Description digital signals and systems in transform (Z, DFT) domain

(1)

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

(2)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 2

Description digital signals and systems in transform (Z, DFT) domain

Digitális jelek és rendszerek leírása és analízise transzformált tartományokban

János Levendovszky, András Oláh, Dávid Tisza, Kálmán Tornai, Gergely Treplán

Digitális- neurális-, és kiloprocesszoros architektúrákon alapuló jelfeldolgozás

Digital- and Neural Based Signal Processing &

Kiloprocessor Arrays

(3)

• Introduction

• Review of basic definitions and operations on discrete time signals

• Review of LTI system definition by difference equation

• Motivation of using the z-transform for analysis of LTI systems.

• Introduction of z-transform

• Definition

• RoC

• Properties of z-transform

• Poles-zeros (review from complex calculus)

• Inverse z-transform

• z-transform of elementary discrete time signals

(4)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 4

• z-transform of elementary operations on discrete time signals

• z-transform of convolution

• LTI systems in z-domain

• Transfer function

• Poles, zeroes of the transfer function

• Z-transform and Fourier transform

• Polynomial manipulation review

• Polynomial long division

• Partial fraction expansion

• BIBO stability

(5)

• Convolution properties in the z-domain

• Serial combination of LTI systems

• Parallel combination of LTI systems

• Outline of special filter types

• Linear phase filters

• Minimum phase filters

• All pass filters

• Examples of z-transform in system analysis

(6)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 6

Every LTI system can be described in the time domain by its system equation which is a discrete time difference equation.

This equation can be analyzed by well known mathematical analytical methods in the time domain.

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 0 0 0

0

1 0 1

· ·

without the loss of generality we usually assume 1.

· 1 · · · 1 ·

· 1 · · · 1

N M

N M N M

i j

k k i j

D D

N M

a y n k b x n k a S y n b S x n

a

y n a y n a y n N b x n b x n b x n M

y n a y n a y n N b x n b x n b

= = = =

− = − =

=

+ − + + − = + − + + −

= − − − − − + + − + +

∑ ∑ ∑ ∑

⋯ ⋯

⋯ ⋯ ^{·x n}

(

⁻ ^M

)

(7)

• Certain transformations enable us to deal with the recurrence relation (difference equation) in the transformed domain as a simple algebraic equation.

• After solving the algebraic equations with simple mathematical tools we can transform back them into the to get the time domain response of the system

Transform

time domain transformed domain

difference equation algebraic equation

→

(8)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 8

• LTI systems can be fully characterized in the transformed domain by numbers (poles and zeroes).

• From these numbers we can simply tell important features of the system, e.g. stability, structural behavior, degree of the system, frequency characteristics.

• In continuous time we use integral transforms in discrete time we use power series, because these transforms have nice properties for derivation and convolution.

• LTI systems in the transformed domain will be represented by division of polynomials.

(9)

• Note that at the time domain analysis we met with geometric sequences (e.g. homogenous solution) and when we apply the Laurent series the basic review of the geometric series will come in handy. We have the following closed forms for the series:

0

1 1

1 ,

· 1

·

, 1

1

, 1

1

n

k

n k

k

c r r

r

c r

r c c r

c r

c r r

r

+

∞

=

∞ −

= −

−

= <

−

= −

≠

>

∑

(10)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 10

• The z-transform (bilateral) of a discrete time signal is defined as an infinite complex power series (Laurent series):

• For compactness we use the notation

for the z-transform and for the inverse z-transform

.

( ) ( )

^· ^,

n

x n z n z X z

=−

∞ −

∞

∈

=

∑

^ℂ

( ) { } ( )

( )

¹

{ ( ) } ^{( )}

¹

^{( )}

Z Z

X z Z x n

x n X z

x n Z⁻ X z ⁻

=  ←→

= 

(11)

• Let’s transform the following series into the z-domain:

Note that the series is convergent only for a region of z values.

This example is convergent for

( )

( ) ( )

( )

⁰ ¹ ² ³

2 3

· 5.2

0 5.2 3.1 15.2 2.4 0 ,

3.1 15.2 2.4

1 1

3.1 15. 1

5.2 2 2.4

n n

x n

X z x n z

z z z

z z

X z

z z

z

∞ −

∞

− −

↑

=

−

 

=  − 

=

+

… …

∈

= − +

= + − +

∑

^ℂ

, 0

z∈ℂ z ≠

(12)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 12

• Let’s transform the following series into the z-domain:

This example is convergent for

• We have to define Region of Convergence.

