home page http://rubiola.org E. Rubiola Outline

Transcription

High resolution
time & frequency counters
E. Rubiola
FEMTO-ST Institute, CNRS and Université de Franche Comté
May 2012
Outline
• Introduction
• Basic counters (RF & microwave)
• The trigger
• Clock interpolation techniques
• Basic statistics
• Advanced statistics
home page http://rubiola.org
start
stop
Counter – main purposes
2
Counter
Experiment
Tx
(⌧ )
start
stop
detector
Frequency, Period
or Time-Interval (TI)
TDC
physical
events
t = ta
input
triggers
data
stop
t = tb
data
process
detector
start
display
⌫c
External
frequency
reference
• Compare a physical quantity (frequency, period, time
interval) to a frequency reference
• Exploit the full accuracy and precision of the reference,
with no degradation
digital
output
Digital hardware
The AND gate
3
Basic flip-flops
Set-Reset (SR) flip-flop
D-type flip-flop (digital sampler)
4
Binary counter
Disambiguation: the word “counter” – is used for both
• the binary / BCD counter – the digital circuit
• the time / frequency counter – the instrument
5
6
1 – Basic counters
Time Interval (TI) counter
arming
pulse
Tx
start
D Q
S Q
Tx
trigger
gate
pulse
+ quant.
error
A
B
+ quant. err.
1/⌫c
binary
counter
C
T = Tx
R
stop
Nc = ⌫ c Tx
T = Tx
+ quant.
error
gate
control
FF
arming
FF
⌫c
gate pulse
clock
t = ta
t = tb
start
t
stop
t
arming pulse
t
gate pulse
1/⌫c
t
clock
t
C
Nc clock cycles
The resolution is set by the clock period 1/νc
7
The gate-control FF is not shown
The (old) frequency counter
1/⌫x
⌫x
A
B
⌫x
Nx = ⌫ x T = Nc
⌫c
binary
counter
C
trigger
⌫c
T = Nc /⌫c
⌧ =T
÷ Nc
clock
A
divider
8
The gate-control
FF is not shown
gate pulse
1/⌫x
B
T = Nx /⌫x
C
0/–1 count
Nx = T ⌫x counts
The resolution is set by the input period 1/νx, which can be poor
Classical reciprocal counter
Nc = ⌫ c T
T = Nx /⌫x
1/⌫c
gate
pulse
A
⌫x
B
÷ Nx
= Nx ⌫c /⌫x
binary
counter
C
divider
trigger
⌫c
clock
gate pulse
T = Nx /⌫x
⌧ =T
Tx = 1/⌫x
input
t
nominal
gate
t
Tw
actual
gate A
t
T = Nx /⌫x (exact)
clock
B
1/⌫c
t
t
C
Nc clock cycles
• Use the highest clock frequency permitted by the hardware
• The measurement time is a multiple of the input period
9
10
Prescaler
Nx = Nx(p) Nx(rc)
input
GaAs prescaler
classical reciprocal counter
T = Nx /⌫x
1/⌫c
gate
pulse
A
⌫x
front
end
level
translat.
÷ Nx(p)
divider
Nc = ⌫ c T
B
binary
counter
C
÷ Nx(rc)
divider
clock
⌫c
= Nx ⌫c /⌫x
gate pulse
T = Nx /⌫x
⌧ =T
frequency
reference
• The prescaler is a n-bit binary divider ÷ 2n
• GaAs dividers work up to at least 20 GHz
• Reciprocal counter => there is no resolution reduction
• Most microwave counters use the prescaler
Transfer-oscillator counter
transfer oscillator
input
⌫x
harmonic
mixer
front
end
⌫x
N ⌫vco
IF amplifier
1
N
⌫vco =
⌫x ⌫ref
in
⌫vco
⌫IF
VCO
in
control
⌫x
⌫ref
metrix
N ⌫vco = ⌫ref
identify N
N
classical
reciprocal
counter
⇥M
frequency
reference
⌫c
• The transfer oscillator is a PLL
• Harmonics generation takes place inside the mixer
• Harmonics locking condition: N νvco = νx
• Frequency modulation Δf is used to identify N
