z transform stability

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    The z-Transform

    Introduction

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    Content Introduction

    z-Transform Zeros and Poles

    Region of Convergence

    Importantz

    -Transform Pairs Inverse z-Transform

    z-Transform Theorems and Properties

    System Function

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    Why z-Transform? A generalization of Fourier transform

    Why generalize it? FT does not convergeon all sequence

    Notationgood for analysis

    Bring the power of complex variable theorydeal with

    the discrete-time signals and systems

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    The z-Transform

    z-Transform

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    Definition Thez-transform of sequencex(n) is defined by

    n

    nznxzX )()(

    Letz = ej.

    ( ) ( )j j n

    n

    X e x n e

    Fourier

    Transform

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    z-Plane

    Re

    Im

    z = ej

    n

    nznxzX )()(

    ( ) ( )

    j j n

    nX e x n e

    Fourier Transform is to evaluate z-transform

    on a unit circle.

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    z-Plane

    Re

    Im

    X(z)

    Re

    Im

    z = ej

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    Periodic Property of FT

    Re

    Im

    X(z)

    X(ej)

    Can you say why Fourier Transform is

    a periodic function with period 2?

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    The z-Transform

    Zeros and Poles

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    Definition Give a sequence, the set of values ofzfor which the

    z-transform converges, i.e., |X(z)|

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    Example: Region of Convergence

    Re

    Im

    n

    n

    n

    n znxznxzX |||)(|)(|)(|

    ROC is an annual ring centered

    on the origin.

    xx RzR ||r

    }|{ xx

    j RrRrezROC

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    Stable Systems

    Re

    Im

    1

    A stable system requires that its Fourier transformis

    uniformly convergent.

    Fact: Fourier transform is to

    evaluatez-transform on a unit

    circle.

    A stable system requires the

    ROC ofz-transform to include

    the unit circle.

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    Example: A right sided Sequence

    )()( nuanx n

    1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

    n

    x(n)

    . . .

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    Example: A right sided Sequence

    )()( nuanx n

    n

    n

    n znuazX

    )()(

    0n

    nnza

    0

    1)(

    n

    naz

    For convergence ofX(z), we

    require that

    0

    1 ||n

    az 1|| 1 az

    |||| az

    az

    z

    azazzX

    n

    n

    10

    1

    1

    1)()(

    |||| az

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    aa

    Example: A right sided Sequence

    ROC forx(n)=anu(n)

    ||||,)( azaz

    zzX

    Re

    Im

    1 aaRe

    Im

    1

    Which one is stable?

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    Example: A left sided Sequence

    )1()( nuanx n

    1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

    n

    x(n)

    ...

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    Example: A left sided Sequence

    )1()( nuanx n

    n

    n

    n znuazX

    )1()(

    For convergence ofX(z), we

    require that

    0

    1 ||n

    za 1|| 1 za

    |||| az

    az

    z

    zazazX

    n

    n

    10

    1

    1

    11)(1)(

    |||| az

    n

    n

    nza

    1

    n

    n

    nza

    1

    n

    n

    nza

    0

    1

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    aa

    Example: A left sided Sequence

    ROC forx(n)=anu(n1)

    ||||,)( azaz

    zzX

    Re

    Im

    1 aaRe

    Im

    1

    Which one is stable?

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    The z-Transform

    Region of

    Convergence

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    Represent z-transform as a

    Rational Function

    )(

    )(

    )( zQ

    zP

    zX

    whereP(z)and Q(z)are

    polynomials inz.

    Zeros:The values ofzs such thatX(z) = 0

    Poles:The values ofzs such thatX(z) =

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    Example: A right sided Sequence

    )()( nuanx n ||||,)( az

    az

    zzX

    Re

    Im

    a

    ROC isbounded by the

    poleand is the exteriorof a circle.

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    Example: A left sided Sequence

    )1()( nuanx n ||||,)( az

    az

    zzX

    Re

    Im

    a

    ROC isbounded by the

    poleand is the interiorof a circle.

