The z-transform. S-domain. Introduction. Fourier Transform, Laplace Transform, DTFT, & z-transform. Region of Convergence.

The z-Transform

— Introduction

z Why do we study them?

„ A generalization of DTFT.

Some sequences that do not converge for DTFT have valid z-transforms.

„ Better notation (compared to FT) in analytical problems (complex variable theory)

„ Solving difference equation. Æ algebraic equation.

z Fourier Transform, Laplace Transform, DTFT, & z-Transform

Fourier Transform

∫

−^∞∞

Ω

=

−

ℑ { x ( t )} x ( t ) e

^{j t}

dt

To encompass a broader class of signals:

)}

( { )

( )

) (

(

x t e ^t e ^{j t}dt ≡

∫

x t e ^stdt ≡L x t

∫

−^∞∞

∞ −

∞

−

Ω

−

−σ

Laplace Transform

σ

j Ω

S-domain

Region of Convergence

∑

^∞

−∞

=

−

=

k

kT t k x t

x( ) [ ]δ( )

Similarly,

) ( ]}

Z{x[

] [ ]

[

) ( ] [ )}

( ] [ { } ) ( ] [ { )}

( {

z X n z

k x e

k x

dt e kT t k

x dt

e kT t k x kT

t k x L t x L

k

k k

skT

k

st k

st k

≡

≡

≡

=

−

=

−

=

−

=

∑

∑

∑ ∫

∑ ∫ ∑

∞

−∞

=

∞ −

−∞

=

−

∞

−∞

=

∞

∞

−

∞ −

∞

−

∞

−∞

=

∞ −

−∞

=

δ δ

δ

z-Transform

z

Eigenfunctions of discrete-time LTI systems

If x

[

n

]

= z₀ⁿ z₀ⁿ: some complex constant

n n

k

k k

k n k

z z H z z k h k

h z k

h k n x n

h n x n

y[ ]= [ ]∗ [ ]=

∑

[ − ] [ ]=

∑

₀ [ ]={

∑

^∞ [ ] ₀ } ₀ = ( ₀) ₀

−∞

=

∞ −

−∞

=

∞ −

−∞

=

Remark:

∑

^∞

−∞

=

−

=

=

n

jnw e

z

x n e

z

X ( )

jw

[ ]

DTFT can be viewed as a special case:

z = e

^jω

Discrete- Time LTI

z

n

H ( z ) z

ⁿ

w X(e^jw)

DTFT

⇔

n x[n]

2π

Ω X(jΩ)

2π/T

⇑

t x(t)

T

⇓

F.T.

⇒

— z-Transform

z (Two-sided) z-Transform (bilateral z-Transform)

Forward:

∑

^∞

−∞

=

− ≡

=

n

n X z

z n x n

x

Z{ [ ]} [ ] ( )

From DTFT viewpoint:

z re n

n j

x r F n x

Z{ [ ]}= { ⁻ [ ]} ω₌

(Or, DTFT is a special case of z-T when z=e^jω, unit circle.)

Inverse:

( ) [ ( ) ]

2 ] 1

[ ∫

^{1 1}

Γ

−

− ≡

= X z z dz Z X z

n j

x ⁿ

π

Note: The integration is evaluated along a counterclockwise circle on the complex z plane with a radius r. (A proof of this formula requires the complex variable theory.)

z Single-sided z-Transform (unilateral) – for causal sequences

∑

^∞

=

= − 0

] [ )

(

n

z n

n x z

X

z Region of Convergence (ROC)

The set of values of z for which the z-transform converges.

„ Uniform convergence

If z=re^jω (polar form), the z-transform converges uniformly if x[n]r⁻ⁿ is absolutely summable; that is,

∑

^∞ <∞

−∞

=

− n

r n

n x[ ] |

|

„ In general, if some value of z, say z1

z = , is in the ROC, then all values of z on the circle defined by

| z | = | z

₁

|

are also in the ROC. Î ROC is a “ring”.

„ If ROC contains the unit circle, |z| =1, then the FT of this sequence converges.

„ By its definition, X(z) is a Laurent series (complex variable) Î X(z) is an analytic function in its ROC

Î All its derivatives are continuous (in z) within its ROC.

„ DTFT v.s. z-Transform

-- = −∞<n<∞ n

n n

x sin ^c ,

]

1[ π

ω

Not absolutely summable; but square summable

Æ z-transform does not exist; DTFT (in m.s. sense) exists.

--

x

₂

[ n ] = cos ω

₀

n , − ∞ < n < ∞

Not absolutely summable; not square summable

Æ z-transform does not exist; “useful” DTFT (impulses) exists.

