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280 chapter 11N SamplesInputN/2(1)cCoefficientsLHN/2(1)dCoefficientsN/4(2)cCoefficientsLHN/4(2)dCoefficientsN/8(3)cCoefficientsLHN/8(3)dCoefficients2(n)cCoefficientsLH2(n)dCoefficientsFigure 11.9 The original signal is processed by high- and low-band filters, and the outputsare downsampled with every other point kept. The process continues until there are onlytwo output points of high-band filtering and two points of low-band filtering. The totalnumber of output data equals the total number of signal points. It may be easier tounderstand the output of such an analysis as the signal passed through various filtersrather than as a set of Fourier-like coefficients.the integrations or convolutions. Then there is a decimation of the output by onehalf,and finally an interleaving of the output for further filtering. This processsimultaneously cuts down on the number of points in the data set and changesthe scale and the resolution. The decimation reduces the number of values of theremaining signal by one half, with the low-frequency part discarded because thedetails are in the high-frequency parts.As indicated in Figure 11.9, the pyramid algorithm’s DWT successively (1)applies the (soon-to-be-derived) c matrix (11.35) to the whole N-length vector,⎛ ⎞ ⎛⎞ ⎛ ⎞Y 0 c 0 c 1 c 2 c 3 y 0Y 1⎜ ⎟⎝Y 2 ⎠ = c 3 −c 2 c 1 −c 0y 1⎜⎟ ⎜ ⎟⎝c 2 c 3 c 0 c 1 ⎠ ⎝y 2 ⎠ , (11.25)Y 3 c 1 −c 0 c 3 −c 2 y 3−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 280