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ECE 460 – Introduction to Communication Systems Homework #4

1. For a , x(t) = 0.5 u(t) - 1.5 u(t - 1)

and a 1 volt peak carrier with , fc, where fc >> 1 Hz, sketch the modulated

waveform xAM(t) when: a. m = 0.1 b. m = 0.5 c. m = 0.9 d. m = 1.0 e. m = 1.4 Answers:

1.25 1.05 0.9 0.5 m=0.1 m=0.5 -0.5 -0.9 -1.05 -1.25 0 1 2 3 0 1 2 3

1.45 1.5 m=0.9 m=1 0.1 0 -0.1

-1.45 -1.5 0 1 2 3 0 1 2 3

1.7 phase reversal

0.4

-0.4 m=1.4 -1.7 0 1 2 3 2.

125 x (t) AM 75 25 t -25 -75 -125 For the tone modulated AM waveform shown above: a. Find the index, m. [Answer: 2/3]

b. Write an expression for the waveform in terms of ωm and ωc

[ Answer: 75 [ 1 + (2/3) cos ωmt ] cos ωct] c. Sketch the of the waveform Answer:

37.5 37.5

12.5 12.5 12.5 12.5

f f - c c

3. For a baseband signal

x ( t ) = 6 cos 4π 103t + 4 cos 10π 103t and a 100 KHz : 3 3 a. Find xN(t), the normalized version of x(t). [Answer: 0.6 cos 4π 10 t +0.4 cos 10π 10 t] b. Assuming a carrier wave peak amplitude of 20, determine an expression for the

modulated signal, xAM(t), in terms of the modulation index, m. [Answer: 20 [ 1 +

0.6m cos 4π 103 t + 0.4m cos 10π 103 t ] cos 200π 103 t ] c. Sketch the spectrum for m = 1 and m = 0.5 Answer:

1 0 1 0

3 3 3 3

2 2 2 2

f ( K H z ) -1 0 5 -1 0 0 -9 5 9 5 9 8 1 0 0 1 0 5 -1 0 2 -9 8 10 10 1.5 1.5 1.5 1.5 1 1 1 1

f ( K H z ) -1 0 5 -1 0 0 -9 5 9 5 9 8 1 0 0 1 0 5 -1 0 2 -9 8 d. For m = 1, determine the power contained in: (1) the carrier, and (2) each of the [Answers: 200W, 16W, 36W]

e. Calculate the efficiency of the AM signal with m = 1 using the values obtained in part d and: P η = sidebands Answer: 20.6% Psidebands + Pcarrier

4. An AM modulator has the output

xAM(t) = 20 cos 2π (150) t + 6 cos 2π (160) t + 6 cos 2π (140) t

Determine the modulation index, m, the carrier frequency, fc, the baseband signal frequency,

fm, and the efficiency of the modulator. [Answers: 0.6, 150Hz, 10Hz, 15.25%]

5. An AM has an unmodulated power output of 1KW. When a sinusoidal test tone is applied to the input of the modulator, it is found that the spectral line for each is 40% of the carrier line. Determine the: a. modulation index [Answer: 0.8] b. magnitude of a lower sideband component [Answer: 8.94] c. efficiency [Answer: 24.2%] d. total output power [Answer: 1.32KW)

6. An AM signal, xAM(t),

xAM(t) = A[1 + mx(t)] cos(2πfct + φ) is applied to both systems shown below. The maximum frequency of the baseband signal, x(t)

is fm, which is also the cutoff frequency of the ideal low pass filters. Assume that the square root block outputs the positive square root of its input.

A a. Find the output of system A. [Answer: [1+ mx(t)] ] 2 A b. Find the output of system B. [Answer: [1+ mx(t)] ] 2 c. Compare the two outputs.

7. An AM modulator consists of a nonlinear device with an output given by: 2 i(t) = a1v(t) + a2v (t)

followed by an ideal bandpass filter with center frequency, ωc, and 2ωm. If an input:

v(t) = Ac cos ωct + cos ωmt is applied: a. What is the output of the nonlinear device? [Answer:

2 a2 Ac a2 a1Ac cosωct + a1 cosωmt + [1+ cos2ωct] + [1+ cos2ωmt] + 2a2 Ac cosωct cosωmt ] 2 2 ! $ 2a2 b. What is the output of the bandpass filter? [Answer: a1Ac #1+ cosωmt&cosωct ] " a1 %

2a c. Determine the modulation index of the resulting AM signal. [Answer: 2 ] a1

8. Show that the system below can be used to demodulate an AM signal with carrier frequency,

fc. What is the maximum frequency that the low pass filter should pass? [Answer: 2fc – fm] x x AM LPF

y(t) y(t) 1 t -1

T = 1/f c

9. X(f) 1 x(t) f(Hz) -10000 10000 A baseband signal, x(t), with Fourier transform X(f) as shown above is used to modulate a 10 MHz carrier using AM. Pick a suitable value for the time constant, τ (=RC), of the envelope that will demodulate this AM signal. [Answer: one solution: τ = 5 µS ]

Problems 10 – 11 require the use of MATLAB. Before attempting these problem read and try the examples in MATLAB Tutorial #4. Turn in any graphs, Simulink models and/or MATLAB commands that you used.

10. For the AM waveform given in problem 2, assume that fc = 1000 Hz and fm = 100 Hz. a. Use MATLAB to graph the expression for the AM waveform that you obtained in problem 2b to confirm that it matches the given waveform.

b. Use MATLAB to find and plot the amplitude spectrum of this AM waveform. c. Add the carrier term found in problem 2d to the original waveform and use MATLAB to plot this new waveform. Confirm that the modulation index is 0.2. d. Add the carrier term found in problem 2e to the original waveform and use MATLAB to plot this new waveform. Confirm that the modulation index is 0.8.

11. For the AM waveform given in problem 4: a. Use MATLAB to generate and plot this waveform for 0 ≤ t ≤ 0.2 seconds. b. Create an AM modulator in Simulink with a sinusoidal baseband signal similar to that shown in the tutorial. Set the and amplitudes of the two sine wave generators to match those determined in the problem for the baseband signal and carrier. Set the gain block to the value of m determined in the problem. Use a simout block to collect the output of the modulator. Use a phase angle of π/2 for both sine wave generators so that they will be cos and not sin. c. Plot the simout data on the same graph as part a. How do the waveforms compare? d. Import simout into the SP Tool and plot the spectrum of the AM waveform.