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ESE563 1
                                                                                                      Azlin-Fatta-Fad

4a Exercise      2. What is the z-transform of K3 u[k]?


                                ������
              ������ ������[������]   (−������ ������������ ) ������[������]
                                                             ������ ������ = 1 ⋅ ������ ������
                                                                                  ������
                                                             ∴ ������ ������ =
                                                                               ������ − 1

                                ������          ������               ������
                          (−������ ������������ ) [−������ ������������                          ]
                                                        ������−1




                                               ������           ������������ ′ − ������ ������ ′
                                                        =                          >>       ������ = ������      ������′ = 1
                                               ������                 ������ 2

                                                                                            ������ = ������ − 1 ������′ = 1

                                 ������            ������−1 1 − ������ 1
                          (−������ ) −������[                                                   }
                                ������������                          ������−1 2

                                ������          −������
                          (−������ ������������ ) [   ������−1 2
                                                    ]

                                 ������           ������              −������
                          (−������ ) [−������                                          ]
                                ������������         ������������       (������−1)2




                                               ������           ������������ ′ − ������ ������ ′
                                                        =                          >>       ������ = −������       ������′ = −1
                                               ������                 ������ 2

                                                                                            ������ = (������ − 1)2 ������′ = 2(������ − 1)(1)
ESE563 2
                                                                                Azlin-Fatta-Fad


       ������                       ������−1 2 −1 − −������ 2 ������−1 (1)
−������            −������[                                                                   }
      ������������                                              ������−1 4
                                      1
         ������               ������−1 (−������ 2 +������+2������ 2 )
 −������                [
        ������������                     (������−1)(������−1)3

             ������            ������ 2 +������
(−������)               [                        ]
             ������������          ������−1 3



             ������         ������������ ′ − ������ ������ ′
                    =                         >>        ������ = ������ 2 + ������   ������′ = 2������ + 1
             ������               ������ 2

                                                        ������ = (������ − 1)3   ������′ = 3(������ − 1)2(1)


             ������−1 3 2������+1 − ������ 2 +������ 3 ������−1 2 (1)
−������ {                                                                                }
                                                     ������−1 6
                            1
             ������−1         2[         ������−1 2������+1)−(������ 2 +������ 3 ]
−������ {                                                                           }
                                           ������−1 2 ������−1 4
                                                 1
         2������ 2 −2������+������−1−3������ 2 −3������
−������ [                                                           ]
                                     ������−1 4

������ 3 +4������ 2 +������
   ������−1 4

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Ese563

  • 1. ESE563 1 Azlin-Fatta-Fad 4a Exercise 2. What is the z-transform of K3 u[k]? ������ ������ ������[������] (−������ ������������ ) ������[������] ������ ������ = 1 ⋅ ������ ������ ������ ∴ ������ ������ = ������ − 1 ������ ������ ������ (−������ ������������ ) [−������ ������������ ] ������−1 ������ ������������ ′ − ������ ������ ′ = >> ������ = ������ ������′ = 1 ������ ������ 2 ������ = ������ − 1 ������′ = 1 ������ ������−1 1 − ������ 1 (−������ ) −������[ } ������������ ������−1 2 ������ −������ (−������ ������������ ) [ ������−1 2 ] ������ ������ −������ (−������ ) [−������ ] ������������ ������������ (������−1)2 ������ ������������ ′ − ������ ������ ′ = >> ������ = −������ ������′ = −1 ������ ������ 2 ������ = (������ − 1)2 ������′ = 2(������ − 1)(1)
  • 2. ESE563 2 Azlin-Fatta-Fad ������ ������−1 2 −1 − −������ 2 ������−1 (1) −������ −������[ } ������������ ������−1 4 1 ������ ������−1 (−������ 2 +������+2������ 2 ) −������ [ ������������ (������−1)(������−1)3 ������ ������ 2 +������ (−������) [ ] ������������ ������−1 3 ������ ������������ ′ − ������ ������ ′ = >> ������ = ������ 2 + ������ ������′ = 2������ + 1 ������ ������ 2 ������ = (������ − 1)3 ������′ = 3(������ − 1)2(1) ������−1 3 2������+1 − ������ 2 +������ 3 ������−1 2 (1) −������ { } ������−1 6 1 ������−1 2[ ������−1 2������+1)−(������ 2 +������ 3 ] −������ { } ������−1 2 ������−1 4 1 2������ 2 −2������+������−1−3������ 2 −3������ −������ [ ] ������−1 4 ������ 3 +4������ 2 +������ ������−1 4