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# sm9_18

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```Problem 9.18. Suppose that the spectrum of a modulating signal occupies the frequency
band 1 2f f f\u2264 \u2264 . To accommodate this signal, the receiver of an FM system (without
pre-emphasis) uses an ideal band-pass filter connected to the output of the frequency
discriminator; the filter passes frequencies in the interval 1 2f f f\u2264 \u2264 . Determine the
output signal-to-noise ratio and figure of merit of the system in the presence of additive
white noise at the receiver input.

Solution
Since the post detection filter is no longer an ideal brickwall filter, we must revert to Eq.
(9.58) to compute the post-detection noise power. For this scenario (similar to Problem
9.17)

[ ]313220
22
2
0
3
2
powernoisedetection-postAvg.
2
1
1
2
ff
A
N
dffdff
A
N
c
f
f
f
fc
\u2212=
\u23a5\u23a5\u23a6
\u23a4
\u23a2\u23a2\u23a3
\u23a1 += \u222b\u222b
\u2212
\u2212

Since the average output power is still , the post detection SNR is given by Pk f
2

( )31320
22
2
3
ffN
PkA
SNR fcFMpost \u2212=

For comparison purposes, the reference SNR is

2
0 2 12 ( )
c
ref
ASNR
N f f
= \u2212

The corresponding figure of merit is

( )
2
112
2
2
2
120
2
3
1
3
20
22
ref
FM
post
3
)(22
3
SNR
SNR
meritofFigure
ffff
Pk
ffN
A
ffN
PkA
f
cfc
++=
\u2212\u2212=
=

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