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Chapter Three: Digital Modulation Schemes 149

show that

12 + 32 + 52 + · · · + (M − 1)2 = M(M2 − 1)

6

and derive Equation 3.2–5.

3.2 Determine the signal space representation of the four signals sk(t), k = 1, 2, 3, 4, shownin Figure P3.2, by using as basis functions the orthonormal functions φ1(t) and φ2(t). Plotthe signal space diagram, and show that this signal set is equivalent to that for a four-phasePSK signal.

E E

E

E

�2(t)�1(t)

FIGURE P3.2

3.3 π/4-QPSK may be considered as two QPSK systems offset by π/4 rad.1. Sketch the signal space diagram for a π/4-QPSK signal.2. Using Gray encoding, label the signal points with the corresponding data bits.

3.4 Consider the octal signal point constellations in Figure P3.4.1. The nearest-neighbor signal points in the 8-QAM signal constellation are separated

in distance by A units. Determine the radii a and b of the inner and outer circles,respectively.

2. The adjacent signal points in the 8-PSK are separated by a distance of A units. Determinethe radius r of the circle.

r 45�

8-PSK 8-QAM

a

b

FIGURE P3.4

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150 Digital Communications

3. Determine the average transmitter powers for the two signal constellations, and comparethe two powers. What is the relative power advantage of one constellation over the other?(Assume that all signal points are equally probable.)

3.5 Consider the 8-point QAM signal constellation shown in Figure P3.4.1. Is it possible to assign 3 data bits to each point of the signal constellation such that the

nearest (adjacent) points differ in only 1 bit position?2. Determine the symbol rate if the desired bit rate is 90 Mbits/s.

3.6 Consider the two 8-point QAM signal constellations shown in Figure P3.6. The minimumdistance between adjacent points is 2A. Determine the average transmitted power for eachconstellation, assuming that the signal points are equally probable. Which constellation ismore power-efficient?

(a) (b)

FIGURE P3.6

3.7 Specify a Gray code for the 16-QAM signal constellation shown in Figure P3.7.

7

5

3

1

�1�1�7 �5 �3 1 753

�3

�5

�7

FIGURE P3.7

3.8 In an MSK signal, the initial state for the phase is either 0 or π rad. Determine the terminalphase state for the following four input pairs of input data:1. 002. 013. 104. 11

3.9 Determine the number of states in the state trellis diagram for1. A full-response binary CPFSK with h = 2

3 or 34 .

2. A partial-response L = 3 binary CPFSK with h = 23 or 3

4 .

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Chapter Four: Optimum Receivers for AWGN Channels 267

4.4 A binary digital communication system employs the signals

s0(t) = 0 0 ≤ t ≤ Ts1(t) = A 0 ≤ t ≤ T

for transmitting the information. This is called on-off signaling. The demodulator cross-correlates the received signal r (t) with s(t) and samples the output of the correlator att + T .a. Determine the optimum detector for an AWGN channel and the optimum threshold,

assuming that the signals are equally probable.b. Determine the probability of error as a function of the SNR. How does on-off signaling

compare with antipodal signaling?

4.5 A communication system transmits one of the three messages m1, m2, and m3 using signalss1(t), s2(t), and s3(t). The signal s3(t) = 0, and s1(t) and s2(t) are shown in Figure P4.5.The channel is an additive white Gaussian noise channel with noise power spectral densityequal to N0/2.

s1(t)

T�3 Tt

A

�A

s2(t)

T�3 Tt

2A

�2A

FIGURE P4.5

1. Determine an orthonormal basis for this signal set, and depict the signal constellation.2. If the three messages are equiprobable, what are the optimal decision rules for this

system? Show the optimal decision regions on the signal constellation you plotted inpart 1.

3. If the signals are equiprobable, express the error probability of the optimal detector interms of the average SNR per bit.

4. Assuming this system transmits 3000 symbols per second, what is the resulting trans-mission rate (in bits per second)?

4.6 Suppose that binary PSK is used for transmitting information over an AWGN with a powerspectral density of 1

2 N0 = 10−10 W/Hz. The transmitted signal energy is Eb = 12 A2T ,

where T is the bit interval and A is the signal amplitude. Determine the signal amplituderequired to achieve an error probability of 10−6 when the data rate is1. 10 kilobits/s2. 100 kilobits/s3. 1 megabit/s

4.7 Consider a signal detector with an input

r = ±A + n