You can find here some selected publications:

### 2018

- D. G. -, S. Boldo, and T. Hilaire, “A coq formalization of digital filters,” in Intelligent computer mathematics – 11th international conference, CICM 2018, hagenberg, austria, august 13-17, 2018, proceedings, 2018, p. 87–103.

[BibTeX] [Abstract] [PDF]

Digital filters are small iterative algorithms, used as basic bricks in signal processing (filters) and control theory (controllers). They receive as input a stream of values, and output another stream of values, computed from their internal state 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 time-domain. In particular, we formally proved in Coq some error analysis theorems about digital filters, namely the Worst-Case Peak Gain theorem and the existence of a filter characterizing the difference between the exact filter and the implemented one. Moreover, the digital signal processing literature provides us with many equivalent algorithms, called realizations. We formally defined and proved the equivalence of several realizations (Direct Forms and State-Space).

`@inproceedings{Gall18a, author = {Diane Gallois{-}Wong and Sylvie Boldo and Thibault Hilaire}, title = {A Coq Formalization of Digital Filters}, booktitle = {Intelligent Computer Mathematics - 11th International Conference, {CICM} 2018, Hagenberg, Austria, August 13-17, 2018, Proceedings}, pages = {87--103}, year = {2018}, Abstract = {Digital filters are small iterative algorithms, used as basic bricks in signal processing (filters) and control theory (controllers). They receive as input a stream of values, and output another stream of values, computed from their internal state 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 time-domain. In particular, we formally proved in Coq some error analysis theorems about digital filters, namely the Worst-Case Peak Gain theorem and the existence of a filter characterizing the difference between the exact filter and the implemented one. Moreover, the digital signal processing literature provides us with many equivalent algorithms, called realizations. We formally defined and proved the equivalence of several realizations (Direct Forms and State-Space). } }`

### 2017

- F. Qureshi, J. Takala, A. Volkova, and T. Hilaire, “Multiplierless unified architecture for mixed radix-2/3/4 ffts,” in Proc. european signal processing conference (eusipco’17), 2017.

[BibTeX] [Abstract] [PDF]

This paper presents a novel runtime-reconfigurable, mixed radix core for computation 2−, 3−, 4− point fast Fourier transforms (FFT). The proposed architecture is based on radix- 3 Wingorad Fourier transform, however multiplication is per- formed by constant multiplication instead of general multiplier. The complexity is equal to multiplierless 3-point FFT in terms of adders/subtractors with the exception of a few additional multiplexers. The proposed architecture supports all the FFT sizes which can be factorized into 2, 3, 4 point systems. We also show that the proposed architecture has the same bound on the accuracy as the classical one.

`@inproceedings{Faha17, Author = {Qureshi, Fahad and Takala, Jarmo and Volkova, Anastasia and Hilaire, Thibault}, Booktitle = {Proc. European Signal Processing Conference (EUSIPCO'17)}, Date-Added = {2017-10-26 13:34:09 +0000}, Date-Modified = {2017-10-26 16:31:40 +0000}, Title = {Multiplierless Unified Architecture for Mixed Radix-2/3/4 FFTs}, Year = {2017}, Abstract = {This paper presents a novel runtime-reconfigurable, mixed radix core for computation 2−, 3−, 4− point fast Fourier transforms (FFT). The proposed architecture is based on radix- 3 Wingorad Fourier transform, however multiplication is per- formed by constant multiplication instead of general multiplier. The complexity is equal to multiplierless 3-point FFT in terms of adders/subtractors with the exception of a few additional multiplexers. The proposed architecture supports all the FFT sizes which can be factorized into 2, 3, 4 point systems. We also show that the proposed architecture has the same bound on the accuracy as the classical one.}}`

- T. Hilaire and A. Volkova, “Error analysis methods for the fixed-point implementation of linear systems,” in Proc. ieee workshop on signal processing systems (sips), 2017.

[BibTeX] [Abstract] [PDF]

In this paper we propose to perform a complete error analysis of a fixed-point implementation of any linear system described by data-flow graph. The system is translated to a matrix-based internal representation that is used to determine the analytical errors-to-output relationship. The error induced by the finite precision arithmetic (for each sum-of-product) of the implementation propagates through the system and perturbs the output. The output error is then analysed with three different point of view: classical statistical approach (errors modelled as noises), worst-case approach (errors modelled as intervals) and probability density function. These three approaches allow determining the output error due to the finite precision with respect to its probability to occur and give the designer a complete output error analysis. Finally, our methodology is illustrated with numerical examples.

`@conference{Hila17a, Author = {Hilaire, T. and Volkova, A.}, Booktitle = {Proc. IEEE Workshop on Signal Processing Systems (SiPS)}, Title = {Error analysis methods for the fixed-point implementation of linear systems}, Year = {2017}, Abstract = {In this paper we propose to perform a complete error analysis of a fixed-point implementation of any linear system described by data-flow graph. The system is translated to a matrix-based internal representation that is used to determine the analytical errors-to-output relationship. The error induced by the finite precision arithmetic (for each sum-of-product) of the implementation propagates through the system and perturbs the output. The output error is then analysed with three different point of view: classical statistical approach (errors modelled as noises), worst-case approach (errors modelled as intervals) and probability density function. These three approaches allow determining the output error due to the finite precision with respect to its probability to occur and give the designer a complete output error analysis. Finally, our methodology is illustrated with numerical examples.}}`

- A. Volkova, T. Hilaire, and C. Q. Lauter, “Rigorous determination of recursive filter fixed-point implementation with input signal frequency specifications,” in Asilomar conference on signals, systems and computers, 2017.

[BibTeX] [Abstract]

In this article, we focus on the Fixed-Point implementation of Linear Time Invariant (LTI) in state-space representation. For that purpose, we give an algorithm to determine the fixed-point formats of all the involved variables (states and outputs). For the sake of generality, algorithm has been applied to the case of Multiple Inputs Multiple Outputs (MIMO) filters. The computational errors in the intermediate steps of the filter evaluation as well as their accumulation over time is fully taken into account. We handle several rounding modes (round to nearest and truncation) for a two’s-complement based Fixed-Point Arithmetic. Our approach is fully rigorous in the way that the output Fixed-Point formats are shown to avoid overflows and we do not use any (non-exhaustive) filter simulation steps.

`@inproceedings{Volk17b, Author = {Volkova, Anastasia and Hilaire, Thibault and Lauter, Christoph Q.}, Booktitle = {Asilomar Conference on Signals, Systems and Computers}, Month = {November}, Title = {Rigorous determination of recursive filter Fixed-Point Implementation with input signal frequency specifications}, Year = {2017}, Abstract = {In this article, we focus on the Fixed-Point implementation of Linear Time Invariant (LTI) in state-space representation. For that purpose, we give an algorithm to determine the fixed-point formats of all the involved variables (states and outputs). For the sake of generality, algorithm has been applied to the case of Multiple Inputs Multiple Outputs (MIMO) filters. The computational errors in the intermediate steps of the filter evaluation as well as their accumulation over time is fully taken into account. We handle several rounding modes (round to nearest and truncation) for a two's-complement based Fixed-Point Arithmetic. Our approach is fully rigorous in the way that the output Fixed-Point formats are shown to avoid overflows and we do not use any (non-exhaustive) filter simulation steps.}}`

- A. Volkova, C. Lauter, and T. Hilaire, “Reliable verification of digital implemented filters against frequency specifications,” in 2017 ieee 24th symposium on computer arithmetic (arith), 2017, pp. 180-187.

[BibTeX] [Abstract] [PDF]

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.

`@inproceedings{Volk17a, Author = {Volkova, A. and Lauter, C. and Hilaire, T.}, Booktitle = {2017 IEEE 24th Symposium on Computer Arithmetic (ARITH)}, Month = {July}, Pages = {180-187}, Title = {Reliable Verification of Digital Implemented Filters Against Frequency Specifications}, Year = {2017}, Abstract = {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.}, }`

### 2016

- A. Volkova, C. Lauter, and T. Hilaire, “Computing the Worst-Case Peak Gain of Digital Filter in Interval Arithmetic,” in Proceedings of the 17th International Symposium on Scientific Computing, Computer Arithmetics and Verified Numerics (SCAN), 2016.

[BibTeX] [PDF]`@inproceedings{Volk16a, Author = {Volkova, A. and Lauter, C. and Hilaire, T.}, Booktitle = {{Proceedings of the 17th International Symposium on Scientific Computing, Computer Arithmetics and Verified Numerics (SCAN)}}, Title = {{Computing the Worst-Case Peak Gain of Digital Filter in Interval Arithmetic}}, Year = {2016} }`

- T. Hilaire, A. Volkova, and M. Ravoson, “Reliable fixed-point implementation of linear data-flows,” in Proc. IEEE Workshop on Signal Processing Systems (SiPS), 2016.

[BibTeX] [Abstract] [PDF]

In this article, we propose a complete methodology to implement a signal processing or control-engineering algo- rithm described with a linear data-flow into numerical code using fixed-point arithmetic. Our approach is based on a reliable determination of the Worst-Case Peak gain of a filter as well as on rigorous error analysis of roundoff error propagation. It guarantees that no overflow will occur and that the output error due to the finite precision implementation is less than a given bound. Without loss of generality, we consider the linear data- flows given in the form of Simulink block diagram. It is first transposed into an internal matrix-based representation and then the reliable evaluation of the magnitudes of each internal variable is performed. Our approach allows determining the minimum word-length required to achieve a given accuracy. Finally, the methodology is illustrated with numerical examples.

`@inproceedings{Hila16a, Author = {Hilaire, T. and Volkova, A. and Ravoson, M.}, Booktitle = {{Proc. IEEE Workshop on Signal Processing Systems (SiPS)}}, Title = {{Reliable fixed-point implementation of linear data-flows}}, Year = {2016}, Abstract = {In this article, we propose a complete methodology to implement a signal processing or control-engineering algo- rithm described with a linear data-flow into numerical code using fixed-point arithmetic. Our approach is based on a reliable determination of the Worst-Case Peak gain of a filter as well as on rigorous error analysis of roundoff error propagation. It guarantees that no overflow will occur and that the output error due to the finite precision implementation is less than a given bound. Without loss of generality, we consider the linear data- flows given in the form of Simulink block diagram. It is first transposed into an internal matrix-based representation and then the reliable evaluation of the magnitudes of each internal variable is performed. Our approach allows determining the minimum word-length required to achieve a given accuracy. Finally, the methodology is illustrated with numerical examples.} }`

### 2015

- A. Volkova, T. Hilaire, and C. Q. Lauter, “Determining Fixed-Point Formats for a Digital Filter Implementation using the Worst-Case Peak-Gain measure,” in Asilomar Conference on Signals, Systems and Computers, 2015.

[BibTeX] [Abstract] [PDF] [Slides]

In this article, we focus on the Fixed-Point implementation of Linear Time Invariant (LTI) in state-space representation. For that purpose, we give an algorithm to determine the fixed-point formats of all the involved variables (states and outputs). For the sake of generality, algorithm has been applied to the case of Multiple Inputs Multiple Outputs (MIMO) filters. The computational errors in the intermediate steps of the filter evaluation as well as their accumulation over time is fully taken into account. We handle several rounding modes (round to nearest and truncation) for a two’s-complement based Fixed-Point Arithmetic. Our approach is fully rigorous in the way that the output Fixed-Point formats are shown to avoid overflows and we do not use any (non-exhaustive) filter simulation steps.

`@inproceedings{Volk15c, Author = {Volkova, Anastasia and Hilaire, Thibault and Lauter, Christoph Q.}, Booktitle = {{Asilomar Conference on Signals, Systems and Computers}}, Month = {November}, Title = {{Determining Fixed-Point Formats for a Digital Filter Implementation using the Worst-Case Peak-Gain measure}}, Year = {2015}, Abstract = {In this article, we focus on the Fixed-Point implementation of Linear Time Invariant (LTI) in state-space representation. For that purpose, we give an algorithm to determine the fixed-point formats of all the involved variables (states and outputs). For the sake of generality, algorithm has been applied to the case of Multiple Inputs Multiple Outputs (MIMO) filters. The computational errors in the intermediate steps of the filter evaluation as well as their accumulation over time is fully taken into account. We handle several rounding modes (round to nearest and truncation) for a two's-complement based Fixed-Point Arithmetic. Our approach is fully rigorous in the way that the output Fixed-Point formats are shown to avoid overflows and we do not use any (non-exhaustive) filter simulation steps.}, }`

- A. Volkova, T. Hilaire, and C. Q. Lauter, “Reliable evaluation of the Worst-Case Peak Gain matrix in multiple precision,” in 22nd IEEE Symposium on Computer Arithmetic, Jun 2015, Lyon, France. 2015, Lyon, France, 2015.

[BibTeX] [Abstract] [PDF] [Slides]

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.

`@inproceedings{Volk15a, Address = {Lyon, France}, Author = {Volkova, Anastasia and Hilaire, Thibault and Lauter, Christoph Q.}, Booktitle = {{22nd IEEE Symposium on Computer Arithmetic, Jun 2015, Lyon, France. 2015}}, Hal_Id = {hal-01083879}, Hal_Version = {v2}, Title = {{Reliable evaluation of the Worst-Case Peak Gain matrix in multiple precision}}, Year = {2015}, 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. }, }`

- A. Volkova and T. Hilaire, “Fixed-Point implementation of Lattice Wave Digital Filter: comparison and error analysis,” in Proc. European Signal Processing Conference (EUSIPCO’15), 2015.

[BibTeX] [Abstract] [PDF] [Slides]

A consistent analysis of the filter design along with its further implementation in fixed-point arithmetic requires a large amount of work, and this process differs from one filter representation to another. For the unifying purposes of such flow, a Specialized Implicit Form (SIF) had been proposed.Various sensitivity and stability measures have been adapted to it along with an a priori error analysis (quantization of the coefficients and output error). In this paper a conversion algorithm for the widely used Lattice Wave Digital Filters (LWDF) to the SIF is presented, along with a finite precision error analysis. It allows to compare fairly LWDF to other structures, like direct forms and state-space. This is illustrated with a numerical example.

`@inproceedings{Volk15b, Author = {Volkova, A. and Hilaire, T.}, Booktitle = {{Proc. European Signal Processing Conference (EUSIPCO'15)}}, Month = {September}, Title = {{Fixed-Point implementation of Lattice Wave Digital Filter: comparison and error analysis}}, Year = {2015}, Abstract = {A consistent analysis of the filter design along with its further implementation in fixed-point arithmetic requires a large amount of work, and this process differs from one filter representation to another. For the unifying purposes of such flow, a Specialized Implicit Form (SIF) had been proposed.Various sensitivity and stability measures have been adapted to it along with an a priori error analysis (quantization of the coefficients and output error). In this paper a conversion algorithm for the widely used Lattice Wave Digital Filters (LWDF) to the SIF is presented, along with a finite precision error analysis. It allows to compare fairly LWDF to other structures, like direct forms and state-space. This is illustrated with a numerical example.}, }`

### 2014

- B. Lopez, T. Hilaire, and L. Didier, “Formatting bits to better implement signal processing algorithms,” in 4th international Conference on Pervasive and Embedded Computing and Communication Systems (PECCS), Lisbon, Portugal, 2014.

[BibTeX] [Abstract] [PDF] [Slides]

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.

`@inproceedings{Lope14, Address = {Lisbon, Portugal}, Author = {Lopez, Benoit and Hilaire, Thibault and Didier, Laurent-St{\'e}phane}, Booktitle = {{4th international Conference on Pervasive and Embedded Computing and Communication Systems (PECCS)}}, Hal_Id = {hal-01076049}, Hal_Version = {v1}, Month = Jan, Title = {{Formatting bits to better implement signal processing algorithms}}, Year = {2014}, 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. }, }`

### 2013

- T. Hilaire and B. Lopez, “Reliable Implementation of Linear Filters with Fixed-Point Arithmetic,” in Proc. IEEE Workshop on Signal Processing Systems (SiPS), 2013.

[BibTeX] [Abstract] [PDF]

This article deals with the implementation of linear filters or controllers with fixed-point arithmetic. The finite precision of the computations and the roundoff errors induced may have an important impact on the numerical behavior of the implemented system. Moreover, the fixed-point transformation is a time consuming and error- prone task, specially with the objective of minimizing the quantization impact. Based on a formalism able to describe every structure of linear filters/controllers, this paper proposes an automatic method to generate fixed-point version of the inputs-to-outputs algorithm and an analysis of the global error added on the output. An example illustrates the approach.

