11

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

= + = = = + =

\

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 = =System

h[n]

input output

[ ] [ ] [ ] [ ] ( )0 0 0 0 0 0n n k n k nk 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= =

( ) [ ] kk

H z h k z =

=

( )0H z0nz

23

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 kk

x n c z= System

h[n]

input output[ ] ( ) nk k k

ky n c H z z=

( ) [ ] kk

H z h k z =

=

4

Bilateral z TransformThe 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

[ ]{ }( ) ( ) [ ] , , 0jnn

Z x n z X z x n z z r e r =

= = =

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

Z x n z x n r e r x n r =

= = F

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

35

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=

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

Z e x n e x n z x n e z X e z = =

= = =

[ ][ ]

00 0

00

, n

nu

z zz x n X ROCz z

zz x n Xz

[ ] ( )0 0 , j n je x n X e z z ROC

10

19

4. Time reversal

Proof.

[ ]{ } [ ] [ ] 1 1mm nnn m

Z x n x n z x m Xz z

== =

= = =

[ ] ( )1 1, x n X z ROCz

20

5. Differentiation in time

Proof. direct application of the definitions and previous properties. Homework. Prove these properties.

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

1

1

1 1 ;

1 1 1u

x n x n z X z z ROC

x n x n z X z x

11

21

6. Addition in time

Proof.

[ ] ( ) [ ]1

1 ; 11

unk

k

X z x kx k z

z

==

+ >

[ ] ( ) ( )1 ; 11n

k

X z zx k X z z ROC ROCzz

=

=

[ ] [ ] [ ] ( ) ( ) ( ) [ ] [ ] [ ]111 1 1 ; 1uk

x n y n y n X z z Y z y y x k=

= = = [ ] [ ] [ ] [ ] [ ] ( ) ( ) ( )11 1n

ky n x k x n y n y n X z z Y z

== = =

22

7. Differentiation in z domain

Proof.

By direct application of definitions.

Homework. Prove it.

[ ] ( )

[ ] ( );

u

dX znx n z z ROC

dzdX z

nx n zdz

12

23

8. Complex conjugation in time domain

Proof.

By direct application of definitions.

Homework. Prove it.

[ ] ( )[ ] ( )

;

u

x n X z z ROC

x n X z

24

9. Time convolution (convolution theorem)

Proof.

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

[ ] [ ] ( ) ( )

n n

n n k

n kk

n kn k m m k

n k

Z x n y n x n y n z x k y n k z

x k z y n k z

y m z x k z Y z X z

= = = = =

= = =

= =

=

= =

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

,

; at leastx y

u u

x n y n

x n y n X z Y z z ROC ROC

x n y n X z Y z

^

13

25

10. Product theorem or convolution theorem in the complex domain

Proof.

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

1 , 21 ,

2 u u

z dux n y n X u Y ROCj u u

z dux n y n X u Y ROCj u u

vv

: ; : and

:

y y y yx x x xx y

y yx x

zROC R z R ROC R z R R u R R R

u

ROC R R z R R

+ + + +

+ +

< < < < < < < = < =

[ ] ( ) [ ] ( ) [ ]0.5 : 4 0.5 1 4 0.25n nz y n n n< < =

Causal signal y[n]

38

3. Power series expansion of function Y (z)

Expansion of function Y(z) into a power series

Example #a

( )

[ ]

1

1 1 2

0

, : 0

1 1 1 111! 2! ! !1!

n n

n

z

z

Y z e ROC z

e z z z zn n

y nn

=

= >= + + + + + =

=

20

39

Example #b

( )

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

2

00 1

, : 0,

1 1 1 111! 2! ! !

1 1 1! !

1 11! 1 !

m m

m

n n

n n

n

z

z

Y z e ROC z z

e z z z zm m

z zn n

y n n n nn n

=

= =

= = + + + + + =

= = + = =

40

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

[ ] ( )[ ] ( ) [ ]

1

1 1

1

1

ln 1 , :

1 1; 1

initial value theorem:

0 lim ln 1 0

1 1

n nn nn

n

zn

n

Y z az ROC z a

a aY z z y n n

n n

y az

y n a nn

+ + =

= + > = =

= + ==

causal signal

Example #c

21

41

Example #d

( ) [ ] [ ]11 , 1 nY z z a y n a naz= > =

42

Example #e, same transform as #d, different ROC

( )

( ) ( ) [ ] [ ]1

11

1

1 , 1

anticausal signal, z transform contains only powers of z

1m n n n

m n

zY z z az aaz

Y z a z a z y n a n

= =

= =

22

43

Discrete LTI Systems Described by Linear Constant Coefficient Difference Equations

