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Performance of Communication Systems Corrupted by Noise Chap. 7
Let the bandwidth of the filter be B p , where B p is at least as large as the transmission bandwidth
of the OOK signal, so that the signal waveshape is preserved at the filter output. Then
for the case of a binary 1, s 1 (t) = A cos (v c t + u c ), so
or
For a binary 0, s 2 (t) = 0 and
r 1 (t) = A cos (v c t + u c ) + n(t), 0 6 t … T
r 1 (t) = [A + x(t)] cos(v c t + u c ) - y(t) sin(v c t + u c ), 0 6 t … T
(7–49)
r 2 (t) = x(t) cos(v c t + u c ) - y(t) sin(v c t + u c ), 0 6 t … T (7–50)
The BER is obtained by using Eq. (7–8), which, for the case of equally likely
signaling, is
P e = 1 V T
q
2 f(r 0 |s 1 ) dr 0 + 1 2
L L
-q
V T
f(r 0 |s 2 ) dr 0
(7–51)
We need to evaluate the conditional PDFs for the output of the envelope detector, f (r 0 |s 1 ) and
f(r 0 |s 2 ). f(r 0 |s 1 ) is the PDF for r 0 = r 0 (t) = r 01 that occurs when r 1 (t) is present at the input of
the envelope detector, and f(r 0 |s 2 ) is the PDF for r 0 = r 0 (t 0 ) = r 02 that occurs when r 2 (t) is
present at the input of the envelope detector.
We will evaluate f(r 0 |s 2 ) first. When s 2 (t) is sent, the input to the envelope detector,
r 2 (t), consists of bandlimited bandpass Gaussian noise as seen from Eq. (7–50). In
Example 6–13, we demonstrated that for this case, the PDF of the envelope is Rayleigh
distributed. Of course, the output of the envelope detector is the envelope, so r 0 = R = r 02 .
Thus, the PDF for the case of noise alone is
r 0
f(r 0 |s 2 ) = s 2 e-r 0 2 >(2s 2) , r 0 Ú 0
L 0, r 0 otherwise
(7–52)
The parameter s 2 is the variance of the noise at the input of the envelope detector. Thus,
s 2 = (N 0 /2) (2B p ) = N 0 B p , where B p is the effective bandwidth of the bandpass filter and
N 0 /2 is the PSD of the white noise at the receiver input.
For the case of s 1 (t) being transmitted, the input to the envelope detector is given by
Eq. (7–49). Since n(t) is a Gaussian process (that has no delta functions in its spectrum at
f =;f c ) the in-phase baseband component, A + x(t), is also a Gaussian process with a mean
value of A instead of a zero mean, as is the case in Eq. (7–50). The PDF for the envelope
r 0 = R = r 01 is evaluated using the same technique descibed in Example 6–13 and the result
cited in Prob. 6–54. Thus, for this case of a sinusoid plus noise at the envelope detector input,
r 0
f(r 0 |s 1 ) = s2 e-1r 0 2 +A 2 2>(2s 2 )
I 0 a r 0A
s 2 b r 0 Ú 0
L
0, r 0 otherwise
(7–53)