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# HWb - ECE 493 Univ of Illinois HW#11 Version 1.2 Due Tues...

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ECE 493 HW #11 – Version 1.2 February 11, 2011 Spring 2011 Univ. of Illinois Due Tues, Feb 15 Prof. Allen Topic of this homework: Convergence of transforms, ROC, Classification of systems (pole-zero placement). Deliverable: Show your work. 1 Convergence 1. If F ( s ) = 1 / (1 + s ) and the ROC is σ 0 > - 1, find the inverse Laplace transform ( L - 1 ). 2. If F ( s ) = 1 / (1 + s ) and the ROC is σ 0 < - 1, find the inverse Laplace transform. 1. If F ( s ) = 1 / ( s - 1) and the ROC is σ 0 > 1, find the inverse Laplace transform. 2. If F ( s ) = 1 / ( s - 1) and the ROC is σ 0 < 1, find the inverse Laplace transform. 3. Find the inverse Laplace transform of F ( s ) = 1 ( s - 1)( s + 1) (1) (a) if the ROC is between the two poles. (b) if the ROC is to the right of σ = 1. (c) if the ROC is to the left of σ = - 1. 2 Parseval’s Thm What is Parseval’s Theorem for the 1. Fourier Transform? 2. z transform? 3. Fourier series? 4. DFT? 5. Laplace Transform? 6. Derive this result for the case of the FT, starting from the basic definition of the FT transform and its inverse. 1

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3 Filter classes Let s = σ + be the Laplace (complex) frequency. A causal filter h ( t ) H ( s ) is one that is zero for negative time. It necessarily has a Laplace transform L . An finite impulse response (FIR) filter has finite duration, namely if f ( t ) is FIR, then it is zero for t < 0 and for t > T where T is some time. FIR filters only have
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HWb - ECE 493 Univ of Illinois HW#11 Version 1.2 Due Tues...

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