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Performance of Communication Systems Corrupted by Noise Chap. 7
also be a Gaussian process. For baseband signaling, the processing circuits would consist of
linear filters with some gain. For bandpass signaling, as we demonstrated in Chapter 4, a superheterodyne
circuit (consisting of a mixer, IF stage, and product detector) is a linear circuit.
However, if automatic gain control (AGC) or limiters are used, the receiver will be nonlinear,
and the results of this section will not be applicable. In addition, if a nonlinear detector such as
an envelope detector is used, the output noise will not be Gaussian. For the case of a linearprocessing
receiver circuit with a binary signal plus noise at the input, the sampled output is
r 0 = s 0 + n 0
(7–10)
Here the shortened notation r 0 (t 0 ) = r 0 is used. n 0 (t 0 ) = n 0 is a zero-mean Gaussian random
variable, and s 0 (t 0 ) = s 0 is a constant that depends on the signal being sent. That is,
s 0 = e s 01, for a binary 1 sent
(7–11)
s 02 , for a binary 0 sent
where s 01 and s 02 are known constants for a given type of receiver with known input signaling
waveshapes s 1 (t) and s 2 (t). Since the output noise sample n 0 is a zero-mean Gaussian random
variable, the total output sample r 0 is a Gaussian random variable with a mean value of either s 01
or s 02 , depending on whether a binary 1 or a binary 0 was sent. This is illustrated in Fig. 7–2,
where the mean value of r 0 is m r01 = s 01 when a binary 1 is sent and the mean value of r 0 is
m r02 = s 02 when a binary 0 is sent. Thus, the two conditional PDFs are
1
f(r 0 |s 1 ) = e -(r 0-s 01 ) 2 >(2s 2 0 )
(7–12)
12p s 0
and
1
f(r 0 |s 2 ) = e -(r 0-s 02 ) 2 >(2s 2 0 )
(7–13)
12p s 0
s 2 0 = n 2 0 = n 2 0 (t 0 ) = n 2 0 (t) is the average power of the output noise from the receiver processing
circuit where the output noise process is wide-sense stationary.
Using equally likely source statistics and substituting Eqs. (7–12) and (7–13) into
Eq. (7–8), we find that the BER becomes
P e = 1 2 L
V T
-q
1
12p s 0
e -(r 0-s 01 ) 2 >(2s 0 2 ) dr 0 + 1 2 L
q
12p s 0
e -(r 0-s 02 ) 2 >(2s 0 2 ) dr 0
V T
1
(7–14)
This can be reduced to the Q(z) functions defined in Sec. B–7 (Appendix B) and tabulated in
Sec. A–10 (Appendix A). Let l =-(r 0 - s 01 )/s 0 in the first integral and l = (r 0 - s 02 )/s 0 in
the second integral; then
P e = 1 2 L
q
1
-(V T -s 01 )>s 0
12p e-l2 >2 dl + 1 2 L
q
(V T -s 02 )>s 0
1
12p e-l2 >2 dl
or
P e = 1 2 Q a -V T + s 01
b + 1 s 0 2 Q a V T - s 02
b
s 0
(7–15)