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

▶
roundoff noise analysis of finite wordlength
SMART_READER_LITE
LIVE PREVIEW

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

Nov 06, 2022 206 likes •644 views

EUSIPCO07 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.

slide-1
SLIDE 1

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 1/27

Roundoff Noise Analysis of Finite Wordlength Realizations with the Implicit State-Space Framework

  • T. Hilaire, D. M´

enard and O. Sentieys

IRISA, R2D2 Team Lannion, France

EUSIPCO’07 - September 3-7, 2007, Pozna´ n, Poland

slide-2
SLIDE 2

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 2/27

Context

Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) Motivation Evaluate the roundoff noise errors in the implementation Compare various realizations and find an optimal one

slide-3
SLIDE 3

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 2/27

Context

Implementation of Linear Time Invariant controllers/filters Finite Word Length context (fixed-point) Motivation Evaluate the roundoff noise errors in the implementation Compare various realizations and find an optimal one

slide-4
SLIDE 4

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 3/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-5
SLIDE 5

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 4/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-6
SLIDE 6

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 5/27

The need of a unifying framework

Various implementation forms have to be taken into consideration: shift-realizations δ-realizations

  • bserver-state-feedback

direct form I or II cascade or parallel realizations etc...

slide-7
SLIDE 7

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 6/27

The need of a unifying framework

So, we consider all realizations where the outputs are computed from the inputs with operations like: multiplications by a constant additions shifts (value stored and used at the next step) q−1 A

+

mutliplication by a constant additions shift

slide-8
SLIDE 8

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-9
SLIDE 9

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-10
SLIDE 10

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-11
SLIDE 11

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-12
SLIDE 12

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-13
SLIDE 13

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-14
SLIDE 14

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 7/27

The need of a unifying framework

In order to encompass all these implementations, we have proposed a unifying framework to algebraically represent them: Interests macroscopic description of a FWL implementation more general than previous realizations (state-space,...) more realistic with regard to the parameterization directly linked to the in-line computations to be performed We can describe all possible linear graphs (with additions, multiplications and shift operators) and characterize each computational steps.

slide-15
SLIDE 15

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 8/27

Implicit State-Space Framework

All the possible graphs are described by

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Intermediate variables computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-16
SLIDE 16

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 8/27

Implicit State-Space Framework

All the possible graphs are described by

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

State-vector computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-17
SLIDE 17

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 8/27

Implicit State-Space Framework

All the possible graphs are described by

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Output computation Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-18
SLIDE 18

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 8/27

Implicit State-Space Framework

All the possible graphs are described by

1 J.Tk+1 = M.Xk + N.Uk 2 Xk+1 = K.Tk+1 + P.Xk + Q.Uk 3 Yk = L.Tk+1 + R.Xk + S.Uk

Implicit State-Space Framework   J −K I −L I     Tk+1 Xk+1 Yk   =   M N P Q R S     Tk Xk Uk  

slide-19
SLIDE 19

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 9/27

Intermediate variables

The intermediate variables introduced allow to make explicit all the computations done show the order of the computations express a larger parameterization

slide-20
SLIDE 20

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 10/27

Example

A realization with the δ-operator is described by : δXk = AδXk + BδUk Yk = CδXk + DδUk δ q−1

∆

It is computed with    T = AδXk + BδUk Xk+1 = Xk + ∆T Yk = CδXk + DδUk and it corresponds to the following implicit state-space :   I − ∆I I I     Tk+1 Xk+1 Yk   =   Aδ Bδ I Cδ Dδ     Tk Xk Uk  

slide-21
SLIDE 21

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 10/27

Example

A realization with the δ-operator is described by : δXk = AδXk + BδUk Yk = CδXk + DδUk δ q−1

∆

It is computed with    T = AδXk + BδUk Xk+1 = Xk + ∆T Yk = CδXk + DδUk and it corresponds to the following implicit state-space :   I − ∆I I I     Tk+1 Xk+1 Yk   =   Aδ Bδ I Cδ Dδ     Tk Xk Uk  

slide-22
SLIDE 22

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 10/27

Example

A realization with the δ-operator is described by : δXk = AδXk + BδUk Yk = CδXk + DδUk δ q−1