`@conference{Hila13a, Author = {Hilaire, T. and Lopez, B.}, Booktitle = {{Proc. IEEE Workshop on Signal Processing Systems (SiPS)}}, Title = {{Reliable Implementation of Linear Filters with Fixed-Point Arithmetic}}, Year = {2013}, Abstract = {This article deals with the implementation of linear filters or controllers with fixed-point arithmetic. The finite precision of the computations and the roundoff errors induced may have an important impact on the numerical behavior of the implemented system. Moreover, the fixed-point transformation is a time consuming and error- prone task, specially with the objective of minimizing the quantization impact. Based on a formalism able to describe every structure of linear filters/controllers, this paper proposes an automatic method to generate fixed-point version of the inputs-to-outputs algorithm and an analysis of the global error added on the output. An example illustrates the approach.}, }`

### 2012

- B. Lopez, T. Hilaire, and L. Didier, “Sum-of-products Evaluation Schemes with Fixed-Point arithmetic, and their application to IIR filter implementation,” in Conference on Design and Architectures for Signal and Image Processing (DASIP), 2012.

[BibTeX] [Abstract] [PDF] [Slides]

The signal processing and control algorithms are widely based on sum-of-products evaluation. In fixed-point arith- metic, the roundoff errors and coefficient quantization may have an important effect on the application’s performance and characteristics. As part of a global methodology on optimal fixed-point im- plementation of filters/controllers, this paper formalizes the various implementation schemes for sum-of-products in fixed- point arithmetic and automates the fixed-point code production. The order of the operations are considered, as their bit-width and the fixed-point representation of the coefficients, variables and partial results. Applied to linear filters, the output roundoff noise error is then evaluated and used as a criteria to find out interesting evaluation scheme. An example illustrates the approach.

`@inproceedings{Lope12, Author = {Lopez, Benoit and Hilaire, Thibault and Didier, Laurent-St{\'e}phane}, Booktitle = {{C}onference on {D}esign and {A}rchitectures for {S}ignal and {I}mage {P}rocessing ({DASIP})}, Location = {Karlsruhe, Germany}, Month = oct, Title = {{S}um-of-products {E}valuation {S}chemes with {F}ixed-{P}oint arithmetic, and their application to {IIR} filter implementation}, Year = {2012}, Abstract = {The signal processing and control algorithms are widely based on sum-of-products evaluation. In fixed-point arith- metic, the roundoff errors and coefficient quantization may have an important effect on the application’s performance and characteristics. As part of a global methodology on optimal fixed-point im- plementation of filters/controllers, this paper formalizes the various implementation schemes for sum-of-products in fixed- point arithmetic and automates the fixed-point code production. The order of the operations are considered, as their bit-width and the fixed-point representation of the coefficients, variables and partial results. Applied to linear filters, the output roundoff noise error is then evaluated and used as a criteria to find out interesting evaluation scheme. An example illustrates the approach.} }`

- D. Menard, R. Rocher, O. Sentieys, N. Simon, L-S. Didier, T. Hilaire, B. Lopez, E. Goubault, S. Putot, F. Vedrine, A. Najahi, G. Revy, L. Fangain, C. Samoyeau, F. Lemonnier, and C. Clienti, “Design of Fixed-point Embedded Systems (DEFIS),” in 2012 Conference on Design and Architectures for Signal and Image Processing (DASIP), Karlsruhe, Germany, October 23 – 25, 2012, 2012, pp. 365-366.

[BibTeX] [Abstract] [PDF] [HAL]

Embedded applications are usually coming with stringent constraints in term of cost, energy consumption and realtime. Consequently, fixed-point arithmetic is mainstream for their implementation into embedded systems. Hence, the main objective of the french ANR project DEFIS is to provide a complete design flow for fixed-point refinement of complex applications. This tool flow is like the missing link between high-level application specification tools and low-level implementation.

`@inproceedings{Mena12, Author = {Menard, D. and Rocher, R. and Sentieys, O. and Simon, N. and Didier, L-S. and Hilaire, T. and Lopez, B. and Goubault, E. and Putot, S. and Vedrine, F and Najahi, A. and Revy, G. and Fangain, L and Samoyeau, C. and Lemonnier, F and Clienti, C.}, Booktitle = {{2012 Conference on Design and Architectures for Signal and Image Processing (DASIP), Karlsruhe, Germany, October 23 - 25, 2012}}, Editor = {{Chips, ECSI - European Electronic and design Initiative, Systems}}, Isbn = {978-2-9539987-4-0}, Issn = {1966-7116}, Pages = {365-366}, Publisher = {{ECSI - European Electronic Chips and Systems design Initiative}}, Title = {{Design of Fixed-point Embedded Systems ({DEFIS})}}, Year = {2012}, Abstract = {Embedded applications are usually coming with stringent constraints in term of cost, energy consumption and realtime. Consequently, fixed-point arithmetic is mainstream for their implementation into embedded systems. Hence, the main objective of the french ANR project DEFIS is to provide a complete design flow for fixed-point refinement of complex applications. This tool flow is like the missing link between high-level application specification tools and low-level implementation.}, HAL = {https://hal.archives-ouvertes.fr/hal-00822487}, HAL_ID = {hal-00822487}, HAL_VERSION = {v1}, }`

- T. Hilaire and P. Chevrel, “Réalisations optimales pour l’implantation de systèmes LTI paramétrés,” in Septième conférence internationale francophone d’automatique, 2012.

[BibTeX] [Abstract] [PDF] [Slides]

Cet article s’intéresse à la résilience des filtres/régulateurs LTI paramétrés. On entend ici par résilience la robustesse vis-à-vis de l’implantation en précision finie. Précisément, on s’intéresse à l’impact de la quantification des coefficients représentés en virgule fixe. La nouveauté tient au fait que les coefficients implantés ne sont pas supposés définis de manière définitive lors de l’étape d’initialisation du code embarqué, mais calculés à partir de paramètres dont la valeur n’est pas figée a priori. Ces paramètres sont supposés pouvoir être modifiés ultérieurement, à l’intérieur d’une plage prédéfinie, sans reconception du code embarqué. C’est le cas souvent, par exemple dans le domaine automobile, où un dernier réglage ou une adaptation peuvent être réalisés très tard dans le cycle de conception. Nous formaliserons l’erreur occasionnée par une erreur de quantification sur ces paramètres, avant d’étudier le cas d’un filtre du 2ème ordre dont les gain, amortissement et pulsation naturelle ne sont pas supposés fixés a priori.

`@inproceedings{Hila12a, Author = {Hilaire, Thibault and Chevrel, Philippe}, Booktitle = {Septi{\`e}me Conf{\'e}rence Internationale Francophone d'Automatique}, Location = {Grenoble}, Month = July, Title = {R{\'e}alisations optimales pour l'implantation de syst{\`e}mes {LTI} param{\'e}tr{\'e}s}, Year = {2012}, Abstract = {Cet article s'intéresse à la résilience des filtres/régulateurs LTI paramétrés. On entend ici par résilience la robustesse vis-à-vis de l'implantation en précision finie. Précisément, on s'intéresse à l'impact de la quantification des coefficients représentés en virgule fixe. La nouveauté tient au fait que les coefficients implantés ne sont pas supposés définis de manière définitive lors de l'étape d'initialisation du code embarqué, mais calculés à partir de paramètres dont la valeur n'est pas figée a priori. Ces paramètres sont supposés pouvoir être modifiés ultérieurement, à l'intérieur d'une plage prédéfinie, sans reconception du code embarqué. C'est le cas souvent, par exemple dans le domaine automobile, où un dernier réglage ou une adaptation peuvent être réalisés très tard dans le cycle de conception. Nous formaliserons l'erreur occasionnée par une erreur de quantification sur ces paramètres, avant d'étudier le cas d'un filtre du 2ème ordre dont les gain, amortissement et pulsation naturelle ne sont pas supposés fixés a priori. }}`

- T. Hilaire and A. Chapoutot, “Interval-based robustness of linear parametrized filters,” in Proceedings of the 15’th gamm-imacs international symposium on scientific computing, computer arithmetic and verified numerical computations, 2012, 2012.

