Signal Processing 1 - nt.tuwien.ac.at · 4 Univ.-Prof. Dr.-Ing. Markus Rupp Historical Notes The discipline „Digital Signal Processing“ emerged from „Analog Signal Processing“

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Signal Processing 1Part I: Basics

Univ.-Prof.,Dr.-Ing. Markus RuppWS 17/18

Do 14:00-15:30EI3A, Fr 8:45-10:15EI4LVA 389.166

Last change: 12.10.2017

2Univ.-Prof. Dr.-Ing.

Markus Rupp

3Univ.-Prof. Dr.-Ing.

Markus Rupp

Goals Basics: the classics (4U)

Notation: Vector, matrix, random numbers, deterministic numbers

Description of linear systems Convolution Polynomial operators, Bezout Matrix and vector Notation State-space descriptions, time-invariant systems Properties of linear systems

Sampling theorems

4Univ.-Prof. Dr.-Ing.

Markus Rupp

Historical Notes The discipline „Digital Signal Processing“

emerged from „Analog Signal Processing“ by the end of the 60s.

First, quite successfully the concepts of analog techniques based on functional analysis were transferred into the digital domain.

As a consequence, DSP developed into methods for filter design and optimization.

5Univ.-Prof. Dr.-Ing.

Markus Rupp

Historical Notes Midth of the 80s a change in

paradigm took place. Filter and filter bank technology had

found their climax beginning of the 90s: “everything could be optimized with minimal complexity”.

The methods of functional analysis have not improved DSP any further.

6Univ.-Prof. Dr.-Ing.

Markus Rupp

Goals Signals can be described in form of finite

or infinite series (sequences). Finite series can be described as vectors,

linear systems as matrices. Digital signal processing has thus turned

into applied linear algebra. All „Engineer-concepts“ of modern digital

signal processing can be exactly described by methods of linear algebra.

7Univ.-Prof. Dr.-Ing.

Markus Rupp

Notation * conjugate complex T transpose H Hermitian = conjugate and transpose a,b,c deterministic scalar a,b,c Random variable with density fa(a), variance σa

2

and mean ma(1).

a,b,c deterministic column vector 1 Vector with only “Ones” as entries a,b,c Random-column vectors with joint density function

fa(a), variance σa2 and mean ma

(1). Their autocorrelation matrix is given by Raa=E[aaH].

8Univ.-Prof. Dr.-Ing.

Markus Rupp

Notation A,B,C deterministic matrices A,B,C Matrices whose entries are random

variables I Unit matrix, i.e. “Ones” on the main diagonal

and zero otherwise. ak series of scalars f(t) function in variable t T[.] operator (on sequence or function)

9Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Having linear systems is a dominant assumption to describe the input/output relation of systems.

Linear systems are linear by design (sufficiently) linear by observation Linearized by simplification

10Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Definition 1.1: A system is called linear, if

the following (both) is true: If scaling the input signal by any constant α,

the output signal changes by the same amount: y=S[x]; S[αx]=αS[x]= αy.

If the input signal consists of two components, the output signal can also be described by two components, uniquely mapped to the input components: y1=S[x1]; y2=S[x2]; y=S[x1+x2]=y1+y2.

11Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems To describe the input-output relation

S[] of the linear system, we call this a linear transformation or, equivalently, a linear operator.

Note that we have not specified the signals further yet. They can be series as well as vectors or functions.

12Univ.-Prof. Dr.-Ing.

Markus Rupp

Convolutions A convolution is the most important operation for

signals passing linear time-invariant systems that are either time-continuous functions or time-discrete functions:

system causal-non

tat ending system, causal

system causal

0

;)()()](H[

;)()()()()](H[

;)()()()()](H[

0

0

0

0

τττ

ττττττ

ττττττ

dtxhtx

dxthdtxhtx

dthxdtxhtx

t

tt

t

t

−=

−=−=

−=−=

∫∫

∫∫

∞−

∞−

13Univ.-Prof. Dr.-Ing.

Markus Rupp

Convolutions Convolution of series have the following properties:

In communications mostly non-causal series are being considered since for a „very long“ transmission the initial values are of no importance.

system causal-non

1k period of system perodic

kat ending system, causal

system causal

0

0

;]H[

;]H[

;]H[

0

0

0

0

lkl

lk

l

k

kkllklk

k

llk

l

k

llklk

llk

xhx

xhxhx

xhxhx

−∞=

−=−−

=

−∞=−−

=

∑∑

∑∑

=

+

==

==

14Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Example 1.1: Multi-path-propagation (Ger.:

Mehrwegeausbreitung)Wave propagation in a multi-path channel can be described by a linear, time-variant (Ger.: zeitvariant) system h(t,τ) :

Transmit-signal

Receive signalh1(t,τ)

hn(t,τ)

15Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Example 1.1: Multi-path-propagation (Ger.:

Mehrwegeausbreitung)Wave propagation in a multi-path channel can be described by a linear, time-variant (Ger.: zeitvariant) system h(t,τ) :

h(t,τ)Transmit signal Receive signal

x1+x2 H[x1+x2]= H[x1]+H[x2]

16Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Example 1.2: Linear Prediction

q-1Speech signal

Prediction error

q-1 q-1

a1 a2 a3

Σ

+/-Predicted speech signal

statisticalmethod

17Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Systems Example 1.2: Linear prediction is not a linear

system:

Reason: The speech signal x1 causes in general a different predictor A[] than the speech signal x2.

A[]Speech signal Prediction error

x1+x2 A[x1+x2] is not A[x1]+A[x2]

18Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators On top of the convolution we already know

several other linear operators Fourier-transform Laplace-transform Z-transform

However, depending on their ROC=Region of Convergence (Ger.: Existenzbereich) they restrict the class of signals.

Note that the property „linear system“ is a system feature and is thus independent of the input signal.

19Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators If we want to stress the linear property of

the operator without using any particular other property on the transformation, we can introduce a general linear operator to describe a linear system.

Linear Operator: H[] Without defining any particular

transformation, some important properties of such linear operators are already present:Unit delay: H[]=q-1[]; q-1[xk]=xk-1

20Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators The convolution provides us with the hint, that

linear operators can be linearly combined to form new linear operators:

H[]=h0+h1q-1+…+hnq-n ; for a causal systemH[]= h-nqn +…+h0+h1q-1+…+hnq-n; for a non-causal system

Example 1.3: Even more complicated IIR filters can be described that way:

[ ] [ ] [ ]

∑∑−

=−

=−

−−−

+=

+=−

=

1

01

111

1

)()( )(1

)(

BA n

iiki

n

iiki

kkkk

xbya

xqByqAxqA

qBy

21Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators Such description is often called polynomial

description since the linear operators are given in form of (finite or infinite) polynomials.

The advantage of such is that the (complicated) convolution simplifies to a polynomial multiplication:

( ) ( ) ( )( )( ) ( )GH

GH

G

G

H

H

nnnn

nn

nn

qghqghghgh

qgqggqhqhhqHqG+−−

−−−−−−

++++=

++++++=

...

......1

100100

110

110

11

Linear Operators Note that q-1 looks formally like z-1

(z-Transform) but is different: q-1[.] is an operator z-1 is a complex-valued number

If | z-1 |=1, z is defined while | q-1| means nothing, unless we define it.

We apply z-1 on H(z): z-1 H(z)= H(z)z-1

but we apply q-1 on time sequences hk.22

Univ.-Prof. Dr.-Ing. Markus Rupp

23Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear OperatorsZero-Forcing Equalizer (1)

An inverse problem is: given a transfer function in form of a polynomial H(q-1), find the polynomial G(q-1), so that: H(q-1)G(q-1)=q-D.

Example 1.4: Let H be the channel, G the equalizerThe solution for G(q-1) is in general of double infinite length, i.e. G(q-1)=g-∞q∞+...+g-1q1+go+g1q-1+...+g∞q-∞.However, this unfortunate situation changes if we consider more than one transfer function.

24Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators Static channel and its equalizer

H G( ) ( ) ( )( )

( ) ( ) ( )

D

n

g

n

hhghgg

nn

q

qghqghghqghghgh

qgqggqhhqHqGG

Gn

G

G

G

+−

=

−=

==

−−−−−

⋅=

++++++=

++++=

1

...

