Table Of ContentFilter Design With Time Domain Mask Constraints:
Theory and Applications
Applied Optimization
Volume 56
Series Editors:
Panos M. Pardalos
University of Florida, U.S.A.
Donald Hearn
University of Florida, U.S.A.
The titles published in this series are listed at the end of this volume.
Filter Design With Time
Domain Mask Constraints:
Theory and Applications
by
Ba-Ngu Vo
The University of Melbourne, Australia
Antonio Cantoni
The University of Western Australia, Australia
and
KokLay Teo
The Hong Kong Polytechnic University, Hong Kong
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
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ISBN 978-1-4419-4858-8 ISBN 978-1-4757-3409-6 (eBook)
DOI 10.1007/978-1-4757-3409-6
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v
CONTENTS
List of Figures ..................................................... XI
List of Tables ••••••.•••.•.•••.••...••.•••..•••..•..•..•••.•.••.••• xv
Preface .••....•••...••...•.•..••••••••.••••..•••.•..•.••...••••• xvii
CHAPTER 1 INTRODUCTION ..••....•••.....••.•.......••....•..• 1
1.1 Applications .............................................. 2
Pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
TV waveform equalization ................................ 5
Data channel equalization ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Design to meet standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Deconvolution ........................................ 12
1.2 Envelope Constrained Filtering .............................. 13
1.3 Historical Notes .......................................... 18
1.4 Road Map ............................................... 19
CHAPTER 2 FILTERING WITH CONVEX RESPONSE CONSTRAINTS 21
2.1 Analog Filtering with Convexly Constrained Responses .......... 22
The cost functional ..................................... 24
The feasible region ..................................... 27
2.2 Filter Design with Envelope Constraints ....................... 28
2.3 Convex Programming ..................................... 32
2.4 Finite Dimensional Analog CCR Filters ....................... 34
2.5 Discrete-time CCR Filter Design ............................. 37
2.6 Continuous-time CCR Filtering via DSP Approach .............. 40
Problem formulation for hybrid filter ....................... 41
Feasible region ........................................ 42
vi
2.7 Finite Dimensional Hybrid Filter for CCR Filtering .............. 50
FIR digital processor. ................................... 51
2.8 Appendix ................................................ 53
CHAPTER 3 ANALYSIS AND PROBLEM CHARACTERIZATION ..... 61
3.1 Duality of Quadratic Program ................................ 62
3.2 Dual Problems for EC Filtering .............................. 67
Unconstrained dual problem .............................. 71
3.3 Dual Problem for Finite Dimensional Filter ..................... 76
3.4 Semi-Infinite Programming ................................. 79
Optimality conditions ................................... 81
Transformation technique via dual parametrization ............ 84
Getting the primal solution ............................... 87
3.5 Linearly Constrained Quadratic SIP and EC Filters ............... 87
Finite dimensional dual problem with mxn support points ....... 89
Finit~ dimensional dual problem with n support points ......... 90
Finite-dimensional EC filters ............................. 91
3.6 Appendix ................................................ 94
CHAPTER 4 DISCRETE-TIME EC FILTERING ALGORITHMS ...... 103
4.1 Discrete-time EC Filtering Problem .......................... 104
4.2 QP via Active Set Strategy ................................. 106
QP with linear inequality constraints ...................... 106
QP with inequality constraints ........................... 111
4.3 Iterative Algorithm via The Primal-Dual Method ............... 116
Non-smooth dual problem ............ \ .................. 118
Steepest ascent with directional differentials ................ 119
4.4 Iterative Algorithm using Augmented Cost. ................... 126
Approximation results .................................. 129
vii
Update equations ..................................... 130
4.5 Tapped Delay Line FIR Filters .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 136
4.6 Discrete-time Laguerre Networks ........................... 144
4.7 Appendix A ............................................ 149
4.8 Appendix B ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 156
CHAPTER 5 NUMERICAL METHODS FOR CONTINUOUS·T IME EC FIL·
TERING ........................................................ 161
5.1 Continuous-time EC Filtering .............................. 162
Analog filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 162
Hybrid filters ........................................ 163
5.2 Non-iterative Method ..................................... 164
5.3 Primal Dual Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
Discretization of dual problem for analog filters . . . . . . . . . . . .. 167
Discretization of dual problem for hybrid filters ............ , 170
Finite filter structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Steepest ascent algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 172
5.4 Penalty Approach for Semi-Infinite Programming .............. 175
Approximations by conventional constrained problems . . . . . .. 177
Approximations by unconstrained problems . . . . . . . . . . . . . . . . 179
Approximating convex problems ......................... 181
Affine functional inequality constrained problems . . . . . . . . . .. 182
SIP with quadratic cost and affine constraints ............... 184
Application to finite dimensional EC filters ................. 186
5.5 Laguerre Networks in Continuous-time EC Filtering ............ 188
Application to channel equalization ....................... 190
5.6 Hybrid filter with FIR Digital Components .................... 194