[BibTeX] [PDF] [Slides]`@inproceedings{Hila12b, Author = {Hilaire, Thibault and Chapoutot, Alexandre}, Booktitle = {Proceedings of the 15'th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerical Computations, 2012}, Month = Sept., Title = {Interval-based Robustness of Linear Parametrized Filters}, Year = {2012}, }`

### 2011

- J. C. Bajard, L. S. Didier, and T. Hilaire, “rho-Direct Form Transposed and Residue Number Systems for Filter implementations,” in Circuits and Systems (MWSCAS), 2011 IEEE 54th International Midwest Symposium on, 2011. doi:10.1109/MWSCAS.2011.6026263

[BibTeX] [Abstract] [PDF] [Slides]

This paper deals with a new approach for Infinite Impulse Response (IIR) Filter based on specific structure and arithmetic. The $\rho$-Direct Form II transposed, introduced by G. Li, is an numerically efficient structure for FIR or IIR filters. Compared to classical direct forms, it uses more computations but less bits are necessary for the same precision. These properties made it well conditioned for fixed point arithmetic and its implementations are economical and numerically efficient comparing to other forms. In other hand, Residue Number Systems (RNS) offer an interest- ing parallelism where operations are made on small values. RNS are well known for improving the performances of DSP filters. We compare our RNS approach to fixed-point implementations of DFI and $\rho$-DFIIt.

`@inproceedings{Baja11a, Author = {Bajard, J.C. and Didier, L.S. and Hilaire, T.}, Booktitle = {{Circuits and Systems (MWSCAS), 2011 IEEE 54th International Midwest Symposium on}}, Month = {August}, Title = {{rho-Direct Form Transposed and Residue Number Systems for Filter implementations}}, Year = {2011}, Abstract = {This paper deals with a new approach for Infinite Impulse Response (IIR) Filter based on specific structure and arithmetic. The $\rho$-Direct Form II transposed, introduced by G. Li, is an numerically efficient structure for FIR or IIR filters. Compared to classical direct forms, it uses more computations but less bits are necessary for the same precision. These properties made it well conditioned for fixed point arithmetic and its implementations are economical and numerically efficient comparing to other forms. In other hand, Residue Number Systems (RNS) offer an interest- ing parallelism where operations are made on small values. RNS are well known for improving the performances of DSP filters. We compare our RNS approach to fixed-point implementations of DFI and $\rho$-DFIIt.}, Doi = {10.1109/MWSCAS.2011.6026263}}`

- T. Hilaire, “Towards Tools and Methodology for the Fixed-Point Implementation of Linear Filters,” in Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), 2011 IEEE, 2011, pp. 488-493. doi:10.1109/DSP-SPE.2011.5739263

[BibTeX] [Abstract] [PDF]

For embedded systems, Finite Word Length (FWL) effect is still a critical issue in digital filter implementation with fixed-point arithmetic. Few partial solutions exist, but none tackles it in its global nature and consider the complete tradeoff between performance, precision, complexity and hardware cost. This paper proposes to formalize this complex problem and exhibits a unifying approach for the optimal implementation problem of linear filters/controllers. The proposed methodology is based on previously proposed formal description and measures of the FWL effects. A two steps optimization is proposed and a small example emphasizes the process.

`@inproceedings{Hila11a, Author = {Hilaire, T.}, Booktitle = {{Digital Signal Processing Workshop and IEEE Signal Processing Education Workshop (DSP/SPE), 2011 IEEE}}, Month = {January}, Pages = {488-493}, Title = {{Towards Tools and Methodology for the Fixed-Point Implementation of Linear Filters}}, Year = {2011}, Abstract = {For embedded systems, Finite Word Length (FWL) effect is still a critical issue in digital filter implementation with fixed-point arithmetic. Few partial solutions exist, but none tackles it in its global nature and consider the complete tradeoff between performance, precision, complexity and hardware cost. This paper proposes to formalize this complex problem and exhibits a unifying approach for the optimal implementation problem of linear filters/controllers. The proposed methodology is based on previously proposed formal description and measures of the FWL effects. A two steps optimization is proposed and a small example emphasizes the process.}, Doi = {10.1109/DSP-SPE.2011.5739263}}`

- T. Hilaire and P. Chevrel, “Sensitivity-based Pole and Input-Output Errors of Linear Filters as Indicators of the Implementation Deterioration in Fixed-Point Context,” EURASIP Journal on Advances in Signal Processing, vol. {special issue on Quantization of VLSI Digital Signal Processing Systems}, 2011. doi:10.1155/2011/893760

[BibTeX] [Abstract] [PDF]

Input-output or poles sensitivity is widely used to evaluate the resilience of a filter realization to coefficients quantization in an FWL implementation process. However, these measures do not exactly consider the various implementation schemes and are not accurate in general case. This paper generalizes the classical transfer function sensitivity and pole sensitivity measure, by taking into consideration the exact fixed-point representation of the coefficients. Working in the general framework of the specialized implicit descriptor representation, it shows how a statistical quantization error model may be used in order to define stochastic sensitivity measures that are definitely pertinent and normalized. The general framework of MIMO filters and controllers is considered. All the results are illustrated through an example.

`@article{Hila11b, Author = {Hilaire, T. and Chevrel, P.}, Journal = {{EURASIP Journal on Advances in Signal Processing}}, Month = {January}, Title = {{Sensitivity-based Pole and Input-Output Errors of Linear Filters as Indicators of the Implementation Deterioration in Fixed-Point Context}}, Volume = {{special issue on Quantization of VLSI Digital Signal Processing Systems}}, Year = {2011}, Abstract = {Input-output or poles sensitivity is widely used to evaluate the resilience of a filter realization to coefficients quantization in an FWL implementation process. However, these measures do not exactly consider the various implementation schemes and are not accurate in general case. This paper generalizes the classical transfer function sensitivity and pole sensitivity measure, by taking into consideration the exact fixed-point representation of the coefficients. Working in the general framework of the specialized implicit descriptor representation, it shows how a statistical quantization error model may be used in order to define stochastic sensitivity measures that are definitely pertinent and normalized. The general framework of MIMO filters and controllers is considered. All the results are illustrated through an example.}, Doi = {10.1155/2011/893760}}`

- Y. Feng, P. Chevrel, and T. Hilaire, “Generalised modal realisation as a practical and efficient tool for FWL implementation,” International Journal of Control, vol. 84, iss. 1, pp. 66-77, 2011. doi:10.1080/00207179.2010.540714

[BibTeX] [Abstract] [PDF]

Finite word length (FWL) effects have been a critical issue in digital filter implementation for almost four decades. Although some optimisations may be attempted to get an optimal realisation with regards to a particular effect, for instance the parametric sensitivity or the round-off noise gain, the purpose of this article is to propose an effective one, i.e. taking into account all the aspects. Based on the specialised implicit form, a new effective and sparse structure, named q-modal realisation, is proposed. This realisation meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (thanks to sparsity) and simplicity (no tricky optimisation is required to obtain it) as well. Two numerical examples are included to illustrate the q-modal realisation’s interest.

`@article{Feng11a, Author = {Feng, Y. and Chevrel, P. and Hilaire, T.}, Journal = {{International Journal of Control}}, Number = {1}, Pages = {66-77}, Title = {{Generalised modal realisation as a practical and efficient tool for {FWL} implementation}}, Volume = {84}, Year = {2011}, Abstract = {Finite word length (FWL) effects have been a critical issue in digital filter implementation for almost four decades. Although some optimisations may be attempted to get an optimal realisation with regards to a particular effect, for instance the parametric sensitivity or the round-off noise gain, the purpose of this article is to propose an effective one, i.e. taking into account all the aspects. Based on the specialised implicit form, a new effective and sparse structure, named q-modal realisation, is proposed. This realisation meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (thanks to sparsity) and simplicity (no tricky optimisation is required to obtain it) as well. Two numerical examples are included to illustrate the q-modal realisation's interest.}, Doi = {10.1080/00207179.2010.540714}}`

### 2010

- O. Sluciak, T. Hilaire, and M. Rupp, “A General Formalism for the Analysis of Distributed Algorithms,” in Acoustics Speech and Signal Processing (ICASSP), 2010 IEEE International Conference on, 2010, pp. 2890-2893. doi:10.1109/ICASSP.2010.5496169

[BibTeX] [Abstract] [PDF]

The major contribution of this paper is the presentation of a general unifying description of distributed algorithms allowing to map local, node-based algorithms onto a single global, network-based form. As a first consequence the new description offers to analyze their learning and steady-state behavior by classical methods. A further consequence is the analysis of implementation issues as they appear due to quantization in computing and communication links. Exemplarly we apply the new method on several different averaging algorithms: the Push-Sum protocol, Consensus Propagation as well as its quantized form and furthermore examine the effects of quantization noise which is introduced by the bandwidth limited communication links and finite precision computation ability of every node. Statistical properties of these quantization noises are provided and verified by simulations.

