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# HW7sol - ECEN 314 Signals and Systems Solutions to HW 7 1(a...

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Solutions to HW 7 1(a) s + 3 ( s + 2)( s + 4) = 1 2 1 s + 2 + 1 s + 4 · Taking the inverse laplace transform of the above yields h ( t ) = 1 2 ( e - 2 t + e - 4 t ) u ( t ) . 1(b) s 2 ( s + 2)( s + 4) = 1 + 2 s + 2 - 8 s + 4 Taking the inverse laplace transform of the above yields, h ( t ) = δ ( t ) + 2 e - 2 t u ( t ) - 8 e - 4 t u ( t ) . 1(c) 1 s ( s - 5) 2 = 1 25 1 s - 1 s - 5 + 5 ( s - 5) 2 · Taking the inverse laplace transform of the above by keeping in mind the ROC yields h ( t ) = 1 25 - u ( - t ) + e 5 t u ( - t ) - 5 te 5 t u ( - t ) · . 1(d) s s 2 + 2 s + 5 = s ( s + 1) 2 + 2 2 = s + 1 ( s + 1) 2 + 2 2 - 1 ( s + 1) 2 + 2 2 = s + 1 ( s + 1) 2 + 2 2 - 1 2 2 ( s + 1) 2 + 2 2 · Taking the inverse laplace transform of the above yields h ( t ) = e - t cos(2 t ) u ( t ) - 1 2 e - t sin(2 t ) u ( t ) . 2.(a) Taking the Laplace transform of the diﬀerential equation (taking into account zero state response) yields: s 2 Y ( s ) + 3 sY ( s ) + 2 Y ( s ) = X

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HW7sol - ECEN 314 Signals and Systems Solutions to HW 7 1(a...

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