Page 139 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

3. Numerical Linear Algebra 127

14.2 Daubechies- 4 Transformation

Consider the signal consists of N samples. Form the Daubechies
Transformation Matrix [DTM 1] with diagonal matrices filled up with the
matrix ‘D’ given below.

[(1+√3) / 4√2] [(3+√3) / 4√2] [(3-√3) / 4√2] [(1-√3) / 4√2]

[(1-√3) / 4√2] - [(3-√3) / 4√2] [(3+√3) / 4√2] (1+√3) / 4√2]
-

For simplicity matrix D is represented as follows.

a0 a1 a2 a3
D =
b0 b1 b2 b3

For N=8, the matrix ‘DTM1’ is formed as given below.

a0 a1 a2 a3 0 0 0 0 0 0
b0 b1 b2 b3 0 0 0 0 0 0
0 0 a0 a1 a2 a3 0 0 0 0
0 0 b0 b1 b2 b3 0 0 0 0
0 0 0 0 a0 a1 a2 a3 0 0
0 0 0 0 b0 b1 b2 b3 0 0
0 0 0 0 0 0 a0 a1 a2 a3
0 0 0 0 0 0 b0 b1 b2 b3

Note that the matrix ‘D’ are arranged with overlapping in the diagonal of
the matrix ‘DTM 1’. Compare with the Haar transformation in which the
matrix are arranged without overlapping. Also note that the size of the
matrix is of size 8X10 for the signal of size 1x8. So 1x8 sized signal is
th
st
th
extended to the size 1x10 with 9 and 10 samples are filled up with 1 and
nd
2 samples respectively. [Assuming that the signals are repeated cyclically].
The steps involved in obtaining the approximation and detail co-
efficients are as described in the section 14.1 for Haar transformation.