ECS 332: Principles of Communications 2012/1 HW 1 Due: July 13 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems. (5 pt) (c) Late submission will be heavily penalized. (d) 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. In class, we have seen how to use the Euler s formula to show that cos 2 x = 1 (cos (2x) + 1). 2 For this question, use similar technique to show that cos A cos B = 1 (cos (A + B) + cos (A B)). 2 Problem 2. Listen to the Fourier s Song (Fouriers Song.mp3) which can be downloaded from http://sethares.engr.wisc.edu/mp3s/fouriers Song.mp3 Which properties of the Fourier Transform can you recognize from the song? List them here. Problem 3. Derive and plot the signal x(t) whose Fourier transform is given by ( ) 2 sin (5πf) X (f) = sinc 2 (5πf) =. (5πf) 1-1
2.3-2 For the signal g(1) shown in Fig. P2.3-2. ECS 332 HW 1 Due: July 13 2012/1 (a) Sketch the signals (i) g ( -1); (ii) g(1 + 12); (iii) g(31); (iv) g(6-2t). (b) Find the energies of the signals in part (a). Problem 4. For the signal g(t) shown in Figure 1.1, sketch the signals: Figure P.2.3-2 0.5 0 - I g(l) 24,_ Figure 1.1: Problem 4 (a) g( t) (b) g(t + 6) (c) g(3t) (d) g(6 t). Problem 5. Evaluate the following integrals: (a) g (τ) δ (t τ)dτ (b) δ (τ) g (t τ)dτ (c) δ (t) e j2πft dt (d) δ (t 2) sin (πt)dt (e) δ (t + 3) e t dt (f) (t 3 + 4) δ (1 t)dt (g) g (2 t) δ (3 t)dt 1-2
... 3.3-1 Apply the duality property to the appropriate pair in Table 3.1 to show that: (a) 0.5[8(1) + (jjm) ] <====> u(f) (b) 8(t + T ) + 8(t- T) <====> 2 cos 2nfT ECS 332 HW 1 Due: July 13 2012/1 (c) 8(t + T ) - 8(1 - T) <====> 2) sin 2njT Hint: g( -1) (h) e (x 1) cos ( <====> C (- f) and 8( 1) = 8( -t). π (x 5)) δ (x 3)dx 2 3.3-2 The Fourier transform of the triangul ar pulse g (I) in Fig. P3.3-2a is given as Problem 6. The Fourier transform of the triangular pulse g(t) in Figure 1.2a is given as I.? J..7 r C(j) = --(el-7r - j2trfel - lf. 1 ( G(f) = e j2πf j2πfe j2πf 1 ) - I) (2rrf) 2 (2πf) 2 Use this information. and the ti me-shifting and time-scaling properties. to find the Fourier Using this transforms information, of the and signals theshown time-shifting in Fig. P3.3-2b. and time c. d. scaling e, and f. properties, find the Fourier transforms of the signals shown in Figure 1.2b, c, d, e, and f. ure P.3.3-2 0 g(t)..... I (a) (b) (c) - I 0 [ -... 0 1- g/t) (- 2 l.5 g,(t) /.,.. -I 0 [ -... I I 0 I 0 (.,.. 2-2 2 (d) (e) (f) Figure 1.2: Problem 6 1-3
ECS 332 HW 1 Due: July 13 2012/1 Problem 7. Use properties of Fourier transform to evaluate the following integrals. (Do not integrate directly. Recall that sinc(x) = sin(x).) Clearly state the property or properties x that you use. (a) sinc ( 5x ) dx (b) e 2πf 2j 2sinc (2πf) ( e 2πf 5j 2sinc (2πf) ) df (c) sinc ( 5x ) sinc ( 7x ) dx (d) sinc (π (x 5)) sinc ( π ( x 2)) 7 dx 1-4
Q1 Euler's Formula Thursday, November 11, 2010 2:54 PM ECS 332 HW 1 Sol Page 1
Q3 Sinc Function and Triangular Signal Wednesday, July 06, 2011 12:16 PM ECS 332 HW 1 Sol Page 2
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Q4 Manipulation of time Wednesday, July 06, 2011 12:20 PM All the signals are plotted below ECS 332 HW 1 Sol Page 4
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Q5 Sifting Property of the Delta Function Wednesday, July 06, 2011 12:46 PM ECS 332 HW 1 Sol Page 6
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Q6 Using Properties of FT Wednesday, July 06, 2011 1:11 PM ECS 332 HW 1 Sol Page 8
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Q7 Integrations involving sinc function(s) Wednesday, July 27, 2011 8:23 PM ECS 332 HW 1 Sol Page 10