`@conference{Sluc10, Author = {Sluciak, O. and Hilaire, T. and Rupp, M.}, Booktitle = {{Acoustics Speech and Signal Processing (ICASSP), 2010 IEEE International Conference on}}, Month = {March}, Pages = {2890-2893}, Title = {{A General Formalism for the Analysis of Distributed Algorithms}}, Year = {2010}, Abstract = {The major contribution of this paper is the presentation of a general unifying description of distributed algorithms allowing to map local, node-based algorithms onto a single global, network-based form. As a first consequence the new description offers to analyze their learning and steady-state behavior by classical methods. A further consequence is the analysis of implementation issues as they appear due to quantization in computing and communication links. Exemplarly we apply the new method on several different averaging algorithms: the Push-Sum protocol, Consensus Propagation as well as its quantized form and furthermore examine the effects of quantization noise which is introduced by the bandwidth limited communication links and finite precision computation ability of every node. Statistical properties of these quantization noises are provided and verified by simulations. }, Doi = {10.1109/ICASSP.2010.5496169}}`

- C. Reyes, T. Hilaire, S. Paul, and C. Mecklenbräuker, “Evaluation of the Root Mean Square Error Performance of the PAST-Consensus Algorithm,” in International ITG Workshop on Smart Antennas – WSA 2010, 2010. doi:10.1109/WSA.2010.5456452

[BibTeX] [Abstract] [PDF]

In previous work, we developed and investigated a distributed Projection Approximation Subspace Tracking Algo- rithm (PAST-Consensus) based on Consensus Propagation for wireless sensor networks. Preliminary simulation results showing a good tracking capability and still reduced complexity, have motivated us to evaluate the performance of the aforementioned algorithm. In this work, some simulation results will be presented comparing the root mean square error for several signal to noise ratios, as well as the error in the signal subspace given by its angle difference.

`@conference{Reye10, Author = {Reyes, C. and Hilaire, T. and Paul, S. and Mecklenbr{\"a}uker, C.}, Booktitle = {{International ITG Workshop on Smart Antennas - WSA 2010}}, Month = {February}, Title = {{Evaluation of the Root Mean Square Error Performance of the PAST-Consensus Algorithm}}, Year = {2010}, Abstract = {In previous work, we developed and investigated a distributed Projection Approximation Subspace Tracking Algo- rithm (PAST-Consensus) based on Consensus Propagation for wireless sensor networks. Preliminary simulation results showing a good tracking capability and still reduced complexity, have motivated us to evaluate the performance of the aforementioned algorithm. In this work, some simulation results will be presented comparing the root mean square error for several signal to noise ratios, as well as the error in the signal subspace given by its angle difference.}, Doi = {10.1109/WSA.2010.5456452}}`

- T. Hilaire, P. Chevrel, and J. F. Whidborne, “Finite Wordlength Controller Realizations using the Specialized Implicit Form,” Int. Journal of Control, vol. 83, iss. 2, pp. 330-346, 2010.

[BibTeX] [Abstract] [PDF]

A specialised implicit state-space representation is introduced to deal with finite wordlength effects in controller implementations. This specialised implicit form provides a macroscopic description of the algorithm to be implemented. So, it constitutes a unifying framework, allowing to encompass various implementation forms, such as the $\rho$-operator, the $\rho$ Direct Form II transposed, observer-based and many other realisations usually considered separately in the literature. Different measures quantifying the finite wordlength effects on the overall closed-loop behaviour are defined in this new context. They concern both stability and performance. The gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise analysis. The problem of determining a realisation with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods. The approach is illustrated with an example.

`@article{Hila10a, Author = {Hilaire, T. and Chevrel, P. and Whidborne, J.F.}, Journal = {{Int. Journal of Control}}, Month = {February}, Number = {2}, Pages = {330-346}, Title = {{Finite Wordlength Controller Realizations using the Specialized Implicit Form}}, Volume = {83}, Year = {2010}, Abstract = {A specialised implicit state-space representation is introduced to deal with finite wordlength effects in controller implementations. This specialised implicit form provides a macroscopic description of the algorithm to be implemented. So, it constitutes a unifying framework, allowing to encompass various implementation forms, such as the $\rho$-operator, the $\rho$ Direct Form II transposed, observer-based and many other realisations usually considered separately in the literature. Different measures quantifying the finite wordlength effects on the overall closed-loop behaviour are defined in this new context. They concern both stability and performance. The gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise analysis. The problem of determining a realisation with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods. The approach is illustrated with an example.}, }`

### 2009

- C. Reyes, T. Hilaire, and C. Mecklenbräuker, “Distributed Projection Approximation Subspace Tracking based on Consensus Propagation,” in 3rd International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP’09), 2009.

[BibTeX] [Abstract] [PDF] [Slides]

We develop and investigate a distributed algorithm for signal subspace tracking with a wireless sensor network without the need for a fusion center, in order to improve the robustness and scalability. We assume that all sensor nodes may broadcast messages to sensors in their neighborhood defined by a finite (small) communication radius. To this aim, we start from Projection Approximation Subspace Tracking (PAST) which is a well-investigated algorithm suitable for implementation in a fusion center. We arrive at a distributed approximation of the PAST algorithm by letting each sensor broadcast its local observation variable x_n(t) and a filtered observation vector y_n(t) to its neighborhood. Vice versa, the received messages at sensor node n from its neighborhood are fused by employing consensus propagation. Finally, we investigate the proposed distributed algorithm in simulation runs.

`@conference{Reye09, Author = {Reyes, C. and Hilaire, T. and Mecklenbr{\"a}uker, C.}, Booktitle = {{3rd International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP'09)}}, Month = {December}, Title = {{Distributed Projection Approximation Subspace Tracking based on Consensus Propagation}}, Year = {2009}, Abstract = {We develop and investigate a distributed algorithm for signal subspace tracking with a wireless sensor network without the need for a fusion center, in order to improve the robustness and scalability. We assume that all sensor nodes may broadcast messages to sensors in their neighborhood defined by a finite (small) communication radius. To this aim, we start from Projection Approximation Subspace Tracking (PAST) which is a well-investigated algorithm suitable for implementation in a fusion center. We arrive at a distributed approximation of the PAST algorithm by letting each sensor broadcast its local observation variable x_n(t) and a filtered observation vector y_n(t) to its neighborhood. Vice versa, the received messages at sensor node n from its neighborhood are fused by employing consensus propagation. Finally, we investigate the proposed distributed algorithm in simulation runs.}, }`

- T. Hilaire, “FWR Toolbox User’s Guide (v0.99),” , 2009.

[BibTeX] [Download PDF] [PDF]`@manual{Hila09e, url = {http://fwrtoolbox.gforge.inria.fr/}, Author = {Hilaire, T.}, Title = {{FWR Toolbox User's Guide (v0.99)}}, Year = {2009}, }`

- T. Hilaire, “New L2-dynamic-range-scaling Constraints for Low Parametric Sensitivity Realizations,” in Proc. european signal processing conference (eusipco’09), 2009, pp. 988-992.

[BibTeX] [Abstract] [PDF]

This paper presents a new dynamic-range scaling for the implementation of filters/controllers in state-space form. Specific fixed-point considerations allow us to relax the classical L2-scaling constraints while still preserving the implementation from overflows. It gives more degrees of freedom for the optimal L2-parametric sensitivity problem. The underlying constrained problem is converted into an unconstrained problem for which a solution can be provided. This leads to realizations which are still scaled but less sensitive.

`@inproceedings{Hila09d, Author = {Hilaire, T.}, Booktitle = {Proc. European Signal Processing Conference (EUSIPCO’09)}, Pages = {988-992}, Title = {{New L2-dynamic-range-scaling Constraints for Low Parametric Sensitivity Realizations}}, Year = {2009}, Abstract = { This paper presents a new dynamic-range scaling for the implementation of filters/controllers in state-space form. Specific fixed-point considerations allow us to relax the classical L2-scaling constraints while still preserving the implementation from overflows. It gives more degrees of freedom for the optimal L2-parametric sensitivity problem. The underlying constrained problem is converted into an unconstrained problem for which a solution can be provided. This leads to realizations which are still scaled but less sensitive.}, }`

- T. Hilaire, “On the Transfer Function Error of State-Space Filters in Fixed-Point Context,” IEEE Trans. on Circuits & Systems II, vol. 56, iss. 12, p. 936–940, 2009.

