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,
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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