The Wavelet Transform - DSP Using MATLAB and Wavelets - page 317

We can nish the reconstruction for the second octave by looking at the result when
w2 f [n] and z2 f [n] are added together to recreate w d [n]. We will call this recreated
signal w d [n]

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, until we are certain that it is the same as w d [n].

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w d [n] = w2 f [n] + z2 f [n]

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w d [n] = dw2[n] + bw2[n2]az2[n]cz2[n2]; n is even-even

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w d [n] = cw2[n1] + aw2[n3] + bz2[n1] + dz2[n3]; n is odd-even

The important thing to do here is to show that the two signals marked w d [n] in
Figure 9.22 are exactly the same. This means nding the previous expression in
terms of w[n] only. First, we will take the expressions for w2[n] and z2[n], and nd
w2[nk] and z2[nk].

w2[n] = aw d [n] + bw d [n1] + cw d [n2] + dw d [n3]
z2[n] = dw d [n]cw d [n1] + bw d [n2]aw d [n3]
w2[nk] = aw d [nk] + bw d [nk1] + cw d [nk2] + dw d [nk3]
z2[nk] = dw d [nk]cw d [nk1] + bw d [nk2]aw d [nk3]

Now, we can plug these into the previous w d [n] expressions.

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w d [n] = d(aw d [n] + bw d [n1] + cw d [n2] + dw d [n3])

+ b(aw d [n2] + bw d [n3] + cw d [n4] + dw d [n5])
a(dw d [n]cw d [n1] + bw d [n2]aw d [n3])
c(dw d [n2]cw d [n3] + bw d [n4]aw d [n5]); n is even-even

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w d [n] = c(aw d [n1] + bw d [n2] + cw d [n3] + dw d [n4])

+ a(aw d [n3] + bw d [n4] + cw d [n5] + dw d [n6])
+ b(dw d [n1]cw d [n2] + bw d [n3]aw d [n4])
+ d(dw d [n3]cw d [n4] + bw d [n5]aw d [n6]); n is odd-even

Rewriting these expressions and lining them up into columns gives us: