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

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

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

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