Chapter 4: Discrete Systems - nts.uni-duisburg-essen.dents.uni-duisburg-essen.de/downloads/ss1/SS1_Chap4.pdf · Chapter 4 Discrete Systems. Prof. Dr.-Ing. I. Willms Signals and Systems

Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 1

FachgebietNachrichtentechnische Systeme

N T S

Chapter 4

Discrete Systems



N T S

4.1 IntroductionGraphik shows general scheme of a digital signal processing system (used in cound cards, digital sound processors, precise filtering applications etc.).Note: In general additional low-pass filters are needed!

( ) ( ) ( ) where ...s k g k T s k k→ = = −∞ +∞⎡ ⎤⎣ ⎦



N T S

4.1 IntroductionIn practice, such a system is represented by a digital signal processor, essentially consisting of the following elements:

Delay: ( 1) ( 1) [ ( )]s k or s k V s k− − =

α ( )s kα ⋅Multiplier:

+Adder:1 2( ) ( )s k s k+

1( )s k

2 ( )s k



N T S

4.1 IntroductionExample:

The relation of the example from section 2.3.5.1 with

1( 2) ( 1) ( ) ( 1) ( )g k c g k g k s k s k+ + + + = + +

can be rewritten as:

1( 2) ( 1) ( ) ( 1) ( )g k c g k g k s k s k+ = − + − + + +

1( ) ( 1) ( 2) ( 1) ( 2)g k c g k g k s k s k= − − − − + − + −

The corresponding signal flow is represented in the following diagram:



N T S

4.1 Introduction

Signal flow diagram of the example of a digital filter



N T S

4.1 IntroductionThe same basic properties as for analog filters can also be found fordiscrete systems:

1- Real values:

From real-valued s(k) follows that also g(k) is real-valued

2- Time-invariance:

( ) ( )s k g kκ κ+ → + ( ) ( )s k g k→ gives3- Linearity:

1 1( ) ( )s k g k→

1 1( ) ( )

n n

s k g kν ν ν νν ν

α α= =

→∑ ∑ ( ) ( )v vs k g k→

( ) ( )n ns k g k→

Causality and stability are here as important as for analog systems.



N T S

4.2 Linear, Time-Invariant Discrete Systems



N T S

4.2.1 Difference EquationsThe most important class of discrete LTI-systems is the one, describable by an n-th order equation of the following kind:

0 0

( ) ( ) n n

a s k b g kα βα β

α β= =

− = −∑ ∑

From this, one obtains:

0 10

1( ) ( ) ( )n n

g k a s k b g kb α β

α β

α β= =

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦∑ ∑



N T S

4.2.1 Difference Equations

Definition:A discrete-time LTI-system is called recursive if the calculation of each output value g(k) from the preceding output values g(k - ß) with ß > 0is performed.

Definition:A causal digital LTI-system is called non-recursive if the calculation of each output value g(k) is possible without the use of previously calculated output signals g(k - ß) with ß > 0 .



N T S

4.2.1 Difference EquationsExample:

Given is a second order discrete LTI-system, where n = m = 2, thus:

2 2

0 10

1( ) ( ) ( ) g k a s k b g kb α β

α β

α β= =

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦∑ ∑

0One can set b 1 here without any restrictions: =

0 1 2 1 2

1,2 1,2

( ) ( ) ( 1) ( 2) ( 1) ( 2)

In the next slide it is specified: !!

g k a s k a s k a s k b g k b g k

b d

= + − + − − − − −

=



N T S

4.2.1 Difference Equations



N T S

4.2.2 The Discrete Impulse Response

Definition:

0

The impulse response of a discrete system is the response ( ) of the system to ( ) ( )This special sequence to be observed at the output is denoted by: ( )

g ks k

hk

kγ=

The answer of the system to any causal excitation ( ) with ( ) 0 0 thus is:

s ks k for k≡ <

0

( ) ( ) ( ) ( ) ( )g k h s k h k s kν

ν ν+∞

=

= ⋅ − = ∗∑ Discrete convolution

The causality provides: ( ) 0 for 0, thus h k k≡ <

( ) ( ) 0 for 0s k g k k= ≡ <



N T S

4.2.3 The Discrete Transfer Function Hz(z)



N T S


( ) where 0...kpT ks k Ue Uz k= = = ∞

and with being the complex frequency, one gets:p jσ ω= +

0

0 0

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

z

k k kz

Z h H z

g k h k s k h s k

h Uz Uz h z Uz H z

ν

ν ν

ν ν

ν

ν ν

ν ν

+∞

=

+∞ +∞− −

= =

=

= ∗ = ⋅ −

= ⋅ = ⋅ = ⋅

∑

∑ ∑14243

If s(k) as input signal for a discrete system is used with

Discrete transfer function.



