chapter 4: discrete systems -...
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Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 1
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Chapter 4
Discrete Systems
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 2
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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→ = = −∞ +∞⎡ ⎤⎣ ⎦
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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
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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:
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4.1 Introduction
Signal flow diagram of the example of a digital filter
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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.
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4.2 Linear, Time-Invariant Discrete Systems
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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= =
⎡ ⎤= − − −⎢ ⎥
⎣ ⎦∑ ∑
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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 .
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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
= + − + − − − − −
=
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4.2.1 Difference Equations
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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= ≡ <
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4.2.3 The Discrete Transfer Function Hz(z)
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4.2.3 The Discrete Transfer Function Hz(z)
( ) 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.
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4.2.3 The Discrete Transfer Function Hz(z)
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:
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4.2.3 The Discrete Transfer Function Hz(z)
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= ⋅
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4.2.3 The Discrete Transfer Function Hz(z)
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
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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
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4.2.4 General Properties of Hz(z)
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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< < ∞
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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
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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=
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4.2.5 Behaviour of Hz(z) on the unit circle
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
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4.2.5 Behaviour of Hz(z) on the unit circle
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−
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4.2.5 Behaviour of Hz(z) on the unit circle
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:
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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−
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4.2.5 Behaviour of Hz(z) on the unit circle
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
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4.2.5 Behaviour of Hz(z) on the unit circle
( 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
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4.2.5 Behaviour of Hz(z) on the unit circle
Phase of discrete transfer function of 3rd order system
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4.2.5 Behaviour of Hz(z) on the unit circle
Envelope delay of a 3rd order system
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4.2.5 Behaviour of Hz(z) on the unit circle
Locus of the discrete transfer function of 3rd order system
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4.2.5 Behaviour of Hz(z) on the unit circle
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 = ∫
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4.2.5 Behaviour of Hz(z) on the unit circle
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 = ∫
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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.
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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
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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 +
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4.2.6 All-Pass Filters
Pole-Zero diagram of an All-pass in the z-plane
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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∞ <
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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
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4.2.7 Minimum-phase Systems
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=
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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
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4.2.7 Minimum-phase Systems
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4.2.7 Minimum-phase Systems
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4.2.7 Minimum-phase Systems
Minimum phase system All pass system
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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:
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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
=
=∞
=
−= = − ⇒
−
∏∏
∏
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 47
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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 +
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4.2.8 Systems with Linear Phase
Pole-Zero diagram of Non-Recursive System with Linear Phase
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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
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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:
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 51
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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!
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4.3 System Structures for Discrete LTI-Systems
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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
g k a s k b g kb α β
α β
α β= =
⎡ ⎤= − − −⎢ ⎥
⎣ ⎦∑ ∑From chapter 4.2.1:
by setting m = n one obtains:
0 10
1( ) ( ) ( )n n
g k a s k b g kb α β
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
− −
=
⎡ ⎤= + −⎢ ⎥
⎣ ⎦∑
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 54
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4.3.1 The First Canonical Form of a DiscreteSystem
First Canonical form of a digital filter
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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
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 56
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4.3.2 The Second Canonical Form of a DiscreteFilter
The Second Canonical Form of a Digital Filter
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4.3.2 The Second Canonical Form of aDiscrete Filter
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
+ − + − += =
−
=
= − ⇔ = −
= −
∑ ∑
∑
Prof. Dr.-Ing. I. Willms Signals and Systems 1 WS 03/04S. 58
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4.3.2 The Second Canonical Form of aDiscrete Filter
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
−− +
= =
−
− =
−=
=
= =
= =
∑ ∑
∑∑
∑
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4.3.2 The Second Canonical Form of aDiscrete Filter
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
−
=
−
=
= =∑
∑
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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
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4.3.4 The Fourth Canonical form of a Digital Filter
Cascade of a 1st and 2nd Order Digital Filter
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4.3.4 The Fourth Canonical form of a Digital Filter
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.
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4.3.4 The Fourth Canonical form of a Digital Filter
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4.3.4 The Fourth Canonical form of a Digital Filter
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 =
= + =∑
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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