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    Example: Sum of Two Right Sided Sequences

    )()()()()(31

    21 nununx nn

    31

    21

    )(

    z

    z

    z

    zzX

    Re

    Im

    1/2

    ))((

    )(2

    31

    21

    121

    zz

    zz

    1/3

    1/12

    ROC isbounded by poles

    and is the exterior of a circle.

    ROC does not include any pole.

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    Example: A Two Sided Sequence

    )1()()()()(21

    31 nununx nn

    21

    31

    )(

    z

    z

    z

    zzX

    Re

    Im

    1/2

    ))((

    )(2

    21

    31

    121

    zz

    zz

    1/3

    1/12

    ROC isbounded by poles

    and is a ring.

    ROC does not include any pole.

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    Example: A Finite Sequence

    10,)( Nnanx n

    nN

    n

    nN

    n

    n zazazX )()( 11

    0

    1

    0

    Re

    Im

    ROC: 0

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    Properties of ROC

    A ringor diskin the z-plane centered at the origin.

    The Fourier Transform ofx(n) is converge absolutely iff the ROC

    includes the unit circle. The ROC cannot include any poles

    Finite Duration Sequences:The ROC is the entirez-plane except

    possiblyz=0 orz=.

    Right sided sequences:The ROC extends outward from the outermost

    finite pole inX(z) toz=.

    Left sided sequences:The ROC extends inward from the innermost

    nonzero pole inX(z) toz=0.

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    More on Rational z-Transform

    Re

    Im

    a b c

    Consider the rationalz-transform

    with the pole pattern:

    Find the possibleROCs

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    More on Rational z-Transform

    Re

    Im

    a b c

    Consider the rationalz-transform

    with the pole pattern:

    Case 1: A right sided Sequence.

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    More on Rational z-Transform

    Re

    Im

    a b c

    Consider the rationalz-transform

    with the pole pattern:

    Case 2: A left sided Sequence.

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    More on Rational z-Transform

    Re

    Im

    a b c

    Consider the rationalz-transform

    with the pole pattern:

    Case 3: A two sided Sequence.

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    More on Rational z-Transform

    Re

    Im

    a b c

    Consider the rationalz-transform

    with the pole pattern:

    Case 4: Another two sided Sequence.

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    The z-Transform

    Important

    z-Transform Pairs

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    Z-Transform Pairs

    Sequence z-Transform ROC

    )(n 1 Allz

    )( mn mz Allz except 0 (if m>0)or (if m

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    Z-Transform Pairs

    Sequence z-Transform ROC

    )(][cos 0 nun 210

    1

    0

    ]cos2[1

    ][cos1

    zz

    z1|| z

    )(][sin 0 nun 210

    1

    0

    ]cos2[1

    ][sin

    zz

    z1|| z

    )(]cos[ 0 nunrn 221

    0

    1

    0

    ]cos2[1

    ]cos[1

    zrzr

    zrrz||

    )(]sin[ 0 nunrn 221

    0

    1

    0

    ]cos2[1

    ]sin[

    zrzr

    zrrz||

    otherwise0

    10 Nnan

    11

    1

    az

    za NN

    0|| z

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    The z-Transform

    Inverse z-Transform

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    The z-Transform

    z-Transform Theorems

    and Properties

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    LinearityxRzzXnx ),()]([Z

    yRzzYny ),()]([Z

    yx RRzzbYzaXnbynax ),()()]()([Z

    Overlay of

    the above two

    ROCs

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    ShiftxRzzXnx ),()]([Z

    x

    nRzzXznnx )()]([ 00Z

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    Multiplication by an Exponential Sequence

    xx- RzRzXnx ||),()]([Z

    x

    n RazzaXnxa ||)()]([ 1Z

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    Differentiation of X(z)