( )

( ) ( )

( )

³ ² ¹ ⁰

0 5.2 3.1 15.2 2.4 0 ,

3.1 15.2 2.4

· 5.2

n

x n

X z x n z

z z

z

X z z z

↑

=

∞ −

∞

−

 

=  − 

=

… …

∈

+

= + −

∑

^ℂ

,

z∈ℂ z ≠ ∞

(13)

• The region of convergence is defined as follows:

all those numbers for which the series are convergent.

• Example 3:

( )

RoC= : ·

n

z x n z n

∞ −

∞

=−

∈ ∞

 

<

 

 ^ℂ

∑



( )

( ) ( )

( )

{ { } }

2 1 0 1 2

0 5.2 3.1 15.2 2.4 4.6 0 ,

3.1 15.2 2.4 4.6

RoC : \

· 5.2

0,

n n

z X

x n

X z x n z

z z z

z z

z z z

↑

=−

−

∞ −

∞

−

 

=  − 

=

+ − + +

=

… …

∈

=

∈ ∞

∑

^ℂ

ℂ

(14)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 14

• We can decompose a signal to an entrant and to a leaving part:

The z-transform of a signal can be also decomposed to two parts:

( ) ( ) ( )

( ) ( )

⁰⁰

( ) ( )

⁰

,

0 0

0 ,

E L

L E

x n x n x n

x n n n

x n x n

x n n

n ≥ ≥

= +

<



 <

=  = 

 

( ) ( ( ) ( ) )

¹

^{( )} ^{( )}

0

· ⁿ · ⁿ · ⁿ

L L E

n E

n n

z z z

X z x n x n x n x n

∞ ∞

− − −

=−∞ =− =

−

∞

=

∑

+ =

∑

+

∑

(15)

• The two parts have two different region of convergence

( ) ( ) ( ) ( ) ( )

{ }

² ¹

1 1

0 0

1

0 1

if if

2

0

RoC : 1 is an annulus for the general

· · · ·

case

· · · ·

L L

n n n n

L L

n n

E E

n n n n

E E

n n n n

r r r r

X z x n x n x n x n

X

z z z z

r r r r

z x n x n x n x n

z r z r

∞ − ∞ −

− −

∞ ∞

∞ ∞ ∞

− − −

∞

− −

=− = =− =

−

=− = = =

< < <∞ >

= +

= −

= > >

≤ +

≤ + +

∑ ∑ ∑ ∑

(16)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 16

{ }

1 2

2

RoC : is a region outside of a circle for entrant signals RoC : is a region inside of a circle for leaving signals for the general case the RoC is the intersection of the above two

RoC :

z z r z r z

z r z

= >

=

{

> > r¹

}

is an annulus for the general case

(17)

( ) ( ) ( ) ( )

( ) ( ) { }

{ }

( )

1 1 2 2 1 1 2 2 1 2

1

2 1

1

2 1

property time domain Z-domain RoC

linearity RoC RoC

RoC \ 0 , 0 time shift

RoC \ , 0

scaling in the z-domain

time reversal 1/ 1 /

Differentiation in

· ·

z

k

n

x n x n X z X z

x n k z X z k

k

a X a z a r z a r

x n X z z

n

r r

x

α α α α

−

+ + ∩

− >

<

∞

> >

− > >

( ) ( )

( )

( ) ( ) ( ) ( )

1

-domain RoC

accumulation ( ) 1

1

Convolution RoC

·

· C

·

Ro

n

k

n z

x k X z

z x n x n

x n d X z

dz

X z X z

−∞ −

=

−

∗ ∩

∑

(18)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 18

( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( )

0 0

0

0 0

0

0, 1

then is a zero of order

0, 1

t

Let's assume that is analytic

zero: if 0 and we can write ,

analytic,

pole: if we can write 1 ,

analy hen is

tic a

,

n

f z

f z f z z z f z

f z f z

f z h z

z z n

h z h z

z n

n z

≠ ∃ ≥

= = −

= −

ɶ

ɶ ɶ

pole of order n

(19)

( )

{ }

( ) ( ) ( )

( ( )) ¹ ⁰

1 0 2 1 2

0

Laurent series:

if is analytic in an annulus : , then

can be expanded into a pow

1 , where is a closed path encircling , and 2

er series _k ^k, where

k

k k

C

f z D z r z z r r r

f z f

X z dz C

z c z z

z

c j z

π z

∞

= ∞

+

−

= < − < <

= −

=

∫

∑

( )

{

( )

}

^{( )}

( )

1 1

in

inverse z-transform : 1

2

is a closed path in the complex plane encircling the origin, all the poles of and it must lie eintirely within the RoC.

n

j C

D

x n Z X z X z z dz

C

X z

π

− −

= =

∫

(20)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 20

• In an LTI system we expect system exponential type responses (see time domain analysis homogenous solution) and excitation like answers (see time domain analysis particular part).

