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Problems 563
(a) Draw a block diagram of your design and explain how it works.
(b) Give the values for the design parameters R, C, and V T .
(c) Calculate the PSD level for the noise N 0 that is allowed if P e is to be less than 10 -6 .
7–19 A BER of 10 -5 or less is desired for an OOK communication system where the bit rate is R = 10
Mb/s. The input to the receiver consists of the OOK signal plus white Gaussian noise.
(a) Find the minimum transmission bandwidth required.
(b) Find the minimum E b /N 0 required at the receiver input for coherent matched-filter detection.
(c) Rework part (b) for the case of noncoherent detection.
7–20 Rework Prob. 7–19 for the case when FSK signaling is used. Let 2∆F = f 2 - f 1 = 1.5R.
★ 7–21 In this chapter, the BER for a BPSK receiver was derived under the assumption that the coherent
receiver reference (see Fig. 7–7) was exactly in phase with the received BPSK signal. Suppose that
there is a phase error of u e between the reference signal and the incoming BPSK signal. Obtain
new equations that give the P e in terms of u e , as well as the other parameters. In particular,
(a) Obtain a new equation that replaces Eq. (7–36).
(b) Obtain a new equation that replaces Eq. (7–38).
(c) Plot results from part (b) where the log plot of P e is given as a function of u e over a range
-p 6 u e 6 p for the case when E b /N 0 = 10 dB.
★ 7–22 Digital data are transmitted over a communication system that uses nine repeaters plus a
receiver, and BPSK signaling is used. The P e for each of the regenerative repeaters (see Sec. 3–5)
is 5 × 10 -8 , assuming additive Gaussian noise.
(a) Find the overall P e for the system.
(b) If each repeater is replaced by an ideal amplifier (no noise or distortion), what is the P e of the
overall system?
★ 7–23 Digital data are to be transmitted over a toll telephone system using BPSK. Regenerative
repeaters are spaced 50 miles apart along the system. The total length of the system is 600 miles.
The telephone lines between the repeater sites are equalized over a 300- to 2,700-Hz band and
provide an E b /N 0 (Gaussian noise) of 15 dB to the repeater input.
(a) Find the largest bit rate R that can be accommodated with no ISI.
(b) Find the overall P e for the system. (Be sure to include the receiver at the end of the system.)
7–24 A BPSK signal is given by
s(t) = A sin [v c t + u c + (;1)b p ], 0 6 t … T
The binary data are represented by (;1), where (+1) is used to transmit a binary 1 and (-1) is used
to transmit a binary 0. b p is the phase modulation index as defined by Eq. (5–47).
(a) For b p = p/2, show that this BPSK signal becomes the BPSK signal as described by
Eq. (7–34).
(b) For 0 6 b p 6 p/2, show that a discrete carrier term is present in addition to the BPSK signal
as described by Eq. (7–34).
7–25 Referring to the BPSK signal described in Prob. 7–24, find P e as a function of the modulation
index b p , where 0 6 b p … p/2. Find P e as a function of A, b p , N 0 , and B for the receiver that uses
a narrowband filter.
7–26 Rework Prob. 7–25 and find P e as a function of E b , N 0 , and b p for the receiver that uses a matched
filter. (E b is the average BPSK signal energy that is received during one bit.)
7–27 Referring to the BPSK signal described in Prob. 7–24, let 0 6 b p 6 p/2.
(a) Show a block diagram for the detection of the BPSK signal where a PLL is used to recover
the coherent reference signal from the BPSK signal.