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2015 joint Conference of the IEEE International Frequency Symposium & European Frequency and Time Forum Denver The measurement of frequency fluctuations with counters and AVAR-like least-square-fit wavelets .-Y. Bourgeois and E.

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2015 joint Conference of the IEEE International Frequency Symposium & European Frequency and Time Forum – Denver

The measurement of frequency fluctuations with Ω counters and AVAR-like least-square-fit wavelets

  • F. Vernotte†, M. Lenczner‡, P

.-Y. Bourgeois‡ and E. Rubiola‡

† UTINAM – Observatory THETA – University of Franche-Comté/CNRS ‡ Time & Frequency Department – Femto-ST – University of Franche-Comté/CNRS

1

The Ω counters The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

2

The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

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SLIDE 2

The Ω counters The Quadratic Variance

Introduction

Large band instruments ⇒ huge white phase noise What is the best way to kill the beast ⇒ least squares We propose the “Ω counters” ⇒ real time linear regression over phase data for estimating frequency We describe its associated variance, the “Quadratic Variance” (QVAR) ⇒ performances seem very promising Progress of digital electronics ⇒ allows real-time least square estimation of frequency at high rate

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 2

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SLIDE 3

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

The Linear-Regression Estimator

Property of the estimate ˜ A: if x(t) = ω · t + ϕ ⇒ ˜ A = ω The Ω counter: measures N phase-values xk with a sampling step τ0 computes the linear regression ˜ A =

  • k

hΩx(tk)xk = 12 N3τ 2

(N−1)/2

  • −(N−1)/2

k · xk transmits the estimate ˜ A and erases the xk measurements performs a new estimation. . . ˜ A is the best frequency estimate in presence of white PM noise

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 3

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SLIDE 4

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

Why “Ω”?

˜ A =

  • k

hΩx(tk)xk =

  • k

hΩy(tk)¯ yk with hΩx(t) = dhΩy(t) dt . Since hΩx(t) = 12t τ 3 then hΩy(t) = 6 τ 3 τ 2 4 − t2

  • which looks like the Greek letter Ω!
  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 4

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SLIDE 5

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

Why “Ω”?

˜ A =

  • k

hΩx(tk)xk =

  • k

hΩy(tk)¯ yk with hΩx(t) = dhΩy(t) dt . Since hΩx(t) = 12t τ 3 then hΩy(t) = 6 τ 3 τ 2 4 − t2

  • which looks like the Greek letter Ω!
  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 4

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SLIDE 6

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

Comparison of Π, Λ and Ω counters

Π counter Λ counter Ω counter Estimate: ¯ A Estimate: ˆ A Estimate: ˜ A Variance (white PM): Variance (white PM): Variance (white PM): V ¯ A

  • = 2σ2

ǫ

τ 2 V

  • ˆ

A

  • = 2τ0σ2

ǫ

τ 3 V ˜ A

  • = 12τ0σ2

ǫ

τ 3 Span: τ Span: 2τ Span: τ With a 2τ-span: V ˜ A

  • = 3τ0σ2

ǫ

2τ 3 = 3 4V

  • ˆ

A

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

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SLIDE 7

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

Comparison of Π, Λ and Ω counters

Π counter Λ counter Ω counter Estimate: ¯ A Estimate: ˆ A Estimate: ˜ A Variance (white PM): Variance (white PM): Variance (white PM): V ¯ A

  • = 2σ2

ǫ

τ 2 V

  • ˆ

A

  • = 2τ0σ2

ǫ

τ 3 V ˜ A

  • = 12τ0σ2

ǫ

τ 3 Span: τ Span: 2τ Span: τ With a 2τ-span: V ˜ A

  • = 3τ0σ2

ǫ

2τ 3 = 3 4V

  • ˆ

A

  • ր We have a good
  • reason. . .
  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

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SLIDE 8

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

Comparison of Π, Λ and Ω counters

Π counter Λ counter Ω counter Estimate: ¯ A Estimate: ˆ A Estimate: ˜ A Variance (white PM): Variance (white PM): Variance (white PM): V ¯ A

