563489578934

Problems 561
7–5 A whole binary communication system can be modeled as an information channel, as shown in
Fig. P7–5. Find equations for the four transition probabilities P (m ' ƒ m), where both m ' and m can
be binary l’s or binary 0’s. Assume that the test statistic is a linear function of the receiver input
and that additive white Gaussian noise appears at the receiver input. [Hint: Look at Eq. (7–15).]
Binary 1
~ P(m =1|m=1)
Binary 1
m
~ P(m =0|m=1)
~ P(m =1|m=0)
m
~
Binary 0
P(m ~ =0|m=0)
Figure P7–5
Binary 0
★ 7–6 A baseband digital communication system uses unipolar signaling (rectangular pulse shape) with
matched-filter detection. The data rate is R = 9,600 bits/sec.
(a) Find an expression for bit error rate (BER), P e , as a function of (S/N) in . (S/N) in is the signalto-noise
power ratio at the receiver input where the noise is measured in a bandwidth corresponding
to the equivalent bandwidth of the matched filter. [Hint: First find an expression of
E b /N 0 in terms of (S/N) in .]
(b) Plot P e vs. (S/N) in in dB units on a log scale over a range of (S/N) in from 0 to 15 dB.
7–7 Rework Prob. 7–6 for the case of polar signaling.
7–8 Examine how the performance of a baseband digital communication system is affected by the
receiver filter. Equation (7–26a) describes the BER when a low-pass filter is used and the bandwidth
of the filter is large enough that the signal level at the filter output is s 01 =+A or s 02 =-A. Instead,
suppose that a RC low-pass filter with a restricted bandwidth is used where T = 1/f 0 = 2p RC.
T is the duration (pulse width) of one bit, and f 0 is the 3-dB bandwidth of the RC low-pass filter as
described by Eq. (2–147). Assume that the initial conditions of the filter are reset to zero at the
beginning of each bit interval.
(a) Derive an expression for P e as a function of E b /N 0 .
(b) On a log scale, plot the BER obtained in part (a) for E b /N 0 over a range of 0 to 15 dB.
(c) Compare this result with that for a matched-filter receiver (as shown in Fig. 7–5).
★ 7–9 Consider a baseband digital communication system that uses polar signaling (rectangular pulse
shape) where the receiver is shown in Fig. 7–4a. Assume that the receiver uses a second-order
Butterworth filter with a 3-dB bandwidth of f 0 . The filter impulse response and transfer function are
h(t) = c 22v 0 e -(v 0>12)t sin a v 0
12 tbdu(t)
H(f) =
where v 0 = 2pf 0 . Let f 0 = 1/T, where T is the bit interval (i.e., pulse width), and assume that the
initial conditions of the filter are reset to zero at the beginning of each bit interval.
(a) Derive an expression for P e as a function of E b /N 0 .
(b) On a log scale, plot the BER obtained in part (a) for E b /N 0 over a range of 0 to 15 dB.
(c) Compare this result with that for a matched-filter receiver (as shown in Fig. 7–5).
7–10 Consider a baseband unipolar communication system with equally likely signaling. Assume that
the receiver uses a simple RC LPF with a time constant of RC = t where t = T and 1/T is the bit
1
(jf>f 0 ) 2 + 12(jf>f 0 ) + 1