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wavelet analysis & data compression 281(2) applies it to the (N/2)-length smooth vector, (3) and then repeats until twosmooth components remain. (4) After each filtering, the elements are ordered, withthe newest two smooth elements on top, the newest detailed elements below, andthe older detailed elements below that. (5) The process continues until there arejust two smooth elements left.To illustrate, here we filter and reorder an initial vector of length N =8:⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛s (1)y1s (1)1s (2)1s (2)11y 2d (1)1s (1)2d (2)1s (2)2⎜y 3 ⎟⎜s (1)2 ⎟ ⎜s (1)3 ⎟ ⎜s (2)2 ⎟ ⎜d (2)1 ⎟⎞y 4y 5y 6⎜ ⎟⎝y 7 ⎠y 8filter−→⎜⎝d (1)2s (1)3d (1)3s (1)4d (1)4⎟⎠order−→⎜⎝s (1)4d (1)1d (1)2d (1)3d (1)4⎟⎠filter−→⎜⎝d (2)2d (1)1d (1)2d (1)3d (1)4⎟⎠order−→⎜⎝d (2)2d (1)1d (1)2d (1)3d (1)4. (11.26)⎟⎠The discrete inversion of a transform vector back to a signal vector is made usingthe transpose (inverse) of the transfer matrix at each stage. For instance,⎛ ⎞ ⎛⎞ ⎛ ⎞y 0 c 0 c 3 c 2 c 1 Y 0y 1⎜ ⎟⎝y 2 ⎠ = c 1 −c 2 c 3 −c 0Y 1⎜⎟ ⎜ ⎟⎝c 2 c 1 c 0 c 3 ⎠ ⎝Y 2 ⎠ . (11.27)y 3 c 3 −c 0 c 1 −c 2 Y 3As a more realistic example, imagine that we have sampled the chirp signaly(t) = sin(60t 2 ) for 1024 times. The filtering process through which we place thissignal is illustrated as a passage from the top to the bottom in Figure 11.9. First theoriginal 1024 samples are passed through a single low band and a single high band(which is mathematically equivalent to performing a series of convolutions). Asindicated by the down arrows, the output of the first stage is then downsampled(the number reduced by a factor of 2). This results in 512 points from the highbandfilter as well as 512 points from the low-band filter. This produces the firstleveloutput. The output coefficients from the high-band filters are called {d (1)i }to indicate that they show details, and {s (1)i } to indicate that they show smoothfeatures. The superscript indicates that this is the first level of processing. Thedetail coefficients {d (1) } are stored to become part of the final output.In the next level down, the 512 smooth data {s (1)i } are passed through newlow- and high-band filters using a broader wavelet. The 512 outputs from each aredownsampled to form a smooth sequence {s (2)i } of size 256 and a detailed sequence} of size 256. Again the detail coefficients {d (2) } are stored to become part of the{d (2)ifinal output. (Note that this is only half the size of the previously stored details.) Theprocess continues until there are only two numbers left for the detail coefficientsand two numbers left for the smooth coefficients. Since this last filtering is done−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 281