Signal Processing and Linear Systems-B.P.Lathi copy

# The same is true of the capacitor voltage hence veo v

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: eration d uring which certain information is lost. To invert t his o peration, one piece of information a bout y (t) m ust b e provided t o r estore t he o riginal y (t). Using a similar argument, we c an show t hat, given d 2 y/dt 2 , we c an determine y (t) u niquely only if two additional pieces of information (constraints) a bout y (t) are given. In general, t o d etermine y (t) u niquely from its n th derivative, we need n a dditional pieces of information (constraints) a bout y (t). T hese constraints are also called auxiliary conditions. W hen t hese conditions are given a t t = 0, t hey a re called initial conditions. 2.2-1 S ome Insights into the Zero-Input Behavior o f a System B y definition, t he z ero-input response is t he s ystem response t o i ts internal conditions, assuming t hat i ts i nput is zero. Understanding this phenomenon provides interesting insight into system behavior. I f a s ystem is d isturbed m omentarily from its rest position a nd if t he d isturbance is t hen removed, t he s ystem will n ot come back t o r est instantaneously. I n general, it will come back t o r est over a period of time a nd only t hrough a special t ype o f motion t hat is c haracteristic of t he s ystem. t For example, if we press on a n a utomobile fender momentarily a nd t hen release it a t t = 0, t here is no external force on t he a utomobile for t &gt; 0:1. T he a uto b ody will eventually come back t o i ts rest (equilibrium) position, b ut n ot t hrough a ny a rbitrary m otion. I t m ust do so using only a form of response which is s ustainable by t he s ystem on its own w ithout a ny external source, because the i nput is zero. O nly c haracteristic modes satisfy this condition. The system uses a proper c ombination o f c haracteristic modes t o come back t o the rest p osition while s atisfying appropriate b oundary (or i nitial) conditions. I f t he shock absorbers of t he a utomobile are in good condition (high damping coefficient), t he c haracteristic modes will b e m onotonically decaying exponentials, a nd t he a uto b ody will come to rest rapidly without oscillation. I n c ontrast, for t This assumes that the system will eventually come back to its original rest (or equilibrium) position. :I: We ignore the force of gravity, which merely causes a constant displacement of the auto body without affecting the other motion. 114 2 Time-Domain Analysis of Continuous-Time Systems 2.3 T he U nit Impulse Response h(t) 115 t his i mportant p henomenon requires a n u nderstanding of t he z ero-state response; for this reason we p ostpone t his topic until Sec. 2.7-7. IH 2.3 The Unit Impulse Response h (t) 20 f (t) F ig. 2.2 Modes always get a free ride. p oor shock a bsorbers (low damping coefficients), t he c haracteristic modes will b e e xponentially decaying sinusoids, a nd t he b ody will come t o r est through oscillatory motion. W hen a series R C c ircuit with a n initial charge on t he c apacitor is shorted, the c apacitor will s tart t o discharge exponentially t hrough t he resistor. T his response o f t he R C c ircuit is caused entirely by its internal conditi...
View Full Document

Ask a homework question - tutors are online