A#13_Solutions-1

# A#13_Solutions-1 - Assignment#13— Solutions — p...

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Unformatted text preview: Assignment #13— Solutions — p. / ECSE-2410 Signals & Systems - Spring 2007 A) ./ W The signal x(t) = 005(101) has been sampled by multiplying it byanimpulse train. The sampled signal is x30) = c(t)x(t)wherex:(t) is the sampled version of x(t) and c(t) is an impulse train described by , [(ch cm = Z 5 (t - kT) . The spectrum of Xs(m) is shown below after sampling and lowpass ﬁltering. kg -ao , a.) What is the sampling rate? Is the Nyquist sampling rate satisfied? WMFLO auqu Ws‘fw‘“: 5 L03 2 54/75%</ we” b.) What is the actual output, xJ (t), after sampling and lowpass ﬁltering? x30) = 604/ {{ Assignment #13— Solutions - p. 2— ECSE-2410 Signals & Systems — Spring 2007 2.) The input to a linear, time—invariant system is x(t) = 3 + sin(10t — 45°) and the output is ya) = 3% sin 10t- If Han) = 5:1), ﬁndA and B. I (pl 7* ’ (19.6.” W- W . ’42 o , 0" I [HM > ’7’! ~ f 6 ABA/0 [0A _' E :27 A» Z ; M , Assignment #13— Solutions — p. v? ECSE-2410 Signals & Systems — Spring 2007 ﬂag» 3 ‘ t A certain ﬁlter has the following characteristics: [H(0) } = 0 dB and 414(0) = 00’ IH(2)I=1dB and 4171(2) = 0°,!H(5)[=—6dB and 411(5) = 0°, md|H(10)|=10dB and AHUO) = 0° . Find the output of this ﬁlter, y (I), when the input is x(t) = 5 + 3 cos(2t) +10 sin(5t). Way/z «4/6 =9 //W/: /0 "’- ’2': f/Q/far? ff/éff y(t)= 5/ ’L 1356 CwZ-f 14 5/f/a? 5/5, Assignment #13— Solutions — p. ECSE—2410 Signals & Systems — Spring 2007 405). (a) (5) Write the {255131 equation for phase, 4H(co) , where, H (co) = (b) (5) Find the exact 4H(aJ) when a) :1. (c) (5) For the frequency response given in (a), ﬁnd the phase, in degrees, as a) -——> 00. Consider the limit of H (co) as a) ——> 00 as opposed to the exact equation. ”[?0” + 2 W" 735 4/ Z #[w/ : fa“ w I / 1M]; 5,,” »[7°*2M 75 / a ([56 3, [?0O #Z—{S7a/f - ﬂ l0/<{° “5/4.? C (ya—\$00] a) “’1, “"J/ 2 0w dw(UL/Zj/JZZ) Assignment #13— Solutions — p. 5/ ECSE«2410 Signals & Systems - Spring 2007 500). Find the value of a in .60 1+j- H(a))=2 ‘1 Mfg 5 so that the magnitude of H(a)) approaches -6 dB as a) ——> 90. / we?” ////w/: ‘2 w (w ,45 we AW) A 43> Assignment #13— Solutions - p. g ECSE~2410 Signals & Systems — Spring 2007 6(20). Sketch the straight-line Bode plot (magnitude and phase) for the system with the frequency response HM» z 5 j a) 10 I + 5 j a) 10 + j a) 1 Make sure all terms are in the form of I + j a) r where —~ = 60,, is the break frequency. 1 & / PM :0 </ 0.: 02, >0; ./ "8043/ cC/w g2. 7/0 / ~20 d3/ d2 Cd Assignment #13— Solutions — p. 7 ECSE-2410 Signals & Systems - Spring 2007 Wu é Cat: 7-) [ﬁg/W‘époop/fafj A JodL/M 6 10 V 4’ ‘2 2 0 2 co 2 am “90% «)6» k “M W 1 / / ~‘iO‘JB «(MO * l #470 at. aha» ' 0M3 ./ ’5 / (v 02. IL 2_ 0 2% Z m wow} ~ A ‘ Assignment #13— Solutions — p. g ECSE-2410 Signals & Systems - Spring 2007 /W« C (60:17” 3 ﬂ?” ,, o/e. em Afﬁfwrwi/op M‘ A” ’7“ w & a; ‘/ “(go/deg 0.0%.sz /o ‘/ ‘ ﬁfe/Me) [Aw 1- Ma / Cu ’9 ’?CJ° I [O [62> 9b" *" ‘ I m ; 1/5’6 . viii: i l -qoo 0 i I _J E ~ 4 ‘15 L ; “Wok. ; Assignment #13— Solutions ~ p. 7 ECSEv24lO Signals & Systems - Spring 2007 MW 7(20). Use the “bode” command in MATLAB to make the Bode plot for the system in Problem 6. Label the axes and put a title on the graph with your name in the title, e.g., “Bode Plot for l/(jw + 10) — A. Desrochers”. Sketch the straight~line approximation on your MA T LAB plots. >> num=[50 0]; >> den=[5 51 10]; >> bode(num,den) >r> title('Bode Plot for H(w) = Sij/(l+5jw) (10+jw) — A.Desrochers') >> Bode Plot for H(w) = 50jw/(1+Sjw)(10+jw) - A.Desrochers Magnitude (dB) Phase (deg) Frequency (rad/sec) ...
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A#13_Solutions-1 - Assignment#13— Solutions — p...

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