High resolution frequency counters E. Rubiola FEMTO-ST Institute, - PowerPoint PPT Presentation

High resolution frequency counters E. Rubiola FEMTO-ST Institute, CNRS and Université de Franche Comté 28 May 2008 Outline 1. Digital hardware 2. Basic counters 3. Microwave counters 4. Interpolation • time-interval amplifier • frequency vernier • time-to-voltage converter • multi-tap delay line 5. Basic statistics 6. Advanced statistics home page http://rubiola.org

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3 1 – Digital hardware

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8 2 – Basic counters

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13 Practical measurement N c counted cycles (edges) clock nominal time T nom = N' c T c the measurement time starts after the internal trigger the measurement time ends after the nominal time is elapsed input measurement time T m = N x T x � T nom N x counted cycles (edges) measurement equation: N x T x = N c T c or N x T x = (N c ± 1) T c , including quantization uncertainty

14 3 – Microwave counters

15 Prescaler input reciprocal ÷ 2 n counter • a prescaler is a n-bit binary divider ÷ 2 n • GaAs dividers work up to ≈ 20 GHz • reciprocal counter => there is no resolution reduction • Most microwave counters use the prescaler

16 Transfer-oscillator counter f x low input pass N f vco f x / N classical counter f vco VCO • The transfer oscillator is a PLL • Harmonics generation takes place inside the mixer • Harmonics locking condition: N f vco = f x • Frequency modulation Δ f is used to identify N (a rather complex scheme, × N => Δ f -> N Δ f )

17 Heterodyne counter f x f b low classical input pass counter N f r multiplier × N oscillator • Down-conversion: f b = | f x – N f c | • f b is in the range of a classical counter (100-200 MHz max) • no resolution reduction in the case of a classical frequency counter (no need of reciprocal counter) • Old scheme, nowadays used only in some special cases (frequency metrology)

18 4 – Interpolation

1 – Time-interval amplifier 19

1 – Time-interval amplifier 20

1 – Time-interval amplifier 21

1 – Time-interval amplifier 22

2 – Frequency vernier 23

2 – Frequency vernier 24

2 – Frequency vernier 25

2 – Frequency vernier 26

3 – Time-to-voltage converter 27

3 – Time-to-voltage converter 28

4 – Multi-tap delay line 29 Interpolation by sampling delayed copies of the clock or of the stop signal reference reference start stop � input event clock 0 0 0 clock 1 array of D-type flip-flops 0 0 clock 2 0 0 clock 3 0 1 clock 4 0 1 clock 5 1 1 clock 6 1 1 clock 7 1 1 sample word 00000111 sample word 00011111 indicates delay = 5 � indicates delay = 3 � The resolution is determined by the delay τ , instead of by the toggling speed of the flip-flops

4 – Multi-tap delay line 30 Sampling circuits J. Kalisz, Metrologia 41 (2004) 17–32

4 – Multi-tap delay line 31 Ring Oscillator used in PLL circuits for clock-frequency multiplication J. Kalisz, Metrologia 41 (2004) 17–32

32 5 – Basic statistics

33 Old Hewlett Packard application notes

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35 Quantization uncertainty p ( x ) σ 2 = T 2 c 1 /T c 12 T c √ 1 / 12 = 0 . 29 Example: 100 MHz clock Tx = 10 ns σ = 2.9 ns

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37 classical reciprocal counter (1) measurement. time M pulses τ = N/ ν Μ = τν c ν ÷N binary counter trigger ν c N ν c readout = ν reference M period measurement (count the clock pulses) is preferred to frequency measurement (count the input pulses) because: • it provides higher resolution in a given measurement time tau (the clock frequency can be close to the maximum switching speed) • interpolation (M is rational instead of integer) can be used to reduce the quantization (interpolators only work at a fixed frequency, thus at the clock freq.)

38 classical reciprocal counter (2) phase time x x 0 x 1 x 2 x 3 x N (i.e., time jitter) v(t) time t t 0 t 1 t 2 t 3 t 4 t 5 t 6 t N period T 00 weight 1/τ w Π 0 measurement time τ = NT � + ∞ E { ν } = ν ( t ) w Π ( t ) dt Π estimator measure: scalar product −∞ � 1 /τ 0 < t < τ w Π ( t ) = weight 0 elsewhere � + ∞ w Π ( t ) dt = 1 normalization −∞ y = 2 σ 2 variance σ 2 x classical variance τ 2

