Biomedical Engineering Reference
In-Depth Information
records
x
(
t
) and
y
(
t
) of unlimited length
T
, the one-sided cross-spectrum is given by (19.1).
T
E
X
(
f
T
)
2
G
xy
(
f
)
=
Lim
T
,
T
)
Y
(
f
,
(19.1)
→∞
where
X
(
f
,
T
) is the conjugate of the input and
Y
(
f
,
T
) are the finite Fourier transforms
of
x
(
t
) and
y
(
t
), respectively.
It should be noted that the raw spectrum (no averages) of a finite length has the
same equation as (19.1), without the limit as given by (19.2) and will have a resolution
bandwidth given by (19.3).
2
T
[
X
(
f
)
Y
(
f
)]
G
xy
(
f
)
=
(19.2)
1
T
B
e
=
f
=
(19.3)
The ordinary coherence function is defined by (19.4).
2
|
G
xy
|
2
xy
γ
=
(19.4)
G
xx
G
yy
The corresponding variance errors for “smooth” estimates of all of the “raw” es-
timates in (19.4) will be reduced by a factor of
n
d
when averages are taken over
n
d
statistically independent “raw” quantities. To be specific, the variance expressions are
given by (19.5), (19.6), and (19.7).
G
xx
n
d
G
xx
]
Var[
=
(19.5)
G
yy
n
d
Var[
G
yy
]
=
(19.6)
G
xy
|
γ
|
G
xy
Var[
|
|
]
=
(19.7)
xy
n
d
Then, the normalized root-mean-square errors, which are the same as the nor-
malized random errors, are given by (19.8), (19.9), and (19.10), respectively.
1
√
n
d
[
G
xx
]
ε
=
(19.8)
1
√
n
d
G
yy
]
ε
[
=
(19.9)
1
G
xy
ε
[
|
|
]
=
(19.10)
|
√
n
d
|
γ
xy
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