Page 132 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

120 Chapter 3

information regarding high frequency content of the signal, which is called
detail co-efficients. The wavelet transformation of the signal using matrix
method is described below.

14.1 Haar Transformation

Consider the signal with number of samples N. Form the Haar matrix with
size NXN and with the diagonal matrices filled up with the matrix given
below.

½ ½
½ - ½

For N=8, the matrix obtained is

½ ½ 0 0 0 0 0 0
½ - ½ 0 0 0 0 0 0
0 0 ½ ½ 0 0 0 0
TM1 0 0 ½ -½ 0 0 0 0
0 0 0 0 ½ ½ 0 0

0 0 0 0 ½ -½ 0 0
0 0 0 0 0 0 ½ ½
0 0 0 0 0 0 ½ -½

Multiply the matrix TM1 with the is the signal data [d0 d1 d2 d3 d4 d5d6
d7 ] to obtain first level decomposition of Haar transformation.

I level Haar decomposition of the signal

[½ (d0+d1) ½ (d2-d3) ½ (d4+d5) ½ (d6-d7) ]

The samples ½ (d0+d1) ½ (d4+d5) are the average samples. They are
called approximated co-efficients of the signal at the first level. The samples
½ (d2-d3) ½ (d6-d7) are called detail co-efficients of the signal at the first
level.

Thus Approximation 1 = [½ (d0+d1) ½(d4+d5)]

Detail 1= [½ (d2-d3) ½(d6-d7) ]