(a rather complex scheme, ×N => Δν –> NΔν )
11
12
Heterodyne counter
input
⌫out =
⌫x N ⌫c
heterodyne down-converter
⌫x
sampling
mixer
⌫x
front
end
N ⌫c
in
N ⌫c
harmonic
selection
filter
control
N
classical
reciprocal
counter
comb
generator
frequency
reference
⌫c
• Down-conversion: fb = | νx – N νc |
• νb 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)
13
Coarse counting
active edge
Tc = 1/⌫c
t
clock
GTc
gate pulse
G = 1/24
Tx = (N + F )Tc
(N = 3, and F = 11/24)
Nc = 3
G = 4/24
Nc = 3
G = 7/24
Nc = 3
G = 10/24
Nc = 3
G = 13/24
Nc = 3
G = 16/24
Nc = 4
G = 19/24
Nc = 4
G = 22/24
Nc = 4
The integer number Nc of clock cycles that falls in
the gate pulse Tx is either N or N+1, depending on
the the fractional part FTc and on the delay GTc
14
2 – Trigger
15
Trigger hysteresis
in
Slope +
vout
out
Slope –
vout
hysteresis
hysteresis
H
H
vin
L
vin
L
vout
VT VT H
vin
threshold
VT L
hysteresis
vin
hysteresis
VT L VT
VT
t
t
vout
VT H
threshold
VT
t
t
Hysteresis is necessary to avoid chatter in the presence of noise
Threshold fluctuation
‘stop’
‘start’
start
( V )b
x=
Sb
systematic
2
x
random
=
(
2
V )a
Sa2
( V )a
Sa
+
(
2
V )b
Sb2
S Q
Tx
trigger
R
stop
vin
threshold
error V
actual threshold
nominal threshold
slope
S = dv/dt
t
error x =
V /S
vout
t
16
17
noise can cause early triggering only.
18
Trigger noise – oversimplified
Noise can occur on either the Start or Stop pulse or b
quently the measurement may be either too long or to
twice the error shown in the example.
( V )b
systematic x =
Sb
=
( V )a
Sa
+
( V2 )b
Sb2
a.
1
0
Noise
Volts
random
2
x
( V2 )a
Sa2
‘start’
1
b.
0
start
S Q
Tx
stop
trigger
1
c.
R
0
Noise
!t
2. Distortion on Input Signal
Time
Agilent, Application Note 200-3, 1997
‘stop’
F
E
b
i
• The effect of noise is often explained with a plot like this
Figure 26 shows how harmonically related noise as w
• Yet, the formula holds in the absence
of spikes!!!
nonharmonically
related noise moves the trigger poin
Also shown is thelooks
effect ofsimple
noise on a signal.
• To the general practitioner, this explanation
Effect of (too) wide-band noise
When the rms slope of
noise is higher than the
signal slope:
• the trigger leads
• systematic error
E. Rubiola & al., Proc. 46 FCS pp. 265-269, May 1992
19
Trigger behavior vs. bandwidth
Noise rms slope
Sn2
Sn2
= 4⇡
2
4⇡
=
3
2
Z
B
2
f SV (f ) df
0
= 2.0 Hz, Noise = 10.0 mVrms
1.0 B
E. Rubiola, 18 oct 2011
= 64.0 Hz, Noise = 10.0 mVrms
1.0 B
= 2048.0 Hz, Noise = 10.0 mVrms
1.0 B
0.5
0.5
0.5
0.0
threshold
0.0
0.5
0.5
1.0
1.0
1.0
time
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.2
time
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.2
B = 1.0 Hz, Noise = 10.0 mVrms
threshold
0.0
0.1
0.1
0.1
0.49
0.50
0.51
0.52
time
0.53
0.2
0.47
0.2
Ss2
4⇡
=
3
2
B2
Signal slope equals rms noise slope
0.50
0.51
0.52
time
0.53
0.2
0.47
0.2
threshold
0.1
0.1
0.1
0.2
0.48
0.49
0.50
0.51
0.2
0.47
0.2
0.48
0.49
0.50
0.51
time
0.52
0.53
0.2
threshold
0.0
0.1
0.1
0.1
0.2
0.49
0.50
0.51
0.52
time
0.53
0.2
0.47
0.2
0.48
0.49
0.50
