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

However t here a re cases where t he two signals f r

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: great deal o f i nformation a nd insight a bout t he s ystem behavior. In Sec. 2.7 we show t hat t he knowledge of impulse response provides much valuable information, such as t he response time, pulse dispersion, a nd filtering properties of t he system. Many other useful insights a bout t he s ystem behavior can b e o btained by inspection of h (t). We have a similar situation in frequency-domain analysis (discussed in later chapters) where we use a n e verlasting exponential (or sinusoid) t o d etermine system response. An everlasting exponential (or sinusoid) has no physical existence, b ut i t provides another effective intermediary for computing t he s ystem response to a n a rbitrary input. Moreover, t he s ystem response t o everlasting exponential (or sinusoid) provides valuable information a nd insight regarding t he s ystem's behavior. 2 .4-3 A Very Special Function For LTI Systems: The Everlasting Exponential e st T here is a very special connection of LTI systems with t he everlasting exponential function e st. We now show t hat t he LTI system's (zero-state) response to everlasting exponential i nput e st is also t he same everlasting exponential (within a multiplicative constant). Moreover, no other function can make t he s ame claim. Such a n i nput for which t he s ystem response is also of t he s ame form is called t he c haracteristic f unction (also eigenfunction) of t he system. Because a sinusoid is a form of exponential, everlasting sinusoid is also a characteristic function of a n LTI system. Note t hat we a re talking here of a n everlasting exponential (or sinusoid), which s tarts a t t = - 00. I f h (t) is t he s ystem's u nit impulse response, then system response y (t) t o a n everlasting exponential e st is given by 138 2 Time-Domain Analysis of Continuous-Time Systems y(t) = h(t) * est = = est I: I: y ( I) ' ''" j zero-state h(T)es(t-r) dT h (T)e- BT dT 's' H (s) = o utput Signalj (2.49) I nput=everlasting e xponential e st T he t ransfer function is defined for, a nd is meaningful to, LTI systems only. I t does not exist for nonlinear or time-varying systems in general. I n t his discussion we a re talking of t he everlasting exponential, which s tarts a t t = - 00, n ot t he causal exponential estu(t), which s tarts a t t = O. For a s ystem specified by Eq. (2.1), t he t ransfer function is given by H( ) s = P(s) Q(s) (2.50) T his follows readily by considering a n everlasting input f (t) = est. According t o Eq. (2.47), t he o utput is y(t) = H (s)e st . S ubstitution of this f (t) a nd y(t) in Eq. (2.1) yields H (s)[Q(D)e st ] = P (D)e st . Moreover, Drest = drest/dt r = sre st . Hence, st P (D)e = P (s)e st a nd Q(D)e st = Q(s)e st . Consequently, H(s) = P (s)/Q(s). 6 E xercise E 2.14 Show t hat t he t ransfer f unction o f a n i deal i ntegrator is H (s) = l /s a nd t hat o f a n i deal differentiator is H (s) = s . F ind t he a nswer in two ways: using Eq. (2.49) a nd E q. (2.50). V' '/~forced zero-input (a) (b) F ig. 2 .14 T otal r esponse a nd i ts components....
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