[BibTeX] [Abstract] [PDF]

This brief presents a new measure used for the imple- mentation of filters/controllers in state-space form. It investigates the transfer function deviation generated by the coefficient quan- tization. The classical L2-sensitivity measure is extended with precise consideration on their fixed-point representation in order to make a more valid measure. By solving the related optimal re- alization problem, fixed-point accurate realizations in state-space form can be found.

`@article{Hila09c, Author = {Hilaire, T.}, Journal = {{IEEE Trans. on Circuits \& Systems II}}, Month = {December}, Number = {12}, Pages = {936--940}, Title = {{On the Transfer Function Error of State-Space Filters in Fixed-Point Context}}, Volume = {56}, Year = {2009}, Abstract = {This brief presents a new measure used for the imple- mentation of filters/controllers in state-space form. It investigates the transfer function deviation generated by the coefficient quan- tization. The classical L2-sensitivity measure is extended with precise consideration on their fixed-point representation in order to make a more valid measure. By solving the related optimal re- alization problem, fixed-point accurate realizations in state-space form can be found.}, }`

- T. Hilaire, “Low Parametric Sensitivity Realizations with relaxed L_2-dynamic-range-scaling constraints,” IEEE Trans. on Circuits & Systems II, vol. 56, iss. 7, p. 590–594, 2009. doi:10.1109/TCSII.2009.2022210

[BibTeX] [Abstract] [PDF]

This brief presents a new dynamic-range scaling for the implementation of filters/controllers in state-space form. Re- laxing the classical L2-scaling constraints by specific fixed-point considerations allows for a higher degree of freedom for the optimal L2-parametric sensitivity problem. However, overflows in the implementation are still prevented. The underlying constrained problem is converted into an unconstrained problem for which a solution can be provided. This leads to realizations that are still scaled but less sensitive.

`@article{Hila09a, Author = {Hilaire, T.}, Journal = {{IEEE Trans. on Circuits \& Systems II}}, Month = {July}, Number = {7}, Pages = {590--594}, Title = {{Low Parametric Sensitivity Realizations with relaxed L_2-dynamic-range-scaling constraints}}, Volume = {56}, Year = {2009}, Abstract = {This brief presents a new dynamic-range scaling for the implementation of filters/controllers in state-space form. Re- laxing the classical L2-scaling constraints by specific fixed-point considerations allows for a higher degree of freedom for the optimal L2-parametric sensitivity problem. However, overflows in the implementation are still prevented. The underlying constrained problem is converted into an unconstrained problem for which a solution can be provided. This leads to realizations that are still scaled but less sensitive.}, Doi = {10.1109/TCSII.2009.2022210}}`

- Y. Feng, P. Chevrel, and T. Hilaire, “A Practical Strategy of an Efficient and Sparse FWL Implementation of LTI Filters,” in European Control Conference (ECC’09), 2009, pp. 1383-1388.

[BibTeX] [Abstract] [PDF] [HAL]

The problem of finite word length implementation is discussed in this paper. Alternatively to the rho-DFIIt recently proposed by G. Li et al., and leaning on the specialized implicit form for a unified analysis, a new effective and sparse structure, named rho-modal realization, is developed. This realization meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (owing to sparsity), and simplicity (no tricky optimization is involved) as well. Two numerical examples are presented to confirm the theoretical results and illustrate the rho-modal realization interest.

`@inproceedings{Feng09a, Author = {Feng, Y. and Chevrel, P. and Hilaire, T.}, Booktitle = {{European Control Conference (ECC'09)}}, Pages = {1383-1388}, Title = {{A Practical Strategy of an Efficient and Sparse {FWL} Implementation of LTI Filters}}, Year = {2009}, Abstract = {The problem of finite word length implementation is discussed in this paper. Alternatively to the rho-DFIIt recently proposed by G. Li et al., and leaning on the specialized implicit form for a unified analysis, a new effective and sparse structure, named rho-modal realization, is developed. This realization meets simultaneously accuracy (low sensitivity, round-off noise gain and overflow risk), few and flexible computational efforts with a good readability (owing to sparsity), and simplicity (no tricky optimization is involved) as well. Two numerical examples are presented to confirm the theoretical results and illustrate the rho-modal realization interest.}, Hal = {http://hal.archives-ouvertes.fr/hal-00413664/}}`

### 2008

- T. Hilaire, P. Chevrel, and J. F. Whidborne, “Finite Wordlength Controller Realizations using the Specialized Implicit Form,” INRIA, RR-6759, 2008.

[BibTeX] [Abstract] [PDF] [HAL]

A specialized implicit state-space representation is introduced to deal with finite wordlength effects in controller implementations. This special- ized implicit form provides a macroscopic description of the algorithm to be implemented. So, it constitutes a unifying framework, allowing to encompass various implementation forms, such as the $\delta$-operator, the $\rho$-Direct Form II trans- posed, observer-based and many other realizations considered usually separately in the literature. Different measures quantifying the finite wordlength effects on the overall closed loop behaviour, are defined in this new context. They con- cern both stability and performance. The gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise anal- ysis. The problem of determining a realization with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods. The approach is illustrated with two examples.

`@techreport{Hila08b, Author = {Hilaire, T. and Chevrel, P. and Whidborne, J.F.}, Institution = {INRIA}, Number = {RR-6759}, Title = {{Finite Wordlength Controller Realizations using the Specialized Implicit Form}}, Year = {2008}, Abstract = {A specialized implicit state-space representation is introduced to deal with finite wordlength effects in controller implementations. This special- ized implicit form provides a macroscopic description of the algorithm to be implemented. So, it constitutes a unifying framework, allowing to encompass various implementation forms, such as the $\delta$-operator, the $\rho$-Direct Form II trans- posed, observer-based and many other realizations considered usually separately in the literature. Different measures quantifying the finite wordlength effects on the overall closed loop behaviour, are defined in this new context. They con- cern both stability and performance. The gap with the infinite precision case is evaluated classically through the coefficient sensitivity and roundoff noise anal- ysis. The problem of determining a realization with minimum finite wordlength effects can subsequently be solved using appropriate numerical methods. The approach is illustrated with two examples.}, Hal = {http://hal.inria.fr/inria-00359004/en/}}`

- T. Hilaire, D. Ménard, and O. Sentieys, “Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers,” in Computer-Aided Control Systems, 2008. CACSD 2008. IEEE International Conference on, 2008, pp. 607-612. doi:10.1109/CACSD.2008.4627366

[BibTeX] [Abstract] [PDF] [Slides]

The analytic evaluation of roundoff noise is an interesting approach to analyze the effects of fixed-point implementation of linear filters or controllers. This paper is based on a generic framework to describe controller algorithms. It exhibits the fixed-point implementations (in different schemes) that are associated and the analytic expression of the output roundoff noise power. Finally, the optimal realization problem, according to the roundoff noise power, is considered.

`@conference{Hila08c, Author = {Hilaire, T. and M{\'e}nard, D. and Sentieys, O.}, Booktitle = {{Computer-Aided Control Systems, 2008. CACSD 2008. IEEE International Conference on}}, Month = {September}, Pages = {607-612}, Title = {{Bit Accurate Roundoff Noise Analysis of Fixed-Point Linear Controllers}}, Year = {2008}, Abstract = {The analytic evaluation of roundoff noise is an interesting approach to analyze the effects of fixed-point implementation of linear filters or controllers. This paper is based on a generic framework to describe controller algorithms. It exhibits the fixed-point implementations (in different schemes) that are associated and the analytic expression of the output roundoff noise power. Finally, the optimal realization problem, according to the roundoff noise power, is considered.}, Doi = {10.1109/CACSD.2008.4627366}}`

### 2007

- T. Hilaire, P. Chevrel, and J. Whidborne, “Low Parametric Closed-Loop Sensitivity Realizations using Fixed-Point and Floating-Point Arithmetic,” in Proc. European Control Conference (ECC’07), 2007.

[BibTeX] [Abstract] [PDF]

The Specialized Implicit Form provides a general framework for the analysis and design of digital controller implementations with minimal finite wordlength effects. This paper proposes a measure of the closed-loop transfer function sensitivity to finite wordlength effects that is generalized for both fixed-point and floating-point arithmetic and can be used with the Specialized Implicit Form to analyze the effects of quantization and rounding on the parameters of a digital controller implementation. The measure is computationally tractable and hence amenable to solving the problem of minimizing the parametric sensitivity FWL effect. Furthermore, the sensitivity to the rounding of each individual parameter can be easily obtained. The use of the measure is illustrated with examples.

