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hwk8_solns - ECE 300 Signals and Systems Homework 8 Due...

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Unformatted text preview: . ECE 300 Signals and Systems Homework 8 Due Date: Matlab/Prelab, Tuesday October 30 at the beginning of class Problems 1—6; Thursday November 1 at the beginning of class Problems 1. ZTF, problem 4—4. /2. Using the duality grogegy, ﬁnd the corresponding Fourier transform for the following: 3) g(t) = sincz(Bt) b) g(t) = sinc(Wt) C) g(t) = 5(1) d) g(t) = cosmot) Do not just look up the pairs from the table (though you can use any other pairs except the one you are trying to ﬁnd). . f 3. Consider a linear time invariant system with transfer function given by —j2w S 2 Hm) = {Se ml 0 else 2(t-1) 72' ‘with inputx(t) = —8—sincz[ ; 7r . The output of the system is y(t). a) Determine X (0)). b) Sketch the spectrum of X (0)) (magnitude and phase) accurately labeling the axes and important points. c) Sketch the spectrum of H ((0) (magnitude and phase) accurately labeling the axes and important points. d) Determine y(t) , the output of the system. Answer y(t) = Esme (t — 3)] + Esine2 (t — 3)] 7r 7: 7r 7r Fall 2007 @onsider a linear time invariant system with impulse response given by h(t) = isinc£££j with input x(t) = isine cos(t). The output of the system 27: 27! 7! 72’ is y(t). a) Determine X (co) . b) Sketch the spectrum of X (cu) (magnitude and phase) accurately labeling the axes and important points. 0) Determine the energy in x(t) d) Determine H ((0) . e) Sketch the spectrum of H (w) (magnitude and phase) accurately labeling the axes and important points. f) Determine y(t), the output of the system. 9) Determine the energy in y(t). /5. Find the fraction of the total signal energy (as a percentage) contained between 100 and 300 Hz in the signalx(t)given below: x(t)=55inc( t +5sinc( t ) Answer 56% 0.002 0.001 A. In this problem we’ll look at a real world situation when we have to truncate a signal. This actually happens more with digital signal processing, but we can get the basic idea using our continuous time abilities. a) Find an expression for the Fourier transform of f (t) = cos(4t)+cos(5t). b) Now assume we look at f (t) for a ﬁnite time, say T seconds. What we see is actually y(t) = f (t)rect(t/T). Determine an expression for the Fourier transform of y(t), and write your answers in terms of sinc functions. c) Plot, using Matlab, Y(a)) for a) between 0 and 10 when T =1, T =6, T=10, T =20, and T =40. Can you clearly tell there are two cosines present when you are looking at Y (w) for all values of T? What happens as T gets larger (you are looking at more and more data)? Think in terms of the width of the sinc function (the distance between the ﬁrst nulls). Note: The sinc function exists in Matlab. Fall 2007 . tf command and Bode commands will be really useful here. Note that you can click on the curve on the Bode plot to read it more accurately. Turn in your plot. c) Matlab’s command r = pole(H), where H is the transfer function of the Buttewvorth ﬁlter, returns the poles of the transfer function in the array r. Using Matlab’s commands abs and angle, relate the magnitude of the poles to (up, then plot the pole locations in the complex plane on a circle with radius mp. (Note that angle returns angles in radians, and you probably want angle in degrees.) Note that the pole locations are all separated by an angleB. What is this angle? d) For mp =15 rad/sec, ms = 35 rad/sec, and A 2 28 dB, determine the required Butterworth ﬁlter order for this ﬁlter. (Remember it must be an integer). Using the Table at the end of this problem, plot the Bode plot of your Butterworth ﬁlter and verify that all frequencies w>ws have magnitude (power) less than Am. Matlab’s tf command and Bode commands will be really useful here. Turn in your plot. e) Matlab’s command r = pole(H), where H is the transfer function of the Butten/vorth ﬁlter, returns the poles of the transfer function in the array r. Using Matlab's commands abs and angle, relate the magnitude of the poles to cop, . then plot the pole locations in the