# soln3 - EECS 451 SOLUTIONS TO PROBLEM SET#3 1a From handout...

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Unformatted text preview: EECS 451 SOLUTIONS TO PROBLEM SET #3 1a. From handout: Z{ u ( n ) } = U ( z ) = 1 1- z- 1 ; Z{ nu ( n ) } =- z dU dz = z- 1 (1- z- 1 ) 2 → Z{ (1 + n ) u ( n ) } = 1 (1- z- 1 ) 2 . ROC: | z | > 1. 1b. From handout: Z{ ( a n + a- n ) u ( n ) } = 1 1- az- 1 + 1 1- 1 a z- 1 = 2- ( a + 1 a ) z- 1 (1- az- 1 )(1- 1 a z- 1 ) . ROC: | z | > max [ | a | , 1 | a | ]. NOTE: a = e jω o → Z{ 2 cos( ω o n ) } agrees with handout. 1c. Z{ (- 1) n 2- n u ( n ) } = Z{ (- 1 2 ) n u ( n ) } = 1 1+ 1 2 z- 1 . ROC: | z | > 1 2 . 1d. Z{ na n sin( ω o n ) u ( n ) } =- z d dz bracketleftBig az- 1 sin( ω o ) 1- 2 az- 1 cos ω o + a 2 z- 2 bracketrightBig = az- 1 sin ω o- a 3 z- 3 sin ω o (1- 2 az- 1 cos ω o + a 2 z- 2 ) 2 . | z | > a . 2a. Z{ ( 1 3 ) n u ( n )+2 n u (- n- 1) } = 1 1- 1 3 z- 1- 1 1- 2 z- 1 =- 5 3 z- 1 (1- 1 3 z- 1 )(1- 2 z- 1 ) . ROC: 1 3 < | z | < 2. 2b. Z{ ( 1 3 ) n u ( n )- 2 n u ( n ) } = 1 1- 1 3 z- 1- 1 1- 2 z- 1 =- 5 3 z- 1 (1- 1 3 z- 1 )(1- 2 z- 1 ) . ROC: | z | > 2....
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