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Sec. 7–8 Output Signal-to-Noise Ratios for Analog Systems 533
For product detection, using Eq. (4–71), we find that the output is
m ' (t) = Re{g T (t)} = A c + A c m(t) + x n (t)
Here A c is the DC voltage at the detector output that occurs because of the discrete AM
carrier, † A c m(t) is the detected modulation, and x n (t) is the detected noise. The output SNR is
a S N b out
= A c 2 m 2 (t)
x n 2 (t)
= A c 2 m 2 (t)
2N 0 B
(7–87)
where, from Eq. (6–133g), x 2 n = n 2 = 2(N 0 >2) (2B). The input signal power is
P s = A c 2
2 [1 + m(t)]2 = A c 2
2 31 + m2 4
where it is assumed that m(t) = 0 (i.e., no DC on the modulating waveform). Thus, the input
SNR is
a S (7–88)
N b = 1A c 2 >2211 + m 2 2
in 2N 0 B
Combining Eqs. (7–87) and (7–88) yields
(S>N) out
(S>N) in
=
(7–89)
For 100% sine-wave modulation, m2 = 1 and (S>N) out >(S>N) in = 2 3 .
2
For comparison purposes, Eq. (7–87) can be evaluated in terms of (S/N) baseband by
substituting for P s :
2 m 2
1 + m 2
(S>N) out
(S>N) baseband
=
m 2
1 + m 2
(7–90)
For 100% sine-wave modulation, (S>N) out >(S>N) baseband = 1 3 . This illustrates that this AM
system is 4.8 dB worse than a baseband system that uses the same amount of signal power
because of the additional power in the discrete AM carrier. ‡ This is shown in Fig. 7–27.
AM Systems with Envelope Detection
The envelope detector produces KR T (t) at its output, where K is a proportionality constant.
Then, for additive signal plus noise at the input, the output is
KR T (t) = K ƒ g s (t) + g n (t) ƒ
† This DC voltage is often used to provide automatic gain control (AGC) of the preceding RF and IF stages.
‡ The AM carrier does not contribute to the information ability of the signal but permits AM receivers to use
economical envelope detectors.