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ECS 332: Principles of Communications 2012/1 HW 2 Due: July 27 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems (5 pt). Question 8 and Question 9 are optional. However, Question 3.c, (c) Late submission will be heavily penalized. (d) 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. 1 Using MATLAB to find the spectrum of a signal: A signal g(t) can often be expressed in analytical form as a function of time t, and the Fourier transform is defined as the integral of g(t) exp( j2πf t). Often however, there is no analytical expression for a signal, that is, there is no (known) equation that represents the value of the signal over time. Instead, the signal is defined by measurements of some physical process. For instance, the signal might be the waveform at the input to the receiver, the output of a linear filter, or a sound waveform encoded as an mp3 file. In all these cases, it is not possible to find the spectrum by analytically performing a Fourier transform. Rather, the discrete Fourier transform (or DFT, and its cousin, the more rapidly computable fast Fourier transform, or FFT) can be used to find the spectrum or frequency content of a measured signal. The MATLAB function plotspec.m, which plots the spectrum of a signal can be downloaded from our course website. Its help file 2 notes % plotspec(x,ts) plots the spectrum of the signal x % Ts = time (in seconds) between adjacent samples in x (a) The function plotspec.m is easy to use. For instance, the spectrum of a rectangular pulse g(t) = 1[0 t 2] can be found using: 1 Based on [Johnson, Sethares, and Klein, 2011, Sec 3.1 and Q3.3]. 2 You can view the help file for the MATLAB function xxx by typing help xxx at the MATLAB prompt. If you get an error such as xxx not found, then this means either that the function does not exist, or that it needs to be moved into the same directory as the MATLAB application. 2-1
ECS 332 HW 2 Due: July 27 2012/1 % specrect.m plot the spectrum of a square wave close all time=20; % length of time Ts=1/100; % time interval between samples t=0:ts:(time Ts); % create a time vector x=[t 2]; % rectangular pulse 1[0 t 2] plotspec(x,ts) % call plotspec to draw spectrum xlim([ 5,5]) The output of specrect.m is shown in Figure 2.1. The top plot shows the first 20 seconds of g(t). The bottom plot shows G(f). Use what we learn in class about the Fourier transform of a rectangular pulse to find a simplified expression for G(f). Does your expression agree with the bottom plot in Figure 2.1. 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Seconds 3 Magnitude 2 1 0 5 4 3 2 1 0 1 2 3 4 5 Frequency [Hz] Figure 2.1: Plots from specrect.m 2-2
ECS 332 HW 2 Due: July 27 2012/1 (b) Mimic the code in specsquare.m to find the spectrum of an exponential pulse s(t) = e t u(t). Note that you may want to change the parameter time to capture most of the content of s(t) in the time domain. You may also use the command xlim to zoom in the spectrum plot. (c) Continue from part (b), find S(f) analytically. Compare your analytical answer with the plot in part (b). (d) MATLAB can also perform symbolic manipulation when symbolic toolbox is installed. Run the file SymbFourier.m. Check whether you have the same result as part (c). Problem 2. 3 (a) Consider the cosine pulse p (t) = (i) Use MATLAB to plot p(t) for 3 t 3. (ii) Find P (f). { cos (10πt), 1 t 1 0, otherwise (iii) Use the expression from part (ii) to plot P (f) in MATLAB. (b) Consider the cosine pulse (i) Find P (f) analytically. p (t) = { cos (10πt), 2 t 4 0, otherwise (ii) Use MATLAB. Mimic the code in specsquare.m to plot the spectrum of p(t). (iii) Compare your analytical answer from part (i) with the plot in part (ii). 3 Inspired by [Carlson and Crilly, 2009, Q2.2-1 and Q2.2-2]. 2-3