39 enhanced-resolution counter n − 1 phase time x E { ν } = 1 x 0 x 1 x 2 x 3 x N � (i.e., time jitter) ν i ν i = N/τ i n v(t) time i =0 t t 0 t 1 t 2 t 3 t 4 t 5 t 6 t N − D t N t N+D meas. no. Λ estimator 1/ τ w 0 0 i = 0 � + ∞ w 1 i = 1 E { ν } = ν ( t ) w Λ ( t ) dt w 2 weight i = 2 −∞ w i weight  w n − 1 t/τ 0 < t < τ i = n − 1   w Λ ( t ) = 2 − t/τ τ < t < 2 τ delay τ 0 = DT measurement time τ = NT = nDT  0 elsewhere  1 n − 1 n − 1 normalization τ n τ n τ � + ∞ 2 2 weight 1 1 w Λ w Λ ( t ) dt = 1 n τ n τ n τ n τ −∞ white noise: the autocorrelation limit τ 0 -> 0 of the weight function function is a narrow pulse, about 1/ τ the inverse of the bandwidth w Λ (t) 2 σ 2 y = 1 t classical the variance is 0 σ 2 τ 2 τ x τ 2 variance divided by n n

40 actual formulae look like this σ y = 1 � 2( δt ) 2 trigger + 2( δt ) 2 (Π) interpolator τ 1 � 2( δt ) 2 trigger + 2( δt ) 2 (Λ) σ y = τ √ n interpolator � ν 0 τ ν 00 ≤ ν I n = ν I τ ν 00 > ν I

41 understanding technical information y = 2 σ 2 classical classical reciprocal σ 2 x τ 2 variance counter enhanced-resolution 2 σ 2 y = 1 classical σ 2 x τ 2 counter variance n low frequency: τ 0 = T = n = ν 00 τ ⇒ full speed 2 σ 2 1 classical σ 2 x y = τ 3 ν 00 variance high frequency: τ 0 = DT with D> 1 = n = ν 00 τ ⇒ housekeeping takes time 2 σ 2 y = 1 classical x σ 2 τ 3 variance ν I the slope of the classical variance tells the whole story 1 /τ 2 = Π estimator (classical reciprocal) ⇒ 1 /τ 3 = Λ estimator (enhanced-resolution) ⇒ look for formulae and plots in the instruction manual

42 examples Stanford SRS-620 � �� 2 � 2 � �� � � � � (25 ps) 2 + short term gate trigger + 2 × � � � × RMS stability time jitter frequency = resolution N gate time (in Hz) RMS resolution σ ν = ν 00 σ y frequency ν 00 gate time τ Agilent 53132A � � ( t res ) 2 + 2 × ( trigger error ) 2 � � � 4 × t jitter � � RMS frequency = + × ( gate time ) × √ or period resolution no. of samples gate time t res = 225 ps t jitter = 3 ps � (gate time) × (frequency) for f < 200 kHz number of samples = (gate time) × 2 × 10 5 for f ≥ 200 kHz RMS resolution σ ν = ν 00 σ y or σ T = T 00 σ y frequency ν 00 period T 00 gate time τ � ν 00 < 200 kHz ν 00 τ no. of samples n = τ × 2 × 10 5 ν 00 ≥ 200 kHz

43 5 – Advanced statistics

44 Allan variance � 1 � 2 � � σ 2 y ( τ ) = E definition y k +1 − y k 2 � ( k +2) τ � ( k +1) τ � � 2 � 1 � 1 y ( t ) dt − 1 σ 2 y ( τ ) = E y ( t ) dt 2 τ τ ( k +1) τ kτ ��� + ∞ � 2 � wavelet-like σ 2 y ( τ ) = E y ( t ) w A ( t ) dt variance −∞ w 1 A  1 0 < t < τ √ − 2 τ 2 τ time   0 1 w A = τ < t < 2 τ √ t 2 τ − 1  0 elsewhere  2 τ 0 τ 2 τ � + ∞ A ( t ) dt = 1 the Allan variance differs from a wavelet variance in w 2 energy E { w A } = the normalization on power, instead of on energy τ −∞

45 modified Allan variance � 1 � ( i +2 n ) τ 0 � ( i + n ) τ 0 n − 1 � �� 2 � 1 � 1 y ( t ) dt − 1 � definition mod σ 2 y ( τ ) = E y ( t ) dt 2 n τ τ ( i + n ) τ 0 iτ 0 i =0 with τ = n τ 0 . ��� + ∞ � 2 � mod σ 2 y ( τ ) = E y ( t ) w M ( t ) dt −∞ wavelet-like  variance 1 0 < t < τ 2 τ 2 t √ −  1  w  1  2 τ 2 (2 t − 3) τ < t < 2 τ M  √ 2 τ w M = time 1 � 2 τ 2 ( t − 3 2 τ < t < 3 τ 0 √ −  3 τ t  τ 2 τ  − 1  0 elsewhere  0 2 τ � + ∞ M ( t ) dt = 1 w 2 E { w M } = energy 2 τ −∞ E { w M } = 1 compare the energy 2 E { w A } this explains why the mod Allan variance is always lower than the Allan variance