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hw07su10 - GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of...

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GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Summer 2010 Problem Set #7 Assigned: June 29, 2010 Due Date: July 12-13, 2010 Please check the T-square chat and announcements daily. All official course announcements will be posted there. ALL of the STARRED problems should be turned in for grading. PROBLEM 7.1 *: Determine the z -transforms of the following sequences. For each sequence, express your answer three different ways in simplest form: (1) a sum of rational functions; (2) a ratio of polynomials in z - 1 ; and (3) a product of factors of the form (1 - az - 1 ). (a) x 1 [ n ] = 1 2 n u [ n ] + 1 3 n - 1 u [ n - 1]. (b) x 2 [ n ] = 1 2 [(0 . 9) n + (0 . 9) n ( - 1) n ] u [ n ] = (0 . 9) n for n 0 and n even. (c) x 3 [ n ] = 1 2 [(0 . 9) n - (0 . 9) n ( - 1) n ] u [ n ] = (0 . 9) n for n 0 and n odd. (d) x 4 [ n ] = x 2 [ n ] + x 3 [ n ] (e) x 5 [ n ] = ( - 0 . 5) n cos( π 3 n ) u [ n ]. PROBLEM 7.2 *: Consider a discrete time LTI system with the system function H ( z ) = 1 - 2 . 7 z - 3 + 1 . 8 z - 4 6 - 0 . 3 z - 1 - 0 . 5 z - 5 (a) Find the input/output equation that relates the output y [ n ] to an arbitrary input x [ n ]. (To
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