HW 6 Due: November 9, 4 PM

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1 Name ID3 ECS 332: Principles of Communications 2018/1 HW 6 Due: November 9, 4 PM Lecturer: Prapun Suksompong, Ph.D. Instructions (a) This assignment has 10 pages. (b) (1 pt) Work and write your answers directly on these sheets (not on other blank sheets of paper). Hard-copies are distributed in class. (c) (1 pt) Write your first name and the last three digits of your student ID on the upper-right corner of this page. (d) (8 pt) Try to solve all non-optional problems. (e) Carefully write down all the steps that you have done to obtain your answers. You may not get full credit even when your answer is correct without showing how you get your answer. Problem 1. Consider an AM transmitter. (a) Suppose the message is m(t) = 4 cos(10πt) and the transmitted signal is x AM (t) = A cos(100πt) + m(t) cos(100πt). Find the value of A which yields the modulation index in each part below. (i) µ = 50% (ii) µ = 100% (iii) µ = 150% (b) Suppose the message is m(t) = α cos(10πt) and the transmitted signal is x AM (t) = 4 cos(100πt) + m(t) cos(100πt). Find the value of α which yields the modulation index in each part below. 6-1

2 (i) µ = 50% (ii) µ = 100% (iii) µ = 150% Problem 2. Recall that, in QAM system, the transmitted signal is of the form x QAM (t) = m 1 (t) 2 cos (2πf c t) + m 2 (t) 2 sin (2πf c t). We want to express x QAM in the form x QAM (t) = 2E(t) cos(2πf c t + φ(t)), where E(t) 0 and φ(t) ( 180, 180 ]. (This shows that QAM can be expressed as a combination of amplitude modulation and phase modulation.) Consider m 1 (t) and m 2 (t) plotted in Figure 6.1. Draw the corresponding E(t) and φ(t). 1 m 1 (t) t m 2 (t) t Figure 6.1: m 1 (t) and m 2 (t) for Problem

3 the following. ECS 332 Observe that e- HW 10 lt- 1 1 is 6 e- IO irl Due: delayed November by l second. For 9, the 4last PM case you need to consider both 2018/1 Extra Questions (a) Sketch the spectrum of m (c). (b) Sketch the spectrum of the DSB-SC signal m(c) cos IO,OOO.rrt. (c) Identify the upper sideband (USB) and the lower sideband (LSB) spectra Repeat Prob if (i) m(t) = sine ( loom); (ii) m(t) = ( l + t 2 )- 1 ; (iii) m(t) = e- IO it- l l _ the amplitude and the phase spectra You are asked to design a DSB-SC modulator to generate a modulated signal km(l) cos (we t +8), where m(t) is a signal band-limited to B Hz. Figure P4.2-3 shows a DSB-SC modulator ava il able in the stockroom. The ca ttier generator avail able generates not cos Wet, but cos 3 Wei. Exp lain whether you would be able to generate the desired signal using onl y thi s equipment. You may use any kind of fil ter you li k~. Here are some optional questions for those who want more practice. (a) What kind of filter is req uired in Fi g. P4.2-3? Problem 3. You are asked to design a DSB-SC modulator to generate a modulated signal km(t) cos(2πf (b) c t), Determ whereine m(t) the signal is aspectra signal at points band-limited b c, to indicate B Hz. the frequency Figure bands 6.2 shows occupied a by DSB-SC these spectra. modulator available in the stockroom. Note that, as usual, ω c = 2πf c. The carrier generator (c) What is the minimum usable value available generates not cos (2πf c t), but cos 3 of eve? (2πf c t). Explain whether you would be able to (d) Wo uld this scheme work if the carrier generator output were sin generate the desired signal using only this equipment. You may 3 cvc l') Explain. use any kind of filter you (f) Wo uld thi s scheme work if the carrier generator out put were cos" cvet for an y integer n ::: 2.? like. [Lathi and Ding, 2009, Q4.2-3] Figure P m (t)., I M(.f) -8 B J-- (a) (b) Figure 6.2: Problem You are asked to design a DSB-SC modulator to generate a modulated signal km(l) cos wet with the carrier frequency f e = 500kHz (we = 2.rr x 500, 000). The following equipment is ava ilable in the stockroom: (i) a signal generator of frequency I 00 k.h z; (ii) a ring modulator; (iii) a bandpass fi Iter tuned to 500 k.hz. (a) We know that a real-valued signal r(t) that is even and periodic with period T 0 can be expanded using Fourier series into r(t) = c 0 + a 1 cos(2πf 0 t) + a 2 cos(2π(2f 0 )t) + a 3 cos(2π(3f 0 )t) + (6.1) where f 0 = 1 T 0. Consider the signal r(t) = cos 3 (2πf c t). (i) Is it periodic? (ii) Is it even? (iii) Expand r(t) = cos 3 (2πf c t) into a linear combination of cos(2π(nf c )t) as in (6.1) above. 6-3

