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Performance of Communication Systems Corrupted by Noise Chap. 7
[n ' 0(t)] 2 = 2a K 2
b N 0 f 2 1 B
A c
(7–137)
for B>f 1 1.
The output signal power for the preemphasis–deemphasis system is the same as that
when preemphasis–deemphasis is not used because the overall frequency response of the system
to m(t) is flat (constant) over the bandwidth of B Hz. † Thus, from Eqs. (7–137) and
(7–124a), the output SNR is
a S N b = s 0 2
'
out
2
n
In addition, D j /(2pB) = b f /V p , so the output SNR reduces to
0
= A c 2 [D f >(2pf 1 )] 2 m 2
2N 0 B
a S N b out
= A c 2 b f 2 (B>f 1 ) 2 (m>V p ) 2
2N 0 B
(7–138)
Using Eq. (7–128), we find that the output to input SNR is
(S>N) out
(S>N) in
= 2b 2 f (b f + 1)a B 2
b a m 2
b
f 1 V p
(7–139)
where b f is the FM index, B is the bandwidth of the baseband (modulation) circuits, f 1 is the
3-dB bandwidth of the deemphasis filter, V p is the peak value of the modulating signal m(t),
and (m>V p ) 2 is the square of the RMS value of m(t)/V p .
The output SNR is expressed in terms of the equivalent baseband SNR by substituting
Eq. (7–84) into Eq. (7–138):
(S>N) out
(S>N) baseband
= b 2 f a B 2
b a m 2
b
f 1 V p
(7–140)
When a sinusoidal test tone is transmitted over this FM system, (m>V p ) 2 = 1 2
and Eq. (7–140)
becomes
(S>N) out
(S>N) baseband
= 1 2 b f 2 a B 2
b (sinusoidal modulation)
f 1
(7–141)
Of course, each of these results is valid only when the FM signal at the receiver input is
above the threshold.
† For a fair comparison of FM systems with and without preemphasis, the peak deviation ∆F needs to be the
same for both cases. With typical audio program signals, preemphasis does not increase ∆F appreciably, because the
low frequencies dominate in the spectrum of m(t). Thus, this analysis is valid. However, if m(t) is assumed to have a
flat spectrum over the audio passband, the gain of the preemphasis filter needs to be reduced so that the peak deviation
will be the same with and without preemphasis. In the latter case, there is less improvement in performance when
preemphasis is used.