Coherence, Measurement Noise and System Identification: SISO
1. Coherence Function(COH)
1
Assume
and
are both different from zero, meaning that both spectra are
rich, and do not contain delta functions, meaning that both spectra have no deterministic
parts with zero means, the coherence function between the input
and the output
of a system is defined by
)
(
f
S
xx
)
(
f
S
yy
)
(
t
x
)
(
t
y
1
)
(
)
(
)
(
)
(
0
2
2
≤
=
γ
≤
f
S
f
S
f
S
f
yy
xx
xy
xy
.
For a constant parameter linear system, free of measurement noise,
1
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
2
2
2
2
2
=
=
=
γ
f
S
f
H
f
S
f
S
f
H
f
S
f
S
f
S
f
xx
xx
xx
yy
xx
xy
xy
.
If
and
are completely unrelated, the coherence function will be zero. For
, there arise such possible situations as:
)
(
t
x
2
γ
<
xy
)
(
t
y
1
<
)
(
0
f
a)
Extraneous noise is present in the measurements.
b)
The system relating
and
is not linear.
)
(
t
x
)
(
t
y
c)
is an output due to an input
as well as to other inputs.
)
(
t
y
)
(
t
x
d)
Resolution bias errors are present in the spectral estimates.
For linear systems, the coherence function
γ
can be interpreted as
)
(
2
f
xy
the fractional
portion of the mean square value at the output
)
(
t
y
, which is contributed by the input
at frequency
f
. The value of the coherence function indicates how much of one
record is linearly related to the other record. It does not necessarily indicate a causeand
effect relationship between the two records.
)
(
t
x
2. Effects of Measurement Noise
The standard singleinput singleoutput system model with extraneous (meaning ‘not go
through the system’) input and output measurement noises,
m
and
, is shown
below.
)
(
t
)
(
t
n
Now we can construct the simple relations:
)
(
)
(
)
(
),
(
)
(
)
(
t
n
t
v
t
y
t
m
t
u
t
x
+
=
+
=
where
are the measured input and output signals, and,
u
are the true
input and output signals which are not measurable. Note that the measurement noises,
are not measurable in the time domain together with
x
or
,
)
(
),
(
t
y
t
x
)
(
t
)
(
),
(
t
v
t
)
(
),
(
t
y
t
),
(
n
t
m
)
(
),
(
t
v
t
u
1
Bendat & Piersol,
Random Data
, section 6.1, 2000
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MAE 591
RANDOM DATA
)
(
t
m
)
(
t
x
)
(
t
v
)
(
t
u
)
(
f
H
+
+
)
(
t
n
)
(
t
y
either, but their autospectral densities
can be estimated separately when
the input
u
is turned off, leading to
)
(
t
0
)
(
=
t
(
)
(
v
. From now on, we assume for simplicity
that extraneous noise terms are uncorrelated with each other and with the signals, i.e.
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 Spring '09
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