The z Transform - 1 1 The z Transform The z transform generalizes the Discrete-time Fourier...

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Transcript of The z Transform - 1 1 The z Transform The z transform generalizes the Discrete-time Fourier...

  • 1

    1

    The z Transform

    The z transform generalizes the Discrete-time Fourier Transform for the entire complex plane. For the complex variable is used the notation:

    { }

    2 2

    ; ,

    arg

    jz x j y r e x y

    z r x y

    z

    Ω= + ⋅ = ⋅ ∈

    = = +

    Ω =

    2

    notation:

    For any discrete-time LTI system:

    -the eigen function:

    -the eigen value:

    The Discrete LTI System Response to a Complex Exponential

    [ ] 00 0 j nn nx n z r e Ω= =

    System

    h[n]

    input output

    [ ] [ ] [ ] [ ] ( )0 0 0 0 0 0n n k n k n k k

    y n h n z h k z z h k z z H z ∞ ∞

    − −

    =−∞ =−∞ = ∗ ⋅ = ⋅ ⋅= =∑ ∑

    [ ] [ ] ( )0 0 0n ny n h n z z H z= ∗ ⋅=

    ( ) [ ] k k

    H z h k z ∞

    =−∞ = ⋅∑

    ( )0H z 0 nz

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    3

    fast computation of the response of any discrete-time LTI system to linear combinations of complex exponentials of the form:

    transfer function, z transform of the impulse response h[n].

    [ ] ( )0 0ny n z H z= ⋅

    [ ] nk k k

    x n c z= ∑ System

    h[n]

    input output [ ] ( ) nk k k

    k y n c H z z= ∑

    ( ) [ ] k k

    H z h k z ∞

    =−∞ = ⋅∑

    4

    Bilateral z Transform The bilateral z transform of the signal x[n] :

    generalization of the discrete-time Fourier transform

    Discrete-time Fourier transform = particular case of z transform on the unit circle |z|=1

    [ ]{ }( ) ( ) [ ] , , 0jn n

    Z x n z X z x n z z r e r ∞

    Ω−

    =−∞ = = ⋅ = ⋅ ≥∑

    [ ]{ }( ) [ ]( ) [ ]{ }( ); 0j nn n n

    Z x n z x n r e r x n r ∞

    − Ω− −

    =−∞ = ⋅ = ⋅ Ω ≥∑ F

    [ ]{ }( ) [ ]{ }( )1 jz r Z x n e x nΩ= = ⇒ = ΩF |z|=1

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    5

    Examples

    region of convergence (ROC) = set of values of z for which X(z) is convergent

    rational function, zeros = roots of numerator ; poles = roots of denominator.

    Pole/zero plot or constellation (PZC).

    [ ] [ ]1. , 1nx n a n a= σ <

    ( ) ( )1 0 0

    nn n

    n n X z a z az

    ∞ ∞ − −

    = = = =∑ ∑ 1 1az− <

    ( ) 1 1 ,

    1 X z z a

    az− = >

    convergent:

    6

    convergent if:

    one pole in a.

    [ ] [ ]2. 1 , 1nx n a n a= − σ − − <

    ( ) ( ) 1 1

    1 1 1 1

    1 0

    1 1 1n nnn n n n n n

    X z a z az az az az

    − − ∞ ∞ − −

    − − − =−∞ =−∞ = =

    ⎛ ⎞ ⎛ ⎞= − = − = − = −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

    ∑ ∑ ∑ ∑

    1 1 1

    a z− <

    ( ) 1 1

    1

    1 1 ; 1 11

    X z z a azaz

    az

    − −

    = − = < ⎛ ⎞ −−⎜ ⎟ ⎝ ⎠

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    same z transform ; different ROCs

    First example Second example

    Region of convergence ROC : set of values of z for which the series X(z) is convergent.

    The Z transform exists where the Fourier transform of x[n]r-n exists, r=|z|

    ( ) 1 1

    1 X z

    az− =

    8

    • ROC of a bilateral z transform can not contain any pole • for right sided signals (including causal signals), the

    ROC extends outward from the outermost pole, • for left sided signals (including anticausal signals), the

    ROC extends inward from the innermost pole • for infinite duration signals, ROC is a ring that doesn’t

    include poles – bounded on the interior and exterior by a pole.

    • for finite duration signals, the ROC is the entire z- plane, except possibly z=0 or z=∞.

    • a causal and stable system’s transfer function has the poles inside the unit circle. ROC is outside the unit circle.

    Properties of the ROC of the Bilateral z Transform

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    [ ] [ ] [ ]1 2x n x n x n= +

    first component of x[n] is right sided; ROC is outside the circle with radius R-.

    second component of x[n] is left sided, ROC is inside the circle with radius R+.

    The ROC of X(z) is the intersection of the ROCs of its components (ring)

    10

    Example [ ]

    [ ] [ ] [ ]1 2 3. , 0 1

    nx n a a

    x n x n x n

    = < <

    = +

    [ ] [ ]

    [ ] [ ]

    [ ]

    1 1

    2

    1

    1 , 1

    1

    1 1 11 , 11

    n

    n

    x n a n z a az

    x n a n

    n z a az

    a

    = σ ⎯⎯→ > −

    = σ − −

    −⎛ ⎞= σ − − ⎯⎯→

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    11

    Inverse z Transform

    • Γ - counterclockwise closed path (contour) included in the region of convergence.

