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Selected Self-study problems (Chapter 4, Carlson, Communication Systems) 4.2-1 Let ( ) ( )cos2 mx t f t u tπ= with cm ff << . Sketch ( )cx t and indicate the

envelope when the modulation is AM with 1µ < , AM with 1µ > , and DSB. Identify locations where any phase reversals occur.

4.2-3 If ( ) cos200x t tπ= , find TB and TS for the AM modulated signal assuming

10cA = and 0.6µ = . Repeat for DSB transmission. 4.2-4 The signal ( ) 2sinc 40x t t= is to be transmitted using AM with 1µ < . Sketch

the double-sided spectrum of ( )cx t and find TB . 4.2-5 Calculate the transmitted power of an AM wave with 100 percent tone

modulation and peak envelope power 32 kW. 4.2-6 Consider a radio transmitter rated for 4kW peak envelope power. Find the

maximum allowable value of µ for AM with tone modulation and 1kWTS = . 4.2-7 The multitone modulating signal ( ) ( )3 cos8 2cos20x t K t tπ π= + is input to

an AM transmitter with 1µ = and 1000cf = . Find K so the ( )x t is properly normalized, draw the positive-frequency line spectrum of the modulated wave, and calculate the upper bound on 2 /sb TP S .

4.2-9 The signal ( ) 24sinx t tπ= is transmitted by DSB. What range of carrier

frequencies can be used? 4.3-1 The signal ( ) 1 1

2 3cos2 70 cos2 120x t t tπ π= + is input to the square-law modulator system given in Fig. 4.3-3a (p. 160) with a carrier frequency of 10 kHz. Assume 2

1 2out in inv a v a v= + . (a) Given the center frequency and bandwidth of the filter such that this system will produce a standard AM signal. (b) Determine values of 1a and 2a such that 10cA = and 1

2µ = . 4.3-2 A modulation system with nonlinear elements produces the signal

( ) ( ) ( )2 22 cos cosc c cx t aK v t A t b v t A tω ω= + − − . If the carrier has

frequency cf and ( ) ( )v t x t= , show that an appropriate choice of K produces DSB modulation without filtering. Draw a block diagram of the modulation system.

4.3-4 A modulator similar to the one in Fig. 4.3-3a (p. 160) has a nonlinear

element of the form 31 3out in inv a v a v= + . Sketch ( )outV f for the input signal in

Fig. 4.1-1 (p. 143). Find the parameter of the oscillator and BPF to produce a DSB signal with carrier frequency cf .

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4.3-5 Design in block-diagram from an AM modulator using the nonlinear element from Problem 4.3-4 and a frequency doubler. Carefully label all components and find a required condition on cf in terms of W to realize this system.

4.3-6 Find the output signal in Fig. 4.3-5 (p. 161) when the AM modulators are

unbalanced, so that one nonlinear element has 2 31 2 3out in in inv a v a v a v= + +

while the other has 2 31 2 3out in in inv b v b v b v= + + .

4.4-5 The system in Fig. 4.4-5 produces USSB with 1 2cf f f= + when the lower

cutoff frequency of the first BPF equals 1f and the lower cutoff frequency of the second BPF equals 2f . Demonstrate the system’s operation by ( )X f as in Fig. 4.4-4a and sketching spectra at appropriate points. How should the system be modified to produce LSSB?

4.4-6 Suppose the system in Fig. 4.4-5 in designed for USSB as described in

Problem 4.4-5. Let ( )x t be a typical voice signal, so ( )X f has negligible

content outside 200 3200 Hzf< < . Sketch the spectra at appropriate points to find the maximum permitted value of cf when the transition regions of the BPFs must satisfy 2 0.01 cofβ ≥ .

4.4-7 The signal ( ) cos2 100 3cos2 200 2cos2 400x t t t tπ π π= + + is input to an

LSSB amplitude modulation system with a carrier frequency of 10 kHz. Sketch the double-sided spectrum of the transmitted signal. Find the transmitted power TS and bandwidth TB .

4.4-8 Draw the block diagram of a system that would generate the LSSB signal in

Problem 4.4-7, giving exact values for filter cutoff frequencies and oscillators. Make sure your filters meet the fractional bandwidth rule.

4.5-1 Given a bandpass amplifier centered at 66 MHz., design a television

transponder that receives a signal on Channel 11 (199.25 MHz) and transmits it on Channel 4 (67.25 MHz). Using only one oscillator.

4.5-3 The system in Fig. 4.4-5 becomes a scrambler when the first BPF passes

only the upper sideband, the second oscillator frequency is 2 1f f W= + , and the second BPF is replaced by an LPF with B W= . Sketch the output spectrum taking ( )X f as in Fig. 4.4-4a, and explain why this output would

be unintelligible when ( )x t is a voice signal. How can the output signal be unscrambled?

4.5-4 Take ( )cx t as in Eq. (2) and find the output of a synchronous detector whose

local oscillator produces ( )2cos ctω φ+ , where φ is a constant phase error. Then write separate answers for AM, DSB, SSB, and VSB by appropriate substitution of the modulation parameters.

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4.5-6 The signal ( ) 2cos4x t tπ= is transmitted by DSB. Sketch the output signal

if envelope detection is used for demodulation. 4.5-7 Suppose the DSB waveform from Problem 4.5-6 is demodulated using a

synchronous detector that has a square wave with a fundamental frequency of cf as the local oscillator. Will the detector properly demodulate the signal? Will the same be true if periodic signals other than the square wave are substituted for the oscillator?