Causality Constraints in Discrete-Time Filter Design [Elektronische Ressource] / Sander Wahls. Betreuer: Thomas Sikora

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• Publié par: technische_universitat_berlin
• Publié le: 01 janvier 2011
• Nombre de lectures: 16
• Langue: Deutsch
• Poids de l'ouvrage: 4 Mo

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C A U S A L I T Y C O N S T R A I N T S I N
D I S C R E T E - T I M E F I LT E R D E S I G N
sander wahls
DissertationCausality Constraints in Discrete-Time Filter Design
Vorgelegt von Diplom-Mathematiker
Sander Wahls
aus Greifswald.
Von der Fakultät IV – Elektrotechnik und Informatik der Technischen
Universität Berlin zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften (Dr.-Ing.)
genehmigte Dissertation.
Promotionsausschuss:
Vorsitzender: Prof. Dr.-Ing. Thomas Wiegand
Berichter: Prof. Dr.-Ing. Thomas Sikora Prof. Dr.-Ing. Dr. rer. nat. Holger Boche
Tag der wissenschaftlichen Aussprache: 15. September 2011
Berlin 2011
D 83To my sonsC O N T E N T S
notation 7
1 introduction 9
1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . 10
1.2 Notes and References . . . . . . . . . . . . . . . . . . . . 11
2 preliminaries 13
2.1 Some Basic Structures . . . . . . . . . . . . . . . . . . . 13
2.2 Rational Matrices . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 18
2.4 Signals and Systems . . . . . . . . . . . . . . . . . . . . 20
2.5 State-Space . . . . . . . . . . . . . . . . . . . . . 20
2.6 Thez-Transform . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Inner-Outer Factorizations . . . . . . . . . . . . . . . . . 31
2.8 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . 36
2.8.1 Full Information Control . . . . . . . . . . . . . 37
2.8.2 Filtering . . . . . . . . . . . . . . . . . . . . . . . 38
2.9 Linear Operators . . . . . . . . . . . . . . . . . . . . . . 39
2.9.1 Moore-Penrose Pseudoinverses . . . . . . . . . . 39
2.9.2 Toeplitz Operators . . . . . . . . . . . . . . . . . 40
2.10 Notes and References . . . . . . . . . . . . . . . . . . . . 42
3 system inversion with weighted H criterion 452
3.1 Parametrization of Inverses . . . . . . . . . . . . . . . . 46
3.1.1 Extension Matrices . . . . . . . . . . . . . . . . . 46
3.1.2 Simple Parametrization . . . . . . . . . . . . . . 50
3.1.3 General . . . . . . . . . . . . . . 51
3.2 Optimal Inverses . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 Optimal Control Reformulation . . . . . . . . . 54
3.2.2 Existence, Uniqueness, and Parametrization . . 57
3.2.3 Efficient Computation . . . . . . . . . . . . . . . 63
3.3 Notes and References . . . . . . . . . . . . . . . . . . . . 69
4 system inversion with H criterion 71
1
4.1 Finite Section Method for the Approximation of the In-
fimum in System Inversion . . . . . . . . . . . . . . . . 72
4.2 Speed of Convergence of the Finite Section Method . . 72
4.3 Connection to the Infima in OtherH Problems . . . . 77
1
4.3.1 Filtering with Probably Unstable Plant . . . . . 77
4.3.2 Full Information Control . . . . . . . . . . . . . 80
4.4 Notes and References . . . . . . . . . . . . . . . . . . . . 81
5 realizable tomlinson-harashima precoders 85
5.1 Review of Tomlinson-Harashima Precoding . . . . . . . 85
5.2 System model and problem statement . . . . . . . . . . 90
5.3 Optimal Filter Design . . . . . . . . . . . . . . . . . . . . 92
5.4 Role of the Scalar Gain . . . . . . . . . . . . . . . . . . . 97
5.5 Efficient Computation . . . . . . . . . . . . . . . . . . . 101
5.5.1 Recursive Computation of theu . . . . . . . . 102k
55.5.2 Successive Optimization of . . . . . . . . . . . 103
5.5.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . 105
5.6 Numerical Experiments . . . . . . . . . . . . . . . . . . 107
5.6.1 Linearization of the Scaling Device . . . . . . . 107
5.6.2 Performance of the Successively Optimized Per-
mutation Order . . . . . . . . . . . . . . . . . . . 109
5.7 Notes and References . . . . . . . . . . . . . . . . . . . . 110
6 noise-shaping subband coders 117
6.1 Noise-Shaping Quantizers . . . . . . . . . . . . . . . . . 117
6.1.1 System Model . . . . . . . . . . . . . . . . . . . . 117
6.1.2 Stability Theory . . . . . . . . . . . . . . . . . . . 118
6.1.3 Stochastic Quantization Error . . . . . . . . . . . 123
6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.1 Polyphase Representation . . . . . . . . . . . . . 126
6.2.2 Stochastic Reconstruction Error . . . . . . . . . . 127
6.3 Optimal Feedback Filter Design . . . . . . . . . . . . . . 128
6.3.1 Constant Quantization Noise Power . . . . . . 128
6.3.2 Proportional Noise Power . . . . . 132
6.4 Notes and References . . . . . . . . . . . . . . . . . . . . 142
7 conclusion 149
prior publications and copyright information 153
bibliography 155
index 167
6N O TAT I O N
9 Existential quantifier
8 Universal
N Natural numbers,N=f0,1,2,:::g
Z Integers,Z=f::: ,-2,-1,0,1,2,:::g
R Real numbers
C Complex numbers
i Imaginary number
e Euler’s number
Re() Real part
Im() Imaginary part
jj Absolute value
¯ Complex conjugate
round() Nearest integer
floor() Nearest smaller or equal integer
ceil() Nearest larger or equal integer
sign() Signum, sign()= lim (+)=j+j&0
qp
C Complexqp matrices
q
Cq1 vectors
I Identity matrix of sizeqqq
I matrix of suitable dimensions
0 Zero matrix of sizeqqq
0 Zero matrix of sizeqpqp
0 Zero matrix of suitable dimensions
T Transpose of a matrix
Conjugate transpose of a matrix, adjoint operator
trace[] Trace of a matrix
[] Largest singular valuemax
[] Smallest valuemin
2 kk Frobenius norm,kk = trace[()() ]F F
kk Spectral norm,kk = []S S max
y Moore-Penrose Pseudoinverse
XY The matrixX-Y is positive semi-definite
XY The matrixX-Y is negative
O() Landau symbol; iff,g:R
!R thenf2O(g)
,9M>0,x 2
8x2
,x>x :jf(x)j6Mjg(x)j0 0
Consult the index on other symbols.
71
I N T R O D U C T I O N
Consider the problem depicted in Figure 1.0.1. Two linear time-
invariant (LTI) systemsM andN are driven by the same input. We
want to design a third LTI systemQ that estimates the output of the
first systemM from the output of the second systemN. This is the
filtering problem. Many well-known problems can be recast in this
framework. For example,
• the Wiener filter corresponds to the situation

