Quiz I ReviewSignals and Systems6.003Massachusetts Institute of TechnologyMarch 1, 2010(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 1 / 15Quiz 1 Details•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) Quiz I Review March 1, 2010 2 / 15Review Outline•Preliminaries•Converting CT to DT•System modeling•Discrete time systems•Feedback, poles, and fundamental modes•Continuous time systems•Laplace transforms•Z transforms•Numerical methods(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 3 / 15Preliminaries: converting CT to DTWhen converting a DT signal to CT, we can use either zero-orderholdxc(t)=∞!n=−∞xd[n]b"t − nTT#(1)where b is a unit square function. Additionally, we can also use apiecewise linear approximationxc(t)=∞!n=−∞xd[n]a"t − nTT#+∞!n=∞xd[n + 1]c"t − nTT#(2)where a and c are the right- and left-sided unit triangles functions,respectively.(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 4 / 15Preliminaries: System modelingKnow the basics: (1) system modeling: spring equations, LRCcircuits, leaky tank models; (2) equations solutions: solving differenceand differential equations; (3) signals: scaling, inverting and shifting.•Leaky tank modeling: The leak rate r(t) is proportional to theheight of the water in the tank h(t),dh(t)dt∝ rin(t) − rout(t) (3)dr(t)dt=rin(t)τ−rout(t)τ(4)•Circuit modeling:•Capacitor: V = CdV /dt•Inductor: V = LdI /dt•Resistor: V = IR :-)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 5 / 15Discrete Time SystemsThe unit sample is given byδ[n]=$1 n =0,0 otherwise .(5)The unit step is given byu[n]=$1 n ≥ 0 ,0 otherwise .(6)•Given a system function equation H(s)=AB, A and B are twosystems running in series•Given a system function equation H(s)=A + B , A and B aretwo systems running in parallelFor systems with feedback, we often use Black’s formulaH(s) = feed through transmission/(1 − looptransmission ) (7)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 6 / 15Poles, and fundamental modes•A pole p is the base of a g eometric sequence•When dealing with a system functional Y /X , use partialfractions to fin d 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) Quiz I Review March 1, 2010 7 / 15Continuous Time SystemsThe unit sample is given byδ(t) = lim!→0$1/2# t ∈ [−#, #]0 otherwise(8)The unit step is given byu(t)=%t−∞δ(λ)d λ =$1 t ≥ 0 ,0 otherwise .(9)•The f undamental mode associated with p converges ifRe(p) < 0 and diverges if Re(p) > 0•Compared to a DT system, the fundamental mode associatedwith p converges if p lies within the unit circle(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 8 / 15Laplace Transforms•Defined byX (s)=%∞−∞x(t)e−stdt (10)•A double-sided LT and its ROC provide a unique system function•Left-sided signals have left-sided ROCs, and right-sided signalshave right-sided R OCs•The ROC is the intersection of each ROC generated by eachpole individually•Go over problem 3 in homework 3 to review ROCs•The sifting prop erty of δ(t)f (0) =%∞−∞f (t)δ(t)dt (11)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 9 / 15Laplace Transforms: PropertiesTable: Key LT propertiesProperty x(t) X (s)Linearity ax1(t)+bx2(t) aX1(s)+bX2(s)Delay by T x(t − T ) e−sTX (s)Multiply by t tx(t)−dX (s)dsMultiply by e−αTx(t)e−αTX (s + α)Differentiatedx (t)dtsX (s)Integration&t−∞x(λ)dλX (s)s(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 10 / 15Initial and Fina l value theorems•Initial value theorem: If x(t) = 0 for t < 0 and x(t) conta ins noimpulses or higher-order singularities at t = 0 thenx(0+) = lims→∞sX (s) (12)•Final value theorem: If x(t) = 0 for t < 0 and x(t) has a finitelimit as t →∞thenx(∞) = lims→0sX (s) (13)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 11 / 15Z Transforms•Defined byX (z)=∞!n=−∞h[n]z−n(14)•ROCs are delimited by circles•Inside and outside circles are given by left- and right-sidedtransforms, respectively.(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 12 / 15Z Transforms: PropertiesTable: Key ZT propertiesProperty x[n] X(z)Linearity ax1[n]+bx2[n] aX1(z)+bX2(z)Delay x[n − 1] z−1X (z)Multiply by n nx[n]−zdX (z )dzMultiply by anx[n]anX (z/a)Unit step u[n]1/(1 − z−1)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 13 / 15Numerical MethodsTo approximate derivatives, we have the following techniques.•Forward Euler:yc(nT ) = (yd[n + 1] − yd[n])/T , (15)where T is the time difference. The pole can often shift out ofthe stability region!•Backward Euler:yc(nT ) = (yd[n] − yd[n − 1])/T . (16)This approximation is more stable than forward Euler.•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. Inparticular, the entire jω axis is mapped onto the unit circle(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 14 / 15End of ReviewGood luck o n Wednesday! :-)(Massachusetts Institute of Technology) Quiz I Review March 1, 2010 15 /
View Full Document