Biomedical Engineering Reference
In-Depth Information
TABLE 19.1:
Normalized Random Errors for Spectral Estimates
ESTIMATE
NORMALIZED RANDOM ERROR,
ε
1
√
n
d
G
xx
(
f
),
G
yy
(
f
)
1
G
xy
(
f
)
|
|
|
√
n
d
|
γ
xy
(
f
)
Q
xy
(
f
)]
1
C
xy
(
f
)
/
[
G
xx
(
f
)
G
yy
(
f
)
+
−
2
C
xy
(
f
)
C
xy
(
f
)
√
2
n
d
C
xy
(
f
)]
1
Q
xy
(
f
)
/
[
G
xx
(
f
)
G
yy
(
f
)
+
−
2
Q
xy
(
f
)
Q
xy
(
f
)
√
2
n
d
The quantity
|
γ
|
is the positive square root of
γ
xy
. Note that
2
ε
for the cross-spectrum
xy
varies inversely with
γ
xy
G
xy
|
1
√
n
d
2
magnitude estimate
|
and approaches
as
γ
xy
ap-
proaches 1.
A summary is given in Table 19.1 on the main normalized random error formulas
for various spectral density estimates. The number of averages
n
d
represents
n
d
distinct
(nonoverlapping) records, which are assumed to contain statistically different information
from record to record. These records may occur by dividing a long stationary ergodic
record into
n
d
parts, or they may occur by repeating an experiment
n
d
times under
similar conditions. With the exception of autospectrum error estimation equation (19.8)
and (19.9), all other error formulas are functions of frequency. Unknown true values
of desired quantities are replaced by measured values when one applies these results to
evaluate the random errors in actual measured data.
Let us work through an example to illustrate calculating the random errors in Fre-
quency Response Transfer Function Estimate. Suppose the frequency response function
between two random signals
x
(
t
) and
y
(
t
) is estimated using
n
d
=
50 averages. Assume
that the coherence function at one frequency of interest is
γ
xy
(
f
1
)
2
=
0
.
10 and at a second
2
frequency of interest is
90, the problem is to determine the random errors
in the frequency response function gain and phase estimates at the two frequencies of
interest.
γ
xy
(
f
2
)
=
0
.
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