Biomedical Engineering Reference
In-Depth Information
include spectral analysis [49], wavelet analysis [50], and matching pursuit [51].
The following sections discuss some of the more recent time-frequency analysis
techniques used to automatically detect the sleep state from EEG using the afore-
mentioned approaches.
10.15.1 Wavelet Analysis
Wavelet analysis is a generalization of the short-term Fourier transform that allows
for basis functions that are more general than a sine or cosine wave. Rather than
considering certain frequency bands present in a given stage of sleep, these can be
used to determine the existence of certain physiological wave forms (K complexes or
sleep spindles) that are introduced in certain stages of sleep [50, 52]. Akin and
Akgul attempt to detect sleep spindles by using a discrete wavelet transform [52].
The discrete wavelet transform is similar to the Fourier transform in that it will
break up any time-varying signal into smaller uniform functions, known as the
basis
functions. The basis functions are created by scaling and translating a single func-
tion of a certain form. This function is known as the Mother wavelet. In the case of
the Fourier transform, the basis functions used are sine and cosine waves of varying
frequency and magnitude. Since a cosine wave is just a sine wave translated by
/2
radians; the mother wavelet in the case of the Fourier transform could be considered
to be the sine wave.
However, for a wavelet transform the basis functions are more general. The
only requirements for a family of functions to be a basis are that the functions are
both complete and orthonormal under the inner product. Consider the family of
functions
π
}, where each
i
value specifies a different scale and each
j
value specifies a different translation based on some mother wavelet function.
Ψ=
{
Ψ
ij
|
−∞ <
i,j
<∞
is
considered to be complete if any continuous function
f
, defined over the real line
x
,
can be defined by some combination of the functions in
Ψ
Ψ
as shown in (10.4) [44]:
∞
∑
()
()
fx
=
c
Ψ
x
(10.4)
ij
ij
ij
,
=−∞
For a family of functions to be orthonormal under the inner product, they must
meet two criteria: It must be the case that for any
i
,
j
,
l,
and
m
where
i
≠
l
and
j
≠
m
that <
1, where <
f
,
g
> is the inner product, defined as in
(10.5), and
f
(
x
)
*
is the complex conjugate of
f
(
x
):
ψ
ij
,
ψ
lm
>
=
0 and <
ij
,
ij
>
=
∞
()()
*
fg
,
=
−∞
∫
f x gxdx
(10.5)
The wavelet basis is very similar to the Fourier basis, with the exception that the
wavelet basis does not have to be infinite. In a wavelet transform the basis functions
can be defined over a certain window and then be zero everywhere else. As long as
the family of functions defined by scaling and translating the mother wavelet is
orthonormally complete, that family of functions can be used as the basis. With the
Fourier transform, the basis is made up of sine and cosine waves that are defined
over all values of
x
where
−∞ <
x
<∞
.
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