MIT6_003S10_assn04

# MIT6_003S10_assn04 - 6.003 Homework 4 Please do the...

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6.003 Homework 4 You need not submit Problems 1. Z transforms Determine the Z transform (including the region of convergence) for each of the following signals: ( 1 ) n a. x 1 [ n ] = 2 u [ n 3] b. x 2 [ n ] = (1 + n ) ( 1 3 ) n u [ n ] ( 1 ) | n Please do the following problems by Wednesday, March 3, 2010 . your answers: they will NOT be graded. Solutions will be posted. c. x 3 [ n ] = n 2 | d. 2 1 0 1 2 3 4 1 1 x 4 [ n ] n 2. Inverse Z transforms Determine and sketch all possible signals with Z transforms of the following forms. For each signal, indicate the associated region of convergence. 1 a. X 1 ( z ) = z 1 1 b. X 2 ( z ) = z ( z 1) 2 1 c. X 3 ( z ) = z 2 + z + 1 d. X 4 ( z ) = 1 z 2 2 z 3. More symmetries Consider the following DT pole-zero diagrams, where the circles have unit radius. X 1 ( z ) X 2 ( z ) X 3 ( z ) X 4 ( z )

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2 6.003 Homework 4 / Spring 2010 c. Which if any of the pole-zero diagrams could represent a system that is causal? d. Which if any of the pole-zero diagrams could represent a system that is both causal and stable? 4. Z transform Let X ( z ) represent the Z transform of x [ n ] , and let r 0 < z < r 1 represent its region of convergence (ROC). | | Let x [ n ] be represented as the sum of even and odd parts x [ n ] = x e [ n ] + x o [ n ] where x e [ n ] = x e [ n ] and x o [ n ] = x o [ n ] .
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