  • = 2σ2

ǫ

τ 2 V

  • ˆ

A

  • = 2τ0σ2

ǫ

τ 3 V ˜ A

  • = 12τ0σ2

ǫ

τ 3 Span: τ Span: 2τ Span: τ With a 2τ-span: V ˜ A

  • = 3τ0σ2

ǫ

2τ 3 = 3 4V

  • ˆ

A

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

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SLIDE 9

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

From Ω counter to Q variance

V ˜ A

  • = E

     1 N − 1

N

  • i=1

  ˜ Ai − 1 N

N

  • j=1

˜ Aj  

2

    In the presence of white PM and for N = 2: V ˜ A

  • = 1

2E ˜ A1 − ˜ A2 2 This define the wavelet shape: The quadratic variance QVAR is defined as: σ2

Q(τ) = 1

2 ˜ A1 − ˜ A2 2 QVAR is an estimator of the variance of the Ω counter estimates

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 6

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SLIDE 10

The Ω counters The Quadratic Variance The Linear-Regression Estimator Comparison of Π, Λ and Ω counters From Ω counter to Q variance

From Ω counter to Q variance

V ˜ A

  • = E

     1 N − 1

N

  • i=1

  ˜ Ai − 1 N

N

  • j=1

˜ Aj  

2

    In the presence of white PM and for N = 2: V ˜ A

  • = 1

2E ˜ A1 − ˜ A2 2 This define the wavelet shape: The quadratic variance QVAR is defined as: σ2

Q(τ) = 1

2 ˜ A1 − ˜ A2 2 QVAR is an estimator of the variance of the Ω counter estimates

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 6

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SLIDE 11

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

QVAR computation weights

From phase data: σ2

Q(τ) =

  • [x(t) ∗ hQx(t)]2

with hQx(t) = 6 √ 2 τ 3

  • t + τ

2

  • if t ∈ [−τ, 0[

= 6 √ 2 τ 3

  • −t + τ

2

  • if t ∈ [0, τ[

From frequency data: σ2

Q(τ) =

  • y(t) ∗ hQy(t)

2 with hQy(t) = 3 √ 2t τ 3 (−t − τ) if t ∈ [−τ, 0[ = 3 √ 2t τ 3 (t − τ) if t ∈ [0, τ[

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 7

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SLIDE 12

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Filter interpretation

QVAR may be calculated either in the direct or in the Fourier domain: Direct domain: σ2

Q(τ) =

  • y(t) ∗ hQy(t)

2 Fourier domain: σ2

Q(τ) =

∞ Sy(f)

  • HQy(f)
  • 2 df

where HQy(f) is the transfer function of QVAR Since HQy(f) is the Fourier transform of hQy(t) :

  • HQy(f)
  • 2 =

9

  • 2 sin2(πτf) − πτf sin(2πτf)

2 2(πτf)6 Convergence properties: for small f:

  • HQy(f)
  • 2 ≈

3 (πτf)2 ⇒ converges for f −2 FM for large f:

  • HQy(f)
  • 2 decreases as f −4 ⇒

converges for f +2 FM

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 8

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SLIDE 13

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Filter interpretation

QVAR may be calculated either in the direct or in the Fourier domain: Direct domain: σ2

Q(τ) =

  • y(t) ∗ hQy(t)

2 Fourier domain: σ2

Q(τ) =

∞ Sy(f)

  • HQy(f)
  • 2 df

where HQy(f) is the transfer function of QVAR Since HQy(f) is the Fourier transform of hQy(t) :

  • HQy(f)
  • 2 =

9

  • 2 sin2(πτf) − πτf sin(2πτf)

2 2(πτf)6 Convergence properties: for small f:

  • HQy(f)
  • 2 ≈

3 (πτf)2 ⇒ converges for f −2 FM for large f:

  • HQy(f)
  • 2 decreases as f −4 ⇒

converges for f +2 FM

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 8

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SLIDE 14

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

QVAR transfer function

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 9

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SLIDE 15

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Responses of QVAR for the different types of noise