N T S


0

( ) ( ) ( ) ( ) ( )g k h s k h k s kν

ν ν+∞

=

= ⋅ − = ∗∑ ( ) ( ) ( )( )( )( )

z z z

zz

z

G z H z S zG zH zS z

= ⋅

⇔ =

0

( ) ( )zH z h z ν

ν

ν+∞

−

=

= ⋅∑

* According to z-transform properties it holds:

* The discrete transfer function is the z-transform of the impulse response h(k)

* For a chain of two discrete LTI-systems it holds:



N T S


With the equations:

1 2( ) ( ) ( )g k g k h k= ∗and1 1( ) ( ) ( )g k h k s k= ∗

[ ]1 2 1 2( ) ( ) ( ) ( ) ( ) ( )g k g k h k h k s k h k= ∗ = ∗ ∗it follows:

1 2( ) ( ) ( ) ( )g k h k h k s k= ∗ ∗ 1 2

( )

( ) ( ) ( ) ( )z

z z z z

H z

G z H z H z S z= ⋅ ⋅1442443

1 2( ) ( ) ( )z z zH z H z H z

or

=( ) ( ) ( )z z zG z H z S z= ⋅



N T S


0 0( ) ( )

n n

a s k b g kα βα β

α β= =

− = −∑ ∑

Considering a difference equation, this relation can be described in the z-domain:

The z-transform of both sides of the equation is then:

0 0

( ) ( )n n

z za S z z b G z zα βα β

α β

− −

= =

⋅ = ⋅∑ ∑

Rewriting this formula gives:

0

0

( )( )( )

n

zz n

z

a zG zH zS z b z

αα

α

ββ

β

−

=

−

=

⋅= =

⋅

∑

∑0

0

( ) where

m

z n

d zH z n m

c z

µµ

µ

νν

ν

=

=

⋅= ≥

⋅

∑

∑

By means of an

index conversion



N T S

4.2.3 The Discrete Transfer Function Hz(z)So this second form of the discrete transfer function Hz(z) is a rational function of the variable z:

Numerator polynom in z ( )( )Denominator ploynom in z ( )z

P zH zQ z

= =

01

1

( )( )

( )

m

mz n

n

z zdH zc z z

µµ

µν

=

∞=

−= ⋅

−

∏

∏

A third form of the discrete transfer function is based on the zeros of the numerator and the denominator polynom:

Roots of the of the numerator or the zeros

Roots of the denominator or the poles



N T S

4.2.4 General Properties of Hz(z)



N T S

4.2.4 General Properties of Hz(z)

1. Properties of system coefficients and of poles and zeros:

The coefficients respectively are real constants. The zeros and poles are either real or conjugated complex:

0 01 1

1 1 2 1 10 0 0 0 0 where j j

z z e z z z eµ µ

µ µ µ µ µ

ψ ψ−∗= = =

1 1

1 1 2 1 1 where

j jz z e z z z eν ν

ν ν ν ν ν

ψ ψ∞ ∞−∗∞ ∞ ∞ ∞ ∞= = =

2. Stability:

A discrete system obviously is stable, if

(BIBO: bounded input bounded output criterion)

1( ) s k M k< < ∞ ∀any bounded input signal causes

a bounded output signal 2( )g k M< < ∞



N T S

4.2.4 General Properties of Hz(z)1 z

νν∞ < ∀BIBO stability is given, if the following relation holds:

Pole- zero plot of a real, causal and stable system



N T S

4.2.5 Behaviour of Hz(z) on the unit circle

If we observe the (z) for any point z on the unit circle with 1 and j TZH z z e ω= =

we obtain the so-called frequency response.