--

x

₃

[ n ] = a

ⁿ

u [ n ], | a | > 1 , − ∞ < n < ∞

Æ z-transform exist (a certain ROC); DTFT does not exists.

z Some Common Z-T Pairs

„ δ [n] ↔ 1^,

δ [ n − m ] ↔ z

^−m

, m > 0 , z > 0

^,

∞

<

>

↔

+ m z m z

n

]

^m

, 0 ,

δ [

„ , 1

1 ] 1

[ ₁ >

↔ − ₋ z

n z

u , , 1

1 ] 1 1

[ ₁ <

↔ −

−

−

− ₋ z

n z u

„ _{z a}

n az u

aⁿ >

↔ − ₋ ,

1 ] 1

[ ₁ ,

_{z a}

n az u

aⁿ <

↔ −

−

−

− ₋ ,

1 ] 1 1

[ ₁

„

[ ] [ ]

[

r

]

z r z ^{z r}

z n r

u n

r ⁿ >

+

−

↔ − ₋ ⁻ ₋ ,

cos 2 1

cos ] 1

[

cos _{1 2 2}

0

1 0

0 ω

ω ω

[ ] [ ]

[

r

]

z r z ^{z r}

z n r

u n

r ⁿ >

+

−

↔ − ₋ ⁻ ₋ ,

sin 2 1

sin ] 1

[

sin _{1 2 2}

0

1 0

0 ω

ω ω

— Properties of ROC for z-Transform

z Rational functions

) (

) ) (

( Q z

z z P

X =

Poles – Roots of the denominator; the z such that X

(

z

)

→∞ Zeros – Roots of the numerator; the z such that X

(

z

)

=

0

„ Properties of ROC

(1) The ROC is a ring or disk in the z-plane centered at the origin.

(2) The F.T. of

x [n ]

converges absolutely ⇔ its ROC includes the unit circle.

(3) The ROC cannot contain any poles.

(4) If x[n] is finite-duration, then the ROC is the entire z-plane except possibly

z = 0

or

∞ z =

^.

(5) If x[n] is right-sided, the ROC, if exists, must be of the form

z > r

_max except possi- bly

z = ∞

^{, where}

r

_maxis the magnitude of the largest pole.

(6) If x[n] is left-sided, the ROC, if exists, must be of the form

z < r

_min except possi- bly

z = 0

^{, where}

r

_minis the magnitude of the smallest pole.

(7) If x[n] is two-sided, the ROC must be of the form r₁ < z <r₂ if exists, where

r

₁ ^and

r

2are the magnitudes of the interior and exterior poles.

(8) The ROC must be a connected region.

In general, if

X (z )

is rational, its inverse has the following form (assuming N poles:

{

d_k

}

⁾

∑

=

= ^N

k

n k k d A n

x

1

) ( ]

[ . For a right-sided sequence, it means

n ≥ N

₁^{, where}

N

₁ is the first nonzero sample.

The nth term in the z-transform is

∑

=

−

− = ^N

k

n k k

n A d r

r n x

1

1) ( ]

[ ^.

This sequence converges if

∑

^∞ <∞

=

−

1

|

|

¹

N n

n kr

d for every pole

k = 1 , K , N

. In order to

be so,

| r | > | d

_k

|, k = 1 , K , N

^.

— Pole Location and Time-Domain Behavior for Causal Signals

Reference: Digital Signal Processing by Proakis & Manolakis

— The Inverse z-Transform

Inverse formula:

∫

Γ

= X z z − dz n j

x ( ) ^{n 1}

2 ] 1

[ π

This formula can be proved using Cauchy integral theorem (complex variable theory).

z Methods of evaluating the inverse z-transform (1) Table lookup or inspection

(2) Partial fraction expansion (3) Power series expansion

z Inspection (transform pairs in the table) – memorized them z Partial Fraction Expansion

N N

M M

z a z

a a

z b z

b z b

X _{− −}

−

−

+ + +

+ +

= +

L L

1 1 0

1 1

) 0

( Æ

) (

) ) (

(

0 0

N N

M

M M

N

a z

a z

b z

b z z

X + +

+

= +

L

L

Hence, it has M zeros (roots of

∑

^bkzM−k ), N poles (roots of

∑

^akzN−k ), and (M-N) poles at zero if M>N (or (N-M) zeros at zero if N>M).

Æ

) 1

( ) 1

(

) 1

( ) 1

) (

( _{1 1}

1 0

1 1

1

0 − −

−

−

−

−

−

= −

z d z

d a

z c z

c z b

X

N M

L

L ;

c

_k, nonzero zeros;

d

_k, nonzero poles.

„ Case 1: M <N, strictly proper Simple (single) poles:

) 1

( )

1 ( ) 1

) (

( _{1 1}

2 2 1

1

1 − − + + − −

+ −

= −

z d A z

d A z

d z A

X

N

L N

where

dk

z k

k d z X z

A =(1− ⁻¹) ( )| ₌

Multiple poles: Assume

d

_i is the sth order pole. (Repeated s times)

s i

s i

i N

i k

k k

k

z d C z

d C z

d C z

d z A

X( ) (1 ) (1 ¹ ₁) (1 ² ₁)₂ (1 ₁)

, 1

1 − − −

≠

= − + + −

+ − + −

=

∑

− L

single-pole terms multiple-pole terms where

1

)]

( ) 1 ) [(

( )!

