The measurement of frequency fluctuations with counters and - PowerPoint PPT Presentation

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 .-Y. Bourgeois ‡ and E. Rubiola ‡ F. Vernotte † , M. Lenczner ‡ , P † UTINAM – Observatory THETA – University of Franche-Comté/CNRS ‡ Time & Frequency Department – Femto-ST – University of Franche-Comté/CNRS The Ω counters 1 The Linear-Regression Estimator Comparison of Π , Λ and Ω counters From Ω counter to Q variance The Quadratic Variance 2 Definition of QVAR Properties of QVAR Comparison of AVAR, MVAR and QVAR

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

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance 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 x k with a sampling step τ 0 computes the linear regression ˜ � A = h Ω x ( t k ) x k k ( N − 1 ) / 2 12 � = k · x k N 3 τ 2 0 − ( N − 1 ) / 2 transmits the estimate ˜ A and erases the x k 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

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance Why “ Ω ”? ˜ � A = h Ω x ( t k ) x k k � h Ω y ( t k )¯ = y k k with h Ω x ( t ) = d h Ω y ( t ) . d t Since h Ω x ( t ) = 12 t τ 3 then � τ 2 h Ω y ( t ) = 6 � 4 − t 2 τ 3 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

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance Why “ Ω ”? ˜ � A = h Ω x ( t k ) x k k � h Ω y ( t k )¯ = y k k with h Ω x ( t ) = d h Ω y ( t ) . d t Since h Ω x ( t ) = 12 t τ 3 then � τ 2 h Ω y ( t ) = 6 � 4 − t 2 τ 3 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

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance Comparison of Π , Λ and Ω counters Π counter Λ counter Ω counter Estimate: ¯ Estimate: ˆ Estimate: ˜ A A A Variance (white PM): Variance (white PM): Variance (white PM): � ¯ = 2 σ 2 = 2 τ 0 σ 2 � ˜ = 12 τ 0 σ 2 � � ǫ ˆ ǫ ǫ � � A A A V V V τ 2 τ 3 τ 3 Span: τ Span: 2 τ Span: τ � ˜ = 3 τ 0 σ 2 = 3 � � ˆ ǫ � With a 2 τ -span: V A 4 V A 2 τ 3 F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance Comparison of Π , Λ and Ω counters Π counter Λ counter Ω counter Estimate: ¯ Estimate: ˆ Estimate: ˜ A A A Variance (white PM): Variance (white PM): Variance (white PM): � ¯ = 2 σ 2 = 2 τ 0 σ 2 � ˜ = 12 τ 0 σ 2 � � ǫ ˆ ǫ ǫ � � A A A V V V τ 2 τ 3 τ 3 Span: τ Span: 2 τ Span: τ ր We have a good � ˜ = 3 τ 0 σ 2 = 3 � � ˆ ǫ reason. . . � With a 2 τ -span: V A 4 V A 2 τ 3 F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance Comparison of Π , Λ and Ω counters Π counter Λ counter Ω counter Estimate: ¯ Estimate: ˆ Estimate: ˜ A A A Variance (white PM): Variance (white PM): Variance (white PM): � ¯ = 2 σ 2 = 2 τ 0 σ 2 � ˜ = 12 τ 0 σ 2 � � ǫ ˆ ǫ ǫ � � A A A V V V τ 2 τ 3 τ 3 Span: τ Span: 2 τ Span: τ � ˜ = 3 τ 0 σ 2 = 3 � � ˆ ǫ � With a 2 τ -span: V A 4 V A 2 τ 3 F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 5

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance From Ω counter to Q variance  2    N N � ˜   1 A i − 1   � �  ˜ ˜ � A = E A j V  N − 1 N   i = 1 j = 1   In the presence of white PM and for N = 2: � ˜ �� ˜ = 1 � 2 � A 1 − ˜ � V A 2 E A 2 This define the wavelet shape: The quadratic variance QVAR is defined as: �� ˜ Q ( τ ) = 1 � 2 � σ 2 A 1 − ˜ A 2 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