010

0 1

( )( )

( )

mnj Tj T

j T mz n n

j T j Tn

e za edH ecb e e z

µ

ν

ωαωα

µω α

βω ωβ

β ν

−

==

−∞

= =

−= =

−

∏∑

∑ ∏

2periodic function with 2 or =TTπω π ω=

In short: ( ) ( )j Tz aH e Hω ω=



N T S


In general :

( ) ( ) ( ) ( ) ( ) ( )j T j T j Tz z z z z zG z H z S z G e H e S eω ω ω= ⇒ =

A normalized representation using 2 , leads to:T or Fω π= Ω = Ω⋅

01

1

( )( ) ( ) ( )

( )

mj

j T j mz z Na n

jn

e zbH e H e Hc e z

µ

ν

µω

ν

Ω

=Ω

Ω∞

=

−= = Ω =

−

∏

∏

Magnitude ( ) and its phase ( ) can be rewritten as:a aH ϕΩ Ω

( )( ) ( ) NajNa NaH H e ϕ ΩΩ = Ω



N T S


For the distance of each zero or each pole to the unit circle, it holds:

( ) ( )2 2

2 22 2 2 2

2

cos sin cos sin

cos cos sin sin

cos 2cos cos cos sin 2sin sin sin

1 2 cos( )

with

j

j

e z j z z j

z z

z z z z

z z

z z e ψ

ψ ψ

ψ ψ

ψ ψ ψ ψ

ψ

Ω − = Ω+ Ω− − ⋅

= Ω− + Ω−

= Ω− Ω + + Ω− Ω +

= − Ω− +

=

For the corresponding angle of the connection of each zero or each pole to the unit circle, it holds:

( ) sin sinarctan

cos cosj z

e zz

ψψ

Ω Ω−− =

Ω−



N T S


01

1

2

0 0 01

2

1

( )

1 2 cos( )

1 2 cos( )

mj

mNa n

jn

m

mn

n

e zbHc e z

z zbc z z

µ

ν

µ µ µ

ν ν ν

µ

ν

µ

ν

ψ

ψ

Ω

=

Ω∞

=

=

∞ ∞ ∞=

−Ω = ⋅

−

− ⋅ Ω− += ⋅

− ⋅ Ω − +

∏

∏

∏

∏

Thus it results:

0 0

1 1 0 0

sin sinsin sin( ) arctan arctan

cos cos cos cos

n m

Na

zz

z zµ µν ν

ν ν µ µν µ

ϕ ∞ ∞

= =∞ ∞

Ω − ΨΩ− ΨΩ = −

Ω− Ψ Ω− Ψ∑ ∑

With 0m

n

bc

> it follows:



N T S

4.2.5 Behaviour of Hz(z) on the unit circleFor the derivative of the angle related to the connections of each pole or each zero it holds:

( )

2 2

2

2 2

2 2

sin sinarctan

cos cos

cos (cos cos ) ( sin )(sin sin )1(sin sin ) (cos cos )

1(cos cos )

cos cos cos sin sin sin

(cos cos ) (sin sin )

j zd de zd d z

z zz zz

z zz z

ψψ

ψ ψψ ψψ

ψ ψψ ψ

Ω Ω−− = =

Ω Ω Ω−

Ω Ω− − − Ω Ω−⋅

Ω− Ω−+

Ω−

Ω− Ω + Ω− Ω=

Ω− + Ω−

2 22 2 2 2

2

1 cos( )

cos 2cos cos cos sin 2sin sin sin

1 cos( )

1 2 cos( )

zz z z z

zz z

ψ

ψ ψ ψ ψ

ψ

ψ

− Ω−=

Ω− Ω + + Ω− Ω +

− Ω−=

+ − Ω−



N T S


Thus the frequency normalised envelope delay results to:

21

0 0

21

0 0 0

1 cos( )( )( )1 2 cos( )

1 cos( )

1 2 cos( )

nNa

Nga

m

zdd z z

z

z z

ν ν

ν ν ν

µ µ

µ µ µ

ν

µ

ϕτ ∞ ∞

=∞ ∞ ∞

=

− ⋅ Ω −ΨΩΩ = =

Ω − ⋅ Ω−Ψ +

− ⋅ Ω−Ψ−

− ⋅ Ω−Ψ +

∑

∑

Note: According to the formulas given above it follows:

( ) ( )Na Naϕ ϕ−Ω = − Ω( ) ( )Na NaH HΩ = −Ω

( ) ( )Nga Ngaτ τ−Ω = Ω



N T S


( 1)( 0.2)( )( 0.3)( 0.3 0.6)( 0.3 0.6)

z zH zz z j z j

+ +=

− − − − +

Example: 3rd order system with the discrete transfer function

Magnitude of discrete transfer function of 3rd order system



N T S


Phase of discrete transfer function of 3rd order system



N T S


Envelope delay of a 3rd order system



N T S


Locus of the discrete transfer function of 3rd order system



N T S


If the regarded system are causal and stable then:

1. ( ) is analytically regular for Re 0 and ( ) is analytically regular for z 1L ZH p p H z> <

Z

2. ( ) ( ) shows the frequency behaviour of analog filter

H ( ) ( ) ( ) shows the frequency behaviour of digital filter.L

j t jZ a

H j H

e H e Hω

ω ω

ωΩ

=

= =

( ) Re ( ) Im ( )j jz z zH e H e j H eΩ Ω= +with jΩ or

( ) Re ( ) Im ( )Na Na NaH H j HΩ = Ω + Ω

It results: 1Re ( ) lim ( ) Im ( ) cot

2 2Na z NazH H z H d

π

π

ηη ηπ

+

→∞−

−ΩΩ = − ∫

1Im ( ) Re ( ) cot2 2Na NaH H d

π

π

ηη ηπ

+

−

−ΩΩ = ∫



N T S


Note: For minimum-phase systems, it applies:

1ln ( ) lim ln ( ) ( ) cot2 2Na z az

H H z dπ

π

ηϕ η ηπ

+

→∞−

−ΩΩ = − ∫

1( ) ln ( ) cot2 2Na NaH d

π

π

ηϕ η ηπ

+

−

−ΩΩ = ∫



N T S

4.2.6 All-Pass FiltersAn all-pass filter is defined according to:

( ) ( ) const. ja zH H e ΩΩ = = ∀Ω

( )( )( )z

P zH zQ z

=For the discrete transfer fuction of an all-pass:one observes:

1

1 1( ) ( )

n

nQ z c z z ν

ν∞

=

=−∏

and looks for a:

01

( ) ( )m

mP z b z z µµ=

= −∏

In such a way that equation is fulfilled.



N T S

4.2.6 All-Pass FiltersThe following possibilities are given:

Case 0 , z ν ν∞ = ∀1( ) where ( ) 1, because 1 where j

Nan

H P z z z ec

ΩΩ = = = =

• Other case it results:

2

0 0 01

2

1

1 2 cos( )( )

1 2 cos( )

m

mNa n

n

z zdHc z z

µ µ µµ

ν ν νν

ψ

ψ

=

∞ ∞ ∞=

− Ω− +Ω =

− Ω− +

∏

∏

21 2 cos( )z zν νψ∞ ∞ ∞− Ω− + νSo every pole makes a contribution to ( )NaH Ω



N T S

4.2.6 All-Pass Filters( ) const. results in case the contribution of every pole is compensated by :NaH Ω =

2

0 0 01 2 cos( )z zµ µ µψ− Ω− +

This is possible, if m=n and:

00 0

1 1 1j jjz z e e

z z e zµ ν

ν

ψ ψµ µ ψ

ν ν ν

∞

∞− ∗∞ ∞ ∞

= = = =

so that:

20

2

1 11 2 cos( )1

1 cos( )

j

j

ze z zze z z z

ννµ ν

νν ν ν ν

∞Ω∞ ∞

Ω∞∞ ∞ ∞ ∞

− Ω−Ψ +−

= =− − Ω−Ψ +



N T S

4.2.6 All-Pass Filters

Pole-Zero diagram of an All-pass in the z-plane



N T S

4.2.6 All-Pass Filters

One gets in case of:

