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A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at s = -2 and s = -4, and one simple zero at s = -1. A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

1. [exp(-2t) + exp(-4t)]u(t)
2. [-4 exp(-2t) + 12 exp(-4t) - exp(-t)]u(t)
3. [-4 exp(-2t) + 12 exp(-4t)]u(t)
4. [-0.5 exp(-2t) + 1.5 exp(-4t)]u(t)

Correct answer: [-4 exp(-2t) + 12 exp(-4t)]u(t)

Solution

With poles at -2,-4 and zero at -1, H(s)=K(s+1)/((s+2)(s+4)). Unit-step steady-state output 1 means H(0)=K/8=1, so K=8. Partial fractions give residues -4 at s=-2 and +12 at s=-4, so h(t)=[-4e^(-2t)+12e^(-4t)]u(t), which is option C, not the stored option D.

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