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Signal Processing 1 - nt.tuwien.ac.at · Resume Consider symmetric impulse response: h 0 =a, h 1...
Transcript of Signal Processing 1 - nt.tuwien.ac.at · Resume Consider symmetric impulse response: h 0 =a, h 1...
Signal Processing 1
Signal Spaces
Univ.-Prof.,Dr.-Ing. Markus RuppWS 18/19
Th 14:00-15:30EI3A, Fr 8:45-10:15EI4
LVA 389.166
Last change: 22.11.2018
2Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider the amplitude function of a filter:
A(Ω)=1+cos(Ω)+2cos(2Ω) Question 1: is this a symmetric filter? Answer: potentially yes, but only if
dφ(Ω)/dΩ =const. Question 2: What is the impulse response
if this is a causal symmetric filter? Answer: h0=1; h1=0,5; h2=1; h3=0,5; h4=1.
or further delayed versions of this
3Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider symmetric impulse response: h0=a, h1=1, h2=a. Which values of |a|<0,5 deliver a linear phase filter?
h0+h1exp(-jΩ)+h2exp(-2jΩ) =exp(-jΩ)[h0exp(jΩ)+h1+h2exp(-jΩ)]
=exp(-jΩ)[a exp(jΩ)+1+a exp(-jΩ)] =exp(-jΩ) [1+2a cos(Ω)]
=exp(jφ(Ω)) A(Ω) φ(Ω)=-Ω
What happens for |a|>0,5? A(Ω)=|1+2a cos(Ω)|Thus, for each value of a from R we have a
linear phase system but the phase function may jump.
4Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider symmetric impulse response: h0=a, h1=1, h2=-a. For which values of a is this a linear phase system?
h0+h1exp(-jΩ)+h2exp(-2jΩ) =exp(-jΩ)[h0exp(jΩ)+h1+h2exp(-jΩ)]
=exp(-jΩ)[a exp(jΩ)+1-a exp(-jΩ)] =exp(-jΩ) [1+2aj sin(Ω)]
Thus, for every value of a from C with Real(a)=0, this is a linear phase system. Since we have:
exp(-jΩ) [1+2aj sin(Ω)] =exp(jφ(Ω)) A(Ω) φ(Ω)=-Ω
5Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Question: Is the sampling of a continuous function
a linear operation? Answer: Yes
[ ] )2()1()2()1(
)1()1(
)()()]([
kk
k
ßfftßftfSftfS
+=+
=
αα
6Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Question: Is the interpolation of sampled values a
linear operation? Answer: Yes
(1) (2) (1) (2)
(1) (2)
(1) (2)
( ) ( )
( ) ( )
( ) ( )
( ) ( )
k kk
k k k kk k
k kk k
I f T f p t kT f t
I f ßf T f p t kT T ßf p t kT
T f p t kT ß T f p t kT
f t f t
α
α α
α
α β
∞
=−∞
∞ ∞
=−∞ =−∞
∞ ∞
=−∞ =−∞
= − =
+ = − + −
= − + −
= +
∑
∑ ∑
∑ ∑
7Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Question: Is the conversion from fk to f(t)
and vice versa unique under equidistant sampling?
Answer: Yes
What is the answer for equidistant sampling at arbitrary start point f(kT+To)?
)(
)()(
kTff
kTtpfTtf
k
kk
=
−= ∑∞
−∞=
8Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Note: Many different sequences fk
(i) ;i=1,2,… can lead to the same time-continuous function f(t). However, always a different interpolator is required.
Example:
)2/();(
)(
)()(
)2()1(
)2()2(
)1()1(
TkTffkTff
kTtpfT
kTtpfTtf
kk
kk
kk
+==
−=
−=
∑
∑∞
−∞=
∞
−∞=
Resume Given f(kT)=fk, how do we get
gk=g(kT)=f(kT+T/2)?
9Univ.-Prof. Dr.-Ing.
Markus Rupp
−+=
+= ∑
∞
−∞=
mTTkTT
fTkTfgm
mk 2sinc
2π
10Univ.-Prof. Dr.-Ing.
Markus Rupp
Learning goals Vector Spaces and Applications of Linear Algebra
in Signal Processing (6Units, Chapter 2) Metric spaces, sequences, Cauchy-sequences, supremum,
infimum, sparsity (Ch 2.1) Groups, Vector spaces, linear combination, linear
independence, basis and dimension, orthogonality,blind channel estimation (Ch 2.2, 2.7)
Norms and normed vector spaces (Ch 2.3-2.5) Applications of norms: robustness descriptions,
feedback systems with nonlinear elements: the small gain theorem
Inner vector products and inner product spaces, Hilbert and Banach spaces (Ch 2.8-2.9)
Induced norms, Cauchy-Schwarz inequality: matched filter and correlation coefficient, time-frequency uncertainty (Ch 2.6)
Sets, Spaces, and Vectors A set is a collection of distinct objects. A vector is an n-dimensional set of n ordered
elements. A space is a set with “some added structure”.
11Univ.-Prof. Dr.-Ing.
Markus Rupp
12Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.1: Metric
Code bookCode word
1,0==⊂∈
BYBCx n
Decoder has to decide which one among the received y is allowed and most probable!
13Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.1: Metric
Code Rate:
)(log2 Mk =# Info bits
# Code bits kn ≥
nM
nkR )(log2==
−),( kn Codexy
k2=
14Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.1: Metric Decision of the „most probable symbol“
based on metric: Compare “distance” of received y to all possible,
allowed x. Symbol x with smallest (Hamming-) distance dH
is interpreted as correct.
( ) ∑ =⊕=
→×=n
i iiH
nnH
yxyxd
NBBdB
1
0
,
:,1,0
15Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.2 We transfer two possible signal forms from two
different sensors: f(t) and g(t). The receiver has to find out whether the
distorted received signal form r(t) is closer to f(t) or to g(t).
We need a distance measure from r(t) to f(t) and from r(t) to g(t):
∫∫ −=−=b
a
b
a
dttgtrgrdvsdttftrfrd 22
22 )()(),(.)()(),(
16Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Consider a set X in R,C,N,Z,B… Definition 2.1: A metric
is a functional mapping to measure distances between objects/elements of the set X. In order to call such a distance a metric, the following properties need to be satisfied:
)(: 00 NRXXd +→×
Xzyxzydyxdzxdyxyxd
xydyxdyxd
∈+≤==
=≥
,, allfor );,(),(),()3 ifonly and if ,0),()2
),(),()10),()0
Metric Note that 0) follows from 1), 2), and
3) since:
17Univ.-Prof. Dr.-Ing.
Markus Rupp
0),( :thus3) and 1) todue
),(2),(),(),(0:have wethen
)2 todue 0),(
≥
=+≤=
=
yxd
yxdxydyxdxxd
xxd
18Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Example 2.3: Consider the following metric
d1(x,y)
defined over the vectors x,y. This metric is called l1 metric. (Manhattan distance)
Example 2.4: Consider the following metric dp(x,y)
( ) ∑ =
+
−=
→×n
i ii
nn
yxyxd
RRRd
11
01
,
:
( ) ( ) pn
i
piip
nnp
yxyxd
RRRd/1
1
0
,
:
∑ =
+
−=
→×
19Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces From the lp metrics the l1 and l2 metric as well as
the l∞ metric are of particular interest:
Example 2.5: Hamming Distance
( ) ( )iini
pn
i
piip
nn
yx
yxyxd
RRRd
−=
−=
→×
≤≤
=∞→∞
∞
∑1
/1
1
max
lim,
:
( ) ∑ =⊕=
→×=n
i iiH
nnH
yxyxd
NBBdB
1
0
,
:,1,0
20Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Definition 2.2: A Metric Space (X,d) consists of
a set X and a metric d, valid on this set.
Example 2.6: The set Rn (vectors with n entries from R) together with the metric d2(x,y) builds a metric space.
21Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Note, a metric defines the distance to the zero
vector for p>=1:
A metric thus allows for statements about sizes (lengths,areas, volumes, etc for p=2) of objects. Tied to this is the existence of such objects.
Example 2.7: Consider the infinite long series of real or complex valued numbers xi, i=0,1,...,oo, with the property
∑∞
=
∞<0i
pix
( ) ( ) ( ) pn
i
pi
pn
i
pip xxxd
/1
1
/1
100, ∑∑ ==
=−=
( ) ( ) ∞<= xdxd pp 0,
22Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Together with an lp metric, such sequences form a
metric space called lp(0,∞) space (Ger.: Folgenraum).
Consider on the other hand the sequence xi, i= -∞,...,-1, 0,1,...,∞ with the same properties and same metric, we have the lp(-∞,∞) space.
Consider the metric p ∞, for sequences for which: |xi| is bounded for every i, i.e., |xi|<M, we obtain the l∞(0,∞) or l∞(-∞,∞) space, respectively.
nnnnn yxyxd −=∞ sup),(
23Univ.-Prof. Dr.-Ing.
Markus Rupp
Supremum and Infimum Definition 2.3. (Supremum): Consider the set S in
R,Q,Z,N… with the elements xi. The smallest number z in R, for which we have:
is called supremum (sup) of the set S. It is the least upper bound.
If there is no number in R that is larger than the largest element in S, then we have sup(S)= ∞.
ixz i allfor ;≥
24Univ.-Prof. Dr.-Ing.
Markus Rupp
Supremum and Infimum Definition 2.4 (Infimum): Consider the set S in R
with the elements xi. The largest number z in R, for which we have:
is called Infimum (inf) of the set S. It is also called the greatest lower bound.
If there is no more number in R smaller than the smallest element in S, then we have inf(S)=- ∞.
ixz i allfor ;≤
25Univ.-Prof. Dr.-Ing.
Markus Rupp
Supremum and Infimum Example 2.8: Let S=(3,6) be an open set,
then: inf(S)=3, sup(S)=6
Let T=[4,8), then: inf(T)=4, sup(T)=8 Let U=[2,∞), then: inf(U)=2, sup(U)=∞
Why do we not simply select the maximum (minimum) of S?
26Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Metric spaces can also be formed with particular
properties:
Example 2.9: Let lh(0,∞) be a metric space of sequences in which sequences exist whose quadratic sum is finite (finite energy sequence). Thus, xi in lh(0,∞)=lp=2(0,∞) means, that
∞<∑∞
=0
2
iix
27Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Metric spaces can also be defined over
functions rather than a set of numbers (note that functions are in a way sets of numbers).
These are called metric spaces over functions (Ger.: Funktionenraum).
28Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Definition 2.5 (p-metric): Let X be a set of real-
or complex-valued functions, defined on the interval [a,b] with b>a, with p>=1, that have the property:
The metric over the functions x(t),y(t) from X is then:
(Lebesgue Integral)
∞<≤
−= ∫ pdttytxyxd
pb
a
pp 1;)()(),(
/1
∞<≤∞<∫ pdttxb
a
p 1;)(
29Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces The space with the metric dp over the functions is
called Lp space.
For p∞ we have for bounded functions (from above and below) :
For more details on Lp, read Ch 2.1.3
btatytxyxd
txbat
≤≤−=
∞<
∞
∈
;)()(sup),(
)(sup ],[
30Univ.-Prof. Dr.-Ing.
Markus Rupp
t
0 1 2 3 4 5 6 7 8 9 10
x(t)
0
1
2
3
4
t
0 1 2 3 4 5 6 7 8 9 10
x(t)
0
1
2
3
4
ε
ε
−
+
)()(),(
)(
txtxtx
tx
o
mo
o
)(
)()(
2
1
tx
txtx
oεε
εε
>>>><<
∞
∞
),(),(),(),(
2
1
22
12
xxdxxdxxdxxd
o
o
o
o
εε
<<
∞ ),(),(2
mo
mo
xxdxxd
Metric: Problems with Pytagoras?
31Univ.-Prof. Dr.-Ing.
Markus Rupp
Under which dp metric is the picture correct?
p-Metrics: consider iso-metricἴ iσος=equal
32Univ.-Prof. Dr.-Ing.
Markus Rupp
p=1
p=2p=4 p=100
p=1/2
p=1/4
inflation
deflation( )
pp
p
xx
xdxx
x
21
2
1
1
+=
=
=
Metrics for 0<p<1 What happens if we select 0<p<1? In this case
does not satisfy the triangle equality (subadditive property)
33Univ.-Prof. Dr.-Ing.
Markus Rupp
( ) ( ) pn
i
piip yxyxd
/1
1, ∑ =
−=
Metrics for 0<p<1 Example 2.10: select the three
twodimensional points x=(1,0),y=(0,1) and z=(0.5,0.3)
compute
34Univ.-Prof. Dr.-Ing.
Markus Rupp
2122
1
2 21 12 22 2
1 1
1.57 2.38
4
3.96
ni ii
n ni i i ii i
x y
x z z y
=
=
= =
= =
− =
> − + − =
∑
∑ ∑
Metrics for 0<p<1 Definition 2.6: For 0<p<1, the
following expression is a metric:
Example 2.10 again:
35Univ.-Prof. Dr.-Ing.
Markus Rupp
( ) ( )∑ =−=
n
i
piip yxyxd
1,
( )122
1 12
1 12 22 2
1 1
1.25 1.54
, 2
2.798
ni iip
n ni i i ii i
d x y x y
x z z y
=
==
= =
= =
= − =
< − + − =
∑
∑ ∑
d0-Metric Is there also a d0 metric? Answer: yes but the definition
causes some mathematical difficulties as discontinuities occur for
36Univ.-Prof. Dr.-Ing.
Markus Rupp
( ) ( )∑ =→ −=n
i
piip yxyxd
100 lim,
=
=elsex
x;1
0;00
Sparseness Definition 2.7: In practise this definition
of d0 is often given in the form of
d0(x)=Σxi0
which is truly a counter for sparseness. But note, this is not a metric (norm)
sometimes it is called a pseudonorm
37Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.11Reconsider Sampling For bandlimited signals we found:
Given the interpolation function, what is the optimal sequence gk to minimize
Answer: gk=fk=f(kT)min=038
Univ.-Prof. Dr.-Ing. Markus Rupp
−
−=
∑
∑
∞
−∞=
∞
−∞=
)(sinc),(min
)(sinc)(
2 kTtT
gtfd
kTtT
ftf
kkg
kk
k
π
π
2/12
)(sinc)(min
−−= ∫ ∑
∞
∞−
∞
−∞=
dtkTtT
gtfk
kgk
π
Example 2.11Reconsider Sampling Given a set of interpolation functions pm(t) :
Find the pair (gk,pm(t));m=1,2,..,M that results in a sparse representation of f(t)!
basic concept for modern speech coding (CELP)
39Univ.-Prof. Dr.-Ing.
Markus Rupp
( )
TktkTtpgtf
pgd
m
k
kk
Mmmkgk
00
,...,2,10
0);()(
|min
0
≤≤−= ∑=
=
that such
CELPCodebook excited linear prediction We can alternatively also select from a set of
excitation sequences g1,k,g2,k,…,gM,k. The interpolator can be constructed in form of a
fixed part with a linear filter (predictor) before:
The task now is to find the following minimum:
40Univ.-Prof. Dr.-Ing.
Markus Rupp
−= ∑
−
=
)(sinc)(1
0kTt
Tptp
P
kk
π
−+ ∑
=
0
,0
,2 )(),(minmink
kkmDpg kTtpgttfd
lkm
prediction for tD>0
CELPCodebook excited linear prediction
Problem is split into a search over a fixed set (gm,k) and the minimization of a d2-metricLeast Squares (Chapter 3)
41Univ.-Prof. Dr.-Ing.
Markus Rupp
21
)(
0
2
0
1
0,
21
)(
0
2
0,
0,2
0 0
,
0 0
,
0
,
)(sinc)(minmin
)()(minmin
)(),(minmin
−−−+=
−−+=
−+
∫ ∑ ∑
∫ ∑
∑
+
=
−
=
+
=
=
TkP k
k
P
l lkmDpg
TkP k
kkmDpg
k
kkmDpg
dtkTlTtT
pgttf
dtkTtpgttf
kTtpgttfd
lkm
lkm
lkm
π
42Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces lp(0,∞) : Space of causal sequences lh(0,∞) : Space of causal sequences with
finite energy (p=2) lp(-∞,∞): Space of non-causal sequences
(this includes lp(0,∞)) Lp(0,∞): Space of causal functions Lp(-∞,∞): Space of non-causal functions (C[a,b],dp): Space of continuous functions
43Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Example 2.12 Filter design: a linear time-discrete filter with linear
phase H(exp(jΩ)) =a0+a1exp(jΩ)+a1exp(-jΩ)=a0+2a1cos(Ω) is to design, so that it follows a desired amplitude function |Hd(exp(jΩ))|=1 for |Ω|< ΩG (zero else) most optimally.
( ) ( )
( )
[ ] [ ]
)(;)sin(;
)sin(42)1(min
)cos(2021)cos(21
21min
)cos(221min
21min
10
121
20
20
20,
210
210,
2
10,
2
10
10
10
series Fourier cmp.
Rest
ππ
ππ
ππ
π
ππ
π
π
π
GG
GG
aa
aa
jdaa
jjdH
aa
aaaaa
daadaa
daaeH
deHeH
G
G
Ω=
Ω=
Ω−++−−Ω
=
ΩΩ−−+ΩΩ−−=
ΩΩ−−=
Ω−
∫∫
∫
∫
Ω
Ω−
−
Ω
−
ΩΩ
44Univ.-Prof. Dr.-Ing.
Markus Rupp
Minimization ofquadratic metric d2
ΩG
Example 2.13. Recall Paley Wiener Given an amplitude function Ad(Ω), what is
a valid filter function F(ejΩ), i.e.Ad
2(Ω) =|F(ejΩ)|2
Different to the previous example the full information of the desired Filter is not known, only the amplitude function is given.
We recall that linear phase filters are well described by their amplitude function.
45Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.13. Recall Paley Wiener Linear phase filters have symmetric
impulse responses
46Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.13. Recall Paley Wiener Problem formulation:
Recall that:
47Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.13. Recall Paley Wiener For which we find:
For l=0,1,…,N. From there we can compute the original coefficients fn.
Thus, given Ad, we now know how to compute F.
48Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume What is a set? What is a vector? What is a space? What is a metric space?
49Univ.-Prof. Dr.-Ing.
Markus Rupp
50Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Definition: Space X Objects (numbers,
sequences, functions…) with structure Definition: Metric space (X,d) space X on
a metric d is defined as
Xzyxzydyxdzxdyxyxd
xydyxdyxd
∈+≤==
=≥
,, allfor );,(),(),()3 ifonly and if ,0),()2
),(),()10),()0
Resume When do sequences in metric spaces
with lp metric exist?
51Univ.-Prof. Dr.-Ing.
Markus Rupp
∑∞
=
∞<0i
pix
52Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume lp(0,∞) :
Metric space of causal sequences with p metric
lh(-∞,∞): Metric space of non-causal sequences with 2 metric
53Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Robustness (Ger.: Robustheit):
Often it is important not to optimize the mean value (e.g., mean data rate, BER) but the value at worst case condition.
If a systems shows to be insensitive to the worst cases, it is called robust.
54Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Robustness is often measured in terms of energy Let xi be the input sequence and yi the output sequence of a
system T. The output is assumed to be distorted by an unknown sequence vi. The form of distortion is not necessarily additive or even known. Furthermore, the M initial states zi of this system are also not known. Assume that there exists a reference system without distortion having an output signal yi
(R). The influence of such distortion can be described be the
following expression:( )
2
0
2
1
2
0
2)(
γ≤+
−
∑∑
∑
==
=N
ii
M
ii
N
ii
Ri
vz
yy
55Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Or somewhat more precise:
Assume now that there exists several possibilities for the realization of this system T(F) depending on a specific strategy. Then the robustness criterion reads:
( )2
0
2
1
2
0
2)(
),0(),,1(sup γ≤
+
−
∑∑
∑
==
=
∈∈N
ii
M
ii
N
ii
Ri
NlvMlz vz
yy
hh
( )
∑∑
∑
==
=
∈∈ +
−
N
ii
M
ii
N
ii
Ri
NlvMlzFvz
yy
hh
0
2
1
2
0
2)(
),0(),,1(supinf
56Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.19: Robustness A power amplifier PA in a cell phone (Ger.: Handy)
can be described by the following nonlinear mapping:
The variable ρ= ρi(T) describes the influence of temperature, aging and more.
In order to define the robustness of the output signal with respect to ρ, the following expression is considered:
iiii
iii vx
xx
y ++
= 2
2
1 ρ
ρ
( )2
0
2
0
2
0
2)(
),0(,sup γ
ρρ≤
+
−
∑∑
∑
==
=
∈N
ii
N
ii
N
ii
Ri
Nlv v
yy
h
PA yiT
57Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.19: Robustness If it is possible to reduce the influence of
ρ by new technologies or improved circuitry, a smaller factor γ is the result:
The improvement is achieved by higher robustness (smaller γ).
Note: Robustness is not sensitivity.
( )22
0
2
0
2,
0
2)(
),0(,sup γγ
ρρ<≤
+
−
∑∑
∑
==
=
∈newN
ii
N
inewi
N
ii
Ri
Nlv v
yy
h
PA yiT
Contr.