`@inproceedings{Hila07e, Author = {Hilaire, T. and Chevrel, P. and Whidborne, J.}, Booktitle = {{Proc. European Control Conference (ECC'07)}}, Month = {July}, Title = {{Low Parametric Closed-Loop Sensitivity Realizations using Fixed-Point and Floating-Point Arithmetic}}, Year = {2007}, Abstract = {The Specialized Implicit Form provides a general framework for the analysis and design of digital controller implementations with minimal finite wordlength effects. This paper proposes a measure of the closed-loop transfer function sensitivity to finite wordlength effects that is generalized for both fixed-point and floating-point arithmetic and can be used with the Specialized Implicit Form to analyze the effects of quantization and rounding on the parameters of a digital controller implementation. The measure is computationally tractable and hence amenable to solving the problem of minimizing the parametric sensitivity FWL effect. Furthermore, the sensitivity to the rounding of each individual parameter can be easily obtained. The use of the measure is illustrated with examples.} }`

- T. Hilaire, D. Ménard, and O. Sentieys, “Roundoff Noise Analysis of Finite Wordlength Realizations with the Implicit State-Space Framework,” in 15th European Signal Processing Conference (EUSIPCO’07), 2007, p. 1019–1023.

[BibTeX] [Abstract] [PDF] [Slides]

The analytic evaluation of the system output roundoff noise is an interesting approach to analyze the effect of a Finite Word Length implementation of linear filters. Previous works have introduced the Roundoff Noise Gain in this context and applied it to shift and delta-realizations. To generalize them, the paper is based on a more general representation and exhibits the output noise power in the general case. Finally, the problem optimal realization problem, according to the Roundoff Noise Gain measure, is considered.

`@inproceedings{Hila07c, Author = {Hilaire, T. and M{\'e}nard, D. and Sentieys, O.}, Booktitle = {{15th European Signal Processing Conference (EUSIPCO'07)}}, Month = {September}, Pages = {1019--1023}, Title = {{Roundoff Noise Analysis of Finite Wordlength Realizations with the Implicit State-Space Framework}}, Year = {2007}, Abstract = {The analytic evaluation of the system output roundoff noise is an interesting approach to analyze the effect of a Finite Word Length implementation of linear filters. Previous works have introduced the Roundoff Noise Gain in this context and applied it to shift and delta-realizations. To generalize them, the paper is based on a more general representation and exhibits the output noise power in the general case. Finally, the problem optimal realization problem, according to the Roundoff Noise Gain measure, is considered. }, }`

- T. Hilaire and P. Chevrel, “On the compact formulation of the derivation of a transfer matrix with respect to another matrix,” INRIA, RR-6760, 2007.

[BibTeX] [Abstract] [PDF] [HAL]

A new operator is considered, allowing compact formulae and proofs in the context of the derivation of a transfer matrix with respect to another matrix. The problem of the parametric sensitivity matrix calculation is chosen for illustration. It consists in deriving a Multiple Input Multiple Output transfer function with respect to a parametric matrix and is central in robust control theory. Efficient algorithms may be straightforwardly got from the compact analytic formulae using the operator introduced.

`@techreport{Hila07d, Author = {Hilaire, T. and Chevrel, P.}, Institution = {INRIA}, Number = {RR-6760}, Title = {{On the compact formulation of the derivation of a transfer matrix with respect to another matrix}}, Year = {2007}, Abstract = {A new operator is considered, allowing compact formulae and proofs in the context of the derivation of a transfer matrix with respect to another matrix. The problem of the parametric sensitivity matrix calculation is chosen for illustration. It consists in deriving a Multiple Input Multiple Output transfer function with respect to a parametric matrix and is central in robust control theory. Efficient algorithms may be straightforwardly got from the compact analytic formulae using the operator introduced.}, Hal = {http://hal.inria.fr/inria-00345508}}`

- T. Hilaire, P. Chevrel, and J. F. Whidborne, “A Unifying Framework for Finite Wordlength Realizations,” IEEE Trans. on Circuits and Systems, vol. 8, iss. 54, pp. 1765-1774, 2007.

[BibTeX] [Abstract] [PDF]

A general framework for the analysis of the finite wordlength (FWL) effects of linear time-invariant digital filter implementations is proposed. By means of a special implicit system description, all realization forms can be described. An algebraic characterization of the equivalent classes is provided, which enables a search for realizations that minimize the FWL effects to be made. Two suitable FWL coefficient sensitivity measures are proposed for use within the framework, these being a transfer function sensitivity measure and a pole sensitivity measure. An illustrative example is presented.

`@article{Hila07b, Author = {Hilaire, T. and Chevrel, P. and Whidborne, J.F.}, Journal = {{IEEE Trans. on Circuits and Systems}}, Month = {August}, Number = {54}, Pages = {1765-1774}, Title = {{A Unifying Framework for Finite Wordlength Realizations}}, Volume = {8}, Year = {2007}, Abstract = {A general framework for the analysis of the finite wordlength (FWL) effects of linear time-invariant digital filter implementations is proposed. By means of a special implicit system description, all realization forms can be described. An algebraic characterization of the equivalent classes is provided, which enables a search for realizations that minimize the FWL effects to be made. Two suitable FWL coefficient sensitivity measures are proposed for use within the framework, these being a transfer function sensitivity measure and a pole sensitivity measure. An illustrative example is presented.}, }`

- T. Hilaire, P. Chevrel, and J-P. Clauzel, “Low parametric sensitivity realization design for FWL implementation of MIMO controllers : theory and application to the active control of vehicle longitudinal oscillations,” International Journal of Tomography & Statistics, vol. 6, iss. 7, pp. 128-133, 2007.

[BibTeX] [Abstract] [PDF]

The implementation of a controller in a Finite Word Length (FWL) context may lead to a deterioration of the global performance, due to parametric errors (quantification of coefficients) and numerical noises (roundoff noises). This deterioration depends on the choise of the realization used to numerically implement the controller. In previous papers, the authors have introduced a new representation allowing to unify different realizations including among others those using q or delta-operators. In this paper, the parametric sensitivity measure is generalized in the MIMO case and used with a specific realization : the Observer-State-Feedback realization. Such a realization is not unique and one problem consists in finding an optimal realization according to that measure. Looking for such a structure is applied on a pratical example : the active control of vehicle longitudinal oscillations.

`@article{Hila07a, Author = {Hilaire, T. and Chevrel, P. and Clauzel, J-P.}, Booktitle = {{Special Issue on Control Applications of Optimisation}}, Journal = {{International Journal of Tomography \& Statistics}}, Month = {Summer07}, Number = {7}, Pages = {128-133}, Title = {Low Parametric Sensitivity Realization Design for {FWL} Implementation of {MIMO} Controllers : Theory and Application to the Active Control of Vehicle Longitudinal Oscillations}, Volume = {6}, Year = {2007}, Abstract = {The implementation of a controller in a Finite Word Length (FWL) context may lead to a deterioration of the global performance, due to parametric errors (quantification of coefficients) and numerical noises (roundoff noises). This deterioration depends on the choise of the realization used to numerically implement the controller. In previous papers, the authors have introduced a new representation allowing to unify different realizations including among others those using q or delta-operators. In this paper, the parametric sensitivity measure is generalized in the MIMO case and used with a specific realization : the Observer-State-Feedback realization. Such a realization is not unique and one problem consists in finding an optimal realization according to that measure. Looking for such a structure is applied on a pratical example : the active control of vehicle longitudinal oscillations.}, }`

### 2006

- T. Hilaire, “Analyse et synthèse de l’implémentation de lois de contrôle-commande en précision finie (étude dans le cadre des applications automobiles sur calculateur embarquée),” PhD Thesis, 2006.

[BibTeX] [PDF] [Slides]`@phdthesis{Hila06c, Author = {Hilaire, T.}, Month = {June}, School = {Universit{\'e} de Nantes}, Title = {Analyse et synth{\`e}se de l'impl{\'e}mentation de lois de contr{\^o}le-commande en pr{\'e}cision finie ({\'E}tude dans le cadre des applications automobiles sur calculateur embarqu{\'e}e)}, Year = {2006}, }`

- T. Hilaire, P. Chevrel, and J-P. Clauzel, “Pole Sensitivity Stability Related Measure of FWL Realization with the Implicit State-Space Formalism,” in 5th IFAC Symposium on Robust Control Design (ROCOND’06), 2006.

