Quiz4Soln - Solution ECE 301: Quiz 4 Purdue University,...

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Unformatted text preview: Solution ECE 301: Quiz 4 Purdue University, Summer 2009 1. Recall that F {x(t)} = −∞ x(t)e−jωt dt. Compute the Fourier transform F {e−αt u(t)}, where α > 0 Solution: ∞ ∞ F {e−αt u(t)} = −∞ ∞ e−αt u(t)e−jωt dt e−t(α+jω) dt 0 = = 1 α + jω 2. Let an LTI system have impulse response h(t) = e−2t u(t). Without using convolution, use your answer from the previous problem to find the output y (t) when x(t) = e−5t u(t) is input. Solution: Work in the frequency domain, then take inverse Fourier transform: y (t) = x(t) ∗ h(t) ⇒ Y (ω ) = X (ω )H (ω ) = F {e−5t u(t)}F {e−2t u(t)} 1 1 = (from problem 1). 5 + jω 2 + jω Using partial fraction expansion, we find 1/3 −1/3 + 5 + jω 2 + jω 1 −2t 1 ⇒ y (t) = e u(t) − e−5t u(t) (from problem 1). 3 3 Y (ω ) = 1 ...
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This note was uploaded on 03/30/2011 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Spring '06 term at Purdue University.

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