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

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

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: n c ontrast, if t he i nput is very different from t he n atural mode, ( - ), is l arge, a nd t he s ystem responds poorly. T his is precisely w hat we s et o ut t o prove. We h ave p roved t he above assertion for a single-mode (first-order) system. I t c an b e generalized t o a n n th-order s ystem, which has n c haracteristic modes. T he i mpulse r esponse h (t) of such a system is a l inear combination of its n modes. Therefore, i f f (t) is similar t o a nyone of t he modes, t he c orresponding response will b e high; if i t is similar t o n one of t he modes, t he response will b e small. Clearly, t he c haracteristic m odes are very influential in determining system response t o a given input. I t w ould b e t empting t o c onclude on t he basis of Eq. (2.66) t hat if t he i nput is i dentical t o t he c haracteristic mode, so t hat ( = )" t hen t he r esponse goes t o infinity. R emember, however, t hat if ( = )" t he n umerator o n t he r ight-hand side of Eq. (2.66) also goes t o zero. We shall s tudy t his complex behavior (resonance p henomenon) l ater in t his section. We s hall now show t hat m ere i nspection o f t he impulse response h (t) (which is composed o f c haracteristic m odes), reveals a g reat deal a bout t he s ystem behavior. 2 .7-2 t- Fig. 2 .18 Effective duration of an impulse response. T here is no single satisfactory definition of effective signal d uration (or width) applicable t o every s ituation. For t he s ituation d epicted in Fig. 2.18, a reasonable definition of t he d uration h (t) w ould be Th, t he w idth of t he r ectangular pulse h(t). T his r ectangular pulse h (t) h as a n a rea i dentical t o t hat of h (t) a nd a h eight identical t o t hat of h (t) a t s ome s uitable i nstant t = to. I n Fig. 2.18, to is chosen as t he i nstant a t which h (t) is maximum. According t o t his d efinition,t or J~oo h (t) dt h(to) Th = (2.67) Now if a system has a single m ode Response Time o f a System: The System Time Constant Like h uman beings, systems have a certain response time. I n o ther words, when a n i nput ( stimulus) is applied t o a s ystem, a certain a mount of t ime e lapses before t he s ystem fully responds t o t hat i nput. T his t ime lag or response t ime is called t he s ystem t ime c onstant. As we shall see, a system's t ime c onstant is equal t o t he w idth o f i ts i mpulse response h(t). A n i nput 8(t) t o a s ystem is i nstantaneous (zero d uration), b ut i ts r esponse h (t) h as a d uration Th. T herefore, t he s ystem requires a t ime Th t o r espond fully t o t his i nput, a nd we are justified in viewing Th as t he s ystem's response t ime o r time w ith), n egative a nd r eal, t hen h (t) is m aximum a t t T herefore, according t o E q. (2.67) 1 o w ith value h(O) A. 00 Th = -1 A 0 Ae At dt 1 = - -, ) (2.68) t T his definition is satisfactory when h (t) is a single, mostly positive (or mostly negative) pulse Such systems a re lowpass systems. T his d efinition should n ot b e a pplied indiscriminat...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online