ECS 332 HW 2 Due: July 27 2012/1 Problem 3. Consider a signal g(t). Recall that G(f) 2 is called the energy spectral density of g(t). Integrating the energy spectral density over all frequency gives the signal s total energy. The energy contained in the frequency band B can be found from the integral B G(f) 2 df. In particular, if the band is simply an interval of frequency from f 1 to f 2, then the energy contained in this band is given by f2 G(f) 2 df. (2.1) f 1 In this problem, assume (a) Find the (total) energy of g(t). g(t) = 1[ 1 t 1]. (b) It is mentioned in class that the main lope of the sinc waveform contains about 90% of the total energy. Check this fact by first computing the energy contained in the frequency band occupied by the main lope and then compare with your answer from part (a). Hint: Find the zeros of the main lope. This give f 1 and f 2. Now, we can apply (2.1). MATLAB or similar tools can then be used to numerically evaluate the integral. (c) Suppose we want to include more energy by considering wider frequency band. Let this band be the interval B = [ f 0, f 0 ]. Find the minimum value of f 0 that allows the band to capture at least 99% of the total energy in g(t). Problem 4. Given a system with input-output relationship of y = f(x) = 2x + 10, is this system linear? [Carlson and Crilly, 2009, Q2.3-10] Problem 5. Signal x(t) = 10 cos(2π 7 10 6 t) is transmitted to some destination. The received signal is y(t) = 10 cos(2π 7 10 6 t π/6). (a) What is the minimum distance between the source and destination? (b) What are the other possible distances? [Carlson and Crilly, 2009, Q2.3-14] 2-4
ECS 332 HW 2 Due: July 27 2012/1 Problem 6. You are given the baseband signals (i) m(t) = cos 1000πt; (ii) m(t) = 2 cos 1000πt+ cos 2000πt; (iii) m(t) = (cos 1000πt) (cos 3000πt). For each one, do the following. (a) Sketch the spectrum of m(t). (b) Sketch the spectrum of the DSB-SC signal m(t) cos 10, 000πt. [Lathi and Ding, 2009, Q4.2-1] Problem 7. This question starts with a square-modulator for DSB-SC similar to the one we discussed in class. Then, the use of the square-operation block is further explored on the receiver side of the system. [Doerschuk, 2008, Cornell ECE 320] F (a) Let x(t) = A c m(t) where m(t) M(f) is bandlimited to B, i.e., M(f) = 0 for F 1 f > B. Consider the block diagram shown in Figure 2.2. xt + ut vt 2 H f BP yt Assume f c B and 2 cos 2 ft c Figure 2.2: Block diagram for Problem 7a 1, f f c B H BP (f) = 1, f + f c B 0, otherwise. The block labeled { } 2 has output v(t) that is the square of its input u(t): Find y(t). v(t) = u 2 (t). (b) The block diagram in part (a) provides a nice implementation of a modulator because it may be easier to build a squarer than to build a multiplier. Based on the successful use of a squaring operation in the modulator, we decide to use the same squaring operation in the demodulator. Let x (t) = A c m (t) 2 cos (2πf c t) 2-5
ECS 332 HW 2 Due: July 27 2012/1 xt + 2 H f LP I y t 2 cos 2 ft c Figure 2.3: Block diagram for Problem 7b + Q xt F y t where m(t) M(f) is bandlimited 2 to B, i.e., HLP M(f) f = 0 for f > B. Again, F 1 assume f c B Consider the block diagram shown in Figure 2.3. Use 2 sin 2 ft c H LP (f) = { 1, f B 0, otherwise. Find y I (t). Does this block diagram work as a demodulator; that is, is y I (t) proportional to m(t)? (c) Due to the failure in part (b), we have to think hard and it seems I xt y natural t to consider also the block diagram with cos replaced 2 by sin. Let HLP f x (t) = A c m (t) 2 cos (2πf c t) 2 cos 2 ft c F where m(t) M(f) is bandlimited to B, i.e., M(f) = 0 for f > B as in part (b). F 1 + Again, assume f c B Consider the block diagram shown in Figure 2.4. xt + 2 HLP f Q y t 2 sin 2 ft c Figure 2.4: Block diagram for Problem 7c As in part (b), use Find y Q (t). H LP (f) = { 1, f B 0, otherwise. (d) Use the results from parts (b) and (c). Draw a block diagram of a successful DSB-SC demodulator using squaring operations instead of multipliers. 2-6