+
58Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Convergence of sequences
Definition 2.8 (limit): If there exists to every distance δ a number no so that d(xn,y)< δ for each n>no at fixed value y, the sequence xn is said to be convergent to y.
y is called the limit (Ger.: Grenzwert) of xn. All points in an arbitrarily small distance to y are called the
neighborhood (Ger.: Nachbarschaft) of y.
nn
n
xyyx
∞→=→lim
59Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Example 2.14: the following two sequences
diverge:
Definition 2.9: If a sequence xn returns infinitely often to a neighborhood of z, then we call this point z a limit point (Ger.: Häufungspunkt, Grenzpunkt). E.g., the sequence bn takes on the limit points z=0 and z=2.
If limit points exist, then there must be subsequences (partial sequences, Ger.: Teilfolgen) xn that converge.
nn
n
bna
)1(1
2
−+=
=
60Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces The largest limit point of a sequence xn is called
limes superior, or
The smallest limit point of a sequence xn is called limes inferior, or
A sequence converges, if:
nn x∞→suplim
nn x∞→inflim
nnnn xx ∞→∞→ = supliminflim
61Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Example 2.15: Consider the sequence
There are two limit points. The subsequence c0,c2,c4,.. takes on the limit 4, while the subsequence c1,c3,c5,.. takes on the limit 0.
12 2 ( 1) ; 1,2,3,...
limsup 4 but : sup 4,5liminf 0
nn
n n n n
n n
c nn
c cc
→∞
→∞
= + + ⋅ − =
= ==
62Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Definition 2.10: A sequence xn in a metric
space (X,d) is called Cauchy-sequence, if there exists for every ε>0 a N(ε)>0 so that d(xn,xm)<ε for every m,n>N.
It is possible to prove that sequences that converge are Cauchy-sequences. The opposite is not always true.
63Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Example 2.16: Let X=C[-1,1] be a set of
continuous functions and fn(t) a sequence of functions, defined by
Consider the metric space (X,d2) with
>≤≤−+
−<=
ntntnnt
nttfn
/11/1/12/12/
/10)(
( )∫−
−=1
1
22 )()(),( dttgtfgfd
64Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces
nn /1/1
1
− t
)(tfn
Example 2.16:
65Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces We find
With d20 for large m and n, m-n=k<oo. We thus conclude that it is a Cauchy sequence.
Note, however, the function in the limit is:
This function is not continuous and thus not in X=C[-1,1]. The sequence is thus not convergent in X!
<−
>−
=nm
mnm
nmn
nm
tftfd mn
;6
)(
;6
)(
))(),((
3
2
3
2
2
>=<
=0102/100
)(ttt
tf
66Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces The reason why the Cauchy-sequence does
not converge in X, can be interpreted as a hole in the set of functions X.
If we had defined X as the set that included discontinuous functions as well, then we had obtained a Cauchy sequence that converges.
Definition 2.11: A metric space (X,d) is called complete (Ger.: vollständig), if every Cauchy-sequence converges in X.
67Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Gibbs‘ Phenomenon You are probably well aware of the fact that
periodic functions can be described by Fourier series.
Example 2.17: Consider the following periodic signal of period 1 in L[-0,5 , 0,5]:
with the Fourier-series:
<<≤≤−−<≤−
=5,025,0025,025,0125,05,00
)(tt
ttf
( )
−−
−+= ∑
=
+
tkk
tfn
k
k
n ππ
)12(2cos12
)1(25,0)(1
1
Josiah Willard Gibbs (11.2.1839 –28.4.1903) was an American scientistin Physics, Chemistry, Mathematics
68Univ.-Prof. Dr.-Ing.
Markus Rupp
69Univ.-Prof. Dr.-Ing.
Markus Rupp
Metric Spaces Let‘s run this to noo Note that in this case we have:
0))(),((lim0))(),((lim 2
>=
∞∞→
∞→
tftfdtftfd
nn
nn
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
)(500 tfn=
Example 2.18 Let us consider the problem of finding a
square root of a number A (e.g. A=3 in N) We thus want to find a solution for x2-A=0 An iterative algorithm (Heron) can be
derived (Newton-style):
70Univ.-Prof. Dr.-Ing.
Markus Rupp
21n
n
nxAx
x+
=+
Heron from Alexandria (Mechanicus; † > 62) Greek mathematician and engineerMethod is known since 1700 BC.
Example 2.18 Obviously, starting with a value from
Q, results in a new value from Q:
For noo, we obtain a value from R!
71Univ.-Prof. Dr.-Ing.
Markus Rupp
QabA
ba
xbax
xAx
x
nn
nn
n
∈+
=→=
+=
+
+
2
2
1
1
Rules in the Vector Space Obviously we want to „do“ something
with the objects in our set(space) We can stretch them
We can add them
72Univ.-Prof. Dr.-Ing.
Markus Rupp
x xα→
,x y x y→ +
Binary operation Definition 2.12: A binary operation *
on a set S is a rule that assigns to each ordered pair (a,b) of elements from S some element from S. Since the operation ends with an element in the same set, we call this also a closed operation.
Example 2.20: S=Z, *: a*b=min(a,b)
73Univ.-Prof. Dr.-Ing.
Markus Rupp
74Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.13: A linear vector space S over a set of
scalars T (C,R,Q,Z,N,B) is a set of (objects) vectors together with a binary additive „+“ and a scalar multiplicative „.“ operation, satisfying the following properties:
1) S is a group under addition. 2)
3) W.r.t. the multiplicative operation there exists an Identity (One) and a Zero element:
yaxayxaxbxaxbaxabxbaSxa
SyxTba
+=++=+=∈
∈
)()(,)()(,
:,, have we in and everyFor
00;1 =•=• xxxIdentity elementw.r.t „+“!
75Univ.-Prof. Dr.-Ing.
Markus Rupp
Groups Definition 2.14 (Group): A set S for which a binary
operation * (operation w.r.t two elements of S) is defined, is called a group if the following holds: 1) for each a,b in S it holds that: (a*b) in S 2) there exists an identity element e in S, so that for every
element a in S: e*a=a*e=a. 3) for each element a in S there exists an inverse element b in
S, so that: a*b=b*a=e. 4) The binary operation * is associative, i.e.,: (a*b)*c=a*(b*c)
The group is denominated by (S, *). If, furthermore, for each pair a,b in S it holds that a*b=b*a
(commutativity), then the group is called commutative or Abelian (Ger.: Abelsch).
76Univ.-Prof. Dr.-Ing.
Markus Rupp
Groups Example 2.21: Let S be the set of vectors of a
particular dimension. S forms a group w.r.t the additive operator, if the following properties are satisfied:
( ) ( )zyxzyxSzyx
xyyx
SySxxxx
S
SyxSyx
++=++∈
−==+
∈∈=+=+
∈+∈
:that holdsit eachFor :eassociativ is operation additive The
that so , second a exists there element eachFor
that so , in element identity an exists There
:that holdsit everyFor
,,)4
;0
)300
0)2
,)1
77Univ.-Prof. Dr.-Ing.
Markus Rupp
Rings Definition 2.15 (Ring): A set S for which two
binary operations + and * are defined, is called a ring if the following holds: (S,+) is a commutative group (Abelian) The operation * is associative. Distributivity holds w.r.t. + for all scalars:
a(b+c)=ab+ac, (a+b)c=ac+bc. A ring is denominated by (S,+, *). Note: for * there does not need to be an identity
or inverse element. If there exists additionally the inverse element to
*, then it is called skew field or division ring (Ger.: Schiefkörper).
78Univ.-Prof. Dr.-Ing.
Markus Rupp
Fields Definition 2.16 (Field): A set S equipped
with two binary operations + and * is called a field (Ger.: Körper), if: 1) (S,+) is an Abelian group 2) (S\0,*) is an Abelian group 3) The operations + and * distribute.
If the set is finite, i.e., |S|<oo, we talk about finite groups, finite rings, finite fields….
79Univ.-Prof. Dr.-Ing.
Markus Rupp
Fields Example 2.22: The Galois Field GF(2) is a
field. The elements of the field are 0 and 1. Define the operations + and *
a b a+b0 0 00 1 11 0 11 1 0
a b a*b0 0 00 1 01 0 01 1 1
80Univ.-Prof. Dr.-Ing.
Markus Rupp
Fields W.r.t. + S needs to be an Abelian Group:
a+b is in S. Identity element is 0: 0+0=0, 1+0=1. Inverse element exists: 0+0=0,1+1=0. Associativity: (a+b)+c=a+(b+c). Check the remaining properties yourself...
81Univ.-Prof. Dr.-Ing.
Markus Rupp
Fields Example 2.23:
(R,+,*), (Q,+,*),(C,+,*) are fields; (N,+,*),(Z,+,*) not!
82Univ.-Prof. Dr.-Ing.
Markus Rupp
Group
Relation to each other
Ring AbelianGroup
Vector space
SkewfieldField
83Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.24: for finite dimensional linear vector
spaces: 1) Consider the linear vector space in R4 (set of
quadruples, Ger.: Menge der Quadrupel)
2) The set of m X n matrices with real-valued elements.
3) The set of polynomials of degree 0..n with real-valued coefficients.
=+
=+
−
=
=
2121913
23;
0476
;
2025
;
2451
yxyxyx
84Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Still Example 2.24: Consider the (3,2)
single parity check code in GF(2)3 with the elements:V=[000],[011],[101],[110]
„+“ now means „exclusive or“
Do the remaining elements W=[111],[100],[010],[001] also form a linear vector space?
85Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.25: for infinite-dimensional linear
vector spaces:
1) Consider the set of infinitely long sequences
xn
2) The set of continuous functions over the
interval [a,b] C[a,b]
3) The set of functions in LpLp[a,b]
86Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.17: Let S be a vector space. If V is a
subset of S such that V itself is a vector space then V is called a subspace (Ger.: Unterraum) of S.
Example 2.26: Let S be the set of polynomials of arbitrary degree (>6) and V the set of polynomials of degree less than 6. Then V is a subspace of S.
Example 2.27: A (n,k) binary linear block code is a k-dimensional subspace of GF(2)n.
87Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.18: Let S be a linear vector space
over R and T a subset of S. A point x of S is called linear combination (Ger.: Linearkombination) of points in T, if there is a set of points pi ;i=1,2,...,m in T and a finite set of scalars ci ;i=1,2,...,m in R, so that:
Note that the set T does not need to be finite.
mm pcpcpcx +++= ...2211
88Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.28: Let S=C(R), the set of continuous
functions over the complex (real) numbers. Let, furthermore, p1(t)=1; p2(t)=t, p3(t)=t2. A linear combination of such functions is given by:
Consider the polynomial x(t)=-1+5t+t2, and the function p4(t)=t2-1.
