Biomedical Engineering Reference
In-Depth Information
∞
1
2
)
e
j
ω
t
d
R
xy
(
τ
)
=
S
xy
(
ω
ω
(15.9)
π
−∞
Equations (15.6) through (15.9) are called the “Wiener-Kinchine Relation.”
The Fourier transform of the correlation functions contain real and imaginary
components of energy, meaning that the spectrum is a two-sided spectrum with half of
the energy containing negative frequencies, that is,
−
ω
; hence, the following equivalent
equations are defined in (15.10) and (15.11).
S
xx
(
−
ω
)
=
S
xx
(
ω
)
=
S
xx
(
ω
)
(15.10)
S
xy
(
−
ω
)
=
S
xy
(
ω
)
=
S
xy
(
ω
)
(15.11)
where the bar over the letter
S
denotes the complex conjugate.
The
S
(
) are the estimate of the “two-sided spectral density function,”
in which the magnitude of the FT for a frequency
−
ω
) and
S
(
ω
ω
in the real plane is equal to the
magnitude of the FT of
in the imaginary plane. In the case when the autocorrelation
function is evaluated over the interval from 0 to
ω
, the autospectra may be obtained by
using the symmetry properties of the function. An alternate equation is given in (15.12),
with its inverse transform as (15.13).
∞
2
∞
0
S
xx
(
ω
)
=
R
xx
(
τ
) cos(
ωτ
)
d
τ
(15.12)
∞
1
π
R
xx
(
τ
)
=
S
xx
(
ω
) cos(
ωτ
)
d
ω
(15.13)
0
For the case of the “one-sided spectral density function” (Fig. 15.1), in which
only the real frequency components of the power spectrum are represented, (15.14)
and (15.15) are the same as doubling the energy from one side of the two-sided
spectra.
G
xx
(
ω
)
=
2
S
xx
(
ω
)
(15.14)
G
xy
(
ω
)
=
2
S
xy
(
ω
)
(15.15)
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