ECE 460 – Introduction to Communication Systems Homework #6

1. The telemetry receiver shown below is designed so that the local oscillator operates at frequencies below the incoming carrier frequencies.

a. What is fLO set to when receiving a 136 MHz carrier signal? [Ans. 106MHz] b. What is the image frequency for this carrier frequency? [Ans. 76 MHz] c. If the receiver is designed to receive signals from 120 MHz to 150 MHz, what tuning range is required for the local oscillator? [Ans: 90 – 120 MHz]

f c RF IF Detector Amplifier x Amplifier

fIF = 30 MHz f LO

2. The telemetry receive from problem 1 is converted to a dual conversion receiver by adding a second IF stage as shown below. Assume that the receiver is designed so that the first local oscillator operates below the carrier frequency and the second local oscillator operates above the first IF frequency.

a. What is fLO2 set to produce fIF2 = 10 MHz? [Ans: 40 MHz]

b. For the value of fLO2 found in part a, what frequency other than 30 MHz mixes with fLO2 to

produce fIF2? [Ans: 50 MHz] c. What two frequencies entering the RF amplifier will produce the frequency found in part

b after the first mixer when fLO1 is set to receive 136 MHz? [Ans: 56 and 156 MHz] Note that the three frequencies found in 1b and 2c are all image frequencies

RF First Second x x Detector Amplifier IF Amp IF Amp

f = 30 MHz f = 10 MHz IF1 IF2

f fLO2 LO1

3. A TV receiver using the superheterodyne principle has a video IF carrier frequency of 45.75

MHz and fLO > fC. Channel 2 is broadcast on a carrier frequency of 55.25 MHz. What is the image frequency for channel 2? [Ans: 146.75 MHz]

4. FM broadcast radio operates in the frequency band from 88 to 108 MHz. If a

superheterodyne receiver with fLO greater than the incoming stations is used and we want to have all possible image frequencies fall outside of the FM band, what is the minimum value

of fIF that can be used? The actual fIF used in FM radios is 10.7 MHz, does that meet this criterion? [Ans: 10MHz, yes]

5. A superheterodyne receiver is tuned to a station at 3.7 MHz. The IF frequency of the receiver is 10.7 MHz. Assume that the local oscillator operates at a frequency above both the station and the IF: a. What is the local oscillator frequency? [Ans: 14.4 MHz] b. If the local oscillator produces a second harmonic, what two additional frequencies will be received? [Ans: 18.1 and 39.5 MHz]

6. Let x(t) be a typical voice signal with X(f) = 0 for |f| < 200 and |f| > 3200 Hz. Sketch the spectrum after each multiplier and each filter. Verify that this system produces USB. Find

the maximum permitted values of fc1 and fc2, if the transition regions of the high pass filters

must satisfy 2β > 0.01 fc.

HPF HPF x(t) x x USB cutoff = f c1 cutoff = f c2

f f c1 c2

[Ans: after the first mixer: After the first HPF: 2fc1 + 400 2β

fc1 fc1

after the second mixer: after the second HPF we have a USB signal: 2β = 2fc1 + 400

f c2 fc2 f c1 ≤ 40KHz f c2 ≤ 8.04 MHz

7. A voice signal occupying the frequency band 0.3 - 3.4 KHz is to be modulated onto a carrier wave of frequency 11.6 MHz. High pass filters such as the one shown below are available. Design a system to generate the USB wave using DSB modulators and these filters.

0.01f H(f) c

f f c Answer:

x(t) x HPF x HPF

60KHz 11.6MHz

8. A baseband signal x(t) = cos 2π (100)t + cos 2π (200)t + cos 2π (400)t is applied to a SSB (USB) modulator with a carrier at 100 KHz. a. What are the frequencies of the sidebands produced by this SSB modulator? [Answer: 100.1, 100.2, and 100.4 KHz] b. A synchronous detector with a local oscillator at 100.02 KHz is used to demodulate this SSB signal. Determine the frequency components of the detector output. [Answer: 80Hz, 180Hz and 380 Hz] c. Repeat for LSB. [Answer: 420Hz, 220Hz, and 120Hz.

