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

# 2 4 3 a very special function for lti systems the

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

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online