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

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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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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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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− =

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• 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)

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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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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 ∫

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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

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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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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

∞ −

= = = ∑

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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

− =∞ ∞ ∞− + −− −

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

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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