4 (b) What kind of filter is required in Figure 6.2? (c) Determine the signal spectra at points (b) and (c) in Figure 6.2, and indicate the frequency bands occupied by these spectra. 6-4

5 (d) What is the minimum usable value of f c? (e) Would this scheme work if the carrier generator output were cos 2 (2πf c t)? Explain. 6-5

6 Problem 4. Consider the basic DSB-SC transceiver with time-delay channel presented in class. Recall that the input of the receiver is F where m(t) that f c B. F 1 x (t τ) = m (t τ) 2 cos (ω c (t τ)) M(f) is bandlimited to B, i.e., M(f) = 0 for f > B. We also assume (a) Suppose that, at the receiver, we multiply by 2 cos ((ω c t) θ) instead of 2 cos (ω c t) as illustrated in Figure 6.3. Assume { 1, f B H LP (f) = 0, otherwise. Find ˆm(t) (the output of the LPF). xt vt HLP f ˆm t 2 cos t c Figure 6.3: Receiver for Problem 4a 6-6

7 HWR ECS 332 HW 6 Due: November 9, 4 PM 2018/1 HWR Figure 6.4: Receiver for Problem 4b (b) Use the same assumptions as part (a). However, at the receiver, instead of multiplying by 2 cos ((ω c t) θ), we pass x(t τ) through a half-wave rectifier (HWR) as shown in Figure 6.4b. Make an extra assumption that m(t) 0 for all time t and that the half-wave rectifier input-output relation is described by a function f( ): { x, x 0, f (x) = 0, x < 0. Find ˆm(t) (the output of the LPF). 6-7

b. (5 pt) Sketch the spectrum of the DSB-SC signal mtcos 5,000 t explanation is needed.. No ECS 332 HW 6 Due: November 9, 4 PM 2018/1 7. (8 pt) Suppose mt M f is bandlimited to W, i.e., M f 0 for F Problem f W5.

Transmitter (modulator) mt xt Delayed by cos c t xt FWR vt LP H f yt Receiver (demodulator) Figure 6.5: A DSB-SC transceiver 1, f W Also assume that fc W and that HLP f { 1, 0, f otherwise.

8 b. (5 pt) Sketch the spectrum of the DSB-SC signal mtcos 5,000 t explanation is needed.. No ECS 332 HW 6 Due: November 9, 4 PM 2018/1 7. (8 pt) Suppose mt M f is bandlimited to W, i.e., M f 0 for F Problem f W5.(M2011Q7). Consider the Suppose following m DSB-SC (t) transceiver. M (f) is bandlimited to W, i.e., M (f) = 0 F 1 for f > W. Consider a DSB-SC transceiver shown in Figure 6.5. Transmitter (modulator) mt xt Delayed by cos c t xt FWR vt LP H f yt Receiver (demodulator) Figure 6.5: A DSB-SC transceiver 1, f W Also assume that fc W and that HLP f { 1, 0, f otherwise. W Also assume that f c W and that H LP (f) = 0, otherwise. Make Make an extra an extra assumption assumption that that m mt (t) 0 for for all all time time t and t and that that the full-wave the full-wave rectifier (FWR) rectifier input-output (FWR) input-output relation is described relation is by described a function by a ffunction F W R ( ): f FWR : { x, x 0, ffwr x x, x 0, f F W R (x) = x x,, x x 0. < 0. (a) (Questions Recall that start theon half-wave the next { page.) rectifier input-output relation is described by a function x, x 0, f HW R ( ) : f HW R (x) = We have seen in Problem 4b that when the 0, x < Page 5 of 12

9 receiver uses half-wave rectifier, where g HW R (t) = 1 [cos (ω c t) 0]. v (t) = x (t τ) g HW R (t τ) (i) The receiver in this question uses full-wave rectifier. Its v(t) can be described in a similar manner; that is v (t) = x (t τ) g F W R (t τ). Find g F W R (t). Hint: g F W R (t) = c 1 g HW R (t) + c 2 for some constants c 1 and c 2. Find these constants. 6-9

10 (ii) Recall that the Fourier series expansion of g HW R (t) is given by g HW R (t) = ( cos ω c t 1 π 3 cos 3ω ct cos 5ω ct 1 ) 7 cos 7ω ct Find the Fourier series expansion of g F W R (t). (b) Find y(t) (the output of the LPF). Problem 6. Would the scheme in Problem 3 work if the carrier generator output were cos n ω c t for any integer n 2? 6-10

HW 6 Due: November 3, 10:39 AM (in class)

ECS 332: Principles of Communications 2015/1 HW 6 Due: November 3, 10:39 AM (in class) Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do