6 003 Signals and Systems CT Frequency Response and Bode Plots October 22 2009 Mid term Examination 2 Wednesday October 28 7 30 9 30pm Walker Memorial No recitations on the day of the exam Coverage cumulative with more emphasis on recent material lectures 1 12 homeworks 1 7 Homework 7 includes practice problems for mid term 2 It will not collected or graded Solutions will be posted Closed book 2 pages of notes 8 21 11 inches front and back Designed as 1 hour exam two hours to complete Review sessions Monday 5 6pm and 8 9pm in 32 044 Conflict Contact freeman mit edu by Friday October 23 5pm Last Time Complex exponentials are eigenfunctions of LTI systems es0 t H s H s0 es0 t H s0 can be determined graphically using vectorial analysis s0 z0 s0 z1 s0 z2 H s0 K s0 p0 s0 p1 s0 p2 s plane s0 s0 s 0 z0 z0 z0 Response of an LTI system to an eternal cosine is an eternal cosine same frequency but scaled and shifted cos 0 t H s H j 0 cos 0 t H j 0 Frequency Response H s s j H j 5 H s s z1 5 s plane 5 0 5 H j 5 5 2 5 5 5 2 Frequency Response H s s j H s H j 5 9 s p1 5 s plane 5 0 5 H j 5 5 2 5 5 5 2 Frequency Response H s s j H s 3 H j 5 s z1 s p1 5 s plane 5 0 5 H j 5 5 2 5 5 5 2 Poles and Zeros Thinking about systems as collections of poles and zeros is an important design concept simple just a few numbers characterize entire system powerful complete information about frequency response Today poles zeros frequency responses and Bode plots Asymptotic Behavior Isolated Zero The magnitude response is simple at low and high frequencies H j 5 H j j z1 5 5 0 5 H j 5 5 2 5 5 5 2 Asymptotic Behavior Isolated Zero The magnitude response is simple at low and high frequencies H j 5 H j j z1 z1 5 5 0 5 H j 5 5 2 5 5 5 2 Asymptotic Behavior Isolated Zero The magnitude response is simple at low and high frequencies H j 5 H j j z1 z1 5 5 0 5 H j 5 5 2 5 5 5 2 Asymptotic Behavior Isolated Zero Two asymptotes provide a good approxmation on log log axes H s s z1 log 2 H j 5 H j z1 1 1 0 5 0 5 2 log 1 lim H j z1 0 lim H j 0 1 2 z1 Asymptotic Behavior Isolated Pole The magnitude response is simple at low and high frequencies H s 9 s p1 9 H j 5 9 p1 5 5 0 5 H j 5 5 2 5 5 5 2 Asymptotic Behavior Isolated Pole Two asymptotes provide a good approxmation on log log axes H s log H j 5 9 s p1 H j 9 p1 0 1 1 2 5 0 5 2 log 1 lim H j 9 p1 lim H j 9 0 0 1 2 p1 Check Yourself Compare log log plots of the frequency response magnitudes of the following system functions H1 s 1 s 1 and H2 s 1 s 10 The former can be transformed into the latter by 1 2 3 4 5 shifting shifting shifting shifting none of horizontally and scaling horizontally both horizontally and vertically and scaling both horizontally and vertically the above Check Yourself Compare log log plots of the frequency response magnitudes of the following system functions H1 s 1 s 1 and H2 s 1 s 10 log H j H1 j 0 1 H2 j 1 2 log 2 1 0 1 2 Check Yourself Compare log log plots of the frequency response magnitudes of the following system functions H1 s 1 s 1 and H2 s 1 s 10 The former can be transformed into the latter by 1 2 3 4 5 shifting shifting shifting shifting none of horizontally and scaling horizontally both horizontally and vertically and scaling both horizontally and vertically the above no scaling in either vertical or horizontal directions 3 Asymptotic Behavior of More Complicated Systems Constructing H s0 Q Y H s0 K q 1 P Y s0 zq product of vectors for zeros s0 pp product of vectors for poles p 1 s0 s 0 z1 s plane s 0 p1 z1 p1 Asymptotic Behavior of More Complicated Systems The magnitude of a product is the product of the magnitudes Q Y q 1 P Y K s0 pp p 1 q 1 P Y s0 zq s 0 pp p 1 s0 s0 z1 p1 s plane p1 s0 z1 H s0 K Q Y s0 zq Bode Plot The log of the magnitude is a sum of logs Q Y H s0 K q 1 P Y Q Y s0 zq K s0 pp p 1 log H j log K q 1 P Y s0 zq s 0 pp p 1 Q X q 1 log j zq P X p 1 log j pp Bode Plot Adding Instead of Multiplying log H s s s 1 s 10 0 1 s plane j log j j 1 j 10 10 2 log 2 1 0 1 2 0 10 10 log 3 1 j 1 1 log 2 1 0 1 2 1 log 10 3 1 j 10 2 log 2 1 0 1 2 3 Bode Plot Adding Instead of Multiplying log H s s s 1 s 10 0 1 s plane j log j j 1 j 1 10 10 2 log 2 10 10 1 0 1 2 3 1 log 10 1 j 10 2 log 2 1 0 1 2 3 Bode Plot Adding Instead of Multiplying log H s s s 1 s 10 1 2 s plane j j 1 j 10 10 3 log 2 10 10 1 0 1 2 3 1 log 10 1 j 10 2 log 2 1 0 1 2 3 Bode Plot Adding Instead of Multiplying log H s s s 1 s 10 1 2 s plane j j 1 j 10 10 3 log 2 10 10 1 0 1 2 3 1 log 10 1 j 10 2 log 2 1 0 1 2 3 Asymptotic Behavior Isolated Zero The angle response is simple at low and high frequencies H j 5 H s s z1 5 s plane 5 0 5 H j 5 5 2 5 5 5 2 Asymptotic Behavior Isolated Zero Three straight lines provide a good approxmation on log log axes H s s z1 H j H j 2 2 5 4 5 2 0 2 log 1 lim H j 0 0 lim H j 2 0 1 2 z1 Asymptotic Behavior Isolated Pole The angle response is …
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