(

1 ₁

= −

−

−

−

− ⎭⎬⎫

⎩⎨

⎧ −

−

= −

di w s

m i s

m s m s i

m dw X w

dw d d

m C s

„ Case 2: M ≥ N

∑ ∑ ∑

= −

≠

= −

−

=

−

+ − + −

= ^s

m

m i

m N

i k

k k

k N

M

r r

r dz

C z

d z A

B z

X

1

1 ,

1

1

0 (1 ) (1 )

) (

z Power Series Expansion

∑

^∞

−∞

=

= − n

z n

n x z

X( ) [ ]

„ Case 1: Right-sided sequence, ROC:

z > r

_max

It is expanded in powers of

z

^−1.

Ex. , | | | |

1 ) 1

( ₁ z a

z az

X >

= − ₋

„ Case 2: Left-sided sequence, ROC:

z < r

_min

It is expanded in powers of

z

^.

Ex. _{, | | | |}

1 ) 1

( ₁ z a

z az

X <

= − ₋

„ Case 3: Two-sided sequence, ROC:

2

1 z r

r < <

X(z)= X₊(z) + X₋(z)

converges for

| z | > r

₁ converges for

| z | < r

₂

Æ x[n]= x₊[n] + x₋[n]

causal sequence anti-causal sequence

— z-Transform Properties

If x[n]↔ X[z] and y[n]↔Y[z], ROC:

R

_X

, R

_Y

„ Linearity:

ax [ n ] + by [ n ] ↔ aX ( z ) + bY ( z )

ROC:

R ' ⊃ R

_X

∩ R

_Y -- At least as large as their intersection; larger if pole/zero can- cellation occurs

„ Time Shifting: x[n−n₀]↔z⁻ⁿ⁰X(z) ROC: ^{R'= R}X ^±

{

^0or∞

}

„ Multiplication by an exponential seqence:

aⁿx[n]↔ X(z a) ROC:

RX

a

R'= -- expands or contracts

„ Differentiation of X(z):

( )

dz z zdX n

nx[ ]↔− , ROC:

R ' = R

_X

„ Conjugation of a complex sequence: x*[n]↔ X*(z*), ROC:

R ' = R

_X

„ Time reversal: x*[−n]↔X*(1/z*),

ROC: R

'

=

1 /

R_X (Meaning: If R_X

:

r_R <

|

z

|

<r_L^{, then}

R ' : 1 / r

_L

< | z | < 1 / r

_R^.

Corollary:x[−n]↔X(1/z)

„ Convolution: x

[

n

]

∗y

[

n

]

↔ X

(

z

)

Y

(

z

)

ROC: R'⊃ R_X ∩R_Y (=, if no pole/zero cancellation)

„ Initial Value Theorem:

If x[n]=0, n<0, then x[0] limX(z)

z→∞

=

„ Final Value Theorem:

If (1) x[n]=0, n<0, and

(2) all singularities of

( 1 − z

⁻¹

) X ( z )

are inside the unit circle, then [ ] lim(1 ¹) ( )

1

z X z x

z

−

→ −

=

∞

Remarks: (1) If all poles of X(z) are inside unit circle, x

[

n

]

→

0 as

n→∞ (2) If there are multiple poles at “1”, x

[

n

]

→∞

as

n→∞ (3) If poles are on the unit circle but not at “1”, x[n]≈cosω₀n

<Supplementary>

z-Transform Solutions of Linear Difference Equations Use single-sided z-transform:

Z{y[n−1]}=z⁻¹Y(z)+ y[−1] ] 2 [ ] 1 [ )

( ]}

2 [

{y n− = z⁻²Y z +z⁻¹y− +y −

Z

] 3 [ ] 2 [ ] 1 [ )

( ]}

3 [

{yn− =z⁻³Y z +z⁻²y − +z⁻¹y− + y− Z

For causal signals, their single-sided z-transforms are identical to their two-sided z-transforms.

Ex., Find y[n] of the difference eqn.

] [ ] 1 [ 5 . 0 ]

[n y n x n

y − − = with x[n]=1,n≥0, and y[−1]=1 (Sol) Take the single-sided z-transform of the above eqn.

Æ

1 1

1 ) 1 ( ]}

1 [ ) ( { 5 . 0 )

( ⁻ ₋

= −

=

− +

− z Y z y X z z

z Y

Æ

) 1 )(

5 . 0 1 (

1 5

. 0 1

5 . 0

1 5 1 . 5 0

. 0 1 ) 1 (

1 1

1

1 1

−

−

−

−

−

− + −

= −

⎭⎬

⎫

⎩⎨

⎧ + −

⎭⎬

⎫

⎩⎨

⎧

= −

z z

z

z z z

Y

Æ

1 1 1 0.5

5 . 0 1

) 2

( _{− −}

− −

= −

z z z

Y

Take the inverse z-transform

Æ y

[

n

]

=

2

−

0 . 5 ( 0 . 5 )

^n,

n ≥ 0