...1

0

12

/

20111

/1

1001000

110

110

11

2012010

Transmit signal

Equalizedsignal

Receive signal

D=1

25Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators Static channel and its equalizer

( ) ( ) ( )( )( )

=

⋅=

+++=

++=

−−

−−−−

010

1

1

0

1

01

0

211

1100100

110

110

11

gg

hhh

hq

qghqghghghqggqhhqHqG

D

D=1 3 equationsfor2 variables!!!

26Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear OperatorsÉtienne Bézout (1730-1783)

Zero-Forcing Equalizer (2) Theorem 1.1 (Bezout): Given the nH transfer

functions H1(q-1),..., HnH(q-1).The following equation

has a finite solution if and only if there is no common prime in all Hl(q-1). The Hl(q-1) are said to be coprime.

Dn

lll qqGqH

H−

=

−− =∑1

11 )()(

27Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators Zero Forcing Equalizer (3)

Example 1.5: Let H1(q-1)=(1+hq-1) C1(q-1) and H2(q-1)=(1+hq-1) C2(q-1), C1(q-1) and C2(q-1) are co-prime. Then we have:

)1/(

2

1

111

2

1

1112

1

11

1

)()()1(

)()()1()()(

−− +≠

=

−−−

=

−−−

=

−−

∑∑

+=

+=

hqq

lll

lll

lll

D

qGqChq

qGqChqqGqH

28Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators

H2 G2( ) ( ) ( ) ( )( )( ) ( )( )( ) ( )

( ) Dqqghghqghghghghghgh

qggqhhqggqhhqHqGqHqG

−−

−−−−

−−−−

⋅=++

+++++=

+++++=

+

12)2(1

)2(1

)1(1

)1(1

1)2(0

)2(1

)2(1

)2(0

)1(0

)1(1

)1(1

)1(0

)2(0

)2(0

)1(0

)1(0

1)2(1

)2(0

1)2(1

)2(0

1)1(1

)1(0

1)1(1

)1(0

12

12

11

11

Equal.

SignalReceive signal2

H1 G1Transmit signal

Receive signal1

+

29Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Operators( ) ( ) ( ) ( )( )( ) ( )( )( ) ( )

( )

=

•=++

+++++=

+++++=

+

−−

−−−−

−−−−

010

1

)2(1

)2(0

)1(1

)1(0

)2(1

)1(1

)2(0

)2(1

)1(0

)1(1

)2(0

)1(0

2)2(1

)2(1

)1(1

)1(1

1)2(0

)2(1

)2(1

)2(0

)1(0

)1(1

)1(1

)1(0

)2(0

)2(0

)1(0

)1(0

1)2(1

)2(0

1)2(1

)2(0

1)1(1

)1(0

1)1(1

)1(0

12

12

11

11

gggg

hhhhhh

hh

qqghghqghghghghghgh

qggqhhqggqhhqHqGqHqG

D

3 equationsfor4 variables!!!OK

30Univ.-Prof. Dr.-Ing.

Markus Rupp

Test

( )

( ) ( )

)(22

012

1)ln(3)3)ln(3(

2/)1(

122

12/)1(22

1,,2/)1(,,,2/)1(,0

0

0

51

1

52

3)ln(3

)ln(3ln3

2

23

3

ABpathpath

duudttt

xxex

ex

ex

xx

dogdog

xxx

xxxx x

−==

==−

+−=−−=∂∂

=∂∂

=∂∂

<+<

<+=+<+=

+−

∫∫−

−−

−− −

ρρρρ

ρρρρρ

ρρρρρρρ

31Univ.-Prof. Dr.-Ing.

Markus Rupp

ρ

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρ

ρ

(1−ρ) ρ/2

ρ3

ρ2(1+ρ) ρ/2

32Univ.-Prof. Dr.-Ing.

Markus Rupp

Test

( )17

)96,0log()5,0log(

)5,0log(96,0log5,096,0

96,004,004,096,004,096,004,096,0

1

02

12

01

0

≈=

=

=

×++=+=

+==

n

wwxwxxwxx

x

n

n

Train your Maths skills Känguru link http://www.kaenguru.at

You also find the tests in English there including solutions.

33Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Consider a bandlimited signal x(t) to

be upconverted by ω0 to Y(jω):

Is up/downconverting/mixing a linear operator?

Answer: Yes, of course!

34Univ.-Prof. Dr.-Ing.

Markus Rupp

))((21))((

21)(

)cos()()(

oo

o

jXjXjY

ttxty

ωωωωω

ω

++−=

=

35Univ.-Prof. Dr.-Ing.

Markus Rupp

Test Results SP1 2017Problem

Correct [%] Mean Time [min]

1 78,78,100,92,53,79, 78,66,88,93,95,94,82

3,4.2,3.5,4.5,3.4,4.1,3.3,3.6,4,4.5,4.5,3.75,2.87

2 78,78,82,75,47,89,68,89,76,79,74,88,91

3.2,4.8,3.5,4.2,4.6,3.3,4,4.7,8.5,6,5,6.8,6

3 100,86,91,92,71,95, 96,100,84,86,84,88,100

4.8,6.2,4.3,7.2,6.6,2.7,5.3,4.8,5.3,3.1,3.5,6.7,5.4

4 100,94,73,67,41,65, 75,56,85,57,58,77,55

4.0,5.9,4.5,12.4,5.5,5.4,10.1, 12.5,9.8,8,6,8.5,9.7

5 56,50,40,75,53,53,62,39,66,72,53,59,50

8.6,7,5.3,9.5,8.5,5.5,12.2,9,7.7,11.5,4.5,6.6,6.9

Problem 1

36Univ.-Prof. Dr.-Ing.

Markus Rupp

percent correct

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

50

100

average time[min]

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

2

4

6

Problem 2

37Univ.-Prof. Dr.-Ing.

Markus Rupp

percent correct

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

50

100

average time[min]

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

5

10

Problem 3

38Univ.-Prof. Dr.-Ing.

Markus Rupp

percent correct

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

50

100

average time[min]

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

2

4

6

8

Problem 4

39Univ.-Prof. Dr.-Ing.

Markus Rupp

percent correct

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

50

100

average time[min]

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

5

10

15

Problem 5

40Univ.-Prof. Dr.-Ing.

Markus Rupp

percent correct

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

20

40

60

80

average time[min]

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017

0

5

10

15

41Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors Let‘s consider again the convolution for causal

systems with finite memory and zero initial conditions:

The latter form shows that for the system hk as well as for the signal xk only a finite number of values is required to describe the operation.

( )[ ] 111101

11

10

1

1

0

......

][

+−−−+−

−−

+−=−−

=

+++=+++

== ∑∑

HH

H

H

H

H

nknkkkn

n

l

k

nkllklk

n

llk

xhxhxhxqhqhh

xhxhxH

42Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors The finite signal xk as well as the system

hk can fill a vector, so that the convolution can be written in the following manner:

[ ][ ]

hxxhy

hhhh

xxxx

Tkk

Tk

Tn

Tnkkkk

H

H

==

=

=

+−−

110

11

,..,,

,..,,

43Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors Consider the linear FIR system h with nH

coefficients:

Let the signal xk be its input:

Thus, the inner vector product hTxk or xkTh

describes the output yk of system h at every time instant k.

The input signal does not need to be of finite duration in order to use a vector description for the convolution.

[ ]TnHhhhh 110 ,...,, −=

[ ]Tnkkkk Hxxxx 11,...,, +−−=

44Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors Let‘s consider the output of such a system (e.g., an FIR

filter) at different time instants k=0,1,...,nK-1 (nK>nH):

For h0 not equal to zero, this system can be inverted!

xHy

xxx

xxx

hhhhhh

hhhhh

hhh

yy

yyy

K

K

K

H

H

H

K

K

n

n

n

n

n

n

n

n

=

=

1

2

3

2

1

0

011

011

011

01

01

0

1

2

2

1

0

45Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors Note that the order of the vectors is sometimes changed so

that the matrix notion takes on the following form:

Again, for h0 not equal to zero, this system can be inverted!

( ) ( ) notation! (reverse) backward

...

0

1

2

3

2

1

0

10

1

0

110

10

110

0

1

3

2

1

←=

=

rTr

n

n

n

n

n

n

n

n

xHy

xxx

xxx

hhh

hh

hhhhh

hhh

yy

yyy

K

K

K

H

H

K

K

K

46Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors Such form of a matrix description thus

contains initial values. Note that the dimensions of the vectors

and the matrix grow with growing time index.