ECS 332 HW 2 Due: July 27 2012/1 Problem 8 (Cube modulator). Consider the block diagram shown in Figure 2.5 where { } 3 indicates a device whose output is the cube of its input. mt + xt yt 3 H f zt 2 cos 2 ft 0 Figure 2.5: Block diagram for Problem 8. Note the use of f 0 instead of f c. F Let m(t) M(f) be bandlimited to B, i.e., M(f) = 0 for f > B. 1 F (a) Plot an H(f) that gives z (t) = m (t) 2 cos (2πf c t). What is the gain in H(f)? What is the value of f c? Notice that the frequency of the cosine is f 0 not f c. You are supposed to determine f c in terms of f 0. (b) Let M(f) be M (f) = { 1, f B 0, otherwise. (i) Plot X(f). (ii) Plot Y (f). Hint: M (f) M (f) = { 2B f, f 2B 0, otherwise. Do not attempt to make an accurate plot or calculation for the Fourier transform of m 3 (t). (c) For your filter of part (a), plot z(t). [Doerschuk, 2008, Cornell ECE 320] 2-7
4.2-2 Repeat Prob. 4.2-1 if (i) m(t) = sine ( loom); (ii) m(t) = ( l + t 2 )- 1 ; (iii) m(t) = e- IO it- l l _ Observe that e- 10 lt- 1 1 is e- IO irl delayed by l second. For the last case you need to consider both the amplitude and the phase spectra. 4.2-3 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~. ECS 332 HW 2 Due: July 27 2012/1 (a) What kind of filter is req uired in Fi g. P4.2-3? (b) Determ ine the signal spectra at points b and c, and indicate the frequency bands occupied by Problem 9. You are asked to design a DSB-SC modulator to generate a modulated signal these spectra. km(t) cos(w c + θ), where m(t) is a signal band-limited to B Hz. Figure 2.6 shows a DSB-SC (c) What is the minimum usable value of eve? modulator available in the stockroom. The carrier generator available generates not cos ω c t, but cos 3 (d) Wo uld this scheme work if the carrier generator output were sin ω c t. Explain whether you would be able to generate the 3 cvc l') Explain. desired signal using only (f) Wo uld thi s scheme work if the carrier generator out put were cos" cvet for an y integer n ::: 2.? this equipment. You may use any kind of filter you like. Figure P.4.2-3 m (t)., I Filter @.., M(.f) -8 B J-- (a) (b) Figure 2.6: Problem 9 4.2-4 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) What kind of filter is required in Figure 2.6? (b) Determine the signal spectra at points (b) and (c), and indicate the frequency bands occupied by these spectra. (c) What is the minimum usable value of f c? (d) Would this scheme work if the carrier generator output were cos 2 ω c t? Explain. (e) Would this scheme work if the carrier generator output were cos n ω c t for any integer n 2? [Lathi and Ding, 2009, Q4.2-3] 2-8
Magnitude Magnitude Q1 Spectrum via MATLAB a. You may recall that the Fourier transform of 1t a Hence, 1 t 1 2sinc2 f. Note that g t 10 t 2 is given by 2asinc 2 fa. is simply 1t 1 time-shifted by 1. As we have discussed in class, time shifting does not change the magnitude of the spectrum. Hence, G f is the same as the magnitude of the Fourier transform of 1t 1. Therefore, 2 sinc2 f G f. In the Figure (i) below, the theoretical expression above is plotted using the x marks on top of the provided plot from specrect.m. The marks match the theoretical plot. 1 1 0.5 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Seconds 3 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Seconds 1 2 1 0.5 0-5 -4-3 -2-1 0 1 2 3 4 5 Frequency [Hz] (i) b. See Figure (ii) above. 0-10 -8-6 -4-2 0 2 4 6 8 10 Frequency [Hz] (ii) c. S j2 ft t j2 ft 1j 2 f t f s t e dt e u t e dt e dt 1j 2 f t e 1 j2 f t0 1 j2 1 1 f S f is plotted in part (c) using the x marks on top of the plots from plotspec.m. They are virtually identical. 0 d. With variable a in the m-file set to 1, we have same result.