• So usually it is enough if we have a table of the most common functions z-transforms.

• And then we need to manipulate them into correct form and reverse them into the time domain from a table. Thus we don’t need to evaluate the complex closed path integral.

(21)

• We will present the z-transform, the visualization and the pole- zero plot of some elementary functions:

• Pole-zero plot of an LTI system ( )

( )

( ) ( )

( )

( ) ( )

( )

( ) ( )

( )

2

2 2

0 0

0 0 0

0 2

time z-domain RoC

1 1 1

1 1

1 1 cos

cos 1

1 2 cos 1

1 1 sin

1 sin 1

2

·

cos 1

·

n

n z z

z z

u n z z a u n z a z a

z z z

u n z

z z

u n z z

z z

n u n

n n δ

ω ω

ω ω ω

ω

− >

> >

− −

− − − − < − − + >

− <

− >

− +

∈ℂ

(22)

• Kronecker delta or unit sample

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 22

( ) ( )

{ }

( ) ^{ ^}

( ) ( ) ⁰

1, if 0

0, otherwise 1, RoC= :

1

· ⁿ

n

x n Z z z

X z n z

n n n

z

δ δ

∞ δ

∞

=

−

=  = =

=  ∈

= = =



∑

ℂ

(23)

• Unit step function

pole-zero plot:

( ) ( )

{ }

( )

{ }

( ) ( )

{ }

geom.series 0 assumption

1

1, if 0 0

1 , ROC= : 1

1 1 1

1 RoC=geometric series closed

, othe

form only valid if 1 ROC r

·

= : 1

wise

n

n n

n

n z n

x n u Z z z

z z u n

z z

X z u n

z

z z z

=− =

−

∞ ∞

− −

∞ −

−

= = = >

− −

= =

≥

= 

= −

< ⇒



>

∑ ∑

( )

{

( )

}

{

( )

}

poles and zeroes of a conplex function pole :

zero : 0

X z z X z

z X z

≅ = ±∞

≅ =

(24)

• Plot of function

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 24

( ) ₁ ¹ ¹ ^z ₁^, ^{red line=}

{

^{( ) :} ¹

}

X z X z z

z⁻ z

= = =

− −

(25)

(note that outside the RoC the function is not determined)

( ) ( )

{ }

( ) ₁ ¹ ¹ ^z ₁^, ^RoC=

{

^: ¹

}

x n u Z z z

u z

n n z₋

= = = >

− −

(26)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 26

( ) ( ¹)

{

( ¹)

}

^z ₁^, ^RoC=

{

^: ¹

}

x n u Z u z z

n n z

= − − =

− − − − <

−

(27)

• Plot of function ( ) ( )

{

( )

}

₁ ¹ ^, ^RoC=

{

^: ¹

}

using the time reversal property

x n u n Z u n z z

= = z <

− − −

(28)

• Unit ramp function

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 28

( ) ( )

{

( )

}

( )

⁽ ⁾

^{ ^}

{ }

( )

{

^{( )}

}

( ) ( )

( )

{

( )

} { }

^{( )}

( )

1

2 2

1

2

1 1 2

1

, RoC= : 1

1 1 we use the property of differentiation in

, if 0 0, oth

z-domain:

1

1 1

1

erwise

·

r r

r

n n

z d Z u n n x n dz

n n u n

d d

n u n z Z u n z z

dz dz

z z

x n u Z u z z

z z

Z u

X z Z z

z z

X z z

−

− −

−

≥

= 



− =

=

 

   

= − = −  = −  

= = = >

− −

= −

−

=

 − 

(

− ¹

)

² ^RoC=

^{

z z^: 1 same as the unit step function.

