Signal Processing 1 Part I: Basics Univ.-Prof.,Dr.-Ing. Markus Rupp WS 17/18 Do 14:00-15:30EI3A, Fr 8:45-10:15EI4 LVA 389.166 Last change: 12.10.2017
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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
=Ω
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−−
=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
=Ω
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
kπ
Note, the interpolation
allows for „re-sampling“:
−= ∑
∞
−∞=
)(sinc)( 11 kTmTT
fmTfk
kπ
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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