Q2 Cosine Pulses Wednesday, July 18, 2012 3:43 PM ECS332 HW 2 Sol Page 1
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Q3 Parseval's Theorem and Energy Calculation Wednesday, July 18, 2012 9:32 PM ECS332 HW 2 Sol Page 4
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ECS332 HW 2 Sol Page 7 Q4 Linear System Wednesday, July 18, 2012 4:57 PM
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Q6 Tone Modulation Thursday, July 14, 2011 4:40 PM ECS332 HW 2 Sol Page 10
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Q7 Square MODEM Thursday, July 14, 2011 2:22 PM ECS332 HW 2 Sol Page 13
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Q8 Cube Modulator Thursday, July 14, 2011 2:11 PM ECS332 HW 2 Sol Page 17
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Q9 Powered Cosine Modulator Sunday, July 03, 2011 6:01 PM ECS332 HW 2 Sol Page 20
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ECS 332: Principles of Communications 2012/1 HW 3 Due: Aug 8 and Aug 10 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems. (5 pt) (c) Late submission will be heavily penalized. (d) 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. (e) [M2011] means the problem is from the 2011 midterm exam which is posted on the course website. (f) Submit Problems 1-3 by Aug 8 and Problems 4-6 by Aug 10. Problem 1. (a) Give a simplified expression for the Fourier transform P (f) of a waveform p(t) when { A, 0 t < T p (t) = 0, otherwise (b) A message m = (m[0], m[1], m[2], m[3]) = (1, 1, 1, 1) is sent via where l is the length of m. l 1 x (t) = m [k] p (t kt ), k=0 Find a simplified expression for the Fourier transform X(f) of the waveform x(t). 3-1
ECS 332 HW 3 Due: Aug 8 and Aug 10 2012/1 (c) Assume T = 2 [ms] and A = 1 [mv]. For X(f) generated by m given below, analytically evaluate X(0). (i) m = (1); (ii) m = (1, 1) (iii) m = (1, 1, 0, 0) (iv) m = (1, 1, 1) (v) m = (1, 1, 1, 1) (vi) m = (1, 1, 1, 1) (vii) m = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) (d) When we know how to find X(f) analytically, we may use its expression to plot X(f) in MATLAB without the help of plotspec.m. With the help of the provided function FTofManyShiftedRect.m, you may run HW3 Q1.m to plot X(f) from part (b). Modify the code in HW3 Q1.m to plot X(f) for the m given in part (c). (e) What did you learn from the plots in part (d)? Problem 2 (Fourier Transform of Digital Transmisson). Solve Question 5 in [M2011]. Problem 3 (QAM). Let x QAM (t) = m 1 (t) 2 cos (ω c t) + m 2 (t) 2 sin (ω c t). In class, we have shown that { LPF x QAM (t) } 2 cos (ω c t) = m 1 (t) Give a similar proof to show that { LPF x QAM (t) } 2 sin (ω c t) = m 2 (t) 3-2
ECS 332 HW 3 Due: Aug 8 and Aug 10 2012/1 Problem 4. In quadrature amplitude modulation (QAM ) or quadrature multiplexing, two baseband signals m 1 (t) and m 2 (t) are transmitted simultaneously via the following QAM signal: x QAM (t) = m 1 (t) 2 cos (ω c t) + m 2 (t) 2 sin (ω c t). An error in the phase or the frequency of the carrier at the demodulator in QAM will result in loss and interference between the two channels (cochannel interference). In this problem, show that { LPF x QAM (t) } 2 cos ((ω c + ω) t + δ) = m 1 (t) cos (( ω) t + δ) m 2 (t) sin (( ω) t + δ) { LPF x QAM (t) } 2 sin ((ω c + ω) t + δ) = m 1 (t) sin (( ω) t + δ) + m 2 (t) cos (( ω) t + δ). Problem 5. 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 3.1. 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 3.1: Receiver for Problem 5a 3-3
HWR ECS 332 HW 3 Due: Aug 8 and Aug 10 2012/1 HWR Figure 3.2: Receiver for Problem 5b (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 3.2. 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). Problem 6. Solve Question 7 in [M2011]. 3-4
