Schaum's Outline of Signals and Systems, 3rd Edition (Schaum's Outlines)

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Utilizing Eq. (2. 951, the enter to the 1st integrator is given by way of considering that w(t) is the enter to the second one integrator in Fig. 2-19, now we have Substituting Eq. (2. ninety nine) into Eq. (2. 98), we get that's the mandatory second-order linear differential equation. word that, often, the order of a continuous-time LTI method which includes the interconnection of integrators and scalar multipliers is the same as the variety of integrators within the procedure. CHAP. 21 eighty five LINEAR TIME-INVARIANT structures 2. 20. think of a continuous-time method whose enter x ( t ) and output y ( t ) are comparable via the place a is a continuing. ( a ) locate y ( t ) with the auxiliary y(0) = y, and (2. 102) x ( t ) =~ e - ~ ' u ( t ) ( b ) show y ( t ) by way of the zero-input and zero-state responses. ( a ) enable the place y p ( t ) is the actual resolution pleasant Eq. (2. one hundred and one) and resolution which satisfies y h ( t ) is suppose that y p ( t )= ~ e - ~ ' t>O Substituting Eq. (2. 104) into Eq. (2. 101), we receive - b ~ e - 6 '+ a ~ e - b = ' ok ~ - ~ ' from which we receive A to procure yh(t), =K/(a - b), and we suppose y h ( t )= BeS' Substituting this into Eq. (2. 103) provides sBe"+aBe"=(s+a)Be"'=O from which we've s = - a and y h ( t ) = Be-"' Combining yp(t)and yh(t),we get y ( t ) =Be-"+ okay -e - b ~ a-b t>O From Eq. (2. 106) and the auxiliary y(O) =yo, we receive ok B=yo- a-b hence, Eq. (2. 106) turns into the homogeneous LINEAR TIME-INVARIANT platforms [CHAP. 2 For t < zero, now we have x(t) = zero, and Eq. (2. a hundred and one turns into Eq. (2. 103). for that reason, y(r) = Bepa' 1

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