^}

z⁻

>

(29)

• Pole-zero plot of the unit ramp function in the z-domain

( ) ( )

( )

2

1 2 2

1

poles and zeroes of the function:

1

1 multiplicative roots of 1 1

note: from time domain analysis 0 root of

X z z

z

p z

p

z z

= −

= 

−

= 

=

resonance

(30)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 30

( ) ( )² , red line=

{

( ): 1

}

1

X z z X z z

= z =

−

(31)

• Plot of function _{( )} _{( )}

{

_{( )}

}

₍ ₎² ^, ^RoC=

{

^: ¹

}

r r 1

x n u Z u z z z

z

n n

= = >

−

(32)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 32

( ) ( )

{ }

( ) ₍ ₎² ^, ^RoC=

{

^: ¹

}

1

r 1

x n u Z z z z

z

n x n

−

= − − = <

−

(33)

• Exponential function

( )

ⁿ^, ^if ^,

( )

^, ^if ^,

( ) ( )

^· ^j ⁿ ⁿ^· ^{j n}

x n = a a∈ →ℝ x n ∈ℝ a∈ℂ → x n = r e ^ϕ = r e ^ϕ

(34)

• z-transform of the exponential function

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 34

( ) ( ) { } ( ) { }

( ) ( )

{ ( ) } ^{( )} ^{( )} ( ) ( )

{ } ( ) { }

{ ( ) }

( ) ^{ ^}

1

1 1

1

1 1

, 1 , RoC= :

1

using the scaling in the z domain proper

·

1

ty

, : 1

1

RoC

:

=

RoC=

, 1

n

n n n n

n n

n

z

x n X a z

Z a x n a x n z x n a z X a z

Z u n z z

z

x n a u n Z x n z z z a

z a a

a

Z a u n z z a

z a a z

−

∞ − ∞ − − −

=−∞ =−∞

−

− −

= = =

= >

−

= = >

= −

−

= >

−

=

∑ ∑

⇌

(35)

• Pole-zero plot of the exponential function in the z-domain

( )

1 1

poles and zeroes of the function:

root of 0 root of X z z

z a

p a z a

z z

= −

=

(36)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 36

( ) z , red line=

{

( ):

}

, 0.5

X z X z z a a

z a

= = =

−

(37)

• Plot of function ^{x n}_{( )}⁼ ^{a u}ⁿ _{( )}ⁿ ^Z

{ }

^{x n}_{( )} ⁼ _z₋^z _a ^, ^RoC=

{

^{z z}^: ^> ^a

}

(38)

• z-transform of sine and cosine functions

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 38

( ) ( ) ( ) { }

{ } ( ) _{( )} ^{( )} _{( )} ^{( )}

( ) ( ) ( ) { }

{ } ( ) _{( )} ^{( )} ^{( )} _{( )}

0

0 0

1 2

1 2 2

0

0 0

1

1 2 2

cos , RoC= : 1

1 cos cos

1 2 cos 2 cos 1

sin , RoC= : 1

sin sin

1 2 cos 2 cos 1

x n u n z z

z z z

Z x n

z z z z

x n u n z z

z z

Z x n

z z z

n

z ω

ω ω

ω

ω ω

−

− −

−

− −

= >

− −

= =

− + − +

= >

= =

− + − +

(39)

• Poles and zeroes sine and cosine functions in the z-domain

( ) ( ) ( )

( ) ( )

( ) {

0

1 0

0

0 0

0

0 0

2

2 2 1

2 1

poles and zeroes:

cos sin

2 cos 1 0

cos sin

cos 0 0

cos

sin 0 0

p i

z z

p i

z z z

z

z z

ω ω

ω ω ω

ω ω

ω

 = +

− + = ⇒ 

= −



=

− = ⇒ 

 =

= ⇒ =

(40)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 40

( ) ( )

( )⁰

{

( )

}

⁰

0 2

cos , red line= : 1 ,

2 c / 4

os 1

z z

X z X z z

z z

ω ω π

ω

= − = =

− +

(41)

• Plot of function _{( )} _{( ) ( )}

{ }

_{( )} _{( )}^{( )}

{ }

2

0 0

cos cos ,

2 cos 1

RoC= : 1 , /4

z z

x n n u Z

n x n z

z z z ω ω

ω ω π

= = −

−

= +

>

(42)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 42

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) { } ( ) ^{( )} ^{( )}

( ) ( ) ( )

1

name time z-domain

addition ( ) ( )

multiplication with constant ( )

multiplication ( )

time shift ( )

