Roundoff Noise Analysis of Finite Wordlength O. Sentieys - PowerPoint PPT Presentation

EUSIPCO’07 — T. Hilaire, D. M´ enard and Roundoff Noise Analysis of Finite Wordlength O. Sentieys Realizations with the Implicit State-Space Introduction Framework Implicit State-Space Framework Output Noise Power T. Hilaire, D. M´ enard and O. Sentieys Roundoff Noise Gain IRISA, R2D2 Team Optimal design Lannion, France Example EUSIPCO’07 - September 3-7, 2007, Pozna´ n, Poland Conclusion 1/27

Context EUSIPCO’07 — T. Hilaire, D. M´ enard and O. Sentieys Implementation of Linear Time Invariant controllers/filters Introduction Finite Word Length context (fixed-point) Implicit State-Space Framework Motivation Output Noise Power Evaluate the roundoff noise errors in the implementation Roundoff Noise Gain Compare various realizations and find an optimal one Optimal design Example Conclusion 2/27

Context EUSIPCO’07 — T. Hilaire, D. M´ enard and O. Sentieys Implementation of Linear Time Invariant controllers/filters Introduction Finite Word Length context (fixed-point) Implicit State-Space Framework Motivation Output Noise Power Evaluate the roundoff noise errors in the implementation Roundoff Noise Gain Compare various realizations and find an optimal one Optimal design Example Conclusion 2/27

Outline EUSIPCO’07 — Macroscopic representation of algorithms through the 1 T. Hilaire, D. M´ enard implicit state-space framework and O. Sentieys Introduction Output Noise Power 2 Implicit State-Space Framework Roundoff Noise Gain 3 Output Noise Power Optimal design 4 Roundoff Noise Gain Optimal Example 5 design Example Conclusion and Perspectives Conclusion 6 3/27

Outline EUSIPCO’07 — Macroscopic representation of algorithms through the 1 T. Hilaire, D. M´ enard implicit state-space framework and O. Sentieys Introduction Output Noise Power 2 Implicit State-Space Framework Roundoff Noise Gain 3 Output Noise Power Optimal design 4 Roundoff Noise Gain Optimal Example 5 design Example Conclusion and Perspectives Conclusion 6 4/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, D. M´ enard and Various implementation forms have to be taken into O. Sentieys consideration: Introduction shift-realizations Implicit State-Space δ -realizations Framework Output Noise observer-state-feedback Power direct form I or II Roundoff Noise Gain cascade or parallel realizations Optimal design etc... Example Conclusion 5/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, So, we consider all realizations where the outputs are computed D. M´ enard and from the inputs with operations like: O. Sentieys multiplications by a constant Introduction additions Implicit State-Space shifts (value stored and used at the next step) Framework Output Noise Power Roundoff Noise Gain q − 1 + A Optimal design mutliplication by shift Example a constant Conclusion additions 6/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

The need of a unifying framework EUSIPCO’07 — T. Hilaire, In order to encompass all these implementations, we have D. M´ enard and proposed a unifying framework to algebraically represent them: O. Sentieys Interests Introduction macroscopic description of a FWL implementation Implicit State-Space Framework more general than previous realizations (state-space,...) Output Noise more realistic with regard to the parameterization Power Roundoff directly linked to the in-line computations to be performed Noise Gain Optimal We can describe all possible linear graphs (with additions, design Example multiplications and shift operators) and characterize each Conclusion computational steps. 7/27

Implicit State-Space Framework EUSIPCO’07 — All the possible graphs are described by T. Hilaire, D. M´ enard and 1 J . T k +1 = M . X k + N . U k O. Sentieys 2 X k +1 = K . T k +1 + P . X k + Q . U k Introduction Implicit 3 Y k = L . T k +1 + R . X k + S . U k State-Space Framework Intermediate variables computation Output Noise Power Roundoff Implicit State-Space Framework Noise Gain Optimal         J 0 0 T k +1 0 M N T k design  = − K I 0 X k +1 0 P Q X k Example        − L 0 I Y k 0 R S U k Conclusion 8/27

Implicit State-Space Framework EUSIPCO’07 — All the possible graphs are described by T. Hilaire, D. M´ enard and 1 J . T k +1 = M . X k + N . U k O. Sentieys 2 X k +1 = K . T k +1 + P . X k + Q . U k Introduction Implicit 3 Y k = L . T k +1 + R . X k + S . U k State-Space Framework State-vector computation Output Noise Power Roundoff Implicit State-Space Framework Noise Gain Optimal         J 0 0 T k +1 0 M N T k design  = − K I 0 X k +1 0 P Q X k Example        − L 0 I Y k 0 R S U k Conclusion 8/27