Linear interpolator ................................... , 195
Butterworth and Bessel post-filtering ...................... 196
viii
Chebyshev and elliptic post-filtering ...................... 199
5.7 Appendix ............................................... 203
CHAPTER 6 ROBUST ENVELOPE CONSTRAINED FILTERING ..... 219
6.1 Constraint Function ....................................... 220
6.2 Transformation to Smooth Problem .......................... 224
Problem conversion .................................... 224
6.3 Application to Analog ECUI Filtering Problem ................. 228
Finite dimensional filter for analog problem ................ 231
Example using Walsh functions .......................... 234
Approximations for finite dimensional filter ................ 237
6.4 Discrete-time ............................................ 238
6.5 Hybrid ECUI Filtering .................................... 239
Approximations for hybrid filters ......................... 241
Finite dimensional hybrid filters .......................... 242
6.6 Constraint Robustness ..................................... 247
Characterization of filter structure ........................ 250
Finite dimensional filter ................................ 252
Numerical example with Laguerre filter .................... 253
6.7 EC Filtering with Uncertain Implementation ................... 256
Examples with finite dimensional filters .................... 259
6.8 Appendix ............................................... 265
APPENDIX A MATHEMA TICAL BACKGROUND .................. 275
A.l Topological Space ........................................ 275
A.2 Metric Spaces ........................................... 279
A.3 Vector Spaces ........................................... 282
A.4 Normed Spaces .......................................... 285
A.5 Inner Product Spaces ..................................... 286
IX
A.6 Linear Operators ........................................ 289
A.7 Linear Functionals and Dual Spaces ......................... 292
A.8 Measures and Integration .................................. 295
APPENDIX B OPTIMIZATION THEORY ......................... 301
B.l Projection Theorem ...................................... 301
B.2 Hahn-Banach Theorem ................................... 302
B.3 Positive Cones and Convex Mappings ........................ 303
B.4 Gateaux and Frechet Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
B.5 Lagrange Multipliers ..................................... 310
References ...................................................... 313
Index ........................................................... 325
xi
List of Figures
Figure 1.1.1. Pulse Compression .............................................................................. 3
Figure 1.1.2. Pulse shape constraints for radar/sonar problem ................................. 4
Figure 1.1.3. K-rating mask for equalization of TV channel. ................................... 6
Figure 1.1.4. Model of a data channel.. ..................................................................... 7
Figure 1.1.5. Mask for constraints at sampling instances ......................................... 8
Figure 1.1.6. Mask for handling timing jitter. ........................................................... 8
Figure 1.1.7. Impulse response of coaxial cable for various lengths ...................... 10
Figure 1.1.8. DSX3 pulse template, coaxial cable response and filter output ........ 11
Figure 1.1.9. Pre- shaping of pulse ......................................................................... 11
Figure 1.1.10. ANSI T1.403 for T1 -1.544 Mb/s ...................................................... 12
Figure 1.2.1. Receiver model and output mask ....................................................... 14
Figure 1.2.2. Magnitude response constraint .......................................................... 15
Figure 1.2.3. Antenna receiver ................................................................................ 17
Figure 2.2.1. EC filtering with uncertain input. ...................................................... 30
Figure 2.4.1. Parallel filter structure ....................................................................... 36
Figure 2.4.2. Transversal filter structure ................................................................. 36
Figure 2.6.1. Configuration for digital processing of continuous-time signal ........ 41
Figure 3.2.1. Configuration of optimal analog EC filter ......................................... 74
Figure 3.2.2. Configuration of optimal hybrid EC filter ......................................... 76
Figure 4.3.1. Configuration for an adaptive EC filter ........................................... 117
Figure 4.4.1. Penalty allocator .............................................................................. 126
Figure 4.4.2. Flow chart for line search ................................................................ 132
Figure 4.5.1. A tapped delay line FIR filter .......................................................... 136
Figure 4.5.2. Optimum EC filter output for a 13-bit Barker-coded input... .......... 139
Figure 4.5.3. Augmented cost and noise gain of Barker coded example .............. 140
Figure 4.5.4. Optimum EC filter output for a rectangular input ........................... 141
Figure 4.5.5. Augmented cost and noise gain of rectangular input example ........ 141
Figure 4.5.6. Optimum and sub-optimum EC filter and their outputs .................. 143
Figure 4.5.7. Augmented cost and noise gain of DSX-3 example ........................ 143
Figure 4.6.1. Discrete-time Laguerre network ...................................................... 144
Figure 4.6.2. Signals and output mask for Laguerre and FIR EC filter ................ 147
Figure 5.2.1. Bound on slopes ............................................................................... 165