Obviously, the description is not unique. The number of required coefficients varies.
2321)( tctcctx ++=
)()(5)(2)()(5)(
)()(5)()(
42
4321
321
tptptptptptp
tptptptx
+=+−+=
++−=
89Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.19: Let S be a linear vector space
and T a subset of S. The subset T (pi ;i=1,2,…,m) is said to be linearly independent (Ger.: linear unabhängig), if for each nonempty linear subset T the only finite set of scalars satisfying the equation
is the trivial solution c1= c2 =...= cm =0.
If the above equation is satisfied by a set of scalars that are not all equal to zero, then the subset (pi ;i=1,2,...,m) is called linearly dependent(Ger.: linear abhängig).
0...2211 =+++
mm pcpcpc
90Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.29: The previously presented
polynomials p1(t)=1; p2(t)=t, p3(t)=t2, p4(t)=t2-1 are linearly dependent, since:
The polynomials p1(t),..., p3(t) are linearly independent!
The vectors p1=[2,-3,4],p2=[-1,6,-2] and p3=[1,6,2] are linearly dependent since:
0)()()( 431 =+− tptptp
0354321
=++ ppp
91Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.30: Consider the complex numbers
z=r+j i in R2 in vector form:
This is a set describing z and z*. Note that both elements are linearly independent as long as i is unequal to zero!
−
=
ir
ir
T ,
92Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.31: Consider band-limited functions or
sequences xk (also band-limited random processes) that only exist in a frequency range S. For a sequence fk (linear time-invariant system) existing in the complementary space of S we find:
Band-limited signals are said to be linearly dependent! Note however, that the property “finite number of
elements” is not necessarily satisfied.
( ) ( )( ) ( ) ( )
knk
kknk
kn
jjjkn
kkn
jjn
ff
ff
eXeFeYf
SeFSeX
−
−
−∞=−
∞
=
ΩΩΩ−
∞
−∞=
ΩΩ
∑∑
∑
−−=
==⇔==
∈Ω=∉Ω=
xxx
xy
for for with xLet
1
010
11
00
0;0
94Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 Consider blind channel estimation (for
equalization):
h1
h2
sk
g1
g2
sk(1)
sk(2)
-0
rk(1)
rk(2)
95Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 Observe N receive values at each sensor:
rk(1)..=[rk
(1),rk-1(1)…,rk-N+1
(1)]T
We obtain the equation:
Which properties must the receive vectors rk(1) and rk
(2)
have, so that the problem can be solved uniquely? Vectors rk
(1,2) need to be linearly dependent!
[ ] 0,...,,,,...,,
......
2
21
1
11
)2()2(2
)2(1
)1()1(2
)1(1
)2(2
)2(222
)2(121
)1(1
)1(212
)1(111
==
−
−
+++=+++
gR
g
gg
g
rrrrrr
rgrgrgrgrgrg
m
mmm
mmmm
96Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces At this point several interesting questions arise:
Under which conditions is the linear combination of vectors unique?
Which is the smallest set of vectors required to describe every vector in S by a linear combination?
If a vector x can be described by a linear combination of pi ;i=1..m, how do we get the linear weights ci ;i=1..m?
Of which form do the vectors pi ;i=1..m need to be in order to reach every point x in S?
If x cannot be described exactly by a linear combination of pi ;i=1..m, how can it be approximated in the best way (smallest error)?
97Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Space Metric complete
Convergence Cauchy-Sequence
Set vs. Vector
98Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider the following sequences in the
metric space l∞(1,∞):
Consider a classical Gaussian noise sequence from N(0,σ2). Is such a sequence in the metric space lp(-∞,∞)?
1),(
1;
11
≥−>
+=
+=
∞ nyxyxd
nny
nx
nnnn
nn
allfor ;that Proof
99Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume What can be said about the convergence of
these causal sequences (n=1,2,..)?
Which of these sequences is in the metric space l2(1,∞), which in l∞(1,∞)?
[ ]
[ ]
n
n
n
n
nn
nn
nd
nc
nb
na
235,0
11
)1(111000
11
)1(111
+=
+=
−−
++=
−−+=
100Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider the following sequence in Q in the
metric space (Q,d∞):
? in sequence-Cauchya thisIs
converge? sequence thisDoes
,...100000141421,
1000014142,
10001414,
100141,
1014,
11
Q
Not in Q
Yes!
Resume Problems in Q: Consider Is this convergent in Q? Similarly, consider
101Univ.-Prof. Dr.-Ing.
Markus Rupp
21
1n
sm
nm ∑
=
=
!1
0 ns
m
nm ∑
=
= en
m
nm =∑
=∞→ !
1lim0
61lim
2
21
π=∑
=∞→ n
m
nm
102Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Which of the following sets of vectors are LI and which not?
−
12363
,
9642
,
4321
137
,1
5,
53
,21
2000
,
3300
,
4220
,
5111
4321
,
0321
,
0021
,
0001
104Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Given the set pi and a vector x, how do we obtain the linear
weights?1 21 2
... m mp p p xα α α+ + + =
( )
1
1 2
11
1 2
1
1 2 1 2 1 2
, ,...,
, ,...,
, ,..., , ,..., , ,...,
m
m
m
m
H H
m m m
p p p x
p p p x
p p p p p p p p p x
α
α
α
α
−
−
= =
=
105Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume: LI Consequence of missing LI in linear systems of
equations:Rows < Columns
equations of system thefor solution a be Let 00
3
2
1
4
3
2
1
3431
1411
≠
=
x
bbb
xxxx
aa
aa
( )
*
11 14
* *
31 34
0 0 *
due to linear dependency in the column vectors, a 0 must exist so that0
0 .0
Then we must also have that: 0 .Thus, an infinite amount of solut
xa a
x A xa a
Ax A x x b bα
≠
= =
= + = + =
ions exists for this system of equations.
106Univ.-Prof. Dr.-Ing.
Markus Rupp
System Identification Problem: The impulse response hk of a
linear time-invariant system is to estimate based on observations of input and output signals. What conditions does the input skhave to satisfy to ensure a uniqueidentification?
hsk rk
107Univ.-Prof. Dr.-Ing.
Markus Rupp
System Identification Solution:
11 2 1 1
2
1
0
MM M
M
hs s s s rh
hh
−−
−
=
108Univ.-Prof. Dr.-Ing.
Markus Rupp
System Identification Solution:
11 2 1 1
22 3 1 2
1
0
MM M
MM M
hs s s s rhs s s s r
hh
−−
−+
=
109Univ.-Prof. Dr.-Ing.
Markus Rupp
System Identification Solution:
For a unique identification, the rows (columns) need to be linearly independent!
11 2 1 1
22 3 1 2
11 1
01 2 2 2 1
MM M
MM M
M M
M M M M M
hs s s s rhs s s s r
hs rhs s s s r
−−
−+
− −
+ − −
=
110Univ.-Prof. Dr.-Ing.
Markus Rupp
System Identification A signal with such a property is said
to be of persistent excitation (Ger.: hartnäckige Anregung).
Question: is a complex-valued exponential harmonic sk=exp(jΩοk) of persistent excitation?
What about a sinusoid sk=sin(Ωοk)?
Persistent Excitation Consider excitation signal
only system of order 1 (constant) can be identified.
[ ][ ] 1
322
)1(201
,...,,,,...,,,
xeeeeexeeeex
es
ooooo
oooo
o
jMjjjj
Mjjjj
kjk
ΩΩΩΩΩ
−ΩΩΩΩ
Ω
==
=
=
Univ.-Prof. Dr.-Ing. Markus Rupp111
Persistent Excitation Consider
excitation signal:
only system of order 2 can be identified.
( )[ ][ ]
( )
b
j
a
j
jkjjkjok
ba
Mjjjjb
Mjjjja
kjkjok
xj
exj
ex
eeeej
ks
xj
xj
x
eeeexeeeex
eej
ks
oo
oooo
oooo
oooo
oo
22
21))1(sin(
21
21
,...,,,,...,,,
21)sin(
2
1
1
)1(20
)1(20
Ω−Ω
Ω−Ω−ΩΩ+
−Ω−Ω−Ω−Ω−
−ΩΩΩΩ
Ω−Ω
−=
−=+Ω=
−=
=
=
−=Ω=
Univ.-Prof. Dr.-Ing. Markus Rupp112
Persistent Excitation How many signals (frequencies Ωo) of the
form exp(jΩok) are required to identify a time-invariant system of order M?
How many signals (frequencies Ωo) of the form sin(Ωok) are required to identify a time-invariant system of order M?
Univ.-Prof. Dr.-Ing. Markus Rupp113
114Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.20: Let T be a set of vectors in a
vector space S over a set of scalars R(C,Q,Z,N,B). The set of vectors V that can be reached by all possible (finite) linear combinations of vectors in T is the span (Ger.: aufgespannte Menge, erzeugte Menge, lineare Hülle) of the vectors. :
)span(TV =
Vector Spaces Note that this is an elegant way of saying something in short
that otherwise would require a lengthy formulation. If T is a set, then span(T) is a set as well!
The number field is typically K=R or K=C. The span is typically defining a vector space!
115Univ.-Prof. Dr.-Ing.
Markus Rupp
∈∈==
==
∈+==
=
∑=
21
22
2
11
1
,|)span(
,....,2,1|,|)span(
,
TxKxTV
NixTKyxTV
yxT
iiii
N
i
i
αα
βαβα
116Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.21: Let S be a vector space, and let
T be a set of vectors from S such that span(T)=S. If the elements in T are linearly independent, then T is said to be a Hamel basis for S.
Example 2.34: The vectors p1=[1,6,5], p2=[-2,4,2], p3=[1,1,0] and p4=[7,5,2] are linearly dependent. Note that T=p1,p2,p3 spans the space R3 and thus is a basis for R3.
Example 2.35: The vectors p1=[1,0,0], p2=[0,1,0] and p3=[0,0,1] are linearly independent and are a basis for R3. This basis is often called natural basis (Ger.: natürliche Basis).
Georg Karl Wilhelm Hamel (12 September 1877–4 October 1954)
117Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.36: Consider the (3,2) single parity
check code in GF(2)3 with the elements:V=[000],[011],[101],[110].A Hamel basis is given by:
=]011[]101[
G
118Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.37 The following 3x3 matrices are a Hamel basis in R3x3:
100000000
,010000000
,001000000
000100000
,000010000
,000001000
000000100
,000000010
,000000001
119Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.38 Are these 3x3 matrices a Hamel basis?:
Answer: These matrices are not a basis for R3x3. However, the are a basis for the subspace in R3x3,
that has a zero row and column sum!