9. The diagram shown below is Weaver's SSB modulator. Analyze its operation by assuming

that x(t) = cos2πfmt, fm < W and find the output. Does this modulator produce upper or lower sideband?

Ideal LPF x x cutoff = W/2

cos 2π(W/2)t cos 2π(fc + W/2)t x(t) + xSSB(t) - 90° - 90°

x Ideal LPF x cutoff = W/2 1 [Answer: the output is x3(t) + x6 (t) = cos2π ( fc + fm )t a USB signal ] 2

10. If x(t) has the spectrum, X(f), shown below. Verify that g(t) is a SSB signal by sketching the spectrum at each point in the system.

X(f)

-B B f

x Ideal LPF x cutoff = B Hz

x(t) 2 cos 2πBt 2 cos 2πfct + g(t) 2 sin 2πBt 2 sin 2πfct x Ideal LPF x cutoff = B Hz

-f Answer: c fc at the output

11. A baseband signal

x(t) = cos 2π (1000) t + cos 2π (5000) t

is input to the VSB modulator shown below.

x(t) x H(f) x (t) VSB

cos 2π(100KHz)t H(f) 1 .5

f(KHz) -102 -100 -98 98 100 102

a. Sketch the spectrum of the resulting VSB signal.

b. Write an expression for xVSB(t). If this VSB signal is input to the following demodulator: x 1 (t) x (t) x H(f) x (t) VSB 0

cos 2π(100KHz)t H(f)

f(KHz) -6 6 c. Write an expression for x1(t) d. Write an expression for xo(t) .25 .25*.75 = 3/16 .25*.25 = 1/16 Answer: a. 9599 101 105 1 3 1 b. x (t) = cos2π(95000)t + cos2π(99000)t + cos2π(101000)t VSB 2 8 8 1 3 x1(t) = [cos2π(195000)t + cos2π(5000)t] + [cos2π(199000)t + cos2π(1000)t] c. 4 16 1 + [cos2π(201000)t + cos2π(1000)t] 16 1 1 d. x (t) = cos2π(5000)t + cos2π(1000)t o 4 4

12. A SSB FDM system is to be implemented using the following design strategy. Channel 1 will be retained directly at baseband. A guard band equal to 25% of the bandwidth of channel 1 will be maintained between the upper edge of channel 1 and the lower edge of channel 2. A guard band equal to 25% of the bandwidth of channel 2 will be maintained between the upper edge of channel 2 and the lower edge of channel 3, and so on. Draw a spectral diagram for the composite baseband spectrum and compute the bandwidth if the system contains four data signals, each having a baseband bandwidth of 4 KHz.

Answer: 45 9 10 1415 19

13. Repeat problem 10 if the system contains four signals with assignments and baseband bandwidths as follows: Channel 1: W = 4 KHz Channel 2: W = 6 KHz Channel 3: W = 12 KHz Channel 4: W = 15 KHz

Answer: 45 11 12.5 24.5 27.5 42.5

Problem 14 - 15 require the use of MATLAB. Refer to MATLAB Tutorial #6 before attempting these problems.

14. A baseband signal x(t) = cos 2π (1000)t modulates a 10 KHz carrier wave to produce a lower sideband (USB) signal. a. Create a Simulink model that generates this baseband signal and multiplies it with the carrier. Display the spectrum of the resulting DSB signal using the Spectrum Analyzer.

b. Design a low pass filter with a cutoff frequency of 10 KHz to eliminate the upper sideband with at least 25 dB of sideband suppression. What is the lowest order filter that you can use? Display the spectrum of the SSB signal with the magnitude in dB.

15. Ideally the LSB signal produced in problem 9 will be of the form: x(t) = cos 2π (9000)t a. Create a Simulink model to generate this LSB signal (using a sine wave). Multiply the LSB signal by a 10 KHz sine wave and plot the spectrum of the output. b. Design a 5th order low pass filter with a cutoff frequency of 5 KHz to recover the baseband signal. Add this filter to the Simulink model and run the simulation again. Plot the spectrum of the system output. Is this the baseband signal?