Digital Signal Processing Reference
In-Depth Information
useful information. Fourth, the scaling function allows us to approximate any given
signal with a variable amount of precision [7]. The scaling function, h, gives us an
approximation of the signal via the following equation. This is also known as the
lowpass output:
N1
X
W[j;n] =
W[j1;m]h[2nm]:
m=0
The wavelet function gives us the detail signal, also called highpass output:
N1
X
W
h
[j;n] =
W[j1;m]g[2nm]:
m=0
The n term gives us the shift, the starting points for the wavelet calculations.
The index 2nm incorporates the scaling, resulting in half the outputs for octave
j compared to the previous octave j1.
Example:
Suppose g =f
1
p
1
p
2
g, and x =f34; 28; 76; 51; 19; 12; 25; 43g. Find the detail signal
produced by the wavelet transform on this signal.
2
;
Answer:
W[0;n] = x[n]
W
h
[1;n] = W[0; 0]g[2n0] + W[0; 1]g[2n1] + W[0; 2]g[2n2] + W[0; 3]g[2n3]
+W[0; 4]g[2n4] + W[0; 5]g[2n5] + W[0; 6]g[2n6] + W[0; 7]g[2n7]
For this example, the value of g with any index value outside of [0, 1] is zero.
Also, the value of W[j;n] = 0 for any value of n less than 0 or greater or equal to
N. So we can nd the individual values for W
h
as follows.
Looking at octave 1:
W
h
[1;n] = 34g[2n0] + 28g[2n1] + 76g[2n2] + 51g[2n3] + 19g[2n4]
+12g[2n5] + 25g[2n6] + 43g[2n7]