Therefore, such a description is often not very practical.

If the initial values are not of importance, another form of matrix description is more useful.

47Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors

knk

k

k

k

nnk

n

n

n

n

k

k

k

nk

n

n

n

n

n

n

n

n

xHy

xxx

x

hhh

hhh

hh

yyy

y

xxx

xxx

hhhhhh

hhhhh

hhh

yy

yyy

N

HN

H

H

H

HN

K

K

K

H

H

H

K

K

=

=

=

+−−

+−

1

2

2

01

0

1

01

01

1

2

1

1

2

2

1

0

011

011

011

01

01

0

1

2

1

0

...

......

48Univ.-Prof. Dr.-Ing.

Markus Rupp

Matrices and Vectors This form of a band-matrix description is very common,

although it does not allow for a matrix inversion. Note the index nN, describing the observation window.

Sometimes, several (two) outputs are considered simultaneously. (SIMO system with Sylvester matrix):

=

+−−

+−−

+−

+−

k

k

k

nnk

nnk

n

n

n

n

n

n

k

k

nk

k

k

nk

xxx

xx

gg

gggg

hh

hhhh

zz

zy

y

y

HN

HN

G

G

G

H

H

H

N

N

1

2

3

2

01

01

01

01

01

01

1

1

1

1

49Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.6 The sequence xk

for k=0,1,...,5 is transmitted via a linear system H=h0+h1q-1 (e.g. wireless channel):

knk

k

k

k

k

k

k

k

k

k

xHyxx

xx

hhhh

hhhh

hh

yyyyy

xxxxxxx

hhhh

hhhh

hhhh

h

yyyyyyy

N 5

5

4

1

01

01

01

01

01

5

4

3

2

1

6

5

4

3

2

1

0

01

01

01

01

01

01

0

6

5

4

3

2

1

0

0

=

+

+

+

+

+

+

+

+

=

=

=

=

Zero initials, steady-state consideration

50Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space DescriptionGer.: Zustandsraumdarstellung The most important matrix description is the state

space form (Ger.: Zustandsraumdarstellung) Consider the canonical (Ger.: kanonisch) transfer

function of an IIR filter using operators:

We select filter order p=max(nA,nB-1) Here, either ap or bp can be zero, but not both.

( ) pp

ppo

nn

nno

qaqaqbqbb

qaqaqbqbb

qHA

A

B

B

−−

−−

−−

+−−

−−

+++

+++=

+++

+++=

...1...

...1...

11

11

11

11

111

51Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description Consider the following signal flow graph of an IIR

filter:

q-1 q-1 q-1 q-1Σ Σbp

bp-1b0

ap

ap-1

a1

-xk

zk(p)

zk(p-1)

zk(1)zk

(2)

yk

52Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description The variables zk

(1)..zk(p) define the state

(=p individual states) of the system at time instant k.

Due to the delay chain we have: zk+1

(1) =zk(2), zk+1

(2) =zk(3),..., zk+1

(p-1)=zk(p)

Moreover, we find:zk+1

(p) =xk-a1zk(p)- a2zk

(p-1)-...- apzk(1)

yk= b1zk(p)+...+bp-1zk

(2) +bpzk(1)+b0zk+1

(p)

This allows for describing an input-output relation from xk to yk.

53Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description

[ ] [ ]

[ ] k

pk

pk

k

pppp

pk

pk

k

ppk

pk

pk

k

ppk

k

pk

pk

k

ppp

k

pk

k

xb

zz

z

abbabbabb

zz

z

aaaxb

zz

z

bbby

x

zz

z

aaazz

z

0

)(

)1(

)1(

1011010

)(

)1(

)1(

110

)(

)1(

)1(

11

)(

)1(

)1(

11)(1

)1(1

)1(1

,,,

10

0

10

10

+

−−−=

−+

=

+

−−−

=

−−−

−−−−

−+

−+

+

kkT

k

kkk

dxzcy

xbzAz

+=

+=+1

54Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description Solution of the equation system

Apply Z-transform w.r.t. signals:

)()()(

)()()(

zdXzZczY

zXbzZAzZzT +=

+=

kkT

k

kkk

dxzcy

xbzAz

+=

+=+1 State equation

Output equation

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

)()();()(

)(

1

1

zXdbAzIczY

zXbAzIzZzXbzZAzI

zH

T

+−=

−==−−

55Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description Advantages of this description Due to the following compact form

The properties of the system can be described separately: A (Eigenvalues of A) describe the dynamic behavior. b describes how the input can impact the state. c describes how the state impacts the output. d describes how the input impacts immediately the

output: passage (Ger.: Durchgriff).

kkT

k

kkk

dxzcy

xbzAz

+=

+=+1

=

+

k

kT

k

k

xz

dcbA

yz 1

56Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description Further Advantages: It is straight-forward to derive a description for

simulation (Matlab, Simulink). It is straight-forward to derive a hardware

description (Moore, Mealy), that is efficient to realize.

The description can be extended to linear time-variant systems.

Most applications in control area (also adaptive equalizers etc.)

57Univ.-Prof. Dr.-Ing.

Markus Rupp

State-Space Description Note: Due to the delay chain we have:

zk+1(1)=zk

(2), zk+1(2)=zk

(3),..., zk+1(p-1)=zk

(p)

Thus, we also can describe the state propagation by:

=

=

+

+

+

+

+

1

1

)3(1

)2(1

)1(1

1

k

k

k

k

k

k

k

zz

z

zzz

z

58Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.7 Let‘s consider a filter with a single pole close to

one in order to average the input signal:

)(1

)()()(

11

11111

−−

−−−−

−=

+=

+=

qXaqd

qdXqqaYqYdxayy kkk

q-1+x

xxk

d

yk

-a

Yk-1

-

)(1

)( 11

1

01

11

10

−−

+

+

+=

−==

qXqa

bqY

zaxzzby

kkk

kk

q-1Σ

b0

a1

-xk

zk

yk

59Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.7 Convertion into state-space form

=

−−

=

−=

+=

+

−−

−−

k

k

k

k

k

k

xz

ddaa

xz

baba

yz

qXaqdqX

qabqY

1

1

)(1

)(1

)(

010

11

11

11

1

01

q-1Σ

b0

a1

-xk

zk

yk

60Univ.-Prof. Dr.-Ing.

Markus Rupp

Similarity Transform One of the advantages of the state space form is

that the dynamic behavior of the system is solely defined by A.

However, it is the eigenvalues of A whose position in the unit circle defines the behavior. In general A is not of diagonal structure, but

−−−

=

− 11

1

1

aaa

A

pp

61Univ.-Prof. Dr.-Ing.

Markus Rupp

Similarity Transform Similarity transform

(Ger.: Ähnlichkeitstransformation): Assume there is a matrix T with an inverse. The following substitutions can be performed:

finding an equivalent description. Description forms in which A’ takes on a

diagonal structure or Jordan structure are of particular interest.

ddcTcbTbATTA T ==== −− ';';';' 11

62Univ.-Prof. Dr.-Ing.

Markus Rupp

Companion Form Note that this particular form of the

matrix A is called

companion form (Ger.: Kardinalform). It can be gained directly from the characteristic equation, respectively from the canonic IIR filter representation.

−−−

=

− 11

1

1

aaa

A

pp

63Univ.-Prof. Dr.-Ing.

Markus Rupp

Companion Form The relation of the matrix A to its diagonal form D with the

eigenvalues λ1,λ2,..., λm is given by the similarity transform with a Vandermonde matrix:

TD=AT

Note: A and D have identical eigenvalues λ1,λ2,..., λm.

=

−−− 112

11

222

21

321

...

...1...111

mm

mm

m

m

T

λλλ

λλλλλλλ

64Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.8: We extend our previous averaging filter by

one additional pole.

−−−−=

++=

+−−=

+

−−−

−−

k

k

k

k

k

k

kkkk

xz

z

ddadaaa

yzz

qXqaqa

dqY

dxyayay

1

12

121

12

21

1

1

2211

1010

)(1

)(

65Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.8: To determine the dynamic behavior of the

filter, we only have to analyze A:

We obtain an equivalent system by the similarity transform:

=⇒

=

−−

=

2

1

21

12

11

10

λλ

λλDT

Taa

TD

ddcTcbTbATTDA T ===== −− ';';';' 11

66Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.8: We thus can convert the original system

into an equivalent system

in which the eigenvalues show directly.

