Quiz 1 Details Quiz I Review Signals and Systems 6 003 Date Wednesday March 3 2010 Time 7 30pm 9 30pm Where 34 101 Content boundaries inclusive Lectures 1 7 Recitations 1 8 Homeworks 1 4 Massachusetts Institute of Technology March 1 2010 Massachusetts Institute of Technology Quiz I Review March 1 2010 1 15 Review Outline Massachusetts Institute of Technology Quiz I Review March 1 2010 2 15 Preliminaries converting CT to DT Preliminaries Converting CT to DT System modeling When converting a DT signal to CT we can use either zero order hold t nT 1 xd n b xc t T n Feedback poles and fundamental modes where b is a unit square function Additionally we can also use a piecewise linear approximation Discrete time systems Continuous time systems Laplace transforms xc t Z transforms xd n a n Numerical methods Massachusetts Institute of Technology t nT T n xd n 1 c t nT T 2 where a and c are the right and left sided unit triangles functions respectively Quiz I Review March 1 2010 3 15 Massachusetts Institute of Technology Quiz I Review March 1 2010 4 15 Preliminaries System modeling Discrete Time Systems Know the basics 1 system modeling spring equations LRC circuits leaky tank models 2 equations solutions solving difference and differential equations 3 signals scaling inverting and shifting The unit sample is given by Leaky tank modeling The leak rate r t is proportional to the 1 n 0 1 u n 0 3 4 Quiz I Review n 0 otherwise 6 Given a system function equation H s AB A and B are two systems running in series Given a system function equation H s A B A and B are Circuit modeling Capacitor V CdV dt Inductor V LdI dt Resistor V IR Massachusetts Institute of Technology 5 The unit step is given by height of the water in the tank h t dh t rin t rout t dt dr t rin t rout t dt n 0 otherwise two systems running in parallel For systems with feedback we often use Black s formula H s feed through transmission 1 looptransmission March 1 2010 5 15 Poles and fundamental modes Massachusetts Institute of Technology Quiz I Review 7 March 1 2010 6 15 Continuous Time Systems The unit sample is given by 1 2 t lim 0 0 A pole p is the base of a geometric sequence When dealing with a system functional Y X use partial fractions to find poles p 1 system does not converge alternating sign p 1 0 magnitude converges alternating sign p 0 1 magnitude converges monotonically p 1 magnitude diverges monotonically Complex poles cause oscillations Massachusetts Institute of Technology t otherwise 8 The unit step is given by u t t d 1 0 t 0 otherwise 9 The fundamental mode associated with p converges if Re p 0 and diverges if Re p 0 Compared to a DT system the fundamental mode associated with p converges if p lies within the unit circle Quiz I Review March 1 2010 7 15 Massachusetts Institute of Technology Quiz I Review March 1 2010 8 15 Laplace Transforms Laplace Transforms Properties Defined by X s x t e st dt 10 Table Key LT properties A double sided LT and its ROC provide a unique system function Left sided signals have left sided ROCs and right sided signals have right sided ROCs The ROC is the intersection of each ROC generated by each pole individually Go over problem 3 in homework 3 to review ROCs The sifting property of t f t t dt 11 f 0 Property Linearity Delay by T Multiply by t Multiply by e T Differentiate Integration x t ax1 t bx2 t x t T tx t x t e T t X s aX1 s bX2 s e sT X s dx t dt x d dX s ds X s sX s X s s Massachusetts Institute of Technology Quiz I Review March 1 2010 9 15 Initial and Final value theorems impulses or higher order singularities at t 0 then x 0 lim sX s March 1 2010 10 15 Defined by 12 s X z h n z n 14 n Final value theorem If x t 0 for t 0 and x t has a finite limit as t then x lim sX s 13 s 0 Quiz I Review Quiz I Review Z Transforms Initial value theorem If x t 0 for t 0 and x t contains no Massachusetts Institute of Technology Massachusetts Institute of Technology March 1 2010 11 15 ROCs are delimited by circles Inside and outside circles are given by left and right sided transforms respectively Massachusetts Institute of Technology Quiz I Review March 1 2010 12 15 Z Transforms Properties Numerical Methods To approximate derivatives we have the following techniques Forward Euler Table Key ZT properties Property Linearity Delay Multiply by n Multiply by an Unit step x n ax1 n bx2 n x n 1 nx n x n an u n yc nT yd n 1 yd n T 15 yc nT yd n yd n 1 T 16 where T is the time difference The pole can often shift out of the stability region Backward Euler X z aX1 z bX2 z z 1 X z zdX z dz This approximation is more stable than forward Euler X z a 1 1 z 1 Trapezoidal rule Use centered differences If y t x t y n y n 1 T x n x n 1 2 17 The entire left half plane is mapped onto the unit circle In particular the entire j axis is mapped onto the unit circle Massachusetts Institute of Technology Quiz I Review March 1 2010 13 15 Quiz I Review March 1 2010 15 15 End of Review Good luck on Wednesday Massachusetts Institute of Technology Massachusetts Institute of Technology Quiz I Review March 1 2010 14 15
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