Biomedical Engineering Reference
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L 2 f
h [ -n ]
2
L 1 f
2
h [ -n ]
f(n )
W 2 f
g [ -n ]
2
g [ -n ]
2
W 1 f
2
downsampling by 2
Figure 6.2: Illustration of orthogonal wavelet transform of a discrete signal
f ( n ) with CMF. A two-level expansion is shown.
Wavelet coefficients W j f at scale s = 2 j have a length of N / 2 j and the largest
decomposition depth J is bounded by the signal length N as (sup( J ) = log 2 N ).
For fast implementation (such as filter bank algorithms), a pair of conjugate
mirror filters (CMF) h and g can be constructed from the scaling function φ and
wavelet function ψ as follows:
1
2 φ
t
2
( t n )
1
2 ψ
t
2
( t n )
h [ n ] =
and
g [ n ] =
(6.12)
.
A conjugate mirror filter k satisfies the following relation:
k ( ω )
k ( ω + π )
2
2
k (0) = 2 .
+
= 2
and
(6.13)
It can be proven that h is a low-pass filter and g is a high-pass filter. The discrete
orthogonal wavelet decomposition in Eq. (6.11) can be computed by applying
these two filters to the input signal and recursively decomposing the low-pass
band, as illustrated in Fig. 6.2. A detailed proof can be found in [15].
For orthogonal basis, the input signal can be reconstructed from wavelet
coefficients computed in Eq. (6.11) using the same pair of filters, as illustrated
in Fig. 6.3.
2
h
2
h
f(n)
L 2 f
L 1 f
2
g
2
g
W 2 f
W 1 f
2
upsampling by 2
Figure 6.3: Illustration of inverse wavelet transform implemented with CMF. A
two-level expansion is shown.
 
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