Biomedical Engineering Reference
In-Depth Information
f
1
(
t
)
t
r
xy
(t)
0 = O
t =
t'
f
2
(
t
) =
f
1
(
t - t'
)
0 = O
t
0 = O
t
=
t'
(a)
(b)
FIGURE 17.3
:
Cross-correlation function. Trace (b) is the cross-spectral estimate of the two
traces in (a). The bottom trace in (a) is the time shifted upper trace
and phase of a system's time response. The next step is to look at the equivalents of these
functions in the frequency domain where computation will be simplified.
17.5 SPECTRAL DENSITY FUNCTIONS
Spectral density functions can be derived in several ways. One method takes the direct
Fourier transform of previously calculated autocorrelation and cross-correlation functions
to yield the two-sided spectral density functions given in (17.6).
∞
)
e
−
j
2
π
f
τ
d
S
xx
(
f
)
=
R
xx
(
τ
τ
−∞
∞
)
e
−
j
2
π
f
τ
d
S
yy
(
f
)
=
R
yy
(
τ
τ
−∞
∞
)
e
−
j
2
π
f
τ
d
S
xy
(
f
)
=
R
xy
(
τ
τ
(17.6)
−∞
These integrals always exist for finite intervals. The quantities
S
xx
(
f
) and
S
yy
(
f
)
are the autospectral density functions of signals
x
(
t
) and
y
(
t
) respectively, and
S
xy
(
f
)is
the cross-spectral density function between
x
(
y
) and
y
(
t
). These quantities are related
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