Multiresolution Analysis and Approximation of Audio Signal: the Haar wavelet

Here we show a finite discrete signal and its multi-resolution analysis of level p=1,2,3,4,5,6,7,8,9.10 based on the Haar wavelet, which is also called the Daubechies 1 wavelet and the simplest wavelet function. A sigal is an element of finite dimensional vector space of real numbers, which is decomposed as a direct sum of subspaces the form

R^N = (level p approximation) + (level p detail) + ... + (level 2 detail) + (level 1 detail).
Higher level subspace consist of elements of lower frequcny, and the approximation is the lowest frequency part.

Level 7 Decomposition
Audio Signal	Basic Information
Level 1 Approximation	Level 1 Detail
Level 2 Approximation	Level 2 Detail
Level 3 Approximation	Level 3 Detail
Level 4 Approximation	Level 4 Detail
Level 5 Approximation	Level 5 Detail
Level 6 Approximation	Level 6 Detail
Level 7 Approximation	Level 7 Detail
Level 8 Approximation	Level 8 Detail
Level 9 Approximation	Level 9 Detail
Level 10 Approximation	Level 10 Detail