=

+

k

kT

k

k

xz

dcbA

yz 1

==

+

k

kT

k

k

xz

dcbDA

yz '

'''' 1

67Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description Extension:

Note that the same description form is possible for more than one input and/or output signal:

kkkk

kTk

T

kkkT

k

kkkk

kk

kkkk

xDzCyxx

dzc

xdxdzcy

xBzAzxx

BzA

xbxbzAz

+=

+=

++=

+=

+=

++=

+

+

;

;

)2(

)1(

)2(2

)1(1

)1(

1)2(

)1(

)2(2

)1(11

68Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description The state-space description is not limited to

linear, time-invariant systems but can equally be used for time-variant systems:

The stability of time-variant systems can in general not be solved explicitly (with the Z-transform) but often it is very much advantageous to match the solution structure (e.g., Kalman filter)

kkkkk

kkkkk

xDzCyxBzAz

+=+=+1

69Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description Solution of the time-variant system

Consider the solution of the state equation without external excitation.

Consider the following transition matrix (Ger.: Übergangsmatrix) Φ(k,j) with the property

jkkkkkk zjkzAAzAz ),(22111 Φ=== −−−−−

IkkAAAjk jkk =Φ=Φ −− ),(,...),( 21

70Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description Solution of the time-variant system

Consider the solution of the state equation with external excitation:

∑−

=

−−−−−−−−

−−−−

+Φ+Φ=

=++=

+=

1

11221221

1111

)1,(),(k

jlllj

kkkkkkkk

kkkkk

xBlkzjk

xBxBAzAAxBzAz

71Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description In the special case of a time-invariant system we

have:

With the similarity transform we can compute the eigenvalues of A. the eigenvalue with largest magnitude defines the behavior.

The general solution for yk is given by the following substitution:

jkAjk jk ≥=Φ − ;),(

kk

k

jlllkjk

kkkkk

xDxBlkCzjkC

xDzCy

++Φ+Φ=

+=

∑−

=

1

)1,(),(

72Univ.-Prof. Dr.-Ing.

Markus Rupp

Extended State-Space Description Example 1.9: Linear, time-variant system (to

decribe for example a static time invariant cable channel) with additive noise wk

Example 1.10: Linear, time-variant system(to describe a mobile that is time-variant wireless channel) with additive noise wk

kkTkk

kkk

wzxy

hzwzIz

+=

=+=+ 01 ;0

kkTkk

kkk

wzxy

hzwzAz

+=

=+=+ 01 ;0

Another wireless channel Now let us assume the channel is not dying

out:

How would the state space description look like?

73Univ.-Prof. Dr.-Ing.

Markus Rupp

=

=

++=

=++=

+

+

k

k

k

TTk

k

k

k

Tkk

k

kkkTkk

kkkk

vwz

dxBA

vwz

xIA

yz

vwzxy

hzvwzAz

010

0

;0

1

01

hzvzAz kkk =+=+ 01 ;

74Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume There are several equivalent forms to

describe linear, time-invariant systems (LTI): Convolution, multiplicative mapping in the image

domain Polynomial description Inner vector product Matrix-vector form State-space form

75Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Example: The concatenation of two LTI systems is again

an LTI system.

Convolution:

H GHG=S=GH

lkl

llkl

lk hgghs −

=−

=∑∑ ==

00

xk y(1)k y(2)

k

Convolution of LTI system is commutative.Does this property show in all descriptions?

76Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Image Domain:

S(z)=H(z)G(z)=G(z)H(z); S(s)=H(s)G(s)=G(s)H(s)

Polynomial form:

( ) ( ) ( )( )( )

( ) ( )

( )

( ) ( )( ) ( ) ( )111

...

...

......

11

110

1100100

110

110

111

++=+⊕+=

+++=

++++=

++++++=

=

−−

+−+

+−−

−−−−

−−−

HGGH

nnnn

nnnn

nn

nn

nnnnqGqH

qsqss

qghqghghgh

qgqggqhqhhqHqGqS

GH

GH

GH

GH

G

G

H

H

:Note

filter order

77Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Matrix-vector form: nG=nH=M

kMNkNMNkNMNkMNk

kNk

k

k

k

MNk

M

M

M

M

M

M

M

k

k

k

MNk

k

k

k

MNk

M

M

M

M

k

k

k

Nk

xSxGHxHGyGy

xHy

xxx

x

hhh

hhh

hh

gg

gggg

yyy

y

xxx

x

hhh

hhh

hh

yyy

y

11111)1(

1)2(

1)1(

1

2

0

0

0

0

0

0

0

)2(

)2(1

)2(2

)2(

1

2

0

0

0

0

)1(

)1(1

)1(2

)1(

...

......

...

......

...

......

+−++−++−+−

+

−−

+−

−−

====

=

=

=

Product of Töplitz matrices is commutative

78Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Matrix-vector form:

kMNkNMNkNMNkMNk

k

k

k

MNk

M

M

M

M

k

k

k

MNk

M

M

M

M

M

M

M

k

k

k

MNk

xSxGHxHGyGy

xxx

x

sss

sss

ss

xxx

x

hhh

hhh

hh

gg

gggg

yyy

y

11111)1(

1)2(

1

2

012

0

12

012

012

1

2

0

0

0

0

0

0

0

)2(

)2(1

)2(2

)2(

...

......

...

......

...

......

+−++−++−+−

−−

−−

+−

====

=

=

Product of Töplitz matrices results inTöplitz matrix!

79Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume: concatenation State-space form:

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

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

)1()1()1()1(

)1()1()1()1(1

HSystem;

GSystem;

kkk

kkk

kkk

kkk

yDzCy

yBzAz

xDzCy

xBzAz

+=

+=

+=

+=

+

+

[ ]

[ ] kk

k

kkkkkk

kk

k

kkkkkk

xDDzzCDC

xDDzCDzCyDzCy

xDBzzCBA

xDBzCBzAyBzAz

)1()2()1(

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

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

)1()2()1(

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

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

,

,

+

=

++=+=

+

=

++=+=+

[ ] kk

kk

kk

k

k

k

xDDzzCDCy

xB

DBzz

ACBA

zz

)1()2()1(

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

)1(

)1()2(

)1(

)2(

)1(

)1()2()2(

)1(1

)2(1

, +

=

+

=

+

+

Not a companion form!

Resume: concatenation We thus have either

or

80Univ.-Prof. Dr.-Ing.

Markus Rupp

[ ] kk

kk

kk

k

k

k

xDDzzCDCy

xB

DBzz

ACBA

zz

)1()2()1(

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

)1(

)1()2(

)1(

)2(

)1(

)1()2()2(

)1(1

)2(1

, +

=

+

=

+

+

[ ] kk

kk

kk

k

k

k

xDDzz

CDCy

xB

DBzz

ACBA

zz

)2()1()1(

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

)2(

)2()1(

)1(

)2(

)2(

)2()1()1(

)1(1

)2(1

''

,

''

''

+

=

+

=

+

+

Concatenation of twoLTI systems in

state-space form does not exhibit commutativity.

81Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume We considered various system descriptions and found that

the state space form

offers many advantages. In particular it can be extended towards linear time-variant systems:

=

+

k

kT

k

k

xz

dcbA

yz 1

=

+

k

k

kTk

kk

k

k

xz

dcbA

yz 1

82Univ.-Prof. Dr.-Ing.

Markus Rupp

ResumeExtended State-Space Description Example 1.9: Linear, time-variant system (to

decribe for example a static time invariant cable channel) with additive noise wk

Example 1.10: Linear, time-variant (to describe a mobile that is time-variant wireless channel) system with additive noise wk

kkTkk

kkk

wzxy

hzwzIz

+=

=+=+ 01 ;0

kkTkk

kkk

wzxy

hzwzAz

+=

=+=+ 01 ;0

83Univ.-Prof. Dr.-Ing.

Markus Rupp

LTI System Properties 1) Stability 2) Causality 3) Linear phase 4) Passivity/Activity (Allpass) 5) Minimum phase

(Ger.: Mindestphasigkeit)Maximum phase (Ger.: Maximalphasigkeit)

84Univ.-Prof. Dr.-Ing.

