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Performance of Communication Systems Corrupted by Noise Chap. 7
or
I =
3f 1 3 c B f 1
B 3
- tan -1 a B f 1
bd
(7–154)
Because I dB = 10 log (I), we get
I dB = 30 log a B f 1
b - 10 log e 3c B f 1
- tan -1 a B f 1
bdf
(7–155)
For f 1 = 2.1 kHz and B = 15 kHz, Eq. (7–155) gives
I dB = 13.2 dB
This checks with the value for I dB that is obtained from Fig. 7–26.
PROBLEMS
★ 7–1 In a binary communication system the receiver test statistic, r 0 (t 0 ) = r 0 , consists of a polar signal
plus noise. The polar signal has values s 01 =+A and s 02 =-A. Assume that the noise has a
Laplacian distribution, which is
f(n 0 ) =
1
12s 0
e -12|n 0| >s 0
where s0 is the RMS value of the noise.
(a) Find the probability of error P e as a function of A/ s 0 for the case of equally likely signaling
and V T having the optimum value.
(b) Plot P e as a function of A/ s 0 decibels. Compare this result with that obtained for Gaussian
noise as given by Eq. (7–26a).
7–2 Using MATLAB, plot the PDF for a Laplacian distribution (see Prob. 7–1) where s0 = 1. Also,
plot the PDF for a Gaussian distribution where the standard deviation is 1 and the mean is 0.
Compare the two PDFs.
7–3 Using Eq. (7–8), show that the optimum threshold level for the case of antipodal signaling with
additive white Gaussian noise is
V T = s 0 2
lnc P(s 2 sent)
2s 01 P(s 1 sent) d
Here the receiver filter has an output with a variance of s 2 0 . s 01 is the value of the sampled binary
1 signal at the filter output. P(s 1 sent) and P(s 2 sent) are the probabilities of transmitting a binary
1 and a binary 0, respectively.
7–4 A baseband digital communication system uses polar signaling with matched filter in
the receiver. The probability of sending a binary 1 is p, and the probability of sending a binary
zero is 1 - p.
(a) For E b /N 0 = 10 dB, plot P e as a function of p using a log scale.
(b) Referring to Eq. (1–8), plot the entropy, H, as a function of p. Compare the shapes of these
two curves.