Consider the transformation of the data vector xx: y=Q⁢x y Q x y0y1⋮yN-1=( q⁢00q⁢01…q⁢0N−1 q⁢10q⁢11…q⁢1N−1 ⋮⋮⋮⋮ q⁢N−10q⁢N−11…q⁢N−1N−1 )⁢x0x1⋮xN-1 y0 y1 ⋮ yN-1 q 0 0 q 0 1 … q 0 N 1 q 1 0 q 1 1 … q 1 N 1 ⋮ ⋮ ⋮ ⋮ q N 1 0 q N 1 1 … q N 1 N 1 x0 x1 ⋮ xN-1 y⁢k=∑nq⁢kn⁢x⁢n y k n q k n x n If QQ is an orthogonal matrix (meaning that QT⁢Q=I Q Q I ) then the data vector xx can be recovered from yy using the transpose of QQ: x=QT⁢y x Q y x0x1⋮xN-1=( q⁢00q⁢10…q⁢N−10 q⁢01q⁢11…q⁢N−11 ⋮⋮⋮⋮ q⁢0N−1q⁢1N−1…q⁢N−1N−1 )⁢y0y1⋮yN-1 x0 x1 ⋮ xN-1 q 0 0 q 1 0 … q N 1 0 q 0 1 q 1 1 … q N 1 1 ⋮ ⋮ ⋮ ⋮ q 0 N 1 q 1 N 1 … q N 1 N 1 y0 y1 ⋮ yN-1 x⁢n=∑kq⁢kn⁢y⁢k x n k q k n y k

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This work is licensed by Ivan Selesnick under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Charlet Reedstrom on Aug 12, 2005 2:12 pm -0500.