Markus Rupp

Bilinear Transform The following statements are true for time-discrete as well

as time-continuous considerations. We will always show the properties only for one of the two classes. The other can be easily derived.

Alternatively, this can be done also in a systematic manner. Starting from the region of convergence of the Laplace transform (left half plane), one can map it inside the unit circle.

Such mapping is achieved by the bilinear transform:

This also maps the imaginary axis onto the unit circle. It is also called the Möbius transform (August Ferdinand

Möbius (1790–1868))

212;

2tan

11

ωωω

+=Ω

Ω

=+−

= Ω

Ω ddjeej j

j

85Univ.-Prof. Dr.-Ing.

Markus Rupp

Bilinear Transform In the initial period of time-discrete

systems (60s-70s) the bilinear transform was used very frequently since it allows to directly map results from the analog domain (e.g., filter design) to the digital domain.

However, by now a multitude of purely digital methods are available, often much better than relying on the analog methods, so that the bilinear transform is rarely being applied any more.

1) Stability Theorem 1.2 [Stability]: Assuming a rational

system function (Laplace domain) of a linear time-invariant continuous time system

We call such a system (BIBO) stable if and only if Re(si)<0.

86Univ.-Prof. Dr.-Ing.

Markus Rupp

=

=

=

−=

+++

+++=

p

i

tsi

p

i i

ip

p

pp

tUedth

ssd

sasasbsbb

sH

i

1

11

10

)()(

:responseimpulsetimeingcorrespondwith...1...

)(

<

=

>

=

00

021

01)(

:stepUnit

t

tt

tU

Stability Proof: consider input signal x(t)

87Univ.-Prof. Dr.-Ing.

Markus Rupp

∑∑∫

∑∫

=

<

==

=

−≤

=≤

−==

p

i

s

i

i

p

i

s

i

ip

i

si

p

i

si

i

ii

i

esdtx

esdtxdedtx

dtxedtxthty

1

0110

10

1)(max

)(max)(max

)()(*)()(

0Resif only bounded,

i

ττ

τ

τ

ττ

Stability Now consider a closed loop system

(in the Fourier Domain)

88Univ.-Prof. Dr.-Ing.

Markus Rupp

H(jω)

G(jω)

Y(jω)

Z(jω)

X(jω)

-

Stability Theorem 1.3 [Stability of closed loop]:

The closed loop system is BIBO stable if the open loop system has:

Note: use the bilinear transform to derive the corresponding statements for time discrete systems.

89Univ.-Prof. Dr.-Ing.

Markus Rupp

1)()(max <ωωω jGjH

Stability Proof: we find that

Thus, by requiring |H(jω)G(jω)|<1 being bounded by one, the system cannot become unstable.

This is however, a weak condition; tighter conditions in the form of „if and only if“ are possible.

90Univ.-Prof. Dr.-Ing.

Markus Rupp

[ ]

)()(1)(

)()(

)()()()(

ωωω

ωω

ωωωω

jGjHjH

jXjY

jZjXjHjY

+=

−=

91Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Ger.: Kausalität

2) Causality (time-continuous)

)(2/1

)(2/1

tf

tf

−−

)(2/1

)(2/1

)(

tf

tf

tf

92Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality We have:

[ ]

[ ])()0()()sign()(

)()(21)(

)()(21)(

)()()(0),(2)(2)(

tftfttf

tftftf

tftftf

tftftfttftftf

oe

o

e

oe

oe

δ+=

−−=

−+=

+=>==

93Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality We have:

[ ]

[ ]

[ ]

)()(

)sin()()cos()(

)exp()()(

)exp()()(

)exp()()(21

)exp()(21)(

ωω

ωω

ω

ωω

ωωωωπ

ωωωπ

jjFjF

dtttjfttf

dttjtftf

dttjtfjF

dtjjjFjF

dtjjFtf

IR

oe

oe

IR

+=

−=

−+=

−=

+=

=

∞−

∞−

∞−

∞−

∞−

FR is even and FI is an odd function in ω

94Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Thus, we find for the amplitude and phase

response (Ger.: Amplituden- und Phasengang):( )

[ ]

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

)()()()()(

)()(

)sin()()cos()(

)(exp)()(

22

ωφωωωωφ

ωωωωω

ωω

ωω

ωφωω

−−=

==

−=+==

+=

−=

=

∫∞

∞−

jFjFjF

AjFjFjFA

f(t)jjFjF

dtttjfttf

jAjF

R

I

IR

IR

oe

atanarc

have we , functions valued-real to ourselves Limiting

A is an even function,

Φ odd

95Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Thus:

∫∫

∫∫

∞∞

∞−

∞∞

∞−

−=

=

−=−=

==

0

0

0

0

)sin()(1)(

)cos()(1)(

)sin()(2)sin()()(

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

ωωωπ

ωωωπ

ωωω

ωωω

dtjFtf

dtjFtf

dtttfdtttfjF

dtttfdtttfjF

Io

Re

ooI

eeR

96Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality For causal functions

we have, due to

f(t)=0 for t<0:

∫∞

−=

=

<−= >=

=

0

0

)sin()(2

)cos()(2)(

0for )()(else;0

0;)(2)(2)(

ωωωπ

ωωωπ

dtjF

dtjFtf

ttftf

ttftftf

I

R

oe

oe

)()0()sign()()(: tfttftf oe δ+=precise more

97Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality One more substitution delivers:

The real and the imaginary part of causal, linear systems is uniquely related to each other.

It is thus sufficient to know either the real or the imaginary part.

dtdttjFjF

dtdttjFjF

RI

IR

∫∫

∫∫∞∞

∞∞

−=

−=

00

00

')sin()'cos()'(2)(

')cos()'sin()'(2)(

ωωωωπ

ω

ωωωωπ

ω

98Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Consider:

( )

)2/()0(0

0

)('')'(2)(

'')'(2)(

4)0(1)()(

21)(

21)(

21)()(

4)0(1)(

21*)()(

.21)0();()0(

21)()()(

π

ωωωω

πω

ωωωω

πω

πωωω

πωωωω

πωωπδ

πωω

δ

f

RI

R

RI

IRIRIR

FdjFjF

djFjF

fj

jjFjFjFjjFjjFjF

fj

jFjF

tUtftUtftf

∞+−

=

−−=

+

∗+++=+

+

+=

==+=

∫∞

that note

convolution

<

=

>

=

00

021

01)(

:stepUnit

t

tt

tU

99Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Alternatively, this can be reformulated to:

This particular form (convolution with 1/(ω-ω‘)) is called the Hilbert-transform.

∫∫

∫∫∞∞

∞−

∞∞

∞−

−−=

−−=

−+∞=

−+∞=

022

022

'')'(2'

')'(1)(

''

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

ωωωω

πωω

ωωω

πω

ωωωωω

πω

ωωω

πω

djFdjFjF

djFFdjFFjF

RRI

IR

IRR

100Univ.-Prof. Dr.-Ing.

Markus Rupp

CausalityRaymond Paley (1907–1933) Norbert Wiener (1894–1964) Theorem 1.4 (Paley-Wiener): Let H(jω)=A(ω)exp(jφ(ω)). If

an even, non negative amplitude function A(ω) with limited energy

has the property:

then and only then the transfer function H(jω) describes a causal system.

If this is true, a non-unique phase function φ(ω) exists, so that H(jω) describes a causal system.

∞<∫∞

∞−

ωω dA )(2

∞<+∫

∞−

ωωω

dA

21))(ln(

Quick Sketch of Proof Paley Wiener base their concept on the fact that

the Fourier Transform F(jω) of holomorphic functions f(t) of finite energy (square integrable)

is bounded by:

101Univ.-Prof. Dr.-Ing.

