ECE 5510: Random Processes
Lecture Notes
Fall 2009
Lecture 21
Today: (1) LTI Filtering, continued; (2) DiscreteTime Filter
ing
•
HW 9 due Thu at 5pm in HW locker. HW 10: posted today,
due Thursday Dec. 10.
•
OH today from 12:151:15pm.
•
Reading: Today: 11.2, 11.3, 11.6. next class: 12.1, 12.2.
•
If you have not presented a discussion item, please sign up
with me for one of the final three classes.
1
LTI Filtering of WSS Signals
Continued from Lecture 20.
1.1
Addition of r.p.s
Figure 1: Continuoustime filtering with the addition of noise.
If
Z
(
t
) =
Y
(
t
)+
N
(
t
) for two WSS r.p.s (which are uncorrelated
with each other), then we also have that
S
Z
(
f
) =
S
Y
(
f
)+
S
N
(
f
). A
typical example is when noise is added into a system at a receiver,
onto a signal
Y
(
t
) which is already a r.p.
1.2
Partial Fraction Expansion
Lets say you come up with a PSD
S
Y
(
f
) that is a product of frac
tions. Eg,
S
Y
(
f
) =
1
(2
πf
)
2
+
α
2
·
1
(2
πf
)
2
+
β
2
This can equivalently be written as a sum of two different fractions.
You can write it as:
S
Y
(
f
) =
A
(2
πf
)
2
+
α
2
+
B
(2
πf
)
2
+
β
2
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ECE 5510 Fall 2009
2
X(t)
Y(t)
R
C
+
+


Figure 2: An RC filter, with input
X
(
t
) and output
Y
(
t
).
Where you use partial fraction expansion (PFE) to find
A
and
B
.
You should look in an another textbook for the formal definition
of PFE. I use the “thumb method” to find
A
and
B
. The thumb
method is:
1. Pull all constants in the numerators out front. The numerator
should just be 1.
2. Go to the first fraction. What do you need (2
πf
)
2
to equal to
make the denominator 0? Here, it is
−
α
2
.
3. Put your thumb on the first fraction, and plug in the value
from (2.)
for (2
πf
)
2
in the second fraction.
The value that
you get is
A
. In this case,
A
=
1

α
2
+
β
2
.
4. Repeat for each fraction. In the second case,
B
=
1
α
2

β
2
Thus
S
Y
(
f
) =
1
−
α
2
+
β
2
bracketleftbigg
1
(2
πf
)
2
+
α
2
−
1
(2
πf
)
2
+
β
2
bracketrightbigg
1.3
Discussion of RC Filters
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 Fall '08
 Chen,R
 Digital Signal Processing, Signal Processing, discretetime Fourier transform

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