0

1zzν

ν∞ = and 0ν ν∞Ψ = Ψ

( )2

21

1 sin( )( ) arctan

(1 )cos( ) 2

n

Na

z

z z

ν ν

ν ν ν ν

ϕ∞ ∞

= ∞ ∞ ∞

− Ω−ΨΩ =

+ Ω−Ψ −∑

phase delay:

envelope delay:

2

21

1( )

1 2 cos( )

n

Nga

zz z

ν

ν ν ν ν

τ ∞

= ∞ ∞ ∞

−Ω =

− Ω−Ψ +∑ where 1 for all z ν ν∞ <



N T S

4.2.7 Minimum-phase Systems

Discrete system can also be devided into all-passes and minimum-phase systems

Let‘s assume an stable LTI discrete system with the transfer funsction:

1

1

(1)0 0

1 1 (2)0

1

1 1

( ) ( )( ) ( )

( ) ( )

mm

mm m

z n nmn n

z z z zd dH z z zc cz z z z

µ µµ µ

µµ

ν νν ν

= =

= +∞ ∞

= =

− −= = −

− −

∏ ∏∏

∏ ∏

(1)0 11 for 1,...,z mµ µ≤ = 1the first zeros in the unit-circlemwith

(2)0 11 for 1,...,z mµ µ> = + m other zeros outside the unit-circle



N T S


1 1

(2) (2)0 0 (2)

1 1 0

1With ( . 1) ( ) , it yields:m m

m m

z z z zzµ µ

µ µ µ= + = +

− = −∏ ∏

1

1 1

1

.

(1) (2) (2)0 0 0

1 1 1

(2)0 (2)

1 1 0

( )

( ) ( . 1) ( )( ) .

1( ) ( )

zM zAllp

m m m

m mmz n m

n

m

H z H

z z z z z zbH zc z z z z

z

µ µ µµ µ µ

ν µν µ µ

= = + = +

∞= = +

− − −=

− −

∏ ∏ ∏

∏ ∏144444424444443 14444244443

Or in a short form:

.( ) ( ). ( )

Allpz ZM zH z H z H z=



N T S

4.2.7 Minimum-phase Systems(2)

01With , one obtains for the all-pass: z

zµµ∞

=

1

.

1 1

1

(2)0

1 1

real constant

1( )1( ) .

( )Allp

m

mz m m

m m

zz

H zz z z

µ µ

µ µµ µ

= + ∞

∞= + = +

−=

−

∏

∏ ∏14243

(2)0 (2)

0

1Because 1 1z zzµ µ

µ∞> ⇔ = <

The poles of this all-pass lie inside the unit-circle



N T S




N T S




N T S


Minimum phase system All pass system



N T S

4.2.8 Systems with Linear PhaseLinear phase means a constant envelope delay with the frequency

( ) ( )Nga fτ Ω ≠ Ωwhere:

0 02 2

1 1 0 0 0

1 cos( )1 cos( )( )

1 2 cos( ) 1 2 cos( )

n m

Nga

zzz z z z

µ µν ν

ν µν ν ν µ µ µ

τ ∞ ∞

= =∞ ∞ ∞

− Ω−Ψ− Ω−ΨΩ = −

− Ω−Ψ + − Ω−Ψ +∑ ∑

The condition is fulfilled in cases:

0 poles and ze1. ros and fo exhibit the same locar tio all nn m z zµ ν ν µ∞= = = ⇒

02. 1 1 andz zµ νµ ν∞= ∀ = ∀ ⇒ leads to only conditional stable systems and therefore is ignored in the following:



N T S

4.2.8 Systems with Linear PhaseA) 1 must be fulfilled, =0 ensures that a frequncy independent contribution from the poles is introduced in the formula for the group delay.

z zν νν ν∞ ∞< ∀ ∀

0B) The zeros have to be located in such a way, that:µz

0 02

1 0 0 0

1 cos( ).

1 2 cos( )

m zconst

z zµ µ

µ µ µ µ=

− Ω−Ψ→

− Ω−Ψ +∑

To fulfill B) condition, it is required:

01

01

1

( )11) ( ) ( ) non-recursive system.