complex plane on a circle with radius cop. (Note that angle returns angles in radians, and you probably want angle in degrees.) Note that the pole locations are all separated by an angle6l. What is this angle? n denominator(s) S 1 +1 (up 2 2 {3—) +1.414[i]+1 mp mp 3 2 3 —S— +2 1 +2 1 +1 mp (OP (0P ' 4 3 2 4 i +2.6131 —s— +3.4142 i +2.6131 :3— +1 (Up (Up mp (0P 5 4 3 2 5 i +3.2361—i— +5.2361i +5.2361—i— +3.2361—S- +1 (up 60’, (Up (OF (UP Table 1: Denominators of Buten/vorth ﬁlter for ﬁlter orders 1—5. The numerator for the Butten/vorth ﬁlter is 1. Fall 2007 7. (Matlab/Prelab Problem) A Butteiworth ﬁlter has the property that it is maximally ﬂat in the passband. An nth order Butterworth ﬁlter has the magnitude squared response 1 —_—2n “[2] (017 where (up is the passband frequency. At this frequency the power has been reduced by one half or 3 dB, |H(w)lz= 1 1 orlolog10 lH(a)p) |2= 101og10 [a] = —3dB |H(60,,)lz=———-= 2n { F] To determine the required order of a ﬁlter we often look at the desired stopband frequency, ms. Usually we want to indicate the minimum required power difference between the passband and the stopband, A. A is the rejection.Hence we have i 2 A = zologm |H(0) l —2010g,0 |H(ws)l or A =—1010g10 a) . . . . 5 Is called the transmon rat/o. A 1:1[1010 -1) n = ——————— mp Note that n must be an integer, so we always round up (to the next larger integer). The ratio (up a) Show that we can write b) For wp =10 rad/sec, ms 2 20 rad/sec, and A = 18 dB, determine the required Butterworth ﬁlter order for this ﬁlter. (Remember it must be an integer). Using the Table at the end of this problem, plot the Bode plot of your Butterworth ﬁlter and verify that all frequencies a2>w5 have magnitude (power) less than Amax .Matlab’s Fall 2007 @ 27F paw/97W 9) 4212(an 77g) ,5 aid/4,. my) 1" f ma aim/é -OU 6Q : fir/de/Mﬂ/é -/ f 1%“) 3/)? £154? -00 - 03 f p)\ mm“ a data” 7» 0 $0 31+er “kg 3““0 : j/rmuoqwﬁﬂ ” o be ' 9mg) ws(wt\l£ + 5?,“er 003 (w E‘Léi- ~03 0 WW: 4: . 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We) 9‘ 7 {£6 2‘ MWWA T=20 10/16/06 1:27 PM C:\Documents and Settin s\throne\M Document.. .\homework5 sinc.m 1 of 1 % . plot for problem 5 of homework 5 % w: T: m Y1 T: a: Y4O linspace(0,lO,lOOO); T/(2*pi); = @(w) (T/2)*(sinc(a*(w+4))+sinc(a*(w—4))+sinc(a*(w+5))+sinc(a*(w—5))) 6; T/(2*pi); = @(w) (T/2)*(sinc(a*(w+4))+sinc(a*(w~4))+sinc(a*(w+5))+sinc(a*(w—5))) 10; T/(2*pi); = @(w) (T/2)*(sinc(a*(w+4))+sinc(a*(w—4))+sinc(a*(w+5))+sinc(a*(w-5))) 20; T/(2*pi); = @(w) (T/2)*(sinc(a*(w+4))+sinc(a*(w—4))+sinc(a*(w+5))+sinc(a*(w—5))) 40; T/(2*pi); = @(w) (T/2)*(sinc(a*(w+4))+sinc(a*(w—4))+sinc(a*(w+5))+sinc(a*(w—5))) orient tall subplot(5,l,l); plot(w,Yl(w)); grid; title('T = 1'); subplot(5,l,2); plot(w,Y6(w)); grid; title('T = 6'); subplot(5,l,3); plot(w,YlO(w)); grid; title('T = 10'); subplot(5,l,4); plot(w,Y20(w)); grid; title('T = 20'); subplot(5,l,5); plot(w,Y40(w)); grid; title('T = 40'); xlabel('\omega '); RC! QHEETS 100 SHEETS 22—144 200 SHEETS 2244? 22-149 1’57 Lama/14mm w? :}Ofa,./Q/ggc W3 3 10 Waco/5,2; 4'3 +(C/Jg , n ‘: ,Qn inju‘g’) q f 2 I M410) I (2%? [email protected] 3w PW” ‘ r ,0 © 1C°r ‘téu'S mPOKQS‘aﬁ—QJ ’(8) “Sidg'é7ao i TKO MK?W{1«CIQS ¥of PacA (‘3 (O raj/gee = (,3? % TL? vl/xaLé’S crime, 455’on -t10 {100 l <3 Z (hog/QC LUS 7 Kg rape/Q0 d 2 +9.? {I ‘5 QW/QD2;:”‘ : 33 [email protected] C29 50t0+ 2 "‘ K 1 Q ﬁbrﬁzs RHEMLA FOLK am paé 4. W031? 338322) .. .g,gsg/m:53.mo%f i m mw2m1£ucgﬁ on} W Bmﬂ/Lec :uﬂp q; 'NJMg; 4, 0 ) _, , Bode Diagram —2o— - 3 i E 3 533,} e r 3- K;.',,_,_, " '3 SystemzH ; ; 3 g ; . 3 :3 _ EFrequency(rad/sec): 20 ; '40— ' ' j' 'j ‘j 'j : 1 j: ' j’ Magnitude MEX-18.2E 1' :‘ Magnitude (dB) 6: O I : ~ fPrOblem‘7b‘,f3r‘d fordeeriufte'MOrth ﬁlter ~100r' 420 i LiiigLLJ igigiiLL 4 iL___iiiil.1 (b 0 Phase (deg) 480 *“ me; ,4; 10' 1o 10‘ 10 10 Frequency (rad/sec) Magnitude (dB) Phase (deg) 50 in o I 400‘” f Problem 2 d, 4th order BuﬁeMorthiﬁlterf Bode Diagram r Tr‘llir‘ r MVTIV‘F‘ v 1 System: H . Frequency (rad/sec): 35 § Magnitude (dB): -29.5 ‘ 480 -270 - ~360 «.— 10' 1o 101 Frequency (rad/sec) 1O ...
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