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Performance of Communication Systems Corrupted by Noise Chap. 7
rate. (By “simple,” it is meant that the initial conditions of the LPF are not reset to zero at the
beginning of each bit interval.)
(a) For signal alone at the receiver input, evaluate the approximate worst-case signal to ISI ratio
(in decibels) out of the LPF at the sampling time t = t 0 = nT, where n is an integer.
(b) Evaluate the signal to ISI ratio (in decibels) as a function of the parameter K, where t = t 0 =
(n + K)T and 0 K 1.
(c) What is the optimum sampling time to use to maximize the signal-to-ISI power ratio out of
the LPF?
(d) Repeat part (a) for the case when the equivalent bandwidth of the RC LPF is 2/T.
7–11 Examine a baseband polar communication system with equally likely signaling and no channel
noise. Assume that the receiver uses a simple RC LPF with a time constant of t = RC. (By
“simple,” it is meant that the initial conditions of the LPF are not reset to zero at the beginning
of each bit interval.) Evaluate the worst-case approximate signal to ISI ratio (in decibels) out of
the LPF at the sampling time t = t 0 = nT, where n is an integer. This approximate result will
be valid for T/t 7 1 1
2 . Plot this result as a function of T/t for
2 … T>t … 5.
★ 7–12 Consider a baseband unipolar system described in Prob. 7–10d. Assume that white Gaussian
noise is present at the receiver input.
(a) Derive an expression for P e as a function of E b /N 0 for the case of sampling at the times
t = t 0 = nT.
(b) Compare the BER obtained in part (a) with the BER characteristic that is obtained when a
matched-filter receiver is used. Plot both of these BER characteristics as a function of
(E b /N 0 ) dB over the range 0 to 15 dB.
7–13 Rework Prob. 7–12 for the case of polar baseband signaling.
7–14 For bipolar signaling, the discussion leading up to Eq. (7–28) indicates that the optimum threshold
at the receiver is V T = A 2 + s 0 2
A ln2.
(a) Prove that this is the optimum threshold value.
(b) Show that A/2 approximates the optimum threshold if P e 6 10 -3 .
★ 7–15 For unipolar baseband signaling as described by Eq. (7–23),
(a) Find the matched-filter frequency response and show how the filtering operation can be
implemented by using an integrate-and-dump filter.
(b) Show that the equivalent bandwidth of the matched filter is B eq = 1/(2T) = R/2.
7–16 Equally likely polar signaling is used in a baseband communication system. Gaussian noise having
a PSD of N 0 /2 W/Hz plus a polar signal with a peak level of A volts is present at the receiver
input. The receiver uses a matched-filter circuit having a voltage gain of 1,000.
(a) Find the expression for P e as a function of A, N 0 , T, and V T , where R = 1/T is the bit rate and
V T is the threshold level.
(b) Plot P e as a function of V T for the case of A = 8 × 10 -3 V, N 0 /2 = 4 × 10 -9 W/Hz, and
R = 1200 bits/sec.
★ 7–17 Consider a baseband polar communication system with matched-filter detection. Assume that the
channel noise is white and Gaussian with a PSD of N 0 /2. The probability of sending a binary 1 is
P(1) and the probability of sending a binary 0 is P(0). Find the expression for P e as a function of
the threshold level V T when the signal level out of the matched filter is A, and the variance of the
noise out of the matched filter is s 2 = N 0 /(2T), where R = 1/T is the bit rate.
7–18 Design a receiver for detecting the data on a bipolar RZ signal that has a peak value of A = 5 volts.
In your design assume that an RC low-pass filter will be used and the data rate is 2,400 (bits/sec).