0.51
0.52
time
0.53
0.2
B = 1024.0 Hz, Noise = 10.0 mVrms
threshold
0.0
0.1
0.1
0.1
0.49
0.50
0.51
0.52
time
0.53
0.48
0.49
0.50
0.51
0.52
time
0.53
0.52
time
0.53
B = 2048.0 Hz, Noise = 10.0 mVrms
threshold
0.0
0.48
0.51
0.1
0.0
0.2
0.47
0.50
0.1
threshold
0.49
0.2
0.47
0.1
0.48
time
0.52
0.53
threshold
0.0
0.48
time
0.53
0.1
0.0
0.2
0.47
0.52
0.1
threshold
0.51
0.2
0.47
0.1
0.50
threshold
0.0
time
0.52
0.53
0.49
0.0
0.2
0.47
0.48
0.1
0.0
2
V
0.49
0.1
threshold
4⇡
=
SV B 3
3
0.48
0.1
Ss2
threshold
0.0
0.2
2
0.1
0.0
0.48
0.1
0.2
0.47
Critical slope
0.2
threshold
time
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1
B2
0.0
0.5
threshold
2
V
threshold
20
0.2
0.47
0.48
0.49
0.50
0.51
0.52
time
0.53
0.2
0.47
0.48
0.49
0.50
0.51
• When the noise slope exceeds the clean-signal slope, the total slope changes sign
• There result spikes, and systematic lead error
21
3 – Interpolation schemes
Clock interpolation – Main idea
Tx
t
gate
Tc = 1/⌫c
t
clock
Ta
counted
edge
N c Tc
not-counted
edge
Tx = Nc Tc + Ta
Ta+ = Ta + Tc
Tb
Tb
Tb+ = Tb + Tc
Too short Ta and Tb are difficult to measure, so we add one Tc to each
Interpolation is made possible by the fact that
the clock frequency is constant and accurately known
22
The frequency Vernier
23
Pierre Vernier, French mathematician
Ornans (Besancon), 1580–1637
t
gate
vernier
clock
Note that Na = Na0
Tc0 = 1/⌫c0
⌫c0
n
=
⌫c
n+1
Na0 Tc0
Tc = 1/⌫c
main
clock
N a Tc
Ta + Na Tc = Na0 Tc0 = Na Tc0
Ta
early
(0→1) ➔ coincidence
late
24
The key elements
Synchronized oscillator
gate pulse
gate pulse
A
A
t
C
B
t
C
delay
idle
run
Coincidence detector
sync clock
coicidence
D Q
vernier
clock
t
main
clock
t
main clock
1
1
1
1
1
1
0
0
0
0
t
coicidence
early
late
Example: Hewlett Packard 5370A
25
26
The Nutt dual-slope interpolator
Na0 = Ta0 /Tc0
Ta0
Vx
C
main
clock
⌫c
R
I2
I1
integrator
S Q
logic
Ta
enable
comp
BCD
counter
⌫c0
reset
gate pulse
+
–
I
aux
clock
cnt-Nanofast
gate
pulse
t
Tc = 1/⌫c
clock
t
Q
t
I1 + I2
I
I2
Actual timing
gate
t
clock
Ta0 =
((I1 + I2 )/I1 )Ta
Ta+ =
Ta + Tc
t
t
enable
Tc0 = 1/⌫c0
Ta0 = Na0 Tc0 (+0/ Tc0 )
t
Ta
discharge
ch
Vx
ar
ge
Ta
Ta0+
Vx
t
charge
smooth start
discharge
Example: Nanofast 536 B
Smithsonian Astrophysical Laboratory
27
28
The ramp interpolator
⌫c
S Q
main
clock
R
VC
reset
gate pulse
I
S Q
Vx
ADC
I0
sample
& hold
C
R
V̂x
aux
clock
integrator
gate
pulse
t
Tc
clock
I0
I
Actual timing
t
gate
t
clock
ge
Ta
ar
ch
VC
t
Vx
hold
ar
ch
t
charge
ge
Ta
Vx
Ta+ =
Ta + Tc
t
smooth start
hold
Example: Stanford SR 620
29
30
Thermometer-code interpolator
Q0
Q2
Q1
Q3
Q4
Q5
Q6
Q7
gate pulse
DQ
main
clock
DQ
DQ
DQ
DQ
DQ
DQ
DQ
etc.
⌫c
✓
✓
C0
C1
✓
C2
✓
C3
✓
C4
✓
✓
C5
C7
C6
gate
pulse
t
C0
clock phases
C1
C2
C3
C4
C5
C6
C7
t
1
0
0
t
1
0
t
1
0
t
0
0
t
0
1
t
0
1
t
0
1
word 0000 0111
indicates delay 5✓
t
0
word 1110 0000
indicates delay 3✓
Also called Multi-tapped delay-line interpolator
Review article: J. Kalisz, Metrologia 41 (2004) 17–32
31
Vernier thermometer-code interpolator
gate pulse
main
clock
✓1
✓1
✓1
✓1
✓1
DQ
DQ
DQ
DQ
DQ
DQ
etc.
