Biomedical Engineering Reference
In-Depth Information
G(f)
S(f)
f
0
FIGURE 17.4
:
One- and two-sided autospectrum
as in (17.7).
H
(
f
)
2
S
yy
(
f
)
=
S
xx
(
f
)
S
xy
(
f
)
=
H
(
f
)
S
xx
(
f
)
(17.7)
In terms of physically measurable one-sided spectral density functions where
f
varies over (0,
∞
), the results are given in (17.8):
G
xx
(
f
)
=
2
S
xx
(
f
)
G
yy
(
f
)
=
2
S
yy
(
f
)
G
xy
(
f
)
=
2
S
xy
(
f
)
(17.8)
Figure 17.4 shows the relation between the two-sided autospectral density function
and the one-sided representation.
The autospectral and cross-spectral density functions are also shown in (17.9).
H
(
f
)
2
G
yy
(
f
)
=
G
xx
(
f
)
G
xy
(
f
)
=
H
(
f
)
G
xx
(
f
)
(17.9)
Note that the top equation of (17.9) is a real-valued relation which contains only
the transfer function gain factor of the system,
. The lower equation is a complex-
valued relation which can be broken down into a pair of equations to give both the gain
factor,
|
H
(
f
)
|
, and the transfer function phase factor,
.
(
f
) of the system. Recall that the
FFT of the cross-correlation yields the complex cross-spectra as given by the following
equation. Note that the cross-spectrum is a complex function with real and imaginary
|
H
(
f
)
|
Search WWH ::
Custom Search