Sy(f) MVAR(τ) QVAR(τ) h−2f −2 11π2h−2τ 20 26π2h−2τ 35 h−1f −1 [27 ln(3) − 32 ln(2)] h−1 8 2 [7 − ln(16)] h−1 5 h0f 0 h0 4τ 3h0 5τ h+1f +1 [24 ln(2) − 9 ln(3)] h+1 8π2τ 2 3 [ln(16) − 1] h+1 2π2τ 2 h+2f +2 3h+2 8π2τ 3 3h+2 2π2τ 3

For a linear frequency drift:

y(t) = D1 · t 1 2 D2

1τ 2

1 2 D2

1τ 2

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 10

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SLIDE 16

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Responses of QVAR for the different types of noise

Sy(f) MVAR(τ) QVAR(τ) h−2f −2 11π2h−2τ 20 26π2h−2τ 35 h−1f −1 [27 ln(3) − 32 ln(2)] h−1 8 2 [7 − ln(16)] h−1 5 h0f 0 h0 4τ 3h0 5τ h+1f +1 [24 ln(2) − 9 ln(3)] h+1 8π2τ 2 3 [ln(16) − 1] h+1 2π2τ 2 h+2f +2 3h+2 8π2τ 3 3h+2 2π2τ 3

For a linear frequency drift:

y(t) = D1 · t 1 2 D2

1τ 2

1 2 D2

1τ 2

T h i s i s t h e g

  • d

r e a s

  • n

! ց

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 10

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SLIDE 17

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Equivalent Degrees of Freedom of QVAR

As the other variances, the QVAR estimates are χ2 distributed: ˜ σ2

Q(τ) = αχ2 ν

α ≡ proportionality coefficient ν ≡ Equivalent Degrees of Freedom (EDF) Properties of a χ2

ν distribution:

E{χ2

ν} = ν

V{χ2

ν} = 2ν

Relative dispersion of the QVAR estimates: ∆˜ σ2

Q(τ)

˜ σ2

Q(τ)

=

  • V{˜

σ2

Q(τ)}

E{˜ σ2

Q(τ)}

= √ 2α2ν αν =

  • 2

ν The EDF are connected to the dispersion of the QVAR estimates

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 11

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SLIDE 18

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Equivalent Degrees of Freedom (EDF)

Examples for N = 2048, τ0 = 1 s

For f +2, f +1, white PM: EDFMVAR < EDFQVAR < EDFAVAR For f −1, f −2 PM: EDFMVAR < EDFAVAR < EDFQVAR

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 12

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SLIDE 19

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

White FM and linear frequency drift detection

Criterion: lowest detected response = response equal to the higher bound of the 95 % confidence interval over QVAR for the largest τ White FM noise detection: QVAR is the first estimator to detect the white FM in a white PM context Linear frequency drift detection: QVAR is the first estimator to detect the linear frequency drift in a white PM context

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 13

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SLIDE 20

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

White FM and linear frequency drift detection

Criterion: lowest detected response = response equal to the higher bound of the 95 % confidence interval over QVAR for the largest τ White FM noise detection: QVAR is the first estimator to detect the white FM in a white PM context Linear frequency drift detection: QVAR is the first estimator to detect the linear frequency drift in a white PM context

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 13

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SLIDE 21

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

White FM and linear frequency drift detection

Criterion: lowest detected response = response equal to the higher bound of the 95 % confidence interval over QVAR for the largest τ White FM noise detection: QVAR is the first estimator to detect the white FM in a white PM context Linear frequency drift detection: QVAR is the first estimator to detect the linear frequency drift in a white PM context

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 13

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SLIDE 22

The Ω counters The Quadratic Variance Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

Summary

The Ω counters have the best white PM rejection ⇒ perform the best frequency estimates in the sense of the least squares Nowadays digital electronics allows the implementation of such counters ⇒ suitable to general purpose hardware Ω counters estimates ⇒ QVAR analysis QVAR converges for all types of noise ⇒ from f −2 to f +2 PM + linear frequency drifts Provides the best identification of white FM as well as linear frequency drift ⇒ suitable for RF frequency standards ⇒ suitable for stabilized lasers

  • F. Vernotte, M. Lenczner, P

.-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 14