Biomedical Engineering Reference
In-Depth Information
FIGURE 2.14:
Ilustrative application of Choi-Williams distribution to the same
set of signals as in
Figure 2.13. a)
Linear chirp—its time-frequency representation
is a straight line, b) two Gabor functions—note the cross-term structure in the time-
frequency representation is highly reduced. c) A real world example: 10 sec. of EEG
recorded during sleep stage 2, with a clear sleep spindle visible as a transient oscil-
latory structure.
filtering kernel given by two-dimensional Gaussian function:
exp
2
f
(
ξ
,
τ
)=
−
α
(
ξτ
)
.
(2.80)
A proper selection of the kernel parameter can significantly reduce cross-terms, as
illustrated in Figure 2.14.
2.4.2.2
Time-frequency signal decompositions
2.4.2.2.1 Short time Fourier transform and spectrogram
Time-frequency rep-
resentation of the signal can also be obtained in a different manner. In this approach
one extracts successive short pieces of signal with a window function and computes
its frequency representation with a short time Fourier transform (STFT):
Z
∞
h
∗
(
e
−
i
2π
uf
du
F
x
(
t
,
f
;
h
)=
x
(
u
)
u
−
t
)
(2.81)
−
∞
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