Reliable Verification of Digital Implemented Filters Against Frequency Specifications

Reliable implementation of digital filters in finiteprecision is based on accurate error analysis. However, a small error in the time domain does not guarantee that the implemented filter verifies the initial band specifications in the frequency domain. We propose a novel certified algorithm for the verification of a filter’s transfer function, or of an existing finite-precision implementation. We show that this problem boils down to the verification of bounds on a rational function, and further to the positivity of a polynomial. Our algorithm has reasonable runtime efficiency to be used as a criterion in large implementation space explorations. We ensure that there are no false positives but false negative answers may occur. For negative answers we give a tight bound on the margin of acceptable specifications.We demonstrate application of our algorithm to the comparison of various finite-precision implementations of filters already fully designed.

[1]  L. Rabiner.  Approximation methods for electronic filter design , 1976 .

[2]  Christoph Quirin Lauter,et al.  Determining fixed-point formats for a digital filter implementation using the worst-case peak gain measure , 2015, 2015 49th Asilomar Conference on Signals, Systems and Computers.

[3]  J. Harrison,et al.  Efficient and accurate computation of upper bounds of approximation errors , 2011, Theor. Comput. Sci..

[4]  Ivan W. Selesnick,et al.  Magnitude squared design of recursive filters with the Chebyshev norm using a constrained rational Remez algorithm , 1994, Proceedings of IEEE 6th Digital Signal Processing Workshop.

[5]  Bruce W. Char,et al.  GCDHEU: Heuristic Polynomial GCD Algorithm Based on Integer GCD Computation , 1984, EUROSAM.

[6]  Wayne Luk,et al.  Roundoff-noise shaping in filter design , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[7]  M. Gevers,et al.  Parametrizations in control problems , 1993 .

[8]  A. Varga,et al.  Computation of transfer-function matrices of generalized state-space models , 1989 .

[9]  Christoph Quirin Lauter,et al.  Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision , 2015, 2015 IEEE 22nd Symposium on Computer Arithmetic.

[10]  John N. Warfield,et al.  Synthesis of linear communication networks , 1958 .

[11]  Z. Zhao,et al.  On the generalized DFIIt structure and its state-space realization in digital filter implementation , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[12]  Gang Li,et al.  Roundoff noise analysis of two efficient digital filter structures , 2006, IEEE Transactions on Signal Processing.

[13]  Wayne Luk,et al.  Wordlength optimization for linear digital signal processing , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[14]  S. Boyd,et al.  On computing the worst-case peak gain of linear systems , 1992 .

[15]  Lajos Gazsi,et al.  Explicit formulas for lattice wave digital filters , 1985 .

[16]  Marie-Françoise Roy.  Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals , 1996 .

[17]  Anastasia Volkova,et al.  Reliable Fixed-Point Implementation of Linear Data-Flows , 2016, 2016 IEEE International Workshop on Signal Processing Systems (SiPS).

[18]  Philippe Chevrel,et al.  Sensitivity-Based Pole and Input-Output Errors of Linear Filters as Indicators of the Implementation Deterioration in Fixed-Point Context , 2011, EURASIP J. Adv. Signal Process..

[19]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[20]  Stephen P. Boyd,et al.  Comparison of peak and RMS gains for discrete-time systems , 1987 .

[21]  Philippe Chevrel,et al.  A Unifying Framework for Finite Wordlength Realizations , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[22]  Anastasia Volkova,et al.  Fixed-point implementation of Lattice Wave Digital Filter: Comparison and error analysis , 2015, 2015 23rd European Signal Processing Conference (EUSIPCO).

[23]  Romuald Rocher,et al.  Analytical Fixed-Point Accuracy Evaluation in Linear Time-Invariant Systems , 2008, IEEE Transactions on Circuits and Systems I: Regular Papers.