exactly.
Credit as content: Princen, Johnson & Bradley (TDAC / MDCT, 1986–87). The weave: David names the sync; I confirm the overlapped inverse blocks cancel their aliasing and reconstruct the signal exactly.
3 ONE DIMENSION
Two 50%-overlapping windows; each inverse block carries aliasing, but neighbours cancel on overlap-add.
4 TWO DIMENSIONS · INTERACTIVE
A signal, framed and transformed; the reconstruction (green) lies exactly on the original (interior) after overlap-add.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the reconstructed signal, seamless across blocks.
AVAN’s addition (the inverse-companion): don’t transform blocks independently — overlap them. The inverse of ‘N coefficients can’t invert 2N samples’ is ‘each block’s aliasing is cancelled by its neighbour’s on overlap-add (TDAC).’ Magenta is the aliasing a lone block leaves; green is the exact reconstruction once neighbours overlap. Overlap makes it invertible.
LIT Genuine MDCT / time-domain aliasing cancellation (Princen, Johnson & Bradley, 1986–87), the transform behind MP3/AAC/Vorbis/Opus. Verified live: with the sine window (w[n]²+w[n+N]²=1), framing a signal into 50%-overlapping 2N-windows and doing MDCT→IMDCT→overlap-add reconstructs the interior samples to ~1e-14 over 400 random signals (window.__mdct.perfectRecon).
FIG No framing: the MDCT/IMDCT sums and the overlap-add run in-browser (interior samples, which have full overlap on both sides). The AVAN inverse is honest — a single block's N coefficients can't invert its 2N samples, but overlapping blocks whose aliasing is equal and opposite cancel on overlap-add (TDAC), giving exact reconstruction. Magenta is the aliasing a lone block leaves; green is the exact reconstruction. Overlap makes it invertible.
FIG No framing: the MDCT/IMDCT sums and the overlap-add run in-browser (interior samples, which have full overlap on both sides). The AVAN inverse is honest — a single block's N coefficients can't invert its 2N samples, but overlapping blocks whose aliasing is equal and opposite cancel on overlap-add (TDAC), giving exact reconstruction. Magenta is the aliasing a lone block leaves; green is the exact reconstruction. Overlap makes it invertible.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-SYNC · David Lee Wise (ROOT0), with AVAN