Reliable implementation of linear filters with fixed-point arithmetic

Hilaire, Thibault; Lopez, Benoit

doi:10.1109/sips.2013.6674540

Cited by 11 publications

(16 citation statements)

References 12 publications

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“…In that case, the Worst-Case Peak Gain matrix indicates by how much the radius of the input interval is amplified on the output [8] (although this is not valid for the transient phase, i.e. for the few first steps).…”

Section: Lti Filters and Worst-case Peak Gainmentioning

confidence: 99%

“…Since the filter is linear, the implemented filter H * can be seen as the exact filter H where the output is corrupted by the vector of errors e(k) occurring at each sum of product through a given linear filter H e (see Figure 1). State-space matrices of H e can be obtained analytically [8] and Proposition 1 can be used to determine the output error ∆y due to finite-precision arithmetic.…”

Section: Lti Filters and Worst-case Peak Gainmentioning

confidence: 99%

“…A solution is proposed in [8], based on a property of bounded-input bounded output systems [1], [2] where the largest possible peak value of the output is determined by the use of the Worst-Case Peak Gain (WCPG) matrix. Therefore, the analysis of error propagation in LTI systems is directly dependent on the reliable evaluation of the WCPG.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

Volkova¹,

Hilaire²,

Lauter³

2015

2015 IEEE 22nd Symposium on Computer Arithmetic

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Abstract-The worst-case peak gain (WCPG) of an LTI filter is an important measure for the implementation of signal processing algorithms. It is used in the error propagation analysis for filters, thus a reliable evaluation with controlled precision is required. The WCPG is computed as an infinite sum and has matrix powers in each summand. We propose a direct formula for the lower bound on truncation order of the infinite sum in dependency of desired truncation error. Several multiprecision methods for complex matrix operations are developed and their error analysis performed. We present a multiprecision complex matrix inversion algorithm using Newton-type iteration, along with its error analysis and proof of convergence. A multiprecision matrix powering method is presented. All methods yield a rigorous solution with an absolute error bounded by an a priori given value. The results are illustrated with numerical examples.

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Section: Lti Filters and Worst-case Peak Gainmentioning

confidence: 99%

Section: Lti Filters and Worst-case Peak Gainmentioning

confidence: 99%

See 1 more Smart Citation

Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

Volkova¹,

Hilaire²,

Lauter³

2015

2015 IEEE 22nd Symposium on Computer Arithmetic

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“…For some very special cases, eq. (4) should be adapted (Hilaire and Lopez, 2013). The position of the least significant bit ℓ is deduced from eq.…”

Section: Conversion From Real To Fixed-pointmentioning

confidence: 99%

“…Since the error e(k) done in the evaluation of the SoP is known to be in a given interval [e; e] (see Proposition 4), then the following proposition (Hilaire and Lopez, 2013) gives the output error interval: filter is the Direct Form I: From these informations, the operands to be summed, p i s, can be obtained with their respective FPF:…”

Section: Output Error Analysismentioning

confidence: 99%

Formatting Bits to Better Implement Signal Processing Algorithms

Lopez¹,

Hilaire²,

Didier

2014

Proceedings of the 4th International Conference on Pervasive and Embedded Computing and Communication Systems

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Abstract:This article deals with the fixed-point computation of the sum-of-products, necessary for the implementation of several algorithms, including linear filters. Fixed-point arithmetic implies output errors to be controlled. So, a new method is proposed to perform accurate computation of the filter and minimize the word-lengths of the operations. This is done by removing bits from operands that don't impact the final result under a given limit. Then, the final output of linear filter is guaranteed to be a faithful rounding of the real output.

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A Coq Formalization of Digital Filters

Gallois-Wong

Boldo

Hilaire

2018

Lecture Notes in Computer Science

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Digital lters are small iterative algorithms, used as basic bricks in signal processing (lters) and control theory (controllers). They receive as input a stream of values, and output another stream of values, computed from their internal states and from the previous inputs. These systems can be found in communication, aeronautics, automotive, robotics, etc. As the application domain may be critical, we aim at providing a formal guarantee of the good behavior of these algorithms. In particular, we formally proved in Coq some error analysis theorems about digital lters, namely the Worst-Case Peak Gain theorem and the existence of a lter characterizing the dierence between the exact lter and the implemented one. Moreover, the digital signal processing literature provides us with many equivalent algorithms, called realizations. We formally dened and proved the equivalence of several realizations (Direct Forms and State-Space).

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Reliable implementation of linear filters with fixed-point arithmetic

Cited by 11 publications

References 12 publications

Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision

Formatting Bits to Better Implement Signal Processing Algorithms

A Coq Formalization of Digital Filters

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