Markus Rupp

( )( )( )

( )∞<≤

+

+≤

+

∞<<≤

∫∞

∞−12

22

1ln

11ln

ln

0;

cdjF

BjF

BjF

BejF B

ωω

ω

ωω

ωω

ωω

ω ω

Note that mathematically it is much more envolved…

..as it holds in both directions

( ) ∞<≤∫∞

∞−0

2 cdjF ωω

102Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.11 Consider the low pass of 1st order hLP(t)

( ) ( )∫∫∫

∫∫

∞−

∞−

∞−

∞−

∞−

∞−

∞<+=+

+=

+

+=

+

∞<==+

=

+=→

+=

cdcdcdA

cxcdc

dA

cA

cjjH

LP

LP

LPLP

1ln41

11ln

21

11

1ln

1)(ln

tana1

1)(

11)(

11)(

2

22

2

22

2

222

22

πωωω

ωωωω

ωω

πωω

ωω

ωω

ωω

103Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.12 Consider an ideal low pass hIDEAL(t)

( ) ( ) ( )

∞=∞

>

=

∞−

∞−

∫∫∫

∫∫

++

+=

+

∞<==

<

=

)0ln( since,

2

0)1ln( since ,0

22

2

1)(ln

1)(ln

1)(ln

2)(

else;0;1

)(

L

L

L

L

L

dA

dA

dA

ddA

A

L

L

ωω

ω

ω

ω

ω

ωωω

ωωω

ωωω

ωωωω

ωωω

104Univ.-Prof. Dr.-Ing.

Markus Rupp

Example 1.12

)(thIDEAL

)(thLP

105Univ.-Prof. Dr.-Ing.

Markus Rupp

Causality Note that this theorem can be turned around: if

there is a causal system, then it has the following properties…

The theorem exists in corresponding form also for time-discrete systems:

Tightly connected with this problem is the problem of factorization: Given an amplitude function A(Ω). Can it be separated A2(Ω)=H(exp(jΩ))H(exp(-jΩ)), so that hk is causal?Answer will follow later…

( ) ∞<ΩΩ∫−

π

π

dA )(ln

106Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear PhaseGer.: Linearphasigkeit 3) Linear phase (time-continuous) Consider firstly the

following basic relations from Fourier transform:

Narrow band system (Ger.: Schmalbandsystem): (∆T ∆B<1)( ) ( )

[ ]( )[ ]( )

[ ]( )[ ]( )

<++−−<−−+

=

<+++−<−−+

=

goooo

goooo

goooo

goooo

jj

A

jj

A

jAjH

ωωωωφωωωφωωωωφωωωφ

ω

ωωωωφωωωφωωωωφωωωφ

ω

ωφωω

;)(')()(exp;)(')()(exp

)(

;)(')()(exp;)(')()(exp

)(

)(exp)(

0

0

( )

( ) ( )

( )( ) ))(exp()()exp()(

)exp()(

)()(21)cos()(

)()exp()(

ooooo

oo

ooo

oo

tjjFtjttftjjFttf

jFjFttf

jFtjtf

ωωωωωωω

ωωωωω

ωωω

−−−⇔−−⇔−

++−⇔

−⇔

107Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Phase Assume narrow band excitation X(jω),

transmitted at ω0 and -ω0, thus X(j(ω-ω0)) and X(j(ω+ω0)).

ω

ω

)()( 00 ωω AA =− )( 0ωA

0ω− 0ω

)( ωjX( ) 2/)( 0ωω +jX ( ) 2/)( 0ωω −jX

108Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Phase Let Y(jω) be the output of such system:

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

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

( )

++=

+−++++

=

+−−++−+−

×=

o

oooo

ooo

oooo

oooo

oooo

o

ttxA

tjtxtjtxAty

jjXjjX

AjY

ωωφωωφω

ωωφωφωωφωφω

ωφωωωφωωωφωωωφωω

ωω

)(cos)(')(

)(exp)(')(exp)('

2)()(

)(')()(exp)()(')()(exp)(

2)()(

109Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Phase

Group delay (Ger.: Gruppenlaufzeit)

Phase delay (Ger.: Phasenlaufzeit)

Attention: Sometimes these terms are being used with a „+“ sign, if the initial term is A(ω)exp(-jφ(ω))!

o

oP

oG

T

o

oo

T

oo

T

T

ttxAty

o

P

G

ωωφ

ωφωωφ

ωωφωωφω

ωω

)(

)(')(

)(cos)(')()(

−=

−=∂

∂−=

+

+=

=

−−

110Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Phase The advantage of the narrow band assumption is

that we do not need a convolution. The group delay defines how fast a small group of

energy parts around ωo runs through the linear system.

If some parts at different frequencies ωo run with different speed, then the signal is distorted.

Required condition: In order to obtain an undistorted (only delayed) signal, the group delay needs to be constant. If this is the case, the system is said to be of linear phase.

111Univ.-Prof. Dr.-Ing.

Markus Rupp

Linear Phase Example 1.13: time discrete system with linear

phase

Consider a time discrete system H(exp(jΩ)), constructed out of:

H(exp(jΩ))= G(exp(jΩ)) G(exp(-jΩ)) exp(-j∆Ω)

Let G(exp(jΩ))=1+a exp(-jΩ), ∆=1 then: H(exp(jΩ))=a+(1+a2) exp(-jΩ)+a exp(-2jΩ)

=exp(-jΩ) [1+ a2+2acos(Ω)]= exp(-jφ(Ω)) A(Ω).

Symmetrical filter have linear phase.

Ω−=ΩΦ→ΩΦΩ )())(exp()( jA

112Univ.-Prof. Dr.-Ing.

Markus Rupp

Passivity and Activity 4) Passive/active systems (Allpass, time discrete) Definition 1.2: A system is called passive, when there

is less energy at its output than at its input (after some memory effect of course) for every input signal.

This is equivalent to require:

( ) 1max <ΩΩ

jeH

113Univ.-Prof. Dr.-Ing.

Markus Rupp

Passivity and Activity Definition 1.3: If for some input signal the energy at

the output is larger than at the input, the system is called active.

Example:

k

T

kk

kTkk

kk

yyy

xxx

xHy

=

=

=

2

2

2

2

output at energy with compare and

input at energyconsider

( ) 1max >ΩΩ

jeH

114Univ.-Prof. Dr.-Ing.

Markus Rupp

Passivity and Activity Question: Can it happen that for some

sequences xk, the system attenuates but for others it amplifies?

Answer: yes Solution by worst case consideration

?1maxmax 2

2

2

212

2

>==

k

k

xk

Tk

k

T

kx

x

y

xx

yykk

115Univ.-Prof. Dr.-Ing.

Markus Rupp

Allpass Example 1.14: Consider a linear, time-invariant, time-

discrete system : Y(ejΩ)=H(ejΩ)X(ejΩ).

Assume the Maximum(Minimum) of the transfer function occurs at a specific frequency Ω+(Ω-). Then, all signals that are not at Ω+(Ω-) will come out with less(more) energy. Thus, in order to have identical energy at in and output, for all signals, we must have:

A system that emits as much energy as it absorbs, thus not consumes any energy, is called an allpass.

( )( )( ) ( ) ( ) 1minmaxmax

222

0 === ΩΩ

ΩΩΩ

Ω

≠Ωjj

j

j

eX eHeHeXeY

j

Allpass Definition 1.4 (Classic Allpass): A

linear time-invariant (finite order) system is called allpass, if the following holds:

or, alternatively, if

116Univ.-Prof. Dr.-Ing.

Markus Rupp

( )( )( ) ( ) ( ) 1minmaxmax

222

0 === ΩΩ

ΩΩΩ

Ω

≠Ωjj

j

j

eX eHeHeXeY

j

( ) Ω=Ω allfor ;12jeH

117Univ.-Prof. Dr.-Ing.

Markus Rupp

Allpass Example 1.15: Let 1/H*(ejΩ) be a causal, stable

system. Consider the following linear time-invariant system:

Obviously, this is an allpass. H*(ejΩ)=h0*+h1*e-jΩ +…+hM-1*e-j(M-1)Ω

=h0 +h1 e-jΩ +…+hM-1 e-j(M-1)Ω=H(e-jΩ), if hi from R!

( ) ( )( )

( )( )

( ))(2exp)(exp)(

)(exp)()exp()exp()exp( *

Ω=Ω−Ω

ΩΩ=

ΩΩ

φφ

φ

jjA

jAjHjHjG

118Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Check whether the following system is an allpass.

1

1

1

1

53145

43

6,0125,175,0)(

+

+=

++

=q

q

qqqH

1

1

6,016,025,1 −

++

=q

qno allpass

119Univ.-Prof. Dr.-Ing.

Markus Rupp

Resume Check with help of the state space form, whether

the following system is an allpass.