    • Γ - encircles the origin and must encircle all poles of X(z).

    • integral along a contour. • z transforms in DSP : rational functions. • partial fraction expansion and tables signal – z

    transform pairs (knowing the pole/zero plot).

    ( ){ }[ ] ( ) [ ]1 11 ; 2

    nZ X z n X z z dz x n ROC j

    − − Γ

    = = Γ ⊂ π ∫

    12

    The Bilateral z Transform computation using its pole/zero plot

    X(z) - rational function

    ( ) ( )

    ( ) 0

    1

    1

    ; N

    M

    k k

    pk k

    z z X z k z ROC

    z z

    =

    =

    − = ∈

    poles and zeros ⇒ Z transform without constant k

    X(z0) – also known ⇒ Z transform and constant k

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    Example

    unit circle ⊂ ROC ⇒ discrete-time Fourier transform exists:

    A ∈ unit circle; ∠(OA, Ox) = Ψ

    the spectrum of signal x[n]

    0pole/zero plot 0; 0.5pz z= = ( ) 1 1 for 1

    0.5 1 0.5 zX z k

    z z− = = =

    − −

    causal signal, ROC: 0.5pz z> =

    ( ) ( ) 10.5 1 0.5 j

    j j j eX X e

    e e

    Ω Ω

    Ω − ΩΩ = = =− −

    ( ) ( ); OA

    X AP

    Ω = Φ Ω = ψ −ϕ

    14

    vectors Azpk , Azok: ∠(AZ0k, Ox) = Ψk ; ∠(AZpk, Ox) = ϕk

    Magnitude and phase spectrum:

    The frequency Ω = length of the arc (of circle) in radians on the unit circle, between its intersection with the positive real axis and the point A, in trigonometric sense.

    General case:

    ( ) 0

    1

    1 1

    1

    ; =

    M

    M Nk k

    k kN k k

    pk k

    Az X k Argk

    Az

    =

    = =

    =

    Ω = Φ + ψ − ϕ ∏

    ∑ ∑ ∏

    ( ) ( )

    ( ) 0

    1

    1

    ; N

    M

    k k

    pk k

    z z X z k z ROC

    z z

    =

    =

    − = ∈

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    15

    The Unilateral z Transform

    •For causal signals, unilateral z transform = bilateral z transform

    •ROC: entire complex plane or the outside region of a disc centered in 0.

    •Useful for causal systems described by difference equations, with non zero initial conditions.

    [ ]{ }( ) ( ) [ ] 0

    n u u

    n Z x n z X z x n z

    ∞ −

    = = = ∑

    16

    z Transform Properties notations:

    1. Linearity

    Proof. Directly, using the definitions. Homework - Prove it.

    [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( )

    , ;

    , ;

    u

    u

    x

    y

    ZZ

    ZZ

    x n X z z ROC x n X z

    y n Y z z ROC y n Y z

    ←⎯→ ∈ ←⎯⎯→

    ←⎯→ ∈ ←⎯⎯→

    [ ] [ ] ( ) ( ) [ ] [ ] ( ) ( )

    , at leastx y

    u u

    ax n by n aX z bY z z ROC ROC

    ax n by n aX z bY z

    + ←⎯→ + ∈ ∩

    + ←⎯→ +

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    2. Time shifting

    Proof.

    bilateral z transform:

    If n0>0, z=0 ∉ ROC. If n00

    n

    n n u

    n n

    x n n z X z z ROC

    x n n z X z x n z n

    − − −

    =−

    − ←⎯→ ∈⎡ ⎤⎣ ⎦ ⎛ ⎞

    − ←⎯→ −⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎜ ⎟ ⎝ ⎠

    { } [ ]

    [ ] [ ]

    0 0

    0

    0

    0

    0 0 0

    1

    0 0

    , 0

    n n m nn m u

    n m n

    n m m

    m m n

    Z x n n x n n z x m z z

    z x m z x m z n

    − =∞ ∞ −− −

    = =−

    ∞ − − − −

    = =−

    − = − =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

    ⎛ ⎞ = + >⎜ ⎟⎜ ⎟

    ⎝ ⎠

    ∑ ∑

    ∑ ∑

    { } [ ] ( ) [ ]0 0 00 0 n n m m n nn m

    n m m Z x n n x n n z x m z z x m z

    − =∞ ∞ ∞− + −− −

    =−∞ =−∞ =−∞ − = − = =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦∑ ∑ ∑

    18

    3. Modulation in time

    Proof.

    More generally:

    Homework - Prove it.

    [ ]{ } [ ] [ ]( ) (0 0 0 0nj n j n j jn n n

    Z e x n e x n z x n e z X e z ∞ ∞ −Ω Ω − Ω − Ω−

    =−∞ =−∞ = = =∑ ∑

    [ ]

    [ ]

    0 0 0

    0 0

    , n

    n u

    z zz x n X ROC z z