Biomedical Engineering Reference
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functions, where
C
xy
(
f
) is the coincident spectral density function (cospectrum) and
Q
xy
(
f
) is the quadrature spectral density function (quad-spectrum) as shown in (17.10).
2
)
e
−
j
2
π
f
τ
G
xy
(
f
)
=
R
xy
(
τ
d
τ
=
C
xy
(
f
)
−
jQ
xy
(
f
)
(17.10)
In complex polar notation, the cross-spectral density becomes 17.11.
=
G
xy
(
f
)
e
−
j
θ
xy
(
f
)
G
xy
(
f
)
=
H
(
f
)
e
−
j
φ
(
f
)
H
(
f
)
(17.11)
where
G
xy
(
f
)
Q
xy
(
f
))
1
(
C
xy
(
f
)
/
=
+
2
tan
−1
Q
xy
(
f
)
C
xy
(
f
)
θ
xy
(
f
)
=
Thus, by knowing the autospectra of the input
G
xx
(
f
) and the cross-spectra
G
xy
(
f
) one can determine the transfer function,
. However, the magnitude of the
transfer function does not provide any phase information. Nevertheless, the complete
frequency response function with gain and phase can be obtained when both
G
xx
(
f
) and
G
xy
(
f
) are known [2]. Figure 17.5 illustrates these relationships.
An alternative direct transform method is widely used and does not require com-
putation of the autocorrelation and cross-correlation functions beforehand. It is based
on finite Fourier transforms of the original data records. For any long, but finite records
of length,
T
, the frequency domain equation is written as (17.12).
|
H
(
f
)
|
Y
(
f
)
=
H
(
f
)
X
(
f
)
(17.12)
where
X
(
f
) and
Y
(
f
) are finite Fourier transforms of
x
(
t
) and
y
(
t
) respectively. It follows
that (17.12) can be rewritten as (17.13).
Y
∗
(
f
)
H
∗
(
f
)
X
∗
(
f
)
=
Y
(
f
)
H
(
f
)
X
(
f
)
2
2
2
=
H
(
f
)
X
(
f
)
2
X
∗
(
f
)
Y
(
f
)
=
(17.13)
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