[ ] [ ] 00 0

; 0N M

k kk k

a y n k b x n k a= =

= ( ) ( ) ( ) ( )( )00 0

0

transfer function

Mk

N M kk k k

k k Nkk k

kk

b z N za z Y z b z X z H z

D za z

=

= ==

= = =

Transfer function of a discrete LTI system : rational function in z or z-1. Zeros : roots of numerator N(z); Poles : roots of denominator D(z)

44

Causal and stable system: poles inside unit circle |z|=1

The initial value theorem can be applied:

The degree of the numerator of the transfer function of a

causal and stable system is smaller or equal with the

degree of its denominator, M N

1pkz Partial impulse response increases in timeInstability.

( )sin p pA n +

1pr =

24

47

#2. A pair of double complex conjugate poles

Contribution of the two poles in the impulse response :

( ) ( ) ( )1 1 2 22 21 1 1 1 and

1 1 1 1

j jp p p p

j j j jp p p p

p p

p p p p

z r e z r e

a a a aH z

r e z r e z r e z r e z

= == + + + + +

( ) ( ) [ ]1 2sin sinn np p p p p pA r n A nr n n + + + 1pr < Partial impulse response decreases in time

No instability from these poles. 1pr > Partial impulse response increases in time

Instability.

48

difference equation, non-zero initial conditions unilateral Z transform

Causal input signal, x[-n]=0 for n >0:

Response of a Discrete LTI System Described by a Difference Equation

[ ] [ ] [ ]{ } [ ]{ }00 0 0 0

, 0 N M N M

k k k u k uk k k k

a y n k b x n k a a Z y n k b Z x n k= = = =

= =

( ) [ ] ( )0 1 0

N k Mk n k

k u k uk n k

a z Y z y n z b z X z = = =

+ =

( ) [ ] ( ) [ ]1 10 0

N k M kk n k n

u uk kn nk k

a z Y z y n z b z X z x n z = == =

+ = +

25

49

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

[ ]

[ ] [ ] ( ) [ ] .1 , 1 ; 1.

11

11

11

11

11

.1 ; 1

1

,01 , , 1

Example

0

0

0

000

0

0

0

0

111

111

111

11

>=+==

+

++

zanea

keea

kayany

azay

azeaka

zeeake

azay

azzekzY

zze

kzyzYazzY

ynkenxnxnayny

j

nj

j

nn

jjj

j

ju

juu

nj

50

First Order Systems0pole/zero plot 0; 0.5pz z= = [ ] [ ] [ ]

( )[ ] [ ]

1

1

; 1

, 1 np

y n ay n kx nk kzH z z ROCaz z a

z a z a h n ka n

= = = > = < =

( ) ( ) ( )1

0 1, max. freq. response for 0, 1 0, max. for ,

Positive low-pass filterNegative high-pass filter

j jOAH e ePA PA

aa

aa

= =

< < = < < =

26

51

Second Order Systems[ ] [ ] [ ] [ ] ( ) 21 2 1 2 2

1 2 1 2

1 2 1

k kzy n a y n a y n kx n H za z a z z a z a

+ + = = =+ + + +

1,2

21 2

21 2

2 21 2 1

2

1 2

4 complex conjugated poles

4 real poles . The system is considered causal and stable system.

41 magnitude

2The conditions imposed on ,

jp

a az e

a a

a a aa

a a

<

=

+ = =

> >

1.

2.

21 1 2

0 1,2

40 ,

2pa a a

z z = =

52

( ) ( ).1 : system theof responsefrequency Thepole. real double single a of existence the toscorrespond parabola The

212_____

2_____

1

= jeAPAP

kH

Magnitude: even; Phase: odd

27

53

Transfer Function of Equivalent System for Serial or Parallel Interconnections of two

discrete LTI Systems

[ ] [ ] [ ]( ) ( ) ( ).zHzHzH

nhnhnh

e

e

21

21 ; +=

+=[ ] [ ] [ ]( ) ( ) ( ).zHzHzH

nhnhnh

e

e

21

21 ; =

=

54

Digital Filters Implementation Forms Obtained Using the z

Transform[ ] [ ] [ ] [ ] [ ]. ,1 ,

100

00====

==== Nk

kM

kk

M

kk

N

kk knyaknxbnyNMaknxbknya

28

55

Direct form I (N+M adders). First form of implementation using z transform.

Each adder has 2 inputs. One adder with 2N inputs (M=N, a0=1).

56

Direct form II (2N adders M=N). Second form of implementation using z transform.

Each adder has 2 inputs. Two adders with N inputs each (M=N, a0=1).