ECS 332: Solution for Problem Set 3 Problem 1 a. Give a simplified expression for the Fourier transform P f of a waveform pt when Solution pt A, 0t T 0, otherwise T j2 ft j2 ft 1 j2 ft T P f c t e dt Ae dt A e j2 f 1 A j2 f 0 A j2 f j2 ft j2 ft e 1 1 e b. A message m m0, m1, m2, m3 1, 1,1,1 is sent via 1 xt mkct kt where is the length of m. k0 Find a simplified expression for the Fourier transform X f of the waveform xt. Solution 1 1 j2 fkt We start with xt mkct kt X f P f mke. Hence, k0 k0 X f P f m m e m e m e 2 3 1 z1 z z z j1 ft j2 fkt j2 fkt 0 1 2 3 ; 4 A 1 zm0 m1 z m2 z m3 z ; z e j2 f A j2 f A j2 f A j2 f 1 2z 2z z 2 4 1 2e 2e e j1 ft j2 ft j4 ft 0 2 3 j1 ft
c. Assume T = 2 [ms] and A = 1 [mv]. Find X 0 for the following m. i. m 1 ii. m 1,1 iii. m 1,1,0,0 iv. m 1,1, 1 v. m 1,1, 1,1 vi. m 1,1, 1, 1 vii. m 1,1, 1,1, 1, 1,1,1,1, 1,1,1 Solution First, we find Then j2 0t 0 X c t e dt c t dt AT 1 1 1 2 0 0 0 j kt 0 X P m k e P m k AT m k After plugging in the numbers, we have X k0 k0 k0 0 2, 4, 4, 2, 4, 0, 8 10 6 V/Hz d. All the plots for X f are shown on the next page
[10-6 ] [10-6 ] [10-6 ] [10-6 ] [10-6 ] [10-6 ] [10-6 ] 2 1 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 4 2 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 4 2 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 6 4 2 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 6 4 2 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 6 4 2 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz] 10 5 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 f [khz]
5. (5 pt) In this question, you are provided with a partial proof of an important result in the study of Fourier transform. Your task is to figure out the quantities/expressions inside the boxes labeled a,b,c, and d. Put your answers in the spaces provided at the end of the question. No explanation is needed. We start with a function that involves expression for gt. Then, we define xt g t T xt in terms of a sum that involves g t. What you will see next is our attempt to find another G f.. It is a sum
To do this, we first write xt as xt g t* t T convolution-in-time property, we know that X We can get f is given by X f G f a f b. xt back from X. Then, by the f by the inverse Fourier transform formula: j2 ft. After plugging in the expression for X x t X f e df we get j2 ft x t e G f a f b df j2 ft a e G f f b df By interchanging the order of summation and integration, we have j2 ft x t a e G f f b df. f from above, We can now evaluate the integral via the sifting property of the delta function and get c x t a e G d.. a = c = b = d =
Q2 Poisson Sum Formula [M2011 Q5] Monday, August 06, 2012 7:49 PM ECS332 HW 3 Sol Page 1
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Q4 QAM with Phase and Frequency Offset Sunday, July 03, 2011 7:26 PM ECS332 HW 3 Sol Page 2
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Q5 (a) Time Delay and Phase Offset (b) HWR Rx with Time Delay Thursday, November 11, 2010 11:17 AM ECS332 HW 3 Sol Page 4
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7. (8 pt) Suppose mt M f is bandlimited to W, i.e., M f 0 for f W. Consider the following DSB-SC transceiver. Transmitter (modulator) mt xt Delayed by cos c t xt FWR vt H f LP yt Receiver (demodulator) Also assume that c f W and that H f LP 1, f W 0, otherwise. Make an extra assumption that mt 0 for all time t and that the full-wave rectifier (FWR) input-output relation is described by a function f FWR : f FWR (Questions start on the next page.) x x, x 0, x, x 0.
a. Recall that the half-wave rectifier input-output relation is described by a x, x 0, function f HWR : fhwr x 0, x 0. We have seen in class and in HW2 that when the receiver uses half-wave rectifier, v t x t g t, HWR where g t 1cos t 0 HWR c. i. (3 pt) The receiver in this question uses full-wave rectifier. Its vt can be described in a similar manner; that is v t x t g t. Find gfwr t. Hint: FWR g t c g t c for some constants c 1 and c 2. Can 1 HWR 2 you find these constants? FWR ii. (2 pt) Recall that the Fourier series expansion of ghwr t is given by 1 2 1 1 1 ghwr t cosct cos3 ct cos5 ct cos7 ct 2 3 5 7. Find the Fourier series expansion of g t. FWR
Name ID b. (3 pt) Find yt (the output of the LPF).
Q6 FWR Rx with Time Delay Sunday, August 05, 2012 9:46 PM ECS332 HW 3 Sol Page 7
ECS332 HW 3 Sol Page 8
ECS332 HW 3 Sol Page 9