Digital Signal Processing Reference
In-Depth Information
Z
W
f
(a;b) =
f(t) (at + b)dt
where f(t) is the function to analyze, is the wavelet, and (at + b) is the shifted
and scaled version of the wavelet at time b and scale a [7]. An alternate form of the
equation is:
Z
1
1
f(t)
p
W
f
(s;u) =
s (s(tu))dt
p
again where is the wavelet, while the wavelet family is shown above as
s (s(t
u)), shifted by u and scaled by s. We can rewrite the wavelet transform as an inner
product [1]:
p
W
f
(s;u) =
f(t);
s (s(tu))
:
This inner product is essentially computed by the lters.
So far, this background focuses on how to get the wavelet transform given a
wavelet, but how does one get the wavelet coecients and implement the transform
with lters? The relationship between wavelets and the lter banks that imple-
ment the wavelet transform is as follows. The scaling function, (t), is determined
through recursively applying the lter coecients, since multiresolution recursively
convolutes the input vector after shifting and scaling. All the information about the
scaling and wavelet functions is found by the coecients of the scaling function and
of the wavelet function, respectively [3]. The scaling function is given as:
X
p
(t) =
2
h[k](2tk):
k
The wavelet function is given as:
X
p
(t) =
2
g[k](2tk):
k
There is a nite set of coecients h[k]. Once these coecients are found, allowing
us to design the lowpass lter, then the highpass lter coecients are easy to nd.
Daubechies [2] chose the following coecients, with many desirable (and required)
properties, and solved the above equations with them.
Lowpass (scaling) coecients: [3]
