Biomedical Engineering Reference
In-Depth Information
3. The signal is also lowpass filtered with filter
H
(
e
j
) and downsampled by 2.
The resultant sequence is the “smoothed” coefficients
C
1
of length
N
/2. The
bandwidth of
C
1
sequence is (0,
f
s
/4).
4. The smoothed sequence
C
1
is further highpass filtered with filter
G
(
e
j
) and
downsampled by 2, and lowpass filtered with filter
H
(
e
j
) and downsampled
by 2, to generate
D
2
and
C
2
of length
N
/4. The bandwidth of
C
2
sequence is
(0,
f
s
/8) and of
D
1
sequence is (
f
s
/8,
f
s
/4).
5. The process of lowpass filtering, highpass filtering, and downsampling is
repeated until the required resolution
j
is reached.
The signal
x
(
n
) can be reconstructed again with the preceding coefficients using
the following formula:
()
()
∑
∑
∑
xn
=
C
⋅
φ
+
D
⋅
ψ
t
(3.39)
jk
,
jk
,
jk
,
jk
,
j
j
k
MATLAB provides several MRWA functions:
[C,L] = wavedec(x,N,'wname')
returns the wavelet decomposition of the signal
x
at level
N
, using
'wname'
. Note
that
N
must be a strictly positive integer. Several wavelets are available in MATLAB
including Haar, Daubechies, biorthogonal, Coiflets, Symlets, Morlet, Mexican hat,
and Meyer. The function
x waverec(C,L,'wname')
reconstructs the signal
x
based
on the multilevel wavelet decomposition structure
[C,L]
and wavelet
'wname'
.
For an EEG sampled at 250 Hz, a five-level decomposition results in a good
match to the standard clinical bands of interest [20]. The basis functions of the
wavelet transform should be able to represent signal features locally and adapt
to slow and fast variations of the signal. Another requirement is that the wavelet
functions should satisfy the finite support constraint and differentiability to recon-
struct smooth changes in the signal symmetry to avoid phase distortions [20, 27,
28]. Figure 3.20 shows the MRWA of the 4,096-point EEG data segment described
earlier and shown in Figure 3.12(a). The signal is decomposed into five levels using
the Daubechies 4 wavelet.
3.2
Nonlinear Description of EEGs
Nonlinear methods of dynamics provide a useful set of tools for the analysis of EEG
signals, which by their very nature are nonlinear. Even though these methods are
less well understood than their linear counterparts, they have proven to generate
new information that linear methods cannot reveal, for example, about nonlinear
interactions and the complexity and stability of underlying brain sites [38]. We sup-
port this assertion by applying some of the well-known methods to EEGs and epi-
lepsy in this chapter. For a reader to further understand and develop an intuition for
these approaches, it is advisable to apply them to simulations with known,
well-defined coupled nonlinear systems. Such systems exist, for example, the logis-
tic and Henon maps (discrete-time nonlinear), and the Lorenz, Rossler, and
Mackey-Glass systems (continuous-time nonlinear).
The dynamics of highly complex, nonlinear systems in nature [53], medicine
[54, 55], and economics [56] has been of much scientific interest recently. A strong
Search WWH ::
Custom Search