    xRzzXnx ),()]([Z

    xRzdz

    zdXznnx

    )()]([Z

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    ConjugationxRzzXnx ),()]([Z

    xRzzXnx *)(*)](*[Z

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    ReversalxRzzXnx ),()]([Z

    xRzzXnx /1)()]([1 Z

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    Real and Imaginary Parts

    xRzzXnx ),()]([Z

    xRzzXzXnxe *)](*)([)]([ 21R

    xj RzzXzXnx *)](*)([)]([

    21Im

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    Initial Value Theorem

    0for,0)( nnx

    )(lim)0( zXxz

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    Convolution of Sequences

    xRzzXnx ),()]([Z

    yRzzYny ),()]([Z

    yx RRzzYzXnynx )()()](*)([Z

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    Convolution of Sequences

    k

    knykxnynx )()()(*)(

    n

    n

    k

    zknykxnynx )()()](*)([Z

    k

    n

    n

    zknykx )()(

    k

    n

    n

    k znyzkx )()(

    )()( zYzX

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    The z-Transform

    System Function

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    Shift-Invariant System

    h(n)

    x(n) y(n)=x(n)*h(n)

    X(z) Y(z)=X(z)H(z)H(z)

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    Shift-Invariant System

    H(z)

    X(z) Y(z)

    )(

    )()(

    zX

    zYzH

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    N

    th

    -Order Difference Equation

    M

    r

    r

    N

    k

    k rnxbknya00

    )()(

    M

    r

    r

    r

    N

    k

    k

    k zbzXzazY00

    )()(

    N

    k

    k

    k

    M

    r

    r

    r zazbzH00

    )(

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    Representation in Factored Form

    N

    k

    r

    M

    r

    r

    zd

    zcA

    zH

    1

    1

    1

    1

    )1(

    )1(

    )(

    Contributespolesat 0 andzerosat cr

    Contributeszerosat 0 andpolesat dr

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    Stable and Causal Systems

    N

    k

    r

    M

    r

    r

    zd

    zcA

    zH

    1

    1

    1

    1

    )1(

    )1(

    )( Re

    Im

    Causal Systems : ROC extends outward from the outermost pole.

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    Stable and Causal Systems

    N

    k

    r

    M

    r

    r

    zd

    zcA

    zH

    1

    1

    1

    1

    )1(

    )1(

    )( Re

    Im

    Stable Systems : ROC includes the unit circle.

    1

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    ExampleConsider the causal system characterized by

    )()1()( nxnayny

    1

    1

    1)(

    az

    zH

    Re

    Im

    1

    a

    )()( nuanh n

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    Determination of Frequency Response

    from pole-zero pattern

    A LTI system is completely characterized by its

    pole-zero pattern.

    ))((

    )(21

    1

    pzpz

    zzzH

    Example:

    ))(()(

    21

    1

    00

    0

    0

    pepe

    zeeH

    jj

    jj

    0je

    Re

    Im

    z1

    p1

    p2

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    Determination of Frequency Response

    from pole-zero pattern

    A LTI system is completely characterized by its

    pole-zero pattern.

    ))((

    )(21

    1

    pzpz

    zzzH

    Example:

    ))(()(

    21

    1

    00

    0

    0

    pepe

    zeeH

    jj

    jj

    0je

    Re

    Im

    z1

    p1

    p2

    |H(ej)|=? H(ej)=?

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    Determination of Frequency Response

    from pole-zero pattern

    A LTI system is completely characterized by its

    pole-zero pattern.

    Example:

    0je

    Re

    Im

    z1

    p1

    p2

    |H(ej)|=? H(ej)=?

    |H(ej)| =| |

    | | | | 1

    2

    3

    H(ej) = 1(2+ 3 )

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    Example1

    1

    1)(

    az

    zH

    Re

    Im

    a

    0 2 4 6 8-10

    0

    10

    20

    0 2 4 6 8-2

    -1

    0

    1

    2

    dB