⌫c
✓2
✓2
✓2
✓2
✓2
θeq = θ2 – θ1
Owing to physical size, both θ1 and θ2 are always present
Figure from J. Kalisz, Metrologia 41 (2004) 17–32
Ring Oscillator
32
Also used in PLL circuits for clock-frequency multiplication
SAW delay-line interpolator
A – Block diagram
B – Pulse waveforms
• Dispersion stretches the input pulse
• Sub-sampling and identification of the alias
P. Panek, I. Prochazka, Rev. Sci. Instrum. 78(9):094701, 2007
33
Sigma Time STX301 counter
34
• Gossips report that this is
none of the above methods
• No information at all, I’m
unable to reverse-engineer
35
4 – Basic statistics
– after all, not that basic! –
36
Old Hewlett Packard application notes
Quantization uncertainty
p(x)
2
1/Tc
=
Tc
1/ 12 = 0.29
Example: 100 MHz clock
Tx = 10 ns
σ = 2.9 ns
2
Tc
12
37
38
Classical (Π) reciprocal counter
x0 x1 x2 x3
phase time x
(i.e., time jitter)
xN
time
t
v(t)
weight
t0 t1 t2 t3 t4 t5 t6
wΠ
tN
1/τ
period T00
0
measurement time τ = NT
the measure is a
scalar product
E{ν} =
!
+∞
ν(t)wΠ (t) dt
Π estimator
−∞
"
1/τ 0 < t < τ
wΠ (t) =
0
elsewhere
! +∞
wΠ (t) dt = 1
weight
normalization
−∞
variance
2
2σ
σy2 = 2x
τ
classical variance
From Π to Λ – key concept
39
40
36
Enhanced-resolution (Λ) counter
x0 x1 x2 x3
phase time x
(i.e., time jitter)
xN
v(t)
time
t
t0 t1 t2 t3 t4 t5 t6
weight
1/τ
0
i=0
i=1
i=2
Λ estimator
" +∞
E{ν} =
ν(t)wΛ (t) dt
−∞
wi
wn−1
i = n−1
delay τ0 = DT
measurement time τ = NT = nDT
weight
ν i = N/τi
tN−D tN tN+D
meas. no.
w0
w1
w2
n−1
1!
E{ν} =
νi
n i=0
wΛ
1
nτ
n−1 1 n−1
nτ τ nτ
2
nτ
2
nτ
weight 

t/τ
wΛ (t) = 2 − t/τ


0
0<t<τ
τ < t < 2τ
elsewhere
normalization
" +∞
wΛ (t) dt = 1
1
nτ
−∞
limit τ0 -> 0 of the weight
function
white noise: the autocorrelation function is a narrow
pulse, about the inverse of the bandwidth
1/τ
wΛ(t)
0
τ
2τ
t
the variance is
divided by n
2
2σ
1
x
σy2 =
n τ2
classical
variance
Actual formulae look like this
(Π)
(Λ)
!
1
σy =
2(δt)2trigger + 2(δt)2interpolator
τ
1
σy = √
2(δt)2trigger + 2(δt)2interpolator
τ n
"
ν0 τ ν00 ≤ νI
n=
νI τ ν00 > νI
!
41
Understanding technical data
2
2σ
σy2 = 2x
τ
classical reciprocal
counter
enhancedresolution counter
low frequency:
full speed
classical
variance
2
2σ
1
x
σy2 =
n τ2
classical
variance
τ0 = T
n = ν00 τ
=⇒
2
1
2σ
x
σy2 =
ν00 τ 3
high frequency:
housekeeping takes time
classical
variance
τ0 = DT with D>1
2
1
2σ
x
σy2 =
νI τ 3
=⇒
n = ν00 τ
classical
variance
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
Stanford
SRS-620
Examples
!
RMS
resolution
(in Hz)
"
N
gate time
RMS resolution
frequency
gate time
!
RMS
resolution
Agilent
53132A
=
#
&'
( '
()2
&
)2
$
$
short term × gate
trigger
2+
(25
ps)
+
2×
%
stability
time
jitter
frequency
"
=
tres = 225 ps
tjitter = 3 ps
#
$
frequency
or period
σν = ν00 σy
ν00
τ
'
&
4× (tres )2 + 2 × (trigger error)2
tjitter
√
×
+
(gate time) × no. of samples
gate time
%
(
(gate time) × (frequency) for f < 200 kHz
number of samples =
(gate time) × 2×105
for f ≥ 200 kHz
RMS resolution
frequency
period
gate time
no. of samples
σν = ν00 σy or σT = T00 σy
ν00
T00
τ (
ν00 τ
ν00 < 200 kHz
n=
τ × 2×105 ν00 ≥ 200 kHz
43
Linear-regression counter
T0
v(t)
✓(t)
Tn
Ti
input signal
t
r(t)
pace
t
picket fence
⌧0
⌧ = n⌧0
measurement time
✓n
✓i
✓0
✓n
tn
ˆ=
!
