MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.003: Signals and Systems—Fall 2002Tutorial for the week of November 25th - November 29thAlex’s Office HoursMonday 3-5pmImportant Due Dates:• Problem Set 9 due on Wednesday.To day1. Pole-Zeros to Bode Plot example2. Z-Transform stuff1More on CT Frequency responseYou should feel comfortable understanding how a system is characterized in many differentforms:1. System Function, H(s)eg:H(s)=1s +1, Re(s) > −12. Impulse response, h(t)h(t)=e−tu(t)3. Graph of impulse response, or step responseImpulse ResponseTime (sec)Amplitude0 1 2 3 4 5 600.10.20.30.40.50.60.70.80.914. Bode plotBode DiagramFrequency (rad/sec)Phase (deg) Magnitude (dB)−25−20−15−10−5010−1100101−90−4505. Pole zero diagram12−1−212−1−2ImRe2[Example]All the fun you can have with pole-zero diagrams and bode plots. Match the pole-zero dia-grams on this page with the the frequency response plots on the next page:Im(s)(a)Re(s)3−3Im(s)Re(s)(b)−10 −1Im(s)Re(s)(c)−10 −1Im(s)Re(s)−10(d)1Im(s)Re(s)−100j100j(e)−0.1Im(s)Re(s)−100j100j−1(f)3(a)Frequency (rad/sec)Phase (deg) Magnitude (dB)12.51313.51414.51510010110204590135180(b)Frequency (rad/sec)Phase (deg) Magnitude (dB)−45−40−35−30−25−20100101102−90−4504590(c)Frequency (rad/sec)Phase (deg) Magnitude (dB)−120−100−80−60−40−20101102103−180−135−90−450(d)Frequency (rad/sec)Phase (deg) Magnitude (dB)−100−80−60−40−20100101102−180−135−90−450(e)Frequency (rad/sec)Phase (deg) Magnitude (dB)−120−100−80−60−40101102103−180−135−90−450(f)Frequency (rad/sec)Phase (deg) Magnitude (dB)−45−40−35−30−25−20100101102−90−60−304[Example]Given the z-transform pairx[n] ↔zz2+4, |z| > 2use z-transform properties to determine the z-transform of the following signals:(a) y[n]=2nx[n](b) y[n]=x[n +1]+x[n − 1](c) y[n]=x[n] ∗ x[n] ∗···∗x[n] m times(d) y[n]=(n − 3)x[n − 2]5Work Space6[Example]Determine all possible signals that can have the following z-transforms with the given con-ditions.(a)11−32z−1+12z−2(b)2−32z−11−32z−1+12z−2, causal(c)31−103z−1+z−2,stable(d)1−12z−11+12z−1, right-handed7Work
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