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wavelet analysis & data compression 283on just the smooth (low-band) part. The resulting output is similar, but with coarserfeatures for the smooth coefficients and larger values for the details. Note that inthe upper graphs we have connected the points to make the output look continuous,while in the lower graphs, with fewer points, we have plotted the output ashistograms to make the points more evident. Eventually the downsampling leadsto just two coefficients output from each filter, at which point the filtering ends.To reconstruct the original signal (called synthesis or transformation) a reversedprocess is followed: Begin with the last sequence of four coefficients, upsamplethem, pass them through low- and high-band filters to obtain new levels of coefficients,and repeat until all the N values of the original signal are recovered. Theinverse scheme is the same as the processing scheme (Figure 11.9), only now thedirection of all the arrows is reversed.11.5.2 Daubechies Wavelets via FilteringWe should now be able to understand that digital wavelet analysis has been standardizedto the point where classes of wavelet basis functions are specified notby their analytic forms but rather by their wavelet filter coefficients. In 1988, theBelgian mathematician Ingrid Daubechies discovered an important class of suchfilter coefficients [Daub 95]. We will study just the Daub4 class containing the fourcoefficients c 0 , c 1 , c 2 , and c 3 .Imagine that our input contains the four elements {y 1 ,y 2 ,y 3 ,y 4 } correspondingto measurements of a signal at four times. We represent a lowpass filter L and ahighpass filter H in terms of the four filter coefficients asL = ( +c 0 +c 1 +c 2 +c 3)(11.28)H = ( +c 3 −c 2 +c 1 −c 0). (11.29)To see how this works, we form an input vector by placing the four signal elementsin a column and then multiply the input by L and H:⎛ ⎞⎛ ⎞y 0y 0L ⎜y 1⎟⎝y 2⎠ = ( )+c 0 +c 1 +c 2 +c 3⎜y 1⎟⎝y 2⎠ = c 0y 0 + c 1 y 1 + c 2 y 2 + c 3 y 3 ,y 3 y 3⎛ ⎞⎛ ⎞y 0y 0H ⎜y 1⎟⎝y 2⎠ = ( )+c 3 −c 2 +c 1 −c 0⎜y 1⎟⎝y 2⎠ = c 3y 0 − c 2 y 1 + c 1 y 2 − c 0 y 3 .y 3 y 3We see that if we choose the values of the c i ’s carefully, the result of L acting on thesignal vector is a single number that may be viewed as a weighted average of thefour input signal elements. Since an averaging process tends to smooth out data,the lowpass filter may be thought of as a smoothing filter that outputs the generalshape of the signal.−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 283