−−
−−
−
−
−
−
110110000
,101101000
,110000110
,101000101
120Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.22: The number of elements in a set
is its cardinality |A| (Ger.: Kardinalität).
Theorem 2.1: If two sets T and U are Hamel bases for the same vector space S, then T and U are of the same cardinality.
Proof: (only for the finite dimensional case). Letbe two bases for S and with at least one coefficient unequal to zero,
,...,,;,...,,2121 nm
qqqUpppT ==
mm pcpcpcq +++= ...22111
4,1,1,1,1 =−−+−−+= AA :QPSK jjjj
Georg Cantor (3.3.1845-6.1.1918), German Mathematician
121Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces say c1, then: Thus
is also a basis for S. Further substitution leads to
( ),...,,
...1
21
2211
1
m
mm
ppq
pcpcqc
p −−−=
,,...,,,
,,...,,,...
,...,,,
,...,,,
1321
1321
321
321
mm
mm
m
m
qqqqq
pqqqq
ppqq
pppq
−
−
122Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Consequently, we must have: Now starting with q1 instead of p1, we
obtain Thus, we must have n=m.
Definition 2.23: Let T be a Hamel basis for S. The cardinality of T is the dimension of S, |T|=dim(S). It equals the number of linearly independent vectors, required to span the space S.
nm ≥
.mn ≥
123Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Note: each vector space has at least one
Hamel basis.
Operations in the vector space are often simpler and of lower complexity in their basis.
124Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 revisited Consider blind channel estimator (blind
equalization):
h1
h2
sk
g1
g2
sk(1)
sk(2)
-0
rk(1)
rk(2)
125Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 revisited Observe N receive values at each sensor:
rk(1)..=[rk
(1),rk-1(1)…,rk-N+1
(1)]T
When are the vectors linearly dependent?
In order to obtain linear dependency, we must have a sufficient number of vectors r1
(1)..rm(1), r1
(2)..rm(2), so that
2m>N since the vectors rk(1,2) of dimension N can maximally
span RN!
[ ] mN
m
mmm
mmmm
Rg
g
gg
g
rrrrrr
rgrgrgrgrgrg
2
2
21
1
11
)2()2(2
)2(1
)1()1(2
)1(1
)2(2
)2(222
)2(121
)1(1
)1(212
)1(111
R;0R,...,,,,...,,
......
×∈==
−
−
+++=+++
126Univ.-Prof. Dr.-Ing.
Markus Rupp
Inner Products Definition 2.24: A vector space for which an
inner vector product is defined is said to be a pre- Hilbert space (Ger.: Innerer Vektorproduktraum, Vor-Hilbertraum).
An inner vector product maps two vectors onto one scalar with the following properties:
zyzxzyx
yxyx
xyyx
xxx
,,,)3
,,)2
,,)1
else. 0 and 0for 0,)0*
+=+
=
=
≠>
αα
127Univ.-Prof. Dr.-Ing.
Markus Rupp
Inner Products Example 2.39: Consider the space of continuous functions
C[a,b] with the two elements x(t) and h(t). Let x(t) be an input signal and h(t) the impulse response of a low pass. We have:
Example 2.40: Consider vectors in C: Matrices can also build inner products:
∫
∫
==
−=
T
T
gxdgx
dThxTy
0
0
,)()(
)()()(
τττ
τττ
BAABH ,)Trace( =
xyyx H=,
128Univ.-Prof. Dr.-Ing.
Markus Rupp
Inner Products Example 2.41: Consider the expectation of a
random variable:
This is also an inner vector product, however with an additional weighting function.
[ ]
f
dxdyyxfxy
yx,
),(xyE xy
=
= ∫∫
Inner Products Consider two 2-dimensional vectors x and y. Their inner product (projection) is a
measure of non-orthogonality:
129Univ.-Prof. Dr.-Ing.
Markus Rupp
000 <=> y,xy,xy,xy
xy
xy
x
130Univ.-Prof. Dr.-Ing.
Markus Rupp
Complete Space Remember:
Definition 2.11: A metric Space (X,d) is called complete (Ger.: vollständig), if every Cauchy-sequence converges.
131Univ.-Prof. Dr.-Ing.
Markus Rupp
Hilbert- and Banach Spaces Definition 2.25: A complete normed
(metric) vector space is said to be a Banach space. Is there additionally an inner vector product (the norm is an induced norm), the space is said to be a Hilbert space.
Example 2.42: The space of continuous functions (C[a,b],d∞) is a Banach space.
Example 2.43: The space of continuous functions (C[a,b],dp) is for finite p not a Banach space since it is not complete.
Stefan Banach, (30.3.1892-31.8.1945) Polish Mathematician
132Univ.-Prof. Dr.-Ing.
Markus Rupp
Hilbert- and Banach Spaces Example 2.44: The space of sequences lp(0,∞) is a
Banach space. For p=2 it is also a Hilbert space. Example 2.45: The space of functions Lp[a,b] is a
Banach space. For p=2 it is also a Hilbert space.This Hilbert space is often denoted as L2(R) for functions and l2(R) for sequences.
In the following part of the lecture we will exclusively consider Hilbert spaces. (if not noticed otherwise)
133Univ.-Prof. Dr.-Ing.
Markus Rupp
134Univ.-Prof. Dr.-Ing.
Markus Rupp
Orthogonality Definition 2.26: Vectors of a pre-Hilbert space
are said to be orthogonal or perpendicular (Ger.: normal) if <x,y>=0.
Definition 2.27: Vectors of a pre-Hilbert space are said to be orthonormal if:
1,
1,
0,
=
=
=
yy
xx
yx
135Univ.-Prof. Dr.-Ing.
Markus Rupp
Orthogonality Example 2.46: The following set of
vectors is orthogonal:
How do we have to modify the set, in order to make the vectors orthonormal?
−−
−−
−
−
1111
,
1111
,
1111
,
1111
Orthogonality Consider a set of orthogonal vectors.
We then have:
Consider a set of orthonormal vectors. We then have:
136Univ.-Prof. Dr.-Ing.
Markus Rupp
ii
m
ii
m
iii
m
iii pppp ,,
1
2
11∑∑∑
===
= ααα
∑∑∑===
=m
ii
m
iii
m
iii pp
1
2
11, ααα
137Univ.-Prof. Dr.-Ing.
Markus Rupp
Orthogonality Example 2.47: The following set of functions is
orthonormal in [-π, π]:1,ejt/4, ej2t/4, ej3t/4
The inner product is defined as:
∫
∫
∫
−
−
−
=−=
−=
==
=
π
π
π
π
π
π
π
π
π
0)4/exp(21
)4/2exp()4/exp(21,
:)4/2exp()( and )4/exp()(Let
)()(21, *
dtjt
dtjtjtgf
jttgjttf
dttgtfgf
138Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces At this point several interesting questions arise:
Under which conditions is the linear combination of vectors unique?
Which is the smallest set of vectors required to describe every vector in S by a linear combination?
If a vector x can be described by a linear combination of pi ;i=1..m, how do we get the linear weights ci ;i=1..m?
Of which form do the vectors pi ;i=1..m need to be in order to reach every point x in S?
If x cannot be described exactly by a linear combination of pi ;i=1..m, how can it be approximated in the best way (smallest error)?
139Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.28: A Hamel basis of dimension m is
said to be orthogonal if for all basis vectors T=p1,p2,p3...,pm:
Definition 2.29: A Hamel basis of dimension m is said to be orthonormal if all basis vectors T=p1,p2,p3...,pm:
=≠≠
=jiji
ppji ;0
;0,
ji
ji jiji
pp
−=
=≠
=
δ;1;0
,
140Univ.-Prof. Dr.-Ing.
Markus Rupp
Orthogonality Let T=p1,p2,...,pn be a set of vectors. How can we
find a set S=q1,q2,...,qm with m smaller or equal to n so that span(S)=span(T) and the vectors in S are orthonormal?
Gram-Schmidt Method: 1) take p1 and construct
2) build:
3) continue:
4) if ei=0, throw away pi+1.
Erhard Schmidt 13.1.1876- 6.12.1959 German Mathematician
2223
22311332
111211221
1111
,/
;,,
,/;,
,/
eeeq
qqpqqppe
eeeqqqppe
pppq
=
−−=
=−=
=
141Univ.-Prof. Dr.-Ing.
Markus Rupp
Orthogonality Advantages of orthogonal bases:
[ ]
[ ] . of tionmultiplicaleft by Proof
basis lorthonorma an be let
T
T
ppp
pf
pf
pf
ppp
ppfppfppff
pppPcbaf
321
3
2
1
321
332211
321
,,
,
,
,
,,
,,,
,,],,,[
=
++=
==
142Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Definition 2.30: If there are two bases,
that span the same space with the additional property:
then these bases are said to be dual or biorthogonal (biorthonormal for ki,j=1).
,...,,;,...,,2121 mm
qqqUpppT ==
jijijikqp −= δ,,
143Univ.-Prof. Dr.-Ing.
Markus Rupp
Vector Spaces Example 2.48: let:
These pairs build a dual basis in R2. Consider
then:
[ ] [ ] [ ] [ ] ;1,0;1,1;1,1;0,12121
TTTT qqpp ==−==
Tbaf ],[=
212211
212211
)(,,
)(,,
qbaqaqpfqpff
pbpbapqfpqff
+−+=+=
++=+=
144Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Space Metric Set vs. Vector complete
Convergence Cauchy-Sequence
Linear vector space Subspace
145Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Consider the following function
in C[-∞,∞]:[ ]
[ ]
( )?,C[- in converge it Doesto? converge sequence this does Where
: for sequence functional the Compute
p
n
n
d
xxnn
xf
/n
xxxf
],
2exp
/21)(
1
21exp
21)(
20
2
2022
∞∞
−−=
=
−−=
π
σ
σπσ
( )?,L[- in converge it Does pd],∞∞
x
-50 -40 -30 -20 -10 0 10 20 30 40 50
f n(x
)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
n=2
n=10
147Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Cardinality of a set of QPSK?
Dimension of a vector space?= cardinality of the Hamel basis
What is the dimension of the vectors that form a Hamel basis for QPSK ?
What is a linear combination? When are vectors linearly independent?
4,1,1,1,1 =−−+−−+= AA :QPSK jjjj
148Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume00 0
1 01 1
1 02 2
1 1 0
2
1 1 01 1
1 1 0
1 1 0
1 0
2 1
1 0
......
M
K
MK K
MK K
k N M
M
k M
k
k M
hy xh hy x
h hy xh h h
xh h hy x
h h hy x
y h hh h
y hy hy h
−
−
−− −
−
− + −
−
− −
−
−
=
=
2
2
1
1 0...
k N M
k
k
k
kNk
x
xx
h xy H x
− − +
−
−
=
Which column vectors of these matrices are LI?
149Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 revisited Consider blind channel estimator (blind
equalization):
h1
h2
sk
g1
g2
sk(1)
sk(2)
-0
rk(1)
rk(2)
150Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.33 revisited Observe N receive values at each sensor:
rk(1)..=[rk
(1),rk-1(1)…,rk-N+1
(1)]T
When are the vectors linearly dependent?
In order to obtain linear dependency, we must have a sufficient number of vectors r1
(1)..rm(1), r1
(2)..rm(2), so that
2m>N since the vectors rk(1,2) of dimension N can maximally
span RN!
[ ] mN
m
mmm
mmmm
Rg
g
gg
g
rrrrrr
rgrgrgrgrgrg
2
2
21
1
11
)2()2(2
)2(1
)1()1(2
)1(1
)2(2
)2(222
)2(121
)1(1
)1(212
)1(111
R;0R,...,,,,...,,
......
×∈==
−
−
+++=+++
Example 2.33 revisited It turns out in fact that the following three
conditions are sufficient for a solution. 2m>N We now understand that we require to have
persistent excitation of the input signal sk. The Bezout condition must be satisfied for the
channels: no common zero! The solution itself we will provide later in the
context of SVD methods.
151Univ.-Prof. Dr.-Ing.
Markus Rupp
152Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume The Hilbert space is a linear vector
space with the following three properties: 1) 2) 3)
Complete (Vollständig)MetricNormed (Normiert)There exists an inner product
153Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Definition 2.31: Let S be a vector space with
elements x. A real-valued function ||x|| is said to be a norm (length) of x, if the following four properties are satisfied:
Note that just as for metrics 0) follows from 1) and 3)! (can you show this property)?
)inequality (triangle )3
Cfor)2
0 ifonly and if 0)1
every for 0)0
yxyx
xx
xx
Sxx
+≤+
∈=
==
∈≥
ααα
154Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Note, norms and metrics are related
concepts:
Example 2.49:
pppp xxdyxyxd =−= )0,(;),(
ii
n
ii
n
ii
xxl
xxl
xxl
max1
2
22
111
=−
=−
=−
∞∞
=
=
∑
∑
:norm
:norm
:norm
[ ] )(sup)(
)()(
)()(
,
2
22
11
txtxL
dttxtxL
dttxtxL
bat
b
a
b
a
∈∞∞ =−
=−
=−
∫
∫
:norm
:norm
:norm
Norms Why is l2 never smaller than l00?
With this method you can prove that
155Univ.-Prof. Dr.-Ing.
Markus Rupp
22
max 221max
1 ....ii
i i
xx x x x xx ∞ ∞
≥
= = = + ≥∑ ∑
1/ pp
n x x x∞ ∞
≥ ≥
156Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Definition 2.32: A normed, linear space is a
pair (S,||.||) in which S is a vector space and ||.|| is a norm on S.
Metrics of a normed linear space are defined by norms.
Definition 2.33: A vector is said to be normalized (Ger.: normiert, unit vector, Ger.: Einheitsvektor) if ||x||=1.
Except of the zero vector all vectors can be normalized.
157Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Note: The elements (vectors) of a linear
and normed space are not necessarily normalized vectors.
Only the space is normed, meaning that a norm exists for this space.
Norms Consider a sequence of vectors yk and a
limit y*. To evaluate whether yk converges to the
limit, we employ a norm:
Or, equivalently we can substitute xk=yk-y* and obtain:
158Univ.-Prof. Dr.-Ing.
Markus Rupp
0lim
0lim *
=
=−
∞→
∞→
kk
kk
x
yy
159Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Norm Equivalence Theorem 2.2: Let ||.|| and
||.||‘ be two norms on finite dimensional spaces Rn
(Cn,Qn,Zn,Nn,Bn), then we have:
Proof: It is to show that Without restricting generality we set (Ger.: oBdA)
||.||‘= ||.||2 and obtain for the lower bound:
0'lim,0lim == ∞→∞→ kkkk xx if only and if
2
21
21 1
1
1max
xx
xexexexx
exx
n
ii
n
i
n
iiiii
n
iii
≤
=≤≤≤
=
∑∑ ∑
∑
== =
=
α
α
:Thus
0,;' >≤≤ βαβα xxx
160Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms For the upper bound β consider the case:
And we obtain:
With the above condition, x cannot be the zero vector and since ||.||>0 there must be a positive lower bound β:
Since the property is true for the l2 norm, it is also true for all other norms!
02
>= cx
0;122
2
>≥= ββ
xxxxx
xxc β≤=2
'1'1' xxxxxxαβ
βα ≤≤→≤≤
161Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Note that the equivalence theorem is defined in terms of
Cauchy sequences. This is a consequence from the proof. Since if a particular norm tends to zero, then it does for every norm. The equivalence is to be understood in this sense.
Norms often find their application in terms of energy relations. It can be in form of average energy (l2-norm) or peak values (l∞-norm).
Thus, norms appear often when we describe systems. (Robustness, nonlinear systems, convergence of learning systems such as adaptive equalizers or controllers).
162Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Example 2.50: Hands-free telephone,
Ger.: Freisprechtelefon
+Far end speaker sk rk
Local speaker vk
h -
h
2
2hh − :energyerror System
163Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Example 2.51: Consider an adaptive equalizer as
part of a receiver.
The equalizer has a cost function of the form
which it tries to minimize (adaptive process). The error at its output is given by an error vector:
h f+sk rkvk kk ss ≈ˆ
( )kk ssf −ˆ
],...,[]ˆ,...,ˆ,ˆ[ 11 kNkkkNkNkNkNkk eesssssse −+−+−−− =−−−=
164Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Example 2.51: Let the adaptive equalizer have
the property that the error signal from k-1 to k is mapped via the learning rule: y=g(x)=x3. Under which condition is the equalizer adaptive?
( )
1......sup
...
...supsup?
21
21
61
61
21
21
22
2
21
2
2
31
31
1
<++++
=++++
=
==
=
−−−
−−−
−−−
−
−∈
−
−−
−
−
Nkk
Nkk
Nkk
Nkk
k
kle
k
Nk
k
k
Nk
k
eeee
eeee
ee
e
eeg
e
ee
hk
For |ek|<1 the error term at the output is always smallerthan at the input.
165Univ.-Prof. Dr.-Ing.
Markus Rupp
Norms Example 2.52: Consider the linear system in matrix-
vector-form.
Consider the ratio of input and output energy. When do we have a passive system, when an amplifier?
1 1 0 2
1 0
2 1 2
1 0 1
1 0
1,,
......
...
k N M k N M
M
k M k
k k
k M k
N M kNN k
y h h xh h
y h xy h xy h h xy H x
− + − − − +
−
− − −
− −
−
+ −
=
=
)(sup?
2
2,1
2
2,N
kMN
kNlx Hf
x
yhk
=−+
∈
Norms on Matrices In principle we can consider matrices
as vectors and apply the same norms, e.g., apply the l2 norm:
and obtain the Frobenius norm.
166Univ.-Prof. Dr.-Ing.
Markus Rupp
),min(
...)trace( 222
21
1 1
22
nmp
AAAA pH
m
i
n
jijF
=
++=== ∑∑= =
σσσ
Ferdinand Georg Frobenius (1849-1917)
Norms on Matrices Similarly
define norms on matrices
167Univ.-Prof. Dr.-Ing.
Markus Rupp
ijij
m
i
n
jij
AA
AA
max'
1 1
'
1
=
=
∞
= =∑∑
168Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Norms Inner vector products are helpful for defining
norms. For example, the (quadratic) l2 norm can be written as inner vector product:
This is thus said to be an induced (Ger.: induzierte) norm. The norm is induced by the inner vector product.
Of the lp norms for vectors only the l2 norm is an induced norm!
xxx
xxxxxx n
,
,...
2
222
21
2
2
=
=+++=
169Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Norms Example 2.53: induced norm in L2[a,b]
Note that for real-valued signals we have:
How can this be formulated for complex-valued signals?
yxyxyx ,22
2
2
2
2
2=−−+
21
221
2)()(),()(
== ∫
b
a
dttxtxtxtx
170Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Norms Vector norms can also be used to induce norms on
multivariate functions. Consider the following example:
( )2,0
expsup
)exp(sup
),(sup
)exp(),(
22
21
2121],[
2
2121],[2
2
21],[
212121
21
21
21
≈+
−=
−=
−
=
−=
=
=
=
xx
xxxx
xxxxx
f
lx
xxff
xxxxxxf
xxx
xxx
xxx
normSelect
171Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Norms
-5-4
-3-2
-10
12
34
5
-5-4
-3-2
-10
12
34
50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
172Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Vector Norms Matrix-norms are typically induced
vector-norms. Consider matrix A
Note that (spectral norm):
This is the largest singular value.
px
ppx
p
pxpp
xAxxA
x
xAAA
p 100ind,supsupsup =≠≠ =
===
( )AAAxAA Hx maxmax21ind,2
)(sup2
λσ === =
nmR ×∈
Induced Vector Norms
173Univ.-Prof. Dr.-Ing.
Markus Rupp
174Univ.-Prof. Dr.-Ing.
Markus Rupp
Spectral Norm Consider firstly a vector with M elements, based on the
sequence exp(jΩk):
In other words, all these vectors are linearly dependent , independent of their dimension M and frequency Ω.
If, however, M different frequencies are selected, M linearly independent vectors can be obtained.
[ ][ ]
[ ] kMT
mkM
kMT
kM
TkM
xmjmMkjmkjmkjx
xjMkjkjkjx
Mkjkjkjx
,,
,1,
,
)exp())1(exp()),...,1(exp()),(exp(
...)exp())2(exp(),...,exp()),1(exp(
))1(exp()),...,1(exp(),exp(
Ω=++−Ω+−Ω+Ω=
Ω=+−ΩΩ+Ω=
+−Ω−ΩΩ=
+
+
175Univ.-Prof. Dr.-Ing.
Markus Rupp
Spectral Norm Consider now the spectral norm for the vector xM,k:
0 1
0
1,
0
0exp( ( ))
...exp( ( 1))exp( ( ))
(exp( )) exp( ) exp(
(exp( ))(exp( )) exp( ( 1))(exp( )) exp( ( ))
M
MM k l
h hj k l
hhAx
j k l mj k l m n
h
H j j l
H jH j j l mH j j l m n
−
−+
Ω +
= Ω + + − Ω + + +
Ω Ω = = Ω Ω Ω + − Ω Ω + +
,
,
22,ind
2
, 2, 2,ind
, 2
)
(exp( ))exp( ( 1))exp( ( ))
sup
sup sup (exp( ))M k
M l
x M
M k l
x M MM k l
j l
H j xj l mj l m n
AxA
x
AxH j A
x
=
+Ω= Ω →∞
+
Ω = Ω Ω + − Ω + +
=
= → Ω =
Further Matrix Norms Recall:
Why is that? Consider an element yi of y=Ax
176Univ.-Prof. Dr.-Ing.
