Quiz3Soln - Name: ECE 301: Quiz 3 Purdue University, Summer...

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Unformatted text preview: Name: ECE 301: Quiz 3 Purdue University, Summer 2009 1. If x[n] = αn u[n] and h[n] = u[n], compute y [n] = x[n] ∗ h[n]. Solution: ∞ y [n] = k =−∞ ∞ x[k ]h[n − k ] αk u[k ]u[n − k ] k =−∞ ∞ = = k =0 αk u[n − k ] = 1 − αn+1 1−α u[n]. 0, n k k =0 α = 1−αn+1 , 1−α n<0 n≥0 ⇒ y [n] = 2. Consider an CT LTI system with impulse response h(t) = 1, −1 ≤ t < 1 0, otherwise. What is the output of the system when the input is x(t) = e−t u(t)? Solution: The system is LTI, so the output ∞ y (t) = x(t) ∗ h(t) = h(t) ∗ x(t) = −∞ 1 h(τ )x(t − τ )dτ e−(t−τ ) u(t − τ )dτ −1 1 = = e−t eτ u(t − τ )dτ. −1 Break into cases of t: 0, e−t y (t) = −t e t < −1 = 1−e , −1 ≤ t < 1 −t+1 −t−1 =e −e , t > 1. −t−1 t eτ dτ −1 1 eτ dτ −1 1 ...
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