1

1

6,0125,175,0)( −

++

=q

qqH

−=

−−

=

+

k

k

k

k

k

k

xz

xz

babba

yz

75,08,016,01

0101

11

( ) ( )22

22221

5625,175,08,06,0

kk

kkkkkk

xzxzxzyz

+=

+++−=++

2

0

20

2

0

21 5625,1 k

N

kk

N

kN xzyz ∑∑

==+ +=+

no allpass

1

1

6,016,025,1 −

++

=q

q

Allpass Theorem 1.5: 1) The poles and zeros of an allpass

system are symmetrical with respect to the unit circle.

2) The phase of an allpass is monotonically decreasing, or equivalently,

120Univ.-Prof. Dr.-Ing.

Markus Rupp

0)(' <Ωφ

A1/A*

Allpass Proof: Consider system of order one:

121Univ.-Prof. Dr.-Ing.

Markus Rupp

1)exp()exp()exp())(exp(

)exp()exp()exp(

)exp(1)exp())(exp(

11

)(

*

**

***

=−Ω−Ω−

Ω−=Ω

−Ω−Ω−

Ω−=−Ω

−Ω=Ω

−−

=−

−=

ajajjjG

ajajj

ajajjG

azza

aza

zazG

Allpass Extensions to any other finite order

allpass is straightforward. Note that we did not say that having

zeros and poles symmetrical is sufficient! (only necessary!)

Now to the second property…

122Univ.-Prof. Dr.-Ing.

Markus Rupp

123Univ.-Prof. Dr.-Ing.

Markus Rupp

Allpass Example 1.16: Consider the rational transfer

function

))(2exp()exp()exp())(exp(

*

Ω=Ω−Ω−−

φjjajajG

Circle of Apollonius:The angle ψ(Ω) decreases (becomes more negative), ifmoving from Ω=-π to +π.This property also holds if several of such systems are concatenated.

N:a=Ra+jIa

P

M:1/a*

Ω=-π

)(Ωψ

)(Ωϕ

124Univ.-Prof. Dr.-Ing.

Markus Rupp

Circle of ApolloniusApollonius of Perga [Pergaeus] ca. 262 BC - 190 BC

M:1/a*=(Ra+jIa)/(Ra2+Ia

2)

N:a=Ra+jIa

Consider Points P, so that PM/PN=const.

Such points are allon the (unit)circle

P

))(2exp(

/1)exp()exp(

))(exp(

)(()

**

*

Ω=

−−

−=

Ω−Ω−−

=

Ω

Ω=

Ω

ΩΩ−

φψ

j

eaeaea

jaja

jG

arc

j

jj

)(ΩΨ

from Ω=-π (P=(-1,0)) to Ω=π (P=(1,0))Ψ(Ω) (and φ(Ω)) decreases.

Ω−= jeaPN

125Univ.-Prof. Dr.-Ing.

Markus Rupp

Circle of Apollonius( )

( ) ( )

( ) ( )1:

))sin()cos((21))sin()cos((2

)sin();cos(:

222

222222

2

22

2

22

222

2222

=+−++

+−+++=

==

+

−+

+

−+−=

=

+

−++

−+−=

RIRRIRR

IRRIRRIRk

RIRRIR

IIIR

RR

IIRRk

k

IRIIj

IRRR

IIjRR

PMPN

aaaa

aaaaaa

pp

aa

ap

aa

ap

apap

aa

ap

aa

ap

apap

thus

Assume

αααα

αα

aa jIRa +=

126Univ.-Prof. Dr.-Ing.

Markus Rupp

Allpass a=0,5+0,3j (still Example 1.16)

Α(Ω)ej2φ(Ω)

=ej2φ(Ω)

Allpass Thus a single stage allpass (order one)

causes the phase to decrease monotonically.

Equivalently, Allowing for multiple stages does not

change the argument.

127Univ.-Prof. Dr.-Ing.

Markus Rupp

0)(' <Ωφ

128Univ.-Prof. Dr.-Ing.

Markus Rupp

Minimum Phase SystemsGer.: Mindestphasen-Systeme

5) Minimum phase systems

Question: Is there only a unique relation between real and imaginary part of linear causal systems or does it also exist between amplitude and phase.

Obviously, this is not the case since we just have seen that an allpass does not change the amplitude but the phase only.

Minimum Phase System Consider simple polynomial

N(ejΩ)=a-e-jΩ

zero is at: a-z-1=0zo=1/a Consider simple polynomial

N*(ejΩ)=a*-ejΩ

zero is at: a*-z=0zo=a* Building the conjugate

complex, mirrors the zeroes

129Univ.-Prof. Dr.-Ing.

Markus Rupp

a*

1/a

130Univ.-Prof. Dr.-Ing.

Markus Rupp

Minimum Phase Systems Consider the following rational transfer function:

))(exp())(exp())(exp(

))(exp())(exp())(exp(

))(exp())(exp(

))(exp())(exp())(exp())(exp(

))(exp())(exp())(exp(

))(exp())(exp())(exp(

*

*

ΩΩ=Ω

ΩΩ−Ω−

Ω=

ΩΩΩ

ΩΩ

=

ΩΩΩ

=ΩΩ

jHjHjD

jNjNjN

jNjD

jNjNjNjN

jDjNjN

jDjNjH

ma

io

o

o

io

o

o

oi

Allpass Minimum phase systemAll poles and zeroes inside the unit

circle

If coeffs are real-valued

N*(ejΩ)=N(e-jΩ)

131Univ.-Prof. Dr.-Ing.

Markus Rupp

Minimum Phase Systems Example 1.17: Phase function and its derivative of a

minimum phase system (from Example 1.16) H(z)=a*-z-1.

Φm‘(Ω)> Φ‘(Ω)

132Univ.-Prof. Dr.-Ing.

Markus Rupp

Minimum Phase Systems Definition 1.5: A time discrete rational minimum phase

system Hm(exp(jΩ)) with phase φm(Ω) has all zeroes and poles inside the unit circle.

Theorem 1.6: Given two systems with the same amplitude function but different phase function, a minimum phase system has the property:

Proof: Consider a second rational transfer function H(exp(jΩ)) with phase φ(Ω) and the condition:

;))(exp())(exp( Ω=Ω jHjHm

)(')(' Ω<Ω mφφ

133Univ.-Prof. Dr.-Ing.

Markus Rupp

Minimum Phase Systems Then we must have

an allpass. For an allpass we have φa‘(Ω)<0. Furthermore, we must have:

φa(Ω)=φ(Ω)-φm(Ω) and φa‘(Ω)=φ‘(Ω)-φm‘(Ω)<0

Thus: φ‘(Ω)<φm‘(Ω).

))(exp())(exp())(exp(

ΩΩ

=ΩjHjHjH

ma

Sampling Theorem

134Univ.-Prof. Dr.-Ing.

Markus Rupp

135Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling TheoremGer: Abtastung Question: Under which circumstances can a

continuous function f(t) uniquely be represented by its samples f(kT)=fk or f(tk) and thus be generated by its samples?

Theorem 1.7 (general sampling theorem): A continuous function f(t) can be represented uniquely by its equidistant samples f(kT), if it is band limited with at least |ω|<π/T .

136Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Proof: Consider equidistant sampling with period T. There

must be an interpolation function p(t) such that f(t) can be recovered from fk.

∑∞

−∞=

−∞=

−=

−=

kk

k

kTtpfT

kTtTpkTftf

)(

)()()(

137Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Consider the Fourier Transform:

)(*

)exp()(

)exp()()(

)()(

ω

ωω

ωωω

jF

kk

kk

kk

TjkTfjP

TjkjTPfjF

kTtTpftf

−∞=

−∞=

−∞=

−=

−=

−=

∑∑

∑∞

−∞=

−∞=

−∞=

+=−=

−=

kk

kk

TkjFTjkkTTfjF

kTtTftf

πωωω

δ

2)exp()()(

)()(

*

*

Only supporting functions!!

138Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Why is this true?:

( )

( )

( ) ∑∑ ∫

∫ ∑

∫ ∑

∑ ∫

∑∑

−∞=

−∞=

∞−

∞−

−∞=

∞−

−∞=

−∞=

∞−

∞−

−∞=

−∞=

+=−−=

−−=

−=

−=

==

+=−=

nn

n

k

k

k

nk

TnjFdTnjF

dTnjF

dTjkTjF

TjkdTjkjFTjF

dkTjjFfkTf

TnjFTjkkTTfjF

πωωπωωδω

ωπωωδω

ωωωπ

ω

ωωωωπ

ω

ωωωπ

πωωω

2'/2')'(

'/2')'(

')'(exp2

)'(

)exp(')'(exp)'(21)(

')'(exp)'(21)(

2)exp()()(

*

*

Fourier series(Poisson sum theorem)

139Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Thus we have:

Practically this can be solved if F(jω) is band limited with |ω|<π/T. In this case F(jω)= F*(jω) for |ω|<π/T and P(jω)=1 in this region.

P(jω)=1 in |ω|<π/T and zero elsewhere is an ideal lowpass.

∑∞

−∞=

+=

=

k TkjFjP

jFjPjF

πωω

ωωω

2)(

)()()( *

140Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem ∑∞

−∞=

+=

=

k TkjFjP

jFjPjF

πωω

ωωω

2)(

)()()( *

141Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Thus we have:

This equation also offers other solutions. Consider a band-limited function F(jω) in |ω+k2π/Τ|<π/T and a bandpass P(jω) for this region.

−=

=

=⇔

∑∞

−∞=

)(sinc)(

sinc1sin

)()(

kTtT

ftf

tTTt

tTtpjP

kk

π

ππ

π

ω

142Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem∑

−∞=

+=

=

k TkjFjP

jFjPjF

πωω

ωωω

2)(

)()()( *

∑∞

−∞=

+=

=

−+

+

k TkjFjP

jFjPT

jFT

jF

πωω

ωωπωπω

2)(

)()(22*

143Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem

−= ∑

−∞=

)(sinc)( kTtT

ftfk

Note, the interpolation

allows for „re-sampling“:

−= ∑

−∞=

)(sinc)( 11 kTmTT

fmTfk

144Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Assume the function f(t) is not directly

observable but only through linear filtering by a system H(jω). Consider the sampled output values:

Is it possible to conclude from gk by interpolation to f(t)?

∫∞

∞−= ωωωω

πdkTjjHjFkTg )exp()()(

21)(

∑∞

−∞=

−=k

k kTtTpgtf )()(

145Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Consider again the Fourier transform:

The equation can be solved by selecting a special bandlimited version for P(jω) = 1/H(jω):

∑∞

−∞=

−∞=

+

+=

+=

k

k

TkjH

TkjFjP

TkjGjPjF

πωπωω

πωωω

22)(

2)()(

=<=⇔= ∫

− else;0

||;)(/1)()()exp(

2)( σπωωωω

ωω

π

π

πT

jHjPdjH

tjTtpT

T

146Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Generalization:

H1(jω) H2(jω) Hm(jω)

f(t)

g1(t) g2(t) gm(t)

∫−

=

==σ

σ

ωωωωπ

ωωω

dtjjFjHtg

mkjFjHjG

kk

kk

)exp()()(21)(

,...,2,1;)()()(

147Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Possible sampling instances:

Have m sampling instances in period T.T

T/m

148Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem What is the impact for the bandwidth of F(jω)?

|F(jω)|

ω

BF=2π/[mT]

BG=2π/T=BF/m

149Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Is it possible by interpolation of gk(nT) ; k=1,2,...,m in the

interval T to reconstruct f(t)?

Example 1.18: Consider Hk(jω) interpolators with Hk(jω)=exp(-jkT/m ω); k=1,2,…,m. This is the standard equidistant sampling.

Example 1.19: Consider Hk(jω) interpolators with non equidistant sampling delays.

Example 1.20: Consider Hk(jω) to be m arbitrary linear systems.

∑∑=

−∞=

−=m

kkk

nnTtpTnTgtf

1)()()(

150Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Reconstruction:

We are seeking interpolators, so that the original f(t) reappears:

Note that:

−=

−+++

−++

−+

=

+++

++

+

=+++

tmT

jjPmT

jHjPmT

jHjPmT

jH

tT

jjPT

jHjPT

jHjPT

jH

lyrespectivejPjHjPjHjPjH

mm

mm

mm

)1(2exp)()1(2...)()1(2)()1(2

...

2exp)(2...)(2)(2

1)()(...)()()()(

2211

2211

2211

πωπωωπωωπω

πωπωωπωωπω

ωωωωωω

)()()(

)()()(1

ωωω jFjHjG

nTtTpnTgtf

kk

m

kkk

n

=

−= ∑∑=

−∞=

−−∈T

mT

m ππω )2(,

152Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling TheoremAthanasios Papoulis (1921 – April 25, 2002) was a Greek-engineer and applied mathematician.

Theorem 1.8: An interpolation for f(t) from the samples gk(nT) by

is possible, if the following system of equations can be solved:

∑∑+−

=

−∞=

==

−=

TTm

Tm

kk

m

kkk

n

mkdtjtYTtp

nTtpnTgtf

ππ

π

ωωωπ

22

2

1

,..,2,1;)exp(),(2

)(

)()()(

153Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem

=

−+

−+

+

+

+

Ttmj

Ttj

Ttj

tjY

tjYtjYtjY

TmjH

TmjH

TjH

TjH

TjH

jHjHjH

mm

m

π

π

π

ω

ωωω

πωπω

πω

πωπω

ωωω

2)1(exp

22exp

2exp1

),(

),(),(),(

2)1(...2)1(

22

22)(......)()(

3

2

1

1

1

21

21

Here, t is arbitrary and ω needs to be in theinterval [-mπ/T,-mπ/T+2π/T].

154Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Proof: The signal gk(t-nT) is on the one hand the response

of the system Hk(jω) when excited with f(t-nT) and on the other hand the response to the system Hk(jω) exp(jnωT) on the excitation f(t). Since we have

Then we have for the special excitation f(τ)=exp(jω(t+τ)):

And due to the previous theorem, their outputs are equal.

∑ ∑

∑ ∑ ∫

∑ ∑ ∫

=

−∞=

=

−∞=

∞−

=

−∞=

∞−

−=

−=

−=+

m

k nkk

m

k nkk

m

k nkk

TjnnTptjjH

nTpdTjnjFjH

nTpdTjnjGtj

1

1

1

)exp()()exp()(

)(')'exp()'()'(21

)(')'exp()'(21))(exp(

ωτωω

τωωωωπ

τωωωπ

τω

∑∑=

−∞=

−=m

kkk

nnTtpnTgtf

1)()()(

155Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem It is thus sufficient to prove the identity

Consider for this the system of equations: The matrix entries Hk(j(ω+m2π/T)) are independent

of t or τ. The right hand side is periodic with duration T. Thus we must have: Yk(jω,τ)=Yk(jω,τ-T) and

thereforeYk(jω,τ)=Yk(jω,τ-mT).

∑ ∑=

−∞=

−=m

k nkk TjnnTpjHj

1)exp()()()exp( ωτωωτ

156Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Furthermore, we have:

−∞=

+−

+−

−=

+

=−=−

==

nkk

TTm

Tm

kk

TTm

Tm

kk

nTjnTpjY

TTm

Tm

mkdnTjYTnTp

mkdtjtYTtp

)exp()()exp(),(

222

,..,2,1;))(exp(),(2

)(

,..,2,1;)exp(),(2

)(

22

2

22

2

ωτωττω

πππ

ωτωτωπ

τ

ωωωπ

ππ

π

ππ

π

:,--interval inseriesFourier :tionInterpreta

:Thus

157Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Substitution in the system of equations (arbitrary

row) results into

and thus the following

is correct as well.

∑ ∑

=

−∞=

=

−=

=

m

k nkk

m

kkk

TjnnTpjH

jYjHj

1

1

)exp()()(

)exp(),()()exp(

ωτω

ωττωωωτ

∑∑=

−∞=

−=m

kkk

nnTtpnTgtf

1)()()(

158Univ.-Prof. Dr.-Ing.

Markus Rupp

Sampling Theorem Interpretation: In order to be solvable, the

determinant of the system must be unequal to zero.

This is for example satisfied, if Hk(jω)=(jω)k. This is also satisfied if Hk(jω)=exp(jωαk),

as long as |αk|<T/2 and all αk are different, corresponding to f(nT+αk).

This is also satisfied if all functions Hk(jω) are band-passes (low-passes), which together sum up to the required bandwidth BF.

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