[BibTeX] [Abstract] [PDF] [Slides]

The pole-sensitivity approach is an interesting way to analyze the stability of discrete-time control system with a Finite Word Length implemented digital controller. Previous works have introduced a pole-sensitivity stability related measure in this context, and applied it to shift or delta-realizations. This paper generalizes then by considering a more general representation, a particular implicit state-space realization, which is recently known to encompass both classical shift or delta-realization, as well as other interesting parametrizations. Finally, the problem consisting in finding realizations optimizing the FWL closed-loop stability related measure is considered.

`@inproceedings{Hila06b, Author = {Hilaire, T. and Chevrel, P. and Clauzel, J-P.}, Booktitle = {{5th IFAC Symposium on Robust Control Design (ROCOND'06)}}, Keywords = {Stability, Optimal Design}, Month = {July}, Title = {{Pole Sensitivity Stability Related Measure of {FWL} Realization with the Implicit State-Space Formalism}}, Year = {2006}, Abstract = {The pole-sensitivity approach is an interesting way to analyze the stability of discrete-time control system with a Finite Word Length implemented digital controller. Previous works have introduced a pole-sensitivity stability related measure in this context, and applied it to shift or delta-realizations. This paper generalizes then by considering a more general representation, a particular implicit state-space realization, which is recently known to encompass both classical shift or delta-realization, as well as other interesting parametrizations. Finally, the problem consisting in finding realizations optimizing the FWL closed-loop stability related measure is considered.}, }`

- T. Hilaire, P. Chevrel, and J-P. Clauzel, “Low Parametric Sensitivity Realization Design for FWL Implementation of MIMO Controllers,” in Proc. of Control Applications of Optimisation CAO’O6, 2006.

[BibTeX] [Abstract] [PDF] [Slides]

The implementation of a controller in a Finite Word Length (FWL) context may lead to a deterioration of the global performance, due to parametric errors (quantification of coefficients) and numerical noises (roundoff noises). This deterioration depends on the choise of the realization used to numerically implement the controller. In previous papers, the authors have introduced a new representation allowing to unify different realizations including among others those using q or delta-operators. In this paper, the parametric sensitivity measure is generalized in the MIMO case and used with a specific realization : the Observer-State-Feedback realization. Such a realization is not unique and one problem consists in finding an optimal realization according to that measure. Looking for such a structure is applied on a pratical example : the active control of vehicle longitudinal oscillations.

`@inproceedings{Hila06a, Author = {Hilaire, T. and Chevrel, P. and Clauzel, J-P.}, Booktitle = {{Proc. of Control Applications of Optimisation CAO'O6}}, Month = {April}, Title = {{Low Parametric Sensitivity Realization Design for {FWL} Implementation of {MIMO} Controllers}}, Year = {2006}, Abstract = {The implementation of a controller in a Finite Word Length (FWL) context may lead to a deterioration of the global performance, due to parametric errors (quantification of coefficients) and numerical noises (roundoff noises). This deterioration depends on the choise of the realization used to numerically implement the controller. In previous papers, the authors have introduced a new representation allowing to unify different realizations including among others those using q or delta-operators. In this paper, the parametric sensitivity measure is generalized in the MIMO case and used with a specific realization : the Observer-State-Feedback realization. Such a realization is not unique and one problem consists in finding an optimal realization according to that measure. Looking for such a structure is applied on a pratical example : the active control of vehicle longitudinal oscillations.}, }`

### 2005

- T. Hilaire, P. Chevrel, and Y. Trinquet, “Implicit State-Space Representation : a Unifying Framework for FWL Implementation of LTI Systems,” in Proc. of the 16th IFAC World Congress, 2005.

[BibTeX] [Abstract] [PDF]

Practically, some intermediary realizations are used in order to simulate, numerically, dynamic systems. One of the most popular is the state-space realization. It reveals to be very useful to study the impact of Finite Word Length implementation, especially in the case of embedded controller. Numerous works concerned the design of the “best” realization concerning parameterisation, numerical noise minimisation or saving computation. This paper points out however that a standard state-space realization is too basic to take into account some interesting realizations. On the contrary, it highlights that implicit state-space realizations allows a more direct link with the macroscopic computations to be performed. It is necessary to describe some popular algorithms simulating LTI systems. Moreover, such a representation has the important property to unify different ways of research considering differently the possibilities offered by using the shift, delta or gamma operators.

`@inproceedings{Hila05a, Author = {Hilaire, T. and Chevrel, P. and Trinquet, Y.}, Booktitle = {{Proc. of the 16th IFAC World Congress}}, Editor = {Piztek, P.}, Month = {July}, Publisher = {Elsevier}, Title = {{Implicit State-Space Representation : a Unifying Framework for {FWL} Implementation of {LTI} Systems}}, Year = {2005}, Abstract = {Practically, some intermediary realizations are used in order to simulate, numerically, dynamic systems. One of the most popular is the state-space realization. It reveals to be very useful to study the impact of Finite Word Length implementation, especially in the case of embedded controller. Numerous works concerned the design of the "best" realization concerning parameterisation, numerical noise minimisation or saving computation. This paper points out however that a standard state-space realization is too basic to take into account some interesting realizations. On the contrary, it highlights that implicit state-space realizations allows a more direct link with the macroscopic computations to be performed. It is necessary to describe some popular algorithms simulating LTI systems. Moreover, such a representation has the important property to unify different ways of research considering differently the possibilities offered by using the shift, delta or gamma operators.}, }`

- T. Hilaire, P. Chevrel, and Y. Trinquet, “Designing Low Parametric Sensitivity FWL Realizations of LTI Controllers/Filters within the Implicit State-Space Framework,” in Proc. of the 44th IEEE Conference on Decision and Control and the European Control Conference (CDC-ECC’05), 2005, pp. 5192-5197.

[BibTeX] [Abstract] [PDF]

The problem of Finite Word Length (FWL) implementation of Linear Time Invariant (LTI) filters or controllers is considered in this paper. A specialized implicit state-space representation enabling a macroscopic description of the algorithm to be implemented is exhibited. It constitutes a unifying framework to encompass various implementation form, such as q, delta, the observer-based and other realizations. This paper formalizes especially the problem consisting to analyze the parametric sensitivity of such realizations, and then to optimize them in order to limit deteriorations along the process of FWL implementation. The sensitivity of some structured realizations with respect to the coefficients involved and the computation effort are compared on an example.

`@inproceedings{Hila05b, Author = {Hilaire, T. and Chevrel, P. and Trinquet, Y.}, Booktitle = {{Proc. of the 44th IEEE Conference on Decision and Control and the European Control Conference (CDC-ECC'05)}}, Month = {December}, Pages = {5192-5197}, Title = {{Designing Low Parametric Sensitivity {FWL} Realizations of {LTI} Controllers/Filters within the Implicit State-Space Framework}}, Year = {2005}, Abstract = {The problem of Finite Word Length (FWL) implementation of Linear Time Invariant (LTI) filters or controllers is considered in this paper. A specialized implicit state-space representation enabling a macroscopic description of the algorithm to be implemented is exhibited. It constitutes a unifying framework to encompass various implementation form, such as q, delta, the observer-based and other realizations. This paper formalizes especially the problem consisting to analyze the parametric sensitivity of such realizations, and then to optimize them in order to limit deteriorations along the process of FWL implementation. The sensitivity of some structured realizations with respect to the coefficients involved and the computation effort are compared on an example.}, }`

### 2003

- T. Hilaire and P. Chevrel, “L’optimisation convexe pour la conception et l’analyse des lois de commande – formalisation grâce aux LMI,” Institut de Recherche en Communications et Cybernétique (IRCCyN) 2003.

[BibTeX] [PDF]`@techreport{Hila03a, Author = {Hilaire, T. and Chevrel, P.}, Institution = {Institut de Recherche en Communications et Cybern{\'e}tique (IRCCyN)}, Month = {august}, Title = {L'optimisation convexe pour la conception et l'analyse des lois de commande - formalisation gr{\^a}ce aux {LMI}}, Year = {2003}, }`

- T. Hilaire, “Élaboration d’un Langage de Description d’Architecture Matérielle – application au processeur Alpha21264,” Master Thesis, 2003.

[BibTeX]`@mastersthesis{Hila03c, Author = {Hilaire, Thibault}, School = {Universit{\'e} de Nantes}, Title = {{\'E}laboration d'un {L}angage de {D}escription d'{A}rchitecture {M}at{\'e}rielle -- Application au processeur {Alpha21264}}, Year = {2003} }`