Biomedical Engineering Reference
In-Depth Information
with the use of a window function
w
(
t
) into the Fourier transform [14]:
+∞
f
(
τ
)
w
(
t
−
τ
)
e
−
i
ω
t
d
τ.
Sf
(
ω,
t
)
=
(6.3)
−∞
The energy of the basis function
g
τ,ξ
(
t
)
=
w
(
t
−
τ
)
e
−
i
ξ
t
is concentrated in the
neighborhood of time
τ
over an interval of size
σ
t
, measured by the standard
2
. Its Fourier transform is
g
τ,ξ
(
ω
)
=
w
(
ω
−
ξ
)
e
−
i
τ
(
ω
−
ξ
)
, with en-
ergy in frequency domain localized around
ξ
, over an interval of size
σ
ω
.Ina
time-frequency plane (
t
,ω
), the energy spread of what is called the atom
g
τ,ξ
(
t
)
is represented by the Heisenberg rectangle with time width
σ
t
and frequency
width
σ
ω
. The uncertainty principle states that the energy spread of a function
and its Fourier transform cannot be simultaneously arbitrarily small, verifying:
deviation of
|
g
|
1
2
.
(6.4)
σ
t
σ
ω
≥
The shape and size of Heisenberg rectangles of a WFT determine the spatial and
frequency resolution offered by such transform.
Examples of spatial-frequency tiling with Heisenberg rectangles are shown in
Fig. 6.1. Notice that for a windowed Fourier transform, the shapes of the time-
frequency boxes are identical across the whole time-frequency plane, which
means that the analysis resolution of a windowed Fourier transform remains
the same across all frequency and spatial locations.
To analyze transient signal structures of various supports and amplitudes in
time, it is necessary to use time-frequency atoms with different support sizes
for different temporal locations. For example, in the case of high-frequency
structures, which vary rapidly in time, we need higher temporal resolution to
accurately trace the trajectory of the changes; on the other hand, for lower
frequency, we will need a relatively higher absolute frequency resolution to give
a better measurement of the value of frequency. We will show in the next section
that wavelet transform provides a natural representation which satisfies these
requirements, as illustrated in Fig. 6.1(d).
6.2.1 Continuous Wavelet Transform
A wavelet function is defined as a function
ψ
∈
L
2
(
R
) with a zero average [3,
14]:
+∞
ψ
(
t
)
dt
=
0
.
(6.5)
−∞