∆

It is computed with    T = AδXk + BδUk Xk+1 = Xk + ∆T Yk = CδXk + DδUk and it corresponds to the following implicit state-space :   I − ∆I I I     Tk+1 Xk+1 Yk   =   Aδ Bδ I Cδ Dδ     Tk Xk Uk  

slide-23
SLIDE 23

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 11/27

Example

One can find the Direct Form II transposed with δ-operator

+ + + + +

U(k) Y (k) βn βi

βn−1

β1 β0 αn

αn−1

αi α1 δ−1 δ−1 δ−1 δ−1

with

Aδ = B B B B B B B B @ −αn 1 . . . . . . ... ... . . . . . . . . . ... ... −α1 . . . 1 −α0 . . . . . . 1 C C C C C C C C A Bδ = B B B B B B B B @ βn . . . . . . β1 β0 1 C C C C C C C C A Cδ = `1 . . . 0´

slide-24
SLIDE 24

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 12/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-25
SLIDE 25

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 13/27

Preleminaries

Let’s consider a MIMO transfer function G defined by G : z → C(zI − A)−1B + D and a noise U(k) with moments

µU E {U(k)} , ΨU E

  • U(k)U⊤(k)
  • ,

σ2

U E

  • U⊤(k)U(k)
  • Filtered noise

U(k) Y (k)

G

Then the filtered noise Y satisfies µY = G(0)µU, σ2

Y = tr

  • ΨU(D⊤D + B⊤WoB)
  • where Wo is the observability Grammian of G, solution of the

Lyapunov equation Wo = A⊤WoA + C ⊤C

slide-26
SLIDE 26

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 13/27

Preleminaries

Let’s consider a MIMO transfer function G defined by G : z → C(zI − A)−1B + D and a noise U(k) with moments

µU E {U(k)} , ΨU E

  • U(k)U⊤(k)
  • ,

σ2

U E

  • U⊤(k)U(k)
  • Filtered noise

U(k) Y (k)

G

Then the filtered noise Y satisfies µY = G(0)µU, σ2

Y = tr

  • ΨU(D⊤D + B⊤WoB)
  • where Wo is the observability Grammian of G, solution of the

Lyapunov equation Wo = A⊤WoA + C ⊤C

slide-27
SLIDE 27

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 14/27

Roundoff Noise Analysis

When implemented, the 3 steps of the computations are modified J.T ∗(k + 1) ← M.X ∗(k) + N.U(k) + BT(k) X ∗(k + 1) ← K.T ∗(k + 1) + P.X ∗(k) + Q.U(k) + BX(k) Y ∗(k) ← L.T ∗(k + 1) + R.X ∗(k) + S.U(k) + BY (k) The noises depends on the way the computations are organized and done the fixed-point representation of the inputs, outputs the fixed-point representation of the states, intermediate variables the fixed-point representation of the constants

slide-28
SLIDE 28

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 14/27

Roundoff Noise Analysis

When implemented, the 3 steps of the computations are modified J.T ∗(k + 1) ← M.X ∗(k) + N.U(k) + BT(k) X ∗(k + 1) ← K.T ∗(k + 1) + P.X ∗(k) + Q.U(k) + BX(k) Y ∗(k) ← L.T ∗(k + 1) + R.X ∗(k) + S.U(k) + BY (k) The noises depends on the way the computations are organized and done the fixed-point representation of the inputs, outputs the fixed-point representation of the states, intermediate variables the fixed-point representation of the constants

slide-29
SLIDE 29

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 15/27

Roundoff Noise Analysis

Let B represent all the noises: B =   BT BX BY   Output noise power The output noise power is given by P = tr

  • ΨB
  • M⊤

2 M2 + M1WoM⊤ 1

  • where

M1 =

  • KJ−1

I

  • ,

M2 =

  • LJ−1

I

  • ΨB depends on the hardware/software considerations, whereas

M1 and M2 depends only on the realization

slide-30
SLIDE 30

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 15/27

Roundoff Noise Analysis

Let B represent all the noises: B =   BT BX BY   Output noise power The output noise power is given by P = tr

  • ΨB
  • M⊤

2 M2 + M1WoM⊤ 1

  • where

M1 =

  • KJ−1

I

  • ,

M2 =

  • LJ−1

I

  • ΨB depends on the hardware/software considerations, whereas

M1 and M2 depends only on the realization

slide-31
SLIDE 31

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 16/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-32
SLIDE 32