1

·

accumulation ( )

·

1

C

k k

n k

y n g n h n Y z G z H z

y n a g n Y z aG z

y n g n Y z G v H z v dv

v y n S g n g n k Y z z G z

y n g

n

k Y z G

z h

z

−

=−∞ −

= + = +

= =

= =   

 

= = − =

= =

−

∫

∑

( ) ( ) ( ) ( ) ( )

DT convolution y n = g n( )∗h n Y z = G z H z·

(43)

( )

{ } ⁽ ⁾ ^{( )} ^{( )} ^{( )}

( ) ( )

{

¹ ²

}

¹

^{( ) (}

²

⁾

Addition, multiplication with constant comes from the linearity property.

Time shift operation:

DT Convolution:

·

n m k k m k

m n k

n m m

k

Z x n k z x n k z x m z z x m z X z

Z x n x n Z x k x n k

− − − − − −

=− = − =− =−

=

∞ ∞ ∞

∞

−∞

− = − = = =

 

∗ =  − 



∑ ∑ ∑

∑ ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

1 2

1 2 1 2

1 2

·

· ·

·

n

n k

n m

k k

k

n m

z x k x n k

x k x n k z x k x m z

X z X z

z

−

=− =−

− −

=− =−

∞ ∞

∞ ∞ ∞ ∞

=−

−

∞ ∞ ∞ =−∞

 = − =



= − = =

=

∑ ∑

∑ ∑ ∑ ∑

(44)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 44

( )

( ) ( ( ) ( ) ( ) )

( ) ( ) ^{( )} ^{( )}

( )

1

1 0

A ccum ulation

( ) using the tim e shift property

( ) 1

1 1

1

alternatively accum ulation can be w ritten as a convoluti

·

on

y (

n

k

n

n n

n k n

k k

y n x k

Y z x k z x n x n z

X z z X z z

z

n x

x

X

z z

k

∞

∞ ∞

∞ ∞ ∞

−∞ ∞

= −

− −

= − = − = −

− −

= −

+

=

= = + − + =

= + + = =

∞

+ −

−

=

∑

∑ ∑ ∑

∑

⋯

( ) ( ) ( )

( ) ( ) ( ) ( )

₁

) ( ) ( ) ( )

· 1 1

n n

k k k

x k u n k x k u n k x n u n

Y z z z

U X z z X

∞

= − = −∞ = −∞

−

∞

= − = − =

=

∗

= −

∑ ∑ ∑

(45)

A LTI system in the time domain performs a discrete time convolution on its input with its impulse response function

An LTI system is fully characterized by its impulse response function (h(n)).

If we examine the same system in the transformed domain, we can use the convolution to multiplication property of the z- transform:

LTI system

( )

x n ^{y n}

( )

( ) ( ) ( )

y n = h n ∗ x n

Input, stimulus

Output,

system response

( ) ( ) ( ) ( )

1 2 ^Z 1 · 2

x n ∗ x n ↽ ⇀ X z X z

(46)

11/27/2011. TÁMOP – 4.1.2-08/2/A/KMR-2009-0006 46

A LTI system in the z-domain performs a multiplication on its input’s z-transform and the system’s transfer function

An LTI system is fully characterized by its transfer function H(z).

LTI system

( )

X z ^{Y z}

( )

( ) ( ) ( )

^·

Y n = H z X z

Input, stimulus

Output,

system response

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

if 1

Z

Z Z

x n X z

y n Y z H z X z x n

H z

h n H z

n y n h n

δ

 =

= 

 =

↽ ⇀

(47)

• Remember the linear constant-coefficient difference equation which described an arbitrary LTI system:

Z-transforming it (using the time shift property and linearity):

( ) ( )

0 0

· ·

N M

k k

a y n k b x n k

= =

− = −

∑ ∑

( )

{ }

^{( )}

{

^{( )} ^{( )}

}

^{( )} ^{( )}

{ }

( ) ^{( )}

{ }

^{( )} ^{( )}

( ) ( )

0

1 1 2

0

0 0

2 1 1 2 2

,

· ·

, ,

N M

k k

N M

k k

Z f n k z F zk Z c f n c f n c F z c F z Z y n Y z Z x

Z a y n k Z

n

b x n k

a z Y z b

X

z

X z

= =

− −

= =

−

   

   

− = −

   

   

   

=

− = + = +

= =