User`s guide

Wavelet Synthesis
5-535
5Wavelet Synthesis
Purpose Reconstruct a signal from its multirate bandlimited components (requires the
Wavelet Toolbox).
Library Filtering / Multirate Filters
Description The Wavelet Synthesis block uses the wfilters function from the Wavelet
Toolbox to reconstruct a signal that was decomposed by the Wavelet Analysis
block. The reconstruction or synthesis process is the inverse of the analysis
process, and restores the original signal by upsampling, filtering, and summing
the bandlimited inputs in stages corresponding to the analysis process. An
n-level synthesis filter bank structure is shown below, where n is specified by
the
Number of levels parameter.
At each level, the two bandlimited inputs (one low-frequency, one
high-frequency, both with the same sample rate) are upsampled by a factor of 2
to match the sample rate of the input to the next stage. They are then filtered
by a highpass (HP) and lowpass (LP) filter pair with coefficients calculated to
cancel (in the subsequent summation) the aliasing introduced in the
corresponding analysis filter stage. The output from each
(upsample-filter-sum) level has twice the bandwidth and twice the sample rate
of the input to that level.
HP
LP
↑2
↑2
HP
↑2
LP
↑2
HP
↑2
LP
↑2
y
HP: highpass filter with f
c
≈ 1/2 Nyquist
LP: lowpass filter with
f
c
≈ 1/2 Nyquist
↑2: upsample by 2
u
1
u
2
u
3
u
n
HP
↑2
LP
↑2
. . .
u
n +1
Wave
l
et Synt
h
esis Fi
l
ter Ban
k
, n Leve
l
s
Σ
Σ
Σ
Σ
T
si
=(2
k
)T
s
for input u
k
, 1 ≤ k ≤ n
T
si
=(2
n
)T
s
for input u
n
+1
T
so
=T
s
2T
s
4T
s
8T
s