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

In practice we a re likely to know t he i nitial

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Unformatted text preview: gration in t he complex plane, a s ubject b eyond t he scope of this book. 2 For our purpose, we c an find t he inverse transforms from t he t ransform table 6.1. All we need is t o express F (s) as a s um o f simpler functions o f t he form listed in t he t able. Most of t he t ransforms F (8) o f practical interest a re r ational f unctions; t hat is, ratios o f polynomials in s . Such functions can be expressed as a s um o f s impler functions by using p artial f raction expansion (see Sec. B.5). Values of 8 for which F (s) = 0 a re called t he z eros o f F (8)j t he values of s for which F (s) - -+ 00 a re called t he p oles o f F (s). I f F (s) is a r ational function of t he form P (s)IQ(s), t he r oots of P (8) a re t he zeros a nd t he r oots of Q (8) a re t he poles of F (8). • E xample 6 .3 Find the inverse Laplace transforms of 78 - 6 ( a) 8 2 -8-6 28 2 + 5 6(8 + 34) 8 s + 10 s 2+38+2 ( e) s (s2+108+34) ( d) ( 8+1)(s+2)3 ( a) 7s - 6 F (s) E xercise E 6.1 B y direct integration, find the Laplace transform F (s) and the region of convergence of F (s) for the signals shown in Fig. 6.3. Answer: ( a) ~(l- e- 2s ) for all s. ( b) } (l- e- 2s )e- 2s for ails. ' 7 T he d efinition of t he L aplace transform is identical to t hat o f t he Fourier transform w ith j w r eplaced by 8 . I t is r easonable t o e xpect F (8), t he Laplace transform of f (t), t o b e t he s ame as F (w ), t he Fourier transform of f (t) w ith j w replaced by s . For example, we found t he F ourier transform of e -atu(t) t o b e 1 /(jw + a ). Replacing j w w ith s in t he Fourier transform results in 1 /(s + a ), which is t he Laplace transform as s een from Eq. (6.16b). Unfortunately this procedure is n ot valid for all f (t). We m ay use it only if t he region of convergence for F (s) includes the imaginary ( jw) axis. For instance, t he Fourier transform of t he u nit s tep function is IT.5 (w) + (1 / j w ). T he c orresponding Laplace transform is 1I 8, a nd its region of convergence, which is R e 8 > 0, does n ot include the imaginary axis. I n t his case the connection between t he Fourier a nd Laplace transforms is n ot so simple. T he reason for this complication is r elated t o t he convergence of the Fourier integral, where the ) The inverse transform of none of the above functions is directly available in Table 6.1. We need to expand these functions into partial fractions discussed in Sec. B.5. /::; Connection t o t he Fourier Transform b ( = (s + 2)(s - 3) kl k2 = s+2+s-3 To determine k l, corresponding to the term (s + 2), we cover up (conceal) the term (s + 2) in F (s) and substitute s = - 2 (the value of s t hat makes s + 2 = 0) in the remaining expression (see Sec. B. 5- 2) 7 s-6 (8 - 3) I = - 14-6 = 4 8 =-2 -2 - 3 Similarly, to determine k2 corresponding to the term (s - 3), we cover up the term (s - 3) in F (s) and substitute s = 3 in the remaining expression Therefore 7s - 6 F (s) = (s + 2)(s _ 4 3) = s +2 + 8 3 - 3 (6.25a) 372 6 C ontinuous-Time S ystem Analysis Using t he Laplace T ransform T ab...
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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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