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31 133 LAF'LACE rework AND CONTINUOUS-TIME LTI structures hence, combining the 2 effects for a > zero and a < zero, we will be able to write those relationships as three. nine. locate the Laplace rework and the linked ROC for every of the subsequent indications: ( a ) x ( t ) = S(t - t o ) ( b ) x ( t ) = u(t - to) ( c ) ~ ( t=)e - " [ u ( t ) - u ( t - five ) ] ffi (dl x(t)= S(t - ok T ) k=O ( e ) x ( t ) = S(at + b ) , a , b genuine constants ( a ) utilizing Eqs. (3. thirteen) and (3. 161, we receive S ( I - I,,) H e-s'fl all s ( b ) utilizing Eqs. (3. 14) and (3. 16), we receive (c) Rewriting x ( l ) as Then, from desk 3-1 and utilizing Eq. (3. 161, we receive (d) utilizing Eqs. (3. seventy one) and (1. 99), we receive m ~ ( s= ) C e-. ~'T= k=O m C (e-sT)li = 1 - 1e s T Re(s) > zero ok -0 ( e ) enable f ( zero= s ( a t ) - Then from Eqs. (3. thirteen) and (3. 18)we have f ( t ) = S(a1) 1 F(s)= l. a. l all s Now utilizing Eqs. (3. sixteen) and (3. 741, we receive 1 X ( s ) = e s b / a ~ (= S )-esh/" los angeles l all s (3. seventy three) 134 LAPLACE remodel AND CONTINUOUS-TIME LTI platforms [CHAP. three three. 10. make sure the time differentiation estate (3. 20),that is, From Eq. (3. 24) the inverse Laplace remodel is given by way of Differentiating each side of the above expression with recognize to t, we receive evaluating Eq. ( three . 7 7 ) with Eq. (3. 76), we finish that h ( t ) / d t is the inverse Laplace remodel of sX(s). hence, R'3R notice that the linked ROC is unchanged except a pole-zero cancellation exists at s = zero. three. eleven. ensure the differentiation in s estate (3. 21), that's, From definition (3. three) - IIj Differentiating each side of the above expression with appreciate to s , we've got therefore, we finish that three. 12. make sure the combination estate (3. 22), that's, allow Then CHAP. 31 LAPLACE remodel AND CONTINUOUS-TIME LTI platforms making use of the differentiation estate (3. 20), we receive X(s) =sF(s) therefore, the shape of the ROC R' follows from the potential creation of an extra pole at s = zero through the multiplying by means of l/s. utilizing some of the Laplace remodel homes, derive the Laplace transforms of the subsequent signs from the Laplace rework of u(t). ( b ) 6'(t) ( d ) e-"'u(t) ( a ) 6(t) tu(t) (e) te-"'u(t) (g) e-"'cos w,tu(t) (c) (a) (f) coso,tu(t) From Eq. (3. 14)we have 1 u(t) H - for Re( s) > zero S From Eq. ( 1. 30) we now have therefore, utilizing the time-differentiation estate (3. 20), we receive S(t) H S - 1 = S all s 1 ( b ) back utilising the time-differentiation estate (3. 20) to the end result from half (a), we receive a'(! ) HS all s (c) utilizing the differentiation in s estate (3. 211, we receive (dl utilizing the transferring within the s-domain estate (3. 17), we've e-a'u(t) (el 1 w s+a ( three. seventy eight) Re(s) > -a From the end result from half (c) and utilizing the differentiation in s estate (3. 21), we receive ( f ) From Euler's formulation we will write 136 LAPLACE remodel AND CONTINUOUS-TIME LTI platforms [CHAP. three utilizing the linearity estate (3. 15) and the transferring within the s-domain estate (3. 17), we receive (g) employing the transferring within the s-domain estate (3.