I H
M= , N= ,
0 I
2where H is the medium and is the variance of an additive
white noise source. See Hassibi et al. [1, Ch. 10.3.4].
• the decision feedback equalizer corresponds to

I H
M= , N= .-10 z I
-1where z denotes the unit delay and>0 specifies the relia-
bility of the feedback path. See Erdogan et al. [2].
Without further constraints, the solution of the filtering problem is
straight-forward. Analogously to the case where M, N and Q are
constant matrices, the optimal solution to the filtering problem with
respect to several popular optimality criteria is given by the concate-
nation of an appropriately defined Moore-Penrose pseudoinverse of
N, and M. Consequently, the computation oferses for
finite-dimensional LTI systems has been investigated a lot. See, e.g.,
Jones [3], Varga [4], Stanimirovic [5], Chai et al. [6, 7], Gan and Ling
[8] and the references therein.
The drawback of the pseudoinverse-based solution is that the re-
sultingQ is non-causal, in general. That is, finding the current out-
put ofQ may require the knowledge of some or even all future inputs.
Such an behavior is acceptable as long as the input is processed of-
fline. A typical example of this situation is image processing. See,
e.g., Motwani and Guillemot [9]. However, non-causal systems can-
not be used for the real-time processing of data. In such situations
u s e
+M -
s
N Q
Figure 1.0.1: Filtering Problem. The error signal e = s-s˜ is to be
minimized by suitable choice ofQ.
9an additional causality constraint has to be employed. The filtering
problem with is more complicated but also well-
investigated. See, e.g., the recent book of Saberi et al. [10]. More
references follow in Section 2.10.
Many applications result in non-standard variants of the filtering
problem. For example, the optimality criterion may have to be modi-
fied or the set of admissible solutions has to be changed. There also
often is a special structure that can be exploited for more effective
algorithms. The goal of this thesis is to discuss how such issues can
be addressed subject to causality constraints. We investigate three
exemplary non-standard filtering problems,
• system (left-)inversion,
• Tomlinson-Harashima precoding and
• feedback filter design for noise-shaping subband coders.
Previous approaches to the considered problems often assume finite
impulse responses in order to simplify the problem. However, this
often leads to suboptimal and/or overly complex solutions. See, e.g.,
Crespo and Honig [11], Bai and Fu [12] or Zhang and Bitmead [13,14].
We derive the optimal solutions without finite impulse response as-
sumptions but subject to overall causality constraints. We also make
several theoretical contributions. The existence and uniqueness of so-
lutions to the considered problems is investigated thoroughly. Where
necessary, standard models are modified in order to ensure well-
posed problems. We also develop some other, previously unknown,
properties of the standard models. Finally, several new algorithms are