Biomedical Engineering Reference
In-Depth Information
However, because the original DWT is basically a noncausal process, a wavelet transform featur-
ing causality should be considered. For this purpose, we develop here the à trous wavelet transform
[
60
] to implement a causal DWT. With this procedure, multiresolution spike train analysis can be
regarded as binning spike trains with multiscale windows.
2.7.3 Multiresolution analysis of Neural Spike Trains
To facilitate the computation, we apply the DWT using dyadic Haar wavelets, which when inte-
grated with the à trous wavelet transform, yields very effective DSP implementations. The Haar
wavelet is the simplest form of wavelet and was introduced in the earliest development of wavelet
transform [
59
]. Here, we only introduce the functional form of the Haar wavelets, whereas the de-
tails can be found in Reference [
59
]. Let us first define the Haar scaling function as
ì
1
if [ , )
otherwise.
x
Î
0 1
í
ï
ï
î
ï
ï
(2.4)
f
( )
x
=
0
Let
V
j
be the set of functions of the form
å
j
(2.5)
a
f
(
2 -
x
k
),
k
k
where
a
k
is a real number and
k
belongs to the integer set.
a
k
is nonzero for only a finite set of
k
.
V
j
is
the set of all piecewise constant functions whose supports are finite, where discontinuities between
these functions belong to a set,
{
}
2
2
1
2
1
2
2
2
(2.6)
,
- -
,
, ,
0
,
,
.
j
j
j
j
V
⊂
V
⊂
V
⊂
Note that
. The Haar wavelet function ψ is defined by
0
1
2
ψ
( )
x
=
φ
(
2
x
)
−
φ
(
2
x
−
1
).
(2.7)
If we define
W
j
as the set of functions of the form
(2.8)
∑
ψ
(
W
=
a
2
j
x
−
k
),
j
k
k
then it follows that
