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

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

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Unformatted text preview: r functIOn T herefore 1 H(s) R es> - 5 = s +5 and the input I (t) = e -'u(t) + e - 2'u(-t) The input I (t) is of the type depicted in Fig. 6.50g, and the region of convergence es not exist. In this case, we must determine separately the system response -t ) d I ( ) _ - 2' ( -t) to each of the two i nput components, / I(t) = e u (t an 2 t - e u . f or F () d0 s 1 R es> - 1 FI(s) = ;+l -1 T ransporting all b ut t he first t erm o n t he r ight-hand side yields S3 y (S) = b3S3F(s) Dividing t hroughout b y s3 + [ -a2Y(s) + b2F(s)]s2 + [ -aIY(s) + b IF(s)]s + [ -aoY(s) + boF(s)] yields R es < - 2 F 2(S)=s+2 I f YI(t) and Y2(t) a re the system responses to / I(t) and f2(t), respectively, then YI(s) = (s + 1)(s + 5) 1 /4 R es> - 1 1/4 = s+l-s+5 so t hat and -1 Y2(S) = (s + 2)(s + 5) _ - 1/3 - s+2 + 1 /3 s +5 - 5<Res<-2 1 + : d-aoY(s) s + boF(s)] (6.112) Therefore, Y (s) c an be generated by a dding four signals a ppearing o n t he r ight-hand side of Eq. (6.112). We shall build Y (s) s tep by s tep, a dding one component a t a t ime. Figure 6.53a shows only t he first component; t hat is, b3F(S). F igure 6.53b shows Y (s) formed by t he first two components, b3F(S) a nd ~[-a2Y(s) + b2F(S)]. Observe t hat t he t erm a 2Y(s) is o btained from Y (s) itself. We a dd - azY(s) t o b2F(s) a nd p ass i t t hrough a n i ntegrator t o o btain ~[-a2Y(s) + b2F(s)]. F igure 6.53c shows Y (s) b uilt u p from t he first t hree c omponents. Finally, Fig. 6.53d shows Y (s) b uilt up from all t he c omponents. This is t he final form, which represents a n a lternative realization of H (s) i n Eq. (6.111). T his r ealization c an b e readily generalized for a n n th-order t ransfer f unction in Eq. (6.70) using n i ntegrators. 6.10 S ummary 6 Continuous-Time System Analysis Using t he Laplace Transform 458 ( a) ( b) ( c) ( d) F (s) F ig. 6 .53 Second (observer) canonical realization of an nth-order transfer function. 6 .10 Summary c; T he F ourier transform cannot be used directly in analysis of u~stable, ev~n m ar inally s table systems. Moreover, t he i nputs must also ? e re~tncted t o o uner tran~formable signals, which leaves o ut exponentially growIng slgnal~. B oth th~se l imitations a re t he r esult o fthe fact t hat t he s pectral components used In t~~ Fo~~~r e t ransform to synthesize a signal f (t) a re ordinary sinusoi~s ~r e xponentlt s ° lane form e jwt , whose frequencies are restricted t o t he JW a xiS In t he comp ex p . 459 These spectral components are incapable o f synthesizing exponentially growing signals. We extend t he Fourier transform by generalizing t he frequency variable from s = j w t o s = (T + j w. T he r esulting transform is t he Laplace transform, which c an analyze all types o f LTIC systems. T he Laplace transform can also handle exponentially growing signals. T he s ystem response t o a n everlasting exponential est is also a n everlasting exponential H (s)e st , where H (s) is t he s ystem transfer function. We c an view t he Laplace transform as a too...
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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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