Biomedical Engineering Reference
In-Depth Information
FIGURE 2.11:
Transfer function of the AR model with four rhythms as a sys-
tem of parallel filters each of them corresponding to a characteristic frequency (see
equation 2.63).
2.4 Non-stationary signals
2.4.1 Instantaneous amplitude and instantaneous frequency
In case of stationary signals the concept of amplitude and frequency spectra is
intuitive. For a non-stationary process it is more difficult to determine signal ampli-
tude or frequency, since these quantities can vary in time, so it is useful to introduce
instantaneous amplitude and frequency. To define the instantaneous amplitude and
instantaneous frequency, first we need to introduce the concept of analytic signal.
Analytic signal is a complex signal
x
a
(
t
)
related to a real signal
x
(
t
)
by formula:
x
a
(
t
)=
x
(
t
)+
ix
h
(
t
)
(2.64)
where
x
h
. The Hilbert transform is a linear operator
that can be thought of as the convolution of
x
(
t
)
is the Hilbert transform of
x
(
t
)
1
π
t
.The
(
t
)
with the function
h
(
t
)=
spectral power for
x
a
is nonzero only for positive
f
.
Thus approximation of the analytic signal can be obtained from the following
algorithm:
(
t
)
1. Calculate the FFT of the input sequence
x
consisting of
n
samples:
X
=
(
x
)
fft
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