Practice+Exam+2_ans - Exam#2 Review Challenge 23 Topical...

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Exam #2 Review Exam #2 Review 1 Challenge 23 Topical coverage: Lessons 15-23
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Exam #2 Review Challenge 23 Design a -3dB 2 nd order Butterworth lowpass filter having a passband width of 1000 r/s. What is the prototype filter H p (s) ( 0 =1 r/s)? What is the resulting Butterworth analog filter H(s)? Where are the poles of the final analog filter H(s) located? (Answer on-line) 2
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Exam #2 Review What is the 2 nd order prototype filter H p (s)? The poles satisfy the geometric conditions shown: or use Table 15-1. 707 0 707 0 707 0 707 0 1 414 1 1 1 1 1 1 2 1 2 2 1 4 2 . . ; . . . ) ( ) ( ) ( ) ( ) ( | ) ( | j p j p s s p s p s s H s s H s H s H 90 3
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Exam #2 Review From first principles and MATLAB >> roots([1,0,0,0,1]) % roots of s 4 +1 -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 + 0.7071 0.7071 - 0.7071i >> s1=[1, 0.707 - 0.707i];s2=[1, 0.707 + 0.707i]; s3=conv(s1,s2); s 3 = 1.0000 1.4140 0.9997 ~ s 2 +1.414s+1 1 414 . 1 1 ) ( 2 s s s H 1 4
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Exam #2 Review Using MATLAB, the Slacker’s friend (bypassing essentially everything) >> [b2,a2]=butter(2,1,'s') % 2 nd order, critical frequency =1.0. b2 = 0 0 1 a2 = 1.0000 1.4142 1.0000 1 414 1 1 2 s s s H . ) ( 5
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Exam #2 Review In all cases the Butterworth analog prototype filter H(s) is: What is the final H(s)? Scale H p (s) by k f =1000 (Lesson 21) producing: ; Where are the poles of the final analog filter H(s) located? 1 414 1 1 2 s s s H . ) ( s = (-0.7071 + 0.7071i) 1000 (-0.7071 - 0.7071i) 1000 6 3 2 6 10 10 414 1 10  s s s H . ) ( 1000 6
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Review Exam #2 Q #1: Convolution and steady-state analysis Consider the system shown. 2 s K Exam #2 Review Assume the transfer  function  H(s) is: K s s s K s s K s K G G G s H 2 8 6 4 4 2 2 1 2 1 2 2 1 1 ) ( U(s) Y(s) 4 2 s - K>0 (real) 7
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1.a- What is the convolution y(t) = h(t) u(t) evaluated at t= , where h(t) is the system’s impulse response and u(t) is the unit step function? Exam #2 Review K K K s s s K s sY y s K s s s k s U K s s s K s U s H s Y s s 8 4 2 8 6 ) 4 ( ) ( ) ( theorem value Final 1 2 8 6 4 ) ( ) 8 ( 6 4 ) ( ) ( ) ( 0 2 0 2 2 8 Could have derived y(t) using a Heaviside expansion (more difficult)
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1.b- Assume: Would K need to be increased or decreased if the DC gain is to be increased. H(j0)=4K/(8+2K); the numerator grows faster (2x) than the denominator. Therefore, increase K. K s s s K s H 2 8 6 4 ) ( 2 Exam #2 Review 9
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1.c- Assume that : where K=1/2. What is the filter’s purpose (i.e., lowpass, highpass, bandpass, etc.).
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