The Linear-Regression Estimator The Ω counters Comparison of Π , Λ and Ω counters The Quadratic Variance From Ω counter to Q variance From Ω counter to Q variance  2    N N � ˜   1 A i − 1   � �  ˜ ˜ � A = E A j V  N − 1 N   i = 1 j = 1   In the presence of white PM and for N = 2: � ˜ �� ˜ = 1 � 2 � A 1 − ˜ � V A 2 E A 2 This define the wavelet shape: The quadratic variance QVAR is defined as: �� ˜ Q ( τ ) = 1 � 2 � σ 2 A 1 − ˜ A 2 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

Definition of QVAR The Ω counters Properties of QVAR The Quadratic Variance Comparison of AVAR, MVAR and QVAR QVAR computation weights From phase data: � [ x ( t ) ∗ h Qx ( t )] 2 � σ 2 Q ( τ ) = with √ 6 2 � t + τ � h Qx ( t ) = if t ∈ [ − τ, 0 [ τ 3 2 √ 6 2 � − t + τ � = if t ∈ [ 0 , τ [ τ 3 2 From frequency data: �� � 2 � σ 2 Q ( τ ) = y ( t ) ∗ h Qy ( t ) with √ 3 2 t h Qy ( t ) = ( − t − τ ) if t ∈ [ − τ, 0 [ τ 3 √ 3 2 t = ( t − τ ) if t ∈ [ 0 , τ [ τ 3 F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 7

Definition of QVAR The Ω counters Properties of QVAR The Quadratic Variance Comparison of AVAR, MVAR and QVAR Filter interpretation QVAR may be calculated either in the direct or in the Fourier domain: �� � 2 � Direct domain: σ 2 Q ( τ ) = y ( t ) ∗ h Qy ( t ) � ∞ � 2 d f Fourier domain: σ 2 � � Q ( τ ) = S y ( f ) � H Qy ( f ) 0 where H Qy ( f ) is the transfer function of QVAR Since H Qy ( f ) is the Fourier transform of h Qy ( t ) : � 2 � 2 sin 2 ( πτ f ) − πτ f sin ( 2 πτ f ) 9 � 2 = � � � H Qy ( f ) 2 ( πτ f ) 6 Convergence properties: � 2 ≈ 3 converges for f − 2 FM � � for small f : � H Qy ( f ) ⇒ ( πτ f ) 2 � 2 decreases as f − 4 ⇒ converges for f + 2 FM � � for large f : � H Qy ( f ) F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 8

Definition of QVAR The Ω counters Properties of QVAR The Quadratic Variance Comparison of AVAR, MVAR and QVAR Filter interpretation QVAR may be calculated either in the direct or in the Fourier domain: �� � 2 � Direct domain: σ 2 Q ( τ ) = y ( t ) ∗ h Qy ( t ) � ∞ � 2 d f Fourier domain: σ 2 � � Q ( τ ) = S y ( f ) � H Qy ( f ) 0 where H Qy ( f ) is the transfer function of QVAR Since H Qy ( f ) is the Fourier transform of h Qy ( t ) : � 2 � 2 sin 2 ( πτ f ) − πτ f sin ( 2 πτ f ) 9 � 2 = � � � H Qy ( f ) 2 ( πτ f ) 6 Convergence properties: � 2 ≈ 3 converges for f − 2 FM � � for small f : � H Qy ( f ) ⇒ ( πτ f ) 2 � 2 decreases as f − 4 ⇒ converges for f + 2 FM � � for large f : � H Qy ( f ) F. Vernotte, M. Lenczner, P .-Y. Bourgeois, E. Rubiola Ω counters and AVAR-like least-square-fit wavelets 8

Definition of QVAR The Ω counters Properties of QVAR The Quadratic Variance 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