sm9_24
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sm9_24


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Problem 9.24. Equation (9.59) is the FM post-detection noise for an ideal low-pass filter. 
Find the post-detection noise for an FM signal when the post-detection filter is a second-
order low-pass filter with magnitude response 
( )( )1/ 24
1( )
1 /
H f
f W
=
+
 
Assume 2( )BP cH f f+ \u22481 for / 2Tf B< and 2TB W>> . 
 
Solution 
We modify Eq. (9.58) to include the effects of a non-ideal post-detection filter in order to 
estimate the average post-detection noise power: 
 
2 2 20 0
2 2
20
2 40
1| ( ) |
1 ( / )
2 1
1 ( / )
W W
BPW W
c c
W
c
N N
4f H f df f dfA A
N
f W
f df
A f W
\u2212 \u2212= \u22c5 +
= \u22c5 +
\u222b \u222b
\u222b
 
 
This can be evaluated by a partial fraction expansion of the integrand but for simplicity, 
we appeal to the formula: 
 
2 2 2
1 4
4 2 2 2 2
1 1 2 2 2log tan , 0,
4 2 2 2 2 2
x dx x kx k kx aab k
a bx bk x kx k k x b
\u2212\u23a1 \u23a4\u2212 += + >\u23a2 \u23a5+ + + \u2212\u23a3 \u23a6\u222b = 
 
Using this result, we get the average post-detection noise power is 
 
Avg. post-detection noise power 
33
0 0
2 2
2 2 2log 0.42
4 2 2 2c c
N NW W
A A
\u3c0\u23a1 \u23a4\u2212= \u22c5 + =\u23a2 \u23a5+\u23a3 \u23a6
 
 
 
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