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 17/27

Roundoff Noise Gain

The RNG is the ouput noise power in a specific computational scheme the noises appear only after multiplication (Roundoff After Multiplication) centered white noise each noise has the same power σ2 The Roundoff Noise Gain is defined by [Mullis76,Gevers93] G = P σ2 (1)

slide-33
SLIDE 33

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 18/27

Roundoff Noise Gain

Let introduce the matrices dJ to dS. They are diagonal matrices such (dX)ii number of non-trivial parameters in the ith row of X where trivial parameters are 0, 1 and −1 because they did not imply a multiplication The RNG is given by G = tr

  • (dM + dN + dJ) J−⊤

L⊤L + K ⊤WoK

  • J−1

+tr

  • (dK + dP + dQ) Wo
  • + tr (dL + dR + dS)
slide-34
SLIDE 34

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 19/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-35
SLIDE 35

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 20/27

Optimal design

It is possible to analytically describe equivalent classes of realization (Inclusion Principle) Equivalent realization Consider a realization R0. All realizations R1 such that

@ −J1 M1 N1 K1 P1 Q1 L1 R1 S1 1 A = @ Y U−1 Ip 1 A @ −J0 M0 N0 K0 P0 Q0 L0 R0 S0 1 A @ W U Im 1 A

are equivalent (with U ∈ Rn×n, Y ∈ Rl×l and W ∈ Rl×l non-singular matrices). State-space : (A, B, C, D) → (T −1AT , T −1B, CT , D)

slide-36
SLIDE 36

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 21/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-37
SLIDE 37

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 22/27

Example

We consider the following low-pass filter H(z) = 0.01594(z + 1)3 z3 − 1.9749z2 + 1.5562z − 0.4538 And the following realizations Z1: direct form I with shift-operator, Z2: RNG-optimal state-space realization, Z3: RNG-optimal implicit state-space realization: we consider all the equivalent realizations described by EX(k + 1) = AX(k) + BU(k), Y (k) = CX(k) + DU(k). Z4: RNG-optimal δ-realization, with ∆ = 2−5.

slide-38
SLIDE 38

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion 23/27

Example

The optimizations are done with Adaptative Simulated Annealing method. realization RNG

  • Nb. operations

Z1 27.53dB 6 + 7× Z2 16.40dB 12 + 16× Z3 12.05dB 15 + 19× Z4 13.35dB 15 + 19×

slide-39
SLIDE 39

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion

Bibliography

24/27

Outline

1

Macroscopic representation of algorithms through the implicit state-space framework

2

Output Noise Power

3

Roundoff Noise Gain

4

Optimal design

5

Example

6

Conclusion and Perspectives

slide-40
SLIDE 40

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion

Bibliography

25/27

Conclusions and Perspectives

Conclusions Implicit State-Space as a Unifying Framework Output noise power analysis (RNG scheme)

  • ptimal design on various forms

Perspectives Other structurations to study (q/δ mixed realizations, ρDFIIt...) More realistic computational scheme Methodology to consider other criteria (L2-sensitivity, pole-sensitivity,...) Toolbox to solve theses problems

slide-41
SLIDE 41

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion

Bibliography

25/27

Conclusions and Perspectives

Conclusions Implicit State-Space as a Unifying Framework Output noise power analysis (RNG scheme)

  • ptimal design on various forms

Perspectives Other structurations to study (q/δ mixed realizations, ρDFIIt...) More realistic computational scheme Methodology to consider other criteria (L2-sensitivity, pole-sensitivity,...) Toolbox to solve theses problems

slide-42
SLIDE 42

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion

Bibliography

26/27

Questions

Any questions ?

slide-43
SLIDE 43

EUSIPCO’07 —

  • T. Hilaire,
  • D. M´

enard and

  • O. Sentieys

Introduction Implicit State-Space Framework Output Noise Power Roundoff Noise Gain Optimal design Example Conclusion

Bibliography

27/27

Bibliography

  • M. Gevers and G. Li.

Parametrizations in Control, Estimation and Filtering Probems. Springer-Verlag, 1993.

  • R. Istepanian and J.F. Whidborne, editors.

Digital Controller implementation and fragility. Springer, 2001.

  • T. Hilaire, P. Chevrel, and Y. Trinquet

Implicit state-space representation : a unifying framework for FWL implementation of LTI systems IFAC05 Wolrd Congress, July 2005.