Digital Signal Processing Reference
In-Depth Information
The 3-dB bandwidth is shown in Fig.
7.2
. and is calculated from
2
pB
3dB
¼
2
l;
(7.67)
or equivalently
B
3dB
¼ l=p:
(7.68)
The RMS bandwidth
B
rms
is infinite
B
rms
¼1;
(7.69)
because the integral in the nominator of (
7.59
) is infinite
1
f
2
S
XX
ð f Þ
d
f ¼1
(7.70)
0
7.3 Cross-Power Spectrum
7.3.1 Definition
Applying a similar procedure (such as the one described in Sect.
7.1
for one
process) for two random processes
X
(
t
) and
Y
(
t
), we arrive at the expressions for
cross-power spectrums S
XY
(
o
) and
S
YX
(
o
)or
cross-spectral density functions
X
T
ð
o
Þ
Y
T
ð
o
Þ
T
S
XY
ðoÞ¼
lim
T!1
:
(7.71)
Y
T
ð
o
Þ
X
T
ð
o
Þ
T
S
YX
ðoÞ¼
lim
T!1
;
(7.72)
where
X
T
(
o
) and
Y
T
(
o
) are the Fourier transforms of processes
x
T
(
t
) and
y
T
(
t
),
respectively, and
X*
T
(
o
) and
Y*
T
(
o
) are corresponding complex-conjugated
values.
xðtÞ
for
T=
2
t T=
2
;
x
T
ðtÞ¼
(7.73)
0
otherwise
:
yðtÞ
for
T=
2
t T=
2
;
y
T
ðtÞ¼
(7.74)
0
otherwise
:
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