( )

m

mm m

z n nn n

z zb bH z z zc c zz z

µµ

µµ

νν

=

=∞

=

−= = − ⇒

−

∏∏

∏



N T S

4.2.8 Systems with Linear Phase02) The zeros have to be located pair-wise symmetrically to each other:µz

0 00 0

0 0

1 1a b

a a

b b

j jz z e ez z

Ψ Ψ

∗= = =

Due to this relation, the total group delay gives:

1

0 0 0 02 2

0 0 0 0 0 0

00 0 0

2

0 0 0 0 20 0

1 cos( ) 1 cos( )

1 2 cos( ) 1 2 cos( )

11 cos( )1 cos( )

12 11 2 cos( ) 1 cos( )

a a b b

a a a b b b

a

a a a

a a a

a a

z z

z z z z

z z

z zz zλ

− Ω −Ψ − Ω−Ψ+

− Ω−Ψ + − Ω−Ψ +

− Ω−Ψ− Ω−Ψ

= + =− Ω−Ψ + − Ω−Ψ +



N T S

4.2.8 Systems with Linear Phase

Pole-Zero diagram of Non-Recursive System with Linear Phase



N T S

4.2.8 Systems with Linear PhaseTherefore discrete systems with linear phase have always a transfer function :

13 2

01 0

1 1( ) ( 1) ( 1) ( )( )m

m mmz n

n

bH z z z z z zc z zµ

µ µ∗

=

= − + − −∏

1 2 32m m m m n+ + = ≤where

Example: 1 2 3for 1 0, i.e., 0n m m m m= ⇒ = = = =

0( ) ( 1)m

n

bh k kcγ= −1( ) m

Zn

bH z zc

−=and

1m

n

bc

=Delay element with



N T S

4.2.9 Non-Recursive Systems (FIR-Filters)Non-recursive systems have been defined by:

0

1( ) ( )n

g k a s kb α

α ε

α=

= −∑0

1( ) ( )n

G z a S z zb

αα

α ε

−

=

= ∑

00

( ) 1( )( )

n

zG zH z a zS z b

αα

α

−

=

= = ∑

01. Because of ( ) ( ) , the impulse response:

m

zH z h z ν

ν

ν −

=

=∑Properties:

or

0

( ) for 0... with ( ) 0 for 0 and kah k k n h k k k nb

= = = < >

finite duration also called FIR (Finite Impulse Response) - systems.

0

( ) ( ) ( ) ( ) ( )n

g k s k h k h s kν

ν ν=

= ∗ = −∑With the output signal:



N T S

4.2.9 Non-Recursive Systems (FIR-Filters)

0 00 0

1 12. From this results: ( )nn n

z n

a zH z a zb b z

αα α

αα α

−−

= =

= =∑ ∑

non-recursive systems have just an nth order pole at z = 0 always stable!



N T S

4.3 System Structures for Discrete LTI-Systems



N T S

4.3.1 The First Canonical Form of a DiscreteSystem

A canonical form is a system structure with a minimized number of memories (delay elements).

0 10

1( ) ( ) ( )n n


α β

α β= =

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦∑ ∑From chapter 4.2.1:

by setting m = n one obtains:

0 10

1( ) ( ) ( )n n


α β

α β= =

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦∑ ∑

or 0

10 0

1( ) ( ) ( ) ( )nag k s k a s k b g k

b b γ γγ

γ γ=

⎡ ⎤= + − − −⎢ ⎥

⎣ ⎦∑

0

10 0

1( ) ( ) ( ) ( )n

z z z zaG z S z a S z z b G z zb b

γ γγ γ

γ

− −

=

⎡ ⎤= + −⎢ ⎥

⎣ ⎦∑



N T S

4.3.1 The First Canonical Form of a DiscreteSystem

First Canonical form of a digital filter



N T S

4.3.2 The Second Canonical Form of aDiscrete Filter

A second canonical form results as follows:

0

0

( ) ( )

n

z zn

d zG z S z

c z

µµ

µ

νν

ν

=

=

=∑

∑

The second form works equal to the first canonical form; this is proved in the following:

• The representation in following figure with

0

0

( )( )( )

n

zz n

z

b zG zH zS z c z

µµ

µ

νν

ν

=

=

= =∑

∑equals the one in figure of the 1st Form



N T S

4.3.2 The Second Canonical Form of a DiscreteFilter

The Second Canonical Form of a Digital Filter



N T S


1( ) ( )s k x k→ 1( ) ( )zS z X z→

1 1( ) ( )nnX z z X z−+= 1 1( ) ( )X z z X zν

ν + =

First, one takes a look at the part underneath the dashed line this system part can obviously described by:

Furthermore:in general

1 11

Due to ( ) ( ) ( ), we obtains:n

n nx k s k b x kν νν

+ − +=

= −∑

1 1 1 11 1

11

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

( ) ( )

n nn

n z n z n

nn

z

X z S z b X z z X z S z b X z

S z b z X z

ν ν ν νν ν

νν

ν

+ − + − += =

−

=

= − ⇔ = −

= −

∑ ∑

∑



N T S


1

11

( ) ( )n

n nzX z z z b z S zν

νν

−−

=

⎡ ⎤⇔ + =⎢ ⎥⎣ ⎦∑

0 1

0

( )and with 1 we obtain: ( ) zn

n

S zb X zz b z ν

νν

−

=

= =

∑For the upper part of the system the difference equation 1

0

( ) ( )n

ng k a x kν νν

− +=

= ∑results:

1 10 0

01

0

0

( ) ( ) ( )

( ) ( )

n nn

z n

nn n

nn

znn

G z a X z a z X z

z a zX z z a z S z

z b z

νν ν ν

ν ν

νν

ν νν

ννν

ν

−− +

= =

−

− =

−=

=

= =

= =

∑ ∑

∑∑

∑



N T S


0

0

( ) ( )

n

z zn

a zG z S z

b z

νν

ν

νν

ν

−

=

−

=

=∑

∑

or:

So the whole system is described by:

0

0

( )( )( )

n

zz n

z

a zG zH zS z b z

νν

ν

νν

ν

−

=

−

=

= =∑

∑



N T S

4.3.3 The Third Canonical Form of a Digital System

3rd Form: cascade of the 1st and 2nd order system.

One can divide from:

0 ...1

1

0

( )( ) ( )

( )

m

mz zn

n

z zdH z H zc z z

µµ

γγ

νν

=

=∞

=

−= =

−

∏∏

∏into

22 1 0

22 1 0

( )z

d z d z dH z

c z c z cγ γ γ

γ γ γ

γ

+ +=

+ +1 0

1 0

( )z

d z dH z

c z cγ γ

γ γ

γ

+=

+and



N T S

4.3.4 The Fourth Canonical form of a Digital Filter

Cascade of a 1st and 2nd Order Digital Filter



N T S


0

1

1

( )( )( )

m

n

z n

n

d zRH z R

z zc z z

µµ

µ ν

ν νν

ν

=∞

= ∞∞

=

= = +−−

∑∑

∏

Another canonical structure can be obtained by converting the transfer function into a partional sum:

The residues in this simple case follow from (where =1):nc

lim ( ) for z nz

R H z d m n∞→∞

= = =

lim ( ) ( ) for single poleszz z

R z z H zν

ν ν∞

∞→

= −

This leads to a parallel connection of the parts.



N T S




N T S


00 0

0

For real poles: ( ) where and z

dH z d R c z

z cγ

γ γ

γ

γ γ ∞= = = −+

For conjugate complex poles, two terms must be combined:

1 0

21 0

( )z

b z bH z

z c z cγ γ

γ γ

γ

+=

+ +

2

0 1 0 12 Re , 2Re , and 2Red R z d R c z c zγ γ γ γ γ γγ γ γ

∗∞ ∞ ∞= − = = = −

where

So in completion:...

1( ) ( ) where 1z n z nH z d H z cγ

γ =

= + =∑



N T S

4.3.5 System Structures for Non-RecursiveSystems (FIR-Filters)

00

1( ) ( )n

g k a s kb α

α

α=

= −∑

For any of the first three canonical forms, an appropriate non-recursive system can be directly derived from equation

First Canonical Form of a FIR - Filter

Documents

Chapter 4: Discrete Systems - nts.uni-duisburg-essen.dents.uni-duisburg-essen.de/downloads/ss1/SS1_Chap4.pdf · Chapter 4 Discrete Systems. Prof. Dr.-Ing. I. Willms Signals and Systems