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

2 4 3 a very special function for lti systems the

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Unformatted text preview: (2.48) Note t hat H(s) i s a c onstant for a given s. T hus, t he i nput a nd t he o utput a re t he same (within a multiplicative constant) for t he everlasting exponential signal. H(s), ~hich is called t he t ransfer f unction o f t he system, is a function o f complex varlable s. We c an define t he t ransfer function H (s) o f a n LTIC system from Eq. (2.47) a s i nput signal natural total h (T)e- sr dT I: y ( I) 0t-~~7-~~------------- T he i ntegral on t he r ight-hand side is a function of s. Let us denote i t by H(s). T hust, y(t) = H (s)e st (2.47) where H(s) = 139 2.5 Classical Solution of Differential Equations For repeated roots, t he zero-input component should be appropriately modified. For the series R LC circuit in Example 2.2 with t he i nput I (t) = l Oe- 3t u(t) a nd t he initial conditions y(O-) = 0, v c(O-) = 5, we determined the zero-input component in Example 2.2 [Eq. (2.15)]. We found the zero-state component in Example 2.5 Using t he results in Examples 2.2 and 2.5, we o btain + 5 e- 2t ) '--..-' Total current = ( -5e- t + ( -5e- t + 2 0e- 2t " z ero-input c urrent t 2 0 (2.51a) ' Figure 2.14a shows the zero-input, t he zero-state, and the total response. N atural and Forced Response For the R LC circuit in Example 2.2, the characteristic modes were found to be e - t a nd e - 2t . As we expected, t he zero-input response is composed exclusively of characteristic modes. Note, however, t hat even the zero-state response [Eq. (2.51a)] contains characteristic mode terms. This observation is generally true of LTI systems. We can now lump together all t he characteristic mode terms in t he t otal response, giving us a component known as the n atural r esponse Yn(t). T he remainder, consisting entirely of noncharacteristic mode terms, is known as the f orced r esponse y</>(t). T he t otal response of the R LC circuit in Example 2.2 can be expressed in terms of natural and forced components by regrouping the terms in Eq. (2.51a) as Total current = ,( - lOe - t 2.4-4 Total Response + 25e - 2t), + n atural r esponse Y n(t) T he t otal r esponse of a linear system can be expressed as t he s um o f its zeroinput and zero-state components: 15e- 3t ) .. z ero-state c urrent ( _ 15e- 3t ) -------- (2.51b) forced r esponse y .(t) Figure 2.14b shows t he n atural, forced, and total response. n T otal Response = L Cje Ajt j =l ' -....-.' z ero-input c omponent + ----f (t) * h(t) 2 .5 Classical Solution o f Differential Equations zero-state c omponent t This r esult is v alid o nly for t he v alues o f s for which t he J~oo h('T)e- s", d'T e xists ( or converges). In t he classical method we solve differential equation to find the natural and forced components rather t han t he zero-input and zero-state components of the 140 2 Time-Domain Analysis of Continuous-Time Systems 2.5 Classical Solution of Differential Equations response. Although this method is relatively simple compared t o t he m ethod discussed so far, as we shall see, it also has several glaring drawbacks. As Sec. 2.4-4 shows, when all o f t he c haracteristic mode terms of t he t otal sys...
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