Ch3.3 Z transformation

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Ch3.3 Z transformation
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Z?????(??????)

• ????????????????,????DTFT???,????????r-n, ??????

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Z??

• Z?????
• Z???FT???,z1?r1??????,

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?????Laplace??
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Laplace Transformation

• ?????Laplace ??

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?????????

• ?????????

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Z???

• r1?, ???z????????????,

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S???z???????
j?
z
S
z
S
1
?
1
1
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s?z???(1)
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s?z???(2)

• r?????, re?T
• ?0(s????)???r1(z??????)
• ?lt0(s?????)???rlt1(z??????)
• ?gt0(s?????)???rgt1(z??????)
• ??????, ??T
• ?0(s?????)????0(z??????)
• ??0???? ?0T(z??????),
• ? -?/T to ?/T, ? - ? to ?
• ?s???z??????????

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s?z???(3)
j?
Im(z)
z??
s??
?
Re(z)
s??gt0???z?rgt1 s??0???z?r1 s??lt0???z?rlt1
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s?z???(4)
j?
Im(z)
3?/T
s??
z??
?/T
-1
?
1
Re(z)
-?/T
-3?/T
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X(z)?X(s)???
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z??????(ROC)

• ROC range of convergence
• Definition z?????????z???,
• ??

a
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Z??????

• ??z?????, ?????????????????,
• Z?????????????,

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????????

• x(n), N1-N2,
• ????, -inf - N2,
• z lt Rx1, ?Rx1??????,
• ????, N1- inf,
• zgtRx2,?Rx2??????,
• ????, -inf to inf,
• Rx2ltzltRx1, ???,

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?????z??
?? z?? ROC







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Z?????

• ?? x(n)y(n)
• ????, x(n-m)
• Z?????? nkx(n)
• ??????? x(-n)
• Z???????? anx(n)
• ?????? x(n)y(n)

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Z?????
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Z?????

• ??????????,

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?????z???

• ?????z-1?????, ???z???????,
• y(n)-0.5y(n-1)x(n)
• ?????????, ??????????????z???????z????,???????????
  ??????,
• Y(z)-0.5z1Y(z)X(z)
• H(z)Y(z)/X(z)1/(1-0.5z1)

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?????z???

• ???????
• ???????

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?????z???

• ?????z-1??????H(z)?h(n)

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H(z)???????h(n)

• ????H(z)??????????h(n)???z???

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LTI?????????

• ??????h(n)(impulse response)
• ????
• ????,
• ????,

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FIR IIR??

• FIR ??, h(n)????????,
• ????,
• IIR??, h(n)????????,
• a(k)???0,???????y(n-k)????,IIR????????,z??????,
  ???????,

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?z??(Inverse z-transform)

• ?? ??X(z)?????????x(n)???,
• ?z???????
• ??????,
• ?????,
• ???,

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??????

• X(z)??????, ?????????????z-1????,????????x(n).
• ?? ???????,

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?????(partial-fraction expansion)

• ??X(z)?z-1?????, ?????

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?????(2)

• ?????, zzp?L???,

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?????(3)

• ??,

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???(residue method)(?????)

• ??Zm?????X(z)zn-1???, C???????????????????,

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?????
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???????h(n)??? (?)

• ???????H(s),??????????????????????
• ??????,????????????, ROC?????, h(inf)?0

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???????h(n) (?)

• ?????, ROC?????,h(inf)?0?

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??????(stability)

• ?? ?????????, ??????????, (BIBO, Bounded input
  bounded output)
• ?????????????
• ?????,???????????
• ?????,??H(z)?????????, ???? (general). The unit
  circle belongs to the ROC.

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??????

• ??????(Causality)
• ????, ???????????????????, ?????????,
• ?????, ????????????????????, ?????????,
• ????h(n)????,
• ????? h(n)?????

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?H(z)????????

• ??????????
• ????,h(n), ????,ROC zgta??if
  ROC?????,?????????????,?
• ?????,h(n)?????, if ROC?????,?????????????,??,????
  ????????????,???h(n)???????????????????

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????????????

• ????,h(n), ????,ROC zgta??if stable,
• ROC ??????a????,??????????????
• ?????, h(n), ????, ROC ?????
• ??????????,??????,
• ??????????,??????,

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??

• Questions What kind of the system is it?

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Matlab???

• ?? conv(x,y),
• x12,3,4,x23,4,5,6, x3conv(x,y)6,17,34,43,
  38,24
• ??? x1deconv(x3,x2),
• ??X(z)??????????,

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Matlab?????
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Matlab?????
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DFT?DTFT?ZT???

• x(n),

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DFT?DTFT?ZT???

• X(ej?) ??z??????,
• X(k)??z????N?????????,
• X(z)??X(k)???,

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DFT?DTFT???

• N???????X(ej?)??N?DFT??X(k)???,

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ZT?Laplace???????