EECS 451______________________PROBLEM SET #3_____________________Fall 2009

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ASSIGNED: Sep. 24, 2009. 1996 Text: Sections 3.4.3-3.6.5.
DUE DATE: Oct. 01, 2009. 2007 Text: 3.4.3-3.5.4; 3.6.1-2.

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Basic z-transforms and inverse z-transforms and ROC.

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Basic z-transforms:
1. [20] (4@5) Compute the z-transforms and their associated ROCs of these 4 signals:
   (a)(1+n)u(n); (b)(aⁿ+a^-n)u(n) (c)(-1)ⁿ2^-nu(n) (d)[naⁿsin(ωn)]u(n)
   [1996 & 2007 Text #3.2abcd]. Skip "pole-zero patterns," but specify ROC's.

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2. [20] (4@5) Compute the z-transforms and their associated ROCs of these 4 signals:
   (a) x(n)=(1/3)ⁿu(n)+(1/2)^-nu(-n-1) (b) [(1/3)ⁿ-(2)ⁿ]u(n) (c) x(n+4)
   [1996 & 2007 Text #3.3abcd]. (d) x(-n) from (a). Note some are two-sided signals.

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3. [20] Compute convolution [(1/2)ⁿu(n)]*x(n) using x(n) from #2a using z-transforms.
   [1996 & 2007 Text #3.7]. Uses result of #3.3a. Do analytically; check using conv.

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   Basic inverse z-transforms:
4. [20] (4@5) Compute the causal inverse z-transforms of these 4 functions:
   (a) (1+3z^-1) ⁄ (1+3z^-1+2z^-2) (b) 1 ⁄ (1-z^-1+(½)z^-2) (c) (z^-6+z^-7) ⁄ (1-z^-1) (d) (1+2z^-2) ⁄ (1+z^-2)
   [1996 & 2007 Text #3.14abcd]. All causal inverse z-transforms; so no ROC issues.
   Plenty of partial fraction expansions. See below on how to compute some of them.
   You may use Matlab to compute these (residue), but turn in your code and printout.

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5. [20] Compute all signals having z-transform (5z^-1) ⁄ ((1-2z^-1)(3-z^-1)).
   [1996 & 2007 Text #3.15]. Straight from handout. Answer: 3 different signals.

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6.