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Table of contents
I Technical Pre-requisites
1 Digital filters
1.1 Discrete-time signals
1.2 Discrete-time systems
1.3 Z-transform
1.4 Design of IIR filters
1.5 Filter Structures
1.6 Conclusion
2 Computer Arithmetic
2.1 Fixed-Point Arithmetic
2.2 Floating-Point Arithmetic
2.3 Finite Precision Effects for IIR filters
2.4 Conclusion
3 Towards reliable implementation of Digital Filters
3.1 Automatic Filter Code Generator
3.2 Specialized Implicit Form
3.3 Conclusion
II Improvements to the Specialized Implicit Form
4 Specialized Implicit Form for Lattice Wave Digital Filters
4.1 Lattice Wave Digital Filters
4.2 A LWDF-to-SIF conversion algorithm
4.3 Conversion example
4.4 Conclusion
5 General algorithms for conversion 65
5.1 Conversion of data-flow graphs to SIF
5.2 Conversion of arbitrary structures to transfer functions
5.3 Numerical examples
5.4 Conclusion
III Reliable Fixed-Point Implementation of Digital Filters
6 Reliable evaluation of the dynamic range of an exact filter
6.1 State of the Art
6.2 Algorithm for Worst-Case Peak Gain evaluation
6.3 Truncation order and truncation error
6.4 Summation
6.5 Basic bricks
6.6 Numerical examples
6.7 Extending the WCPG theorem to the range of the state variables
6.8 WCPG for interval systems
6.9 Conclusion
7 Determining Reliable Fixed-Point Formats
7.1 Determining the Fixed-Point Formats
7.2 Taking rounding errors into account
7.3 Error analysis of the MSB computation formula
7.4 Complete algorithm
7.5 Numerical results
7.6 Application to the Specialized Implicit Form
7.7 Conclusion
7.8 Ongoing work: Off-by-One Problem
7.9 Ongoing work: Taking into account the spectrum of the input signal
8 Rigorous verification of implemented filter against its frequency specification
8.1 Problem statement
8.2 Verifying bounds on a transfer function
8.3 Verifying bounds for any LTI realization
8.4 Numerical examples
8.5 Conclusion
IV Hardware Code Generation
9 LTI filters computed just right on FPGA. Implementation of Direct Form I
9.1 Introduction
9.2 Error analysis of direct-form LTI filter implementations.
9.3 Sum of products computing just right
9.4 Implementation results
9.5 Conclusion
Conclusion and Perspectives
A Appendix
1 Lattice Wave Digital Filter basic brick data-flows
2 Coefficients for the examples
3 Off-by-One problem
Bibliography