Markus Rupp
,ind,ind11
,ind
11
1
111
and:BUT
:generalin
max:maxrow
max:maxcolumn
∞∞
=≤≤∞
=≤≤
==
≠
==
=
∑
∑
AAAA
AA
AAA
AA
pp
Tn
jijmi
m
iijnj
?maxmax1
11
==
=→=
∑
∑∑
=
==
m
jjijxix
m
jjiji
m
jjiji
xAy
xAyxAy
jj
We can maximize the expression if sign(xj)=sign(Aij) as then xjAij =|xjAij|
If we further restrict the elements of vector x, e.g., ||x||=1, then there are two possibilities
177Univ.-Prof. Dr.-Ing.
Markus Rupp
,ind,ind11and
∞∞== AAAA
[ ]
[ ] ijj
jijT
ijj
jijj
T
AxAxx
AxAxx
maxmaxand10,...0,1,0,...,0,0
maxand11,...,1,1
1==→=
==→±±±=
∑
∑∑∞
178Univ.-Prof. Dr.-Ing.
Markus Rupp
,ind,ind11and
∞∞== AAAA
11
1
1111
1,1
111
11,
maxmaxmax
maxmaxmax
maxmaxmaxmax
maxmaxmaxmax
11
AAAxA
xAxAxxA
A
AAAxA
xAxAxxA
A
m
iijnj
iijj
jjij
ij
jjij
ixxxind
n
jijmi
jiji
jjijxi
jjijixxxind
====
===
====
===
∑∑∑∑
∑∑
∑∑∑
∑
=≤≤
==
∞=
≤≤=
=∞=∞
∞∞
∞
∞∞
179Univ.-Prof. Dr.-Ing.
Markus Rupp
Submultiplicative Property Some of the matrix norms (all p-norms) satisfy
the submultiplicative property:
Note that this property holds for arbitrary matrices, for example
But in this case the p-norms are different functions!
∞=≤ ,...2,1;ind,ind,ind,
pBAABppp
4332 , ×× ∈∈ RBRA
pppxByAxAB ≤
Submultiplicative Property Proof:
180Univ.-Prof. Dr.-Ing.
Markus Rupp
indpindpp
pxindp
p
pindpx
p
p
p
px
p
pxindp
BAx
xBA
x
yA
x
y
y
yA
xxAB
AB
p
p
pp
,,1,
,1
11,
sup
sup
supsup
==
≤
==
=
=
==
181Univ.-Prof. Dr.-Ing.
Markus Rupp
Submultiplicative Property Not all matrix norms have this submultiplicative
property: Example 2.54:
∆∆∆
∆
>
==
=
BAAB
BA
AA ij
:have we For 1111
max
Submultiplicative Property A similar property, however, is true
for all unitarily invariant norms:
unitarily invariant norm: (C is unitary)
182Univ.-Prof. Dr.-Ing.
Markus Rupp
ind
ind
ABAB
BAAB
,2
,2
≤
≤
xxC =
183Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms Example 2.55: Consider a causal impulse response hk.
In which case is hk an allpass (amplifier)?
Obviously, the energy of the input sequence must be identical to the energy of the output sequence (for every sequence).
Note that the vectors xN+M-1,k and yN,k have different length!
1 1 0 2
1 0
2 1 2
1 0 1
1 0
1,,
......
...
k N M k N M
M
k M k
k k
k M k
N M kNN k
y h h xh h
y h xy h xy h h xy H x
− + − − − +
−
− − −
− −
−
+ −
=
=
184Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms We must thus have for an allpass:
This in turn requires that N∞. For the spectral norm we have at least:
1
supsup
limlimlim
,
,1
,
0,1
,1
0,
,1,1,
==
==
==
−+≠
−+
−+
≠
−+∞→−+∞→∞→
indpN
pkMN
pkN
x
pkMN
pkMNN
xindpN
pkMNNpkMNNNpkNN
H
x
y
x
xHH
xxHy
least at havemust we
have we Since
( ) ( )indNN
jj HeHeH,2
limmax ∞→Ω
Ω∞
Ω ==
185Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms An allpass is thus difficult to describe by
matrices of finite dimension or by the spectral norm.
The reason for the finite dimension is that the initial values need also to be considered to reflect the lossless property of the allpass. This property can be required over the entire observation horizon.
186Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms Allpasses are better suited for state space forms. Note:
Example 2.56: yk=xk-2
=
+=
+=
+
+
k
kT
k
k
kkT
k
kkk
xz
dcbA
yz
dxzcy
xbzAz
1
1
[ ]
=
+=
+
=
+
+
k
k
k
k
kkk
kkk
xz
yz
xzy
xzz
00110001000,1
10
0010
1
1
187Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms Consider the
instantaneous energy:1
2 2 2 2 2 21 2 1 0 1 1
2 2 2 2 2 21 1
2 2 2 2 2 21 2 1 1 1
2 2 2 2 2 21 1
2 2 2 21 0
1 1
0 1 00 0 11 0 0
k k
k k
k k k k k k
k k k k k k
N N N N N NN N
N k kk k
z zy x
z z y z z x
z z y z z x
z z y z z x
z z y z z x
z y z x
+
+ −
+ + + + +
+ −
+= =
=
+ + = + +
+ + = + +
+ + = + +
+ + = + +
+ = +∑ ∑
⇐=+
++1
2
2
2
0
2
2
2
1
N
NN
xz
yz
For all N
188Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms Allpass as system with feedback
dcbA
q-1
1+kzkz
kx ky
189Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms This more general form also allows for describing allpasses
by linear time-variant systems. Example 2.57: Let b=sqrt(1-a2); 0<a<1
.a-1bfor allpass an also describes
allpass. an describes
2kk =
=
−
=
=
−
=
k
kkk
k
kk
kk
k
k
kk
k
k
k
A
xHxx
abba
yy
A
xHxx
abba
yy
)2(
)1(
)2(
)1(
)2(
)1(
)2(
)1(
190Univ.-Prof. Dr.-Ing.
Markus Rupp
Matrix Norms Allpass
unitary! be to said are conditions these satisfythat , Matrices vectors arbitrary allfor
:havemust we allpass anfor this tocontrast In
have we systems,invariant -timelinear For
N
kMN
pkMN
pkMNN
pkMN
pkN
pkMN
pkN
x
pkMN
pkMNN
xindpN
Hx
x
xH
x
y
x
y
x
xHH
,1
,1
,1
,1
,
,1
,
0,1
,1
0,
1
supsup
−+
−+
−+
−+
−+≠
−+
−+
≠
==
==
Matrix Norms Let us summarize:
The 2 induced matrix norm
tells us whether a system is passive or active
If a matrix is unitary, it describes an allpass.
191Univ.-Prof. Dr.-Ing.
Markus Rupp
( )AAAxAA Hx maxmax21ind,2
)(sup2
λσ === =
2,ind
2,ind
1
1
A amplifier
A attenuator
> ⇒
< ⇒
192Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Norms are…
Metrics in a linear vector space. A norm defined over a linear vector space makes it to a
normed, linear vector space. The knowledge of one norm is in general sufficient due
to the Equivalence Theorem
An induced norm is… a norm, that is build by a „simpler construct“
(example: the inner vector product induces the p=2 norm)
0'lim,0lim == ∞→∞→ kkkk xx if only and if
193Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume An induced matrix p-norm is…
a vector p-norm, applied to a matrix.
The spectral norm is… the vector p=2 norm induced to a matrix. In case of linearly time-invariant systems, it is identical with the
maximum magnitude of the transfer function.
Why is this true? Due to submultiplicative property of vector norms.
The submultiplicative property for matrices reads… .
pindppxAxA
,≤
BAAB ≤
194Univ.-Prof. Dr.-Ing.
Markus Rupp
Induced Vector Norms (Resume) Matrix-norms are typically induced
vector-norms. Consider matrix A
Note that (spectral norm):
This is the largest singular value.
px
ppx
p
pxpp
xAxxA
x
xAAA
p 100ind,supsupsup =≠≠ =
===
( )AAAxAA Hx maxmax21ind,2
)(sup2
λσ === =
nmR ×∈
195Univ.-Prof. Dr.-Ing.
Markus Rupp
Resume Inner vector products can also be written in terms
of induced norms. In R:
In C:
yxyxyx ,22
2
2
2
2
2=−−+
yxyxyx
yxyxyx
,Re4
,Re22
2
2
2
2
2
2
2
2
2
=−−+
=−−+
196Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain TheoremGeorge Zames(1934-1997) in 1966
Consider the following system with feedback.
Is it stable?
1
0,5q-1
yNhN
gNzN
xN
-
Pole at-0,5
stable!
197Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem Consider the following system with feedback.
Is it stable?
y=1/(1+h2)
0,5q-1
yNhN
gNzN
xN
-
198Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem The „Small Gain Theorem“ can be interpreted as
an extension from linear systems to nonlinear systems for the stability problem.
In linear time-invariant systems the gain of the open loop is responsible for stability (BIBO stability).
The Small Gain Theorem can be applied to non-linear as well as to time-variant systems.
199Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem Consider for this the input output relation of a
system described by vectors: Definition 2.34: A mapping H is said to be lp-
stable if there exists two positive constants β,γ so that for all input sequences the following is true:
Definition 2.35: The smallest positive value γ, for which lp-stability is obtained, is said to be the gain of H (Ger.: Verstärkung).
Note: BIBO (Bounded Input Bounded Output) stability is equivalent to l∞- stability.
βγ +≤=pNpNNpN
xxHy
NNNxHy =
200Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem Consider now a system with feedback, comprising
of two systems G and H and corresponding gains γgand γh described by the following signals:
[ ][ ]
NNNNNN
NNNNNN
yuGgGz
zxHhHy
+==
−==
201Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem
HN
GN +
yNhN
gNzN uN
xN
-
202Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem Theorem 2.3: (Small Gain Theorem) If both gains
γg and γh are such that:
then the two signals gN and hN of the system with feedback are bounded by:
1<hgγγ
[ ]
[ ]ghhNhNhg
N
hggNgNhg
N
xug
uxh
βγβγγγ
βγβγγγ
+++−
≤
+++−
≤
11
11
203Univ.-Prof. Dr.-Ing.
Markus Rupp
Small Gain Theorem Proof: Start with:
the norm is given by:
The derivation for gN is analogue.
