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

# With this result eq 630a can be expressed as loco j

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Unformatted text preview: rat), a 2 0 ~F(D h (t) h (t)f2(t) differentiation Frequency integration 00 * f2(t) FI(S)F2(S) (6.38) Initial value 1 - 2 FI(S) , (6.39) * F2(S) lim s F(s) ( n&gt; m ) s~oo F inal value a nd ( frequency c onvolution p roperty) f(O+) F (z)dz FI(S)F2(S) ' If) {=} to 2 0 dF(s) ds Frequency convolution F2(S) - tf(t) Frequency t hen ( time c onvolution p roperty) h (t) f (t) dt F (s - so) T ime convolution T his p roperty s tates t hat if {=} f (t - to)u(t - to) Scaling T ime Convolution and Frequency Convolution h (t) - - 00 (6.37) T he p roof is i dentical to t hat of t he scaling property of the Fourier transform derived in C hapter 4 [Eq. (4.34)J. Note t hat a is restricted to only positive values because if f (t) is c ausal, t hen f rat) is a nticausal (exists only for t &lt; 0) for negative a, and anticausal signals are not p ermitted in t he (unilateral) Laplace transform. Recall t hat f rat) is t he signal f (t) time-compressed by t he factor a, a nd F(~) is F (s) e xpanded along t he s-scale by t he same factor a (see Sec. 1.3-2). T he scaling p roperty s tates t hat time-compression o f a signal b y a factor a causes expansion o f i ts Laplace t ransform in s-scale b y the s ame factor. Similarly, time-expansion f (t) causes compression o f F (s) in s-scale b y the s ame factor. 5. 1 s - F(s) 1 + -11 ; F (s) {=} t hen for a &gt; 0 i~ f (r)dr j (oo) lim s F(s) (poles o f s F(s) in LHP) s~o 390 6 C ontinuous-Time S ystem Analysis Using t he L aplace Transform 6.3 Observe t he s ymmetry (or duality) between t he two properties. Proofs of these properties are s imilar t o t he proofs in C hapter 4 for t he Fourier transform. Equation (2.48) indicates t hat H (8), t he t ransfer function of a n L TIC s ystem, is t he Laplace t ransform o f t he s ystem's impulse response h (t); t hat is, and h (t) &lt;=&gt; H (8) Solution of Differential a nd Integro-Differential Equations. d2 y dt2 Moreover, for f (t) We can apply t he t ime convolution p roperty t o t he LTIC i nput-output r elationship y (t) = f (t) * h (t) t o o btain Y (8) = F (8)H(8) (6.41) F (s) df dt and 8 &lt; =} s F(s) - j(0-) = - - - 0 = _ s_ 8 +4 8+4 2 [8 y(S) - 28 - 1) + 5 [8Y(S) _ 2] + 6 Y(s) = _ s_ + s +4 _ 1_ s +4 (6.43a) Collecti~g all the terms of Y (s) and the remaining terms separately on the left-hand side • E xample 6 .8 Using the time convolution property of the Laplace transform, determine c (t) = eatu(t) * e btu(t). Fro Eq. (6.38), it follows t hat we obtam (8 2+ 5s + 6) Y (s) - (28 + 11) = s + 1 s +4 = ( 2s+ 11) + (S2 + 5s + 6) Y (s) =1 [ 1 - - 1 ] -a)(8 - b) a- b 8- a 8- b s+1 8 +4 = 2 28 + 2 0s+45 s +4 and The inverse transform of the above equation yields Y (s) 6 .3 = 2s2 + 20s + 45 (8 2 + 5s + 6)(s + 4) 2S2 + 208 + 45 (s + 2)(8 + 3)(8 + 4) Solution o f Differential and Integro-Differential Equations Expanding the right-hand side into partial fractions yields T he t ime-differentiation p roperty o f t he L aplace transform has set t he s tage for solving l inear differential (or integro-differential) equations w ith c onstant coe...
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