[ ]
[ ]hggNgNhg
hggNgNNhg
ghNhNgN
gNgN
NNNN
ux
uxh
hux
gx
gGxh
βγβγγγ
βγβγγγ
ββγγ
βγ
+++−
=
++++=
++++≤
++≤
+≤
11
NNNNNN yugzxh +=−= ;
HN
GN +
yNhN
gNzN uN
xN
-
204Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.58 Let the automatic control
of a cell phone power amplifier look like this:
H2
G2 +
y2h2
g2z2 u2=0
x2
-
=
+
+=
37,031
01
5,2
120
2
212
212
221
221
2
G
xx
xx
H
ρρ
ρρ
205Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.58 Is this control stable? We describe G2 and H2 by their gains:
2222
,2
22,2222
222
,2
212
212
221
221
22,2222
37,037,0
31
5,20
15,2
120
gggGz
hh
xx
xx
hHy
ind
ind
ind
ind
=
=≤
=
+
+=≤
ρρ
ρρ
206Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.58
stable.-l thus ,0,372,5:gain loop open the for have thus We
21
37,09137,0
sup37.0
31
5,21
21
5,2
sup
01
5,2
120
sup0
15,2
120
22
21
22
21
2
1
,2
22
21
21
2
212
2122
2
2
221
221
22
2
2
1
212
212
221
221
,2
212
212
221
221
22
21
2
12
<×=
=+
+=
=+
++
+=
+
+
=
+
+
=+
=
γ
ρρ
ρρ
ρρ
ρρ
ρρ
ρρ
xx
xx
xx
xx
xx
xx
h
xx
xx
xx
xx
xx
xx
ind
xx
h
ind
207Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.60 Consider the following problem:
Let the shape of the signal s(t) be known at the receiver.
How is the filter g(t) to design so that the SNR at the output of g(t) is maximum?
matched filter (Ger.: Signalangepasstes Filter)Dwight O. North 1943
g(t)+s(t)
v(t)d(t)
r(t)
208Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality Theorem 2.4: In an inner-product space S in Cn
with induced norm ||.|| we have:
The inequality becomes an equality if and only if x=αy.
Proof: We start with the simple fact that
22, yxyx ≤
02
≥− yx α
209Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality We find further that
The minimum is obtained for: Thus:
−
−+−=
+−=−≤
2
2
*
*2
2
2
22
2
2
2
2
2
2
22
2
2
2
,,,
,Re20
y
yx
y
yxy
y
yxx
yyxxyx
αα
ααα
2
2
,
y
yx=α
2
2
2
2
2
2
2
,min0
y
yxxyx −=−≤ αα
22, yxyx ≤→
210Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality We are not done!
For x=αy we obtain ||x-αy||=0. Due to the norm property , equality to zero
can only be obtained for || x-αy||=0 if the argument is zero.
Thus equality is obtained if and only if x=αy.
0≥x
211Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality The Cauchy-Schwarz inequality implies:
for S in Cn:
for the space of real-valued functions in the interval [a,b]:
( )( )yyxxyx HHH ≤2
∫∫∫ ≤
b
a
b
a
b
a
dttgdttfdttgtf )()()()( 22
2
212Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality Example 2.59 Correlation coefficient
Remember: For the correlation coefficient we have:
Proof: Consider zero mean random variablesx‘=x-mx and y‘=y-my:
( )( )[ ]1
E111 ≤≤−⇒≤≤−
yx
yxxy
-y-xσσ
mmr
( )( )[ ]yx
yxxy
-y-xσσ
mmr
E=
[ ] ( )( )[ ]
[ ] [ ][ ] [ ]
'
EEEE
EE
xy22yx
yx
yx
yx
yxxy
y'x'y'x'y'x'
yxxy
r
mmmmr
===
−−=
−=
σσ
σσσσ
213Univ.-Prof. Dr.-Ing.
Markus Rupp
Chauchy-Schwarz Inequality Chauchy-Schwarz requires that
[ ] [ ] [ ][ ] [ ] [ ]
[ ][ ] [ ] 1
y'Ex'Ey'x'E
y'Ex'Ey'x'E
y'Ex'Ey'x'E
''','
22
22y'x'
222
22
22
≤=
≤
≤
≤
r
yxyx
214Univ.-Prof. Dr.-Ing.
Markus Rupp
Still Example 2.59 Correlation Coefficient Note that E[.] defines an ensemble average. In practical problems this is often replaced by a
time average (ergodic processes). Cauchy Schwarz still holds then:
( )( )
( ) ( )∑∑
∑
==
=
==
=
N
k
N
k
N
k
mN
mN
mmNr
1
22
1
22
1
1;1
1
ykyxkx
yx
ykxk
xy
-y-x
-y-x
σσ
σσ
215Univ.-Prof. Dr.-Ing.
Markus Rupp
Back to Example 2.60 Matched Filter (Ger.: Signalangepasstes Filter):
g(t)+s(t)
v(t)d(t)
r(t)
∫∫∫∫
∫
∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
=−≤
−==
−=
ττττττττ
τττ
τττ
dgdrdtgdr
dtgrttd
dtgrtd
)()()()(
)()()(
)()()(
2222
2
02
216Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.60 Matched filter:
g(t)+s(t)
v(t)d(t)
r(t)
)()()()()(
""
)()()( 222
ταταττατ
ττττ
−≈−=⇔=−
=→
≤ ∫∫∞
∞−
∞
∞−
srgrg
dgdrtd
:if only and if Identity,tpoint time suitable a to energyreceiver Maximize 0
217Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.60 Graphical:
r(τ)g(t-τ)
∫∞
∞−
−= τττ dtgrtd )()()(
d(t)
g(t-τ)
τ
t
218Univ.-Prof. Dr.-Ing.
Markus Rupp
Alternative Example 2.60
g(t)+as(t)
v(t)d(t)
r(t)[ ] [ ]
)(),()(),(')'(
')'()'()()()(
)(),()()()()(
).()()(;1
2222
2
2
2
22
22
ττσττσττσ
τττττδτστττ
ττττττττ
τδστ
ggtgtgdtg
dtgdtgdtg
stgdstgdstg
tt(t) s(t)
vvv
v
v
vEN
:zero is signal if energy Receiver
aES
:zero is noise if energy ReceivervvEaE
with vanda signals random Consider
=−−=−=
−−−=
−=
−=
−=
−=
=+=
∫
∫ ∫∫
∫∫
219Univ.-Prof. Dr.-Ing.
Markus Rupp
Alternative Example 2.60
g(t)+As(t)
v(t)d(t)
r(t)
2v
2v
)()(2v
2
1)(),(
2v
20
1)(),()(),()(),(
maxmax
:SNR Maximize
)(),(:only noise todueenergy Receiver
)(),()(
:only signal todueenergy Receiver
σσττ
ττσττ
ττσ
ττ
ταττ ==−
=
=
−==
=−=
ssggstg
NS
ggN
stgttS
stgstg
220Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.61 A fixed symbol sequence sk in C with L symbols is
transmitted for the purpose of synchronization at the beginning of a TDMA frame.
Design an optimal synchronization filter fk. Received sequence without noise:
s Infodata ik s …[ ],...i,,i, 21 ssrT =
221Univ.-Prof. Dr.-Ing.
Markus Rupp
Example 2.61 Correlation filter:
Because of Chauchy-Schwarz we have:
kkk
kkkkkkf
L
lllkk
srf
ffrrfr
frd
k
αα ≈=
=
= ∑−
=+
:thus
,,,max2
1
0
*
Example 2.62Werner Karl Heisenberg (5.12.1901 – 1.2.1976)Hermann Klaus Hugo Weyl, (9.11.1885 – 8.12.1955)
Time-frequency uncertainty:
for F(jω) being the Fourier transform of f(t).
222Univ.-Prof. Dr.-Ing.
Markus Rupp
21
)(21
)(21
)(
)(
2
22
2
22
≥
∫
∫
∫
∫∞
∞−
∞
∞−∞
∞−
∞
∞−
ωωπ
ωωωπ
djF
djF
dttf
dtttf
21
)(
)(
)(
)(
2
22
2
22
≥
∫
∫
∫
∫∞
∞−
∞
∞−∞
∞−
∞
∞−
ωω
ωωω
djF
djF
dttf
dtttf
Example 2.62 Recall the following properties:
1)
2)
3)
(partial integration with Dirichlet condition that f(t)0 for t+-oo)
223Univ.-Prof. Dr.-Ing.
Markus Rupp
∫∫∞
∞−
∞
∞−
=−
=−
=
dttfdttfttf
djFdtfjt
jFjdt
tfd
n
nn
nn
n
F
F
)()()(2
)()()(
)()()(
2'
ωω
ωω
Peter Gustav Lejeune Dirichlet13. 2.1805 in Düren; † 5.5.1859 in Göttingen)
Example 2.62 Cauchy Schwarz says
224Univ.-Prof. Dr.-Ing.
Markus Rupp
22
2
2 2
2
2 2 2
1 ( )2
( ) ( ) ( ) ( )
( ) '( ) ( ) ' ( )
f t dt
f t g t dt f t dt g t dt
tf t f t dt t f t dt f t dt
∞
−∞
∞ ∞ ∞
−∞ −∞ −∞
∞ ∞ ∞
−∞ −∞ −∞
= − ∫
≤
≤
∫ ∫ ∫
∫ ∫ ∫
Example 2.62
225Univ.-Prof. Dr.-Ing.
Markus Rupp
∫
∫
∫=
∫∫≤
≤
∫−
∞
∞−
∞
∞−∞
∞−
∞
∞−
∞
∞−
∞
∞−∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∫
∫∫
∫∫
ωωπ
ωωωπ
djF
djF
dttf
dttft
dttf
dttf
dttf
dttft
dttfdttftdttf
2
22
2
22
2
2
2
22
2222
2
)(21
)(21
)(
)(
)(
)('
)(
)(
41
)(')()(21
Plancherel
Michel Plancherel (* 16.1.1885; † 4.3.1967 in Zürich) was a Suisse mathematician.
Example 2.62 Time-frequency uncertainty:
Q: for which functions do we obtain equality?
A: tf(t)=f‘(t)=exp(-αt2)
226Univ.-Prof. Dr.-Ing.
Markus Rupp
21
)(
)(
)(
)(
2
22
2
22
≥
∫
∫
∫
∫∞
∞−
∞
∞−∞
∞−
∞
∞−
ωω
ωωω
djF
djF
dttf
dtttf
Alternative form (M.Vetterli et al. 2013) Define 1st and 2nd order moment:
227Univ.-Prof. Dr.-Ing.
Markus Rupp
Also works in the sequence domain:
228Univ.-Prof. Dr.-Ing.
Markus Rupp