⌫ˆ =
t0
ti
✓0
t0
1 !
ˆ
2⇡
tn
t
Linear regression on a sequence of time stamps
provides accurate estimation of frequency
44
45
Linear regression vs. Λ estimator
y
n = 10, τ = 100 µs
n = 100, τ = 1 ms
n = 1000, τ = 10 ms
std dev of Λ
n=1
Parameters
trigger noise
time stamping
std dev of LR–Λ
0, τ
= 100
µs
n=1
00, τ
=1m
s
n=1
000,
τ=
10 m
s
= 1 ns
⌧0 = 10 µs
x
Frequency, Hz
The linear regression estimator is
asymptotically equivalent to the Λ estimator
46
5 – Advanced statistics
Decimation of Λ estimates
How to combine contiguous Λ measures in a way that makes sense
(B) optimally overlapped,
the average converges
to a Π estimate
(A) optimally overlapped,
recursive decimation
⌧B
⌧B
time series
t
2⌧B
(C) separated measures,
multi-triangle average
2⌧B
time series
t
2⌧B
t
⌧ = ⌧B
t
2⌧B
w(1)
w(1)
time series
w(1)
t
⌧ = ⌧B
t
2⌧B
2/4
1/4
1/4
1/2
w
1/2
t
(2)
t
⌧ = 2⌧B
1/2
w
(2)
1/2
t
t
4⌧B
t
w
t
⌧ = 2⌧B
(2)
4/16
1/16
2/16
3/16
3/16
2/16
w
⌧ = 4⌧B
(4)
1/4
1/16
t
t
1/4
1/4
w(4)
⌧ = 4⌧B
1/4
1/4
t
t
1/4
1/4
1/4
t
t
w(4)
8⌧B
47
48
Allan variance
! "
$
#
2
1
2
σy (τ ) = E
y k+1 − y k
2
definition
%
! " #
$
#
2
(k+2)τ
(k+1)τ
1
1
1
σy2 (τ ) = E
y(t) dt −
y(t) dt
2 τ (k+1)τ
τ kτ
wavelet-like
variance
σy2 (τ ) = E
! "#
y(t) wA (t) dt
−∞
wA =
 1

− √2τ
√1
 2τ

energy
+∞
E{wA } =
0
!
0<t<τ
τ < t < 2τ
wA
1
2τ
0
elsewhere
0
+∞
−∞
$2 %
2
wA
(t) dt
1
=
τ
τ
2τ
−1
2τ
time
t
the Allan variance differs from a wavelet variance
in the normalization on power, instead of on
energy
Phase noise & friends
v(t) = Vp [1 + (t)] cos [2⇤⇥0 t + ⌅(t)]
random walk freq.
Sϕ(f)
49
signal sources only
b−4 f−4
random phase fluctuation
S (f ) = PSD of (t)
b−3 f−3
power spectral density
white freq.
b−2 f−2
it is measured as
S (f )
E { (f )
⇥ (f )
x
Sy (f)
Sy =
f2
=E
white phase
h−1 f−1
h0
h1 f
flicker phase
white freq.
f
0
σy2 (τ)
Allan variance
(two-sample wavelet-like variance)
2
y (⇥ )
h2 f2
flicker freq.
2 S (f )
b0
f2/ ν20
h−2 f−2
random
walk freq.
random fractional-frequency fluctuation
white phase
f
(average)
1
L(f ) = S (f ) dBc
2
⇤(t)
˙
y(t) =
2⇥ 0
b−1 f−1
(f )} (expectation)
(f )⇤m
flicker phase.
1⇤
2
y k+1
yk
⌅2 ⇥
.
approaches a half-octave bandpass filter (for white noise),
hence it converges even with processes steeper than 1/f
flicker phase
white phase
white freq.
h0 /2 τ
freq.
drift
flicker freq.
2ln(2)h −1
random walk freq.
(2π ) 2
h−2 τ
6
τ
Figures are from E. Rubiola, Phase noise and frequency
stability in oscillators, © Cambridge University Press
S (f ) =
1
T
1
T
both signal sources
and two-port devices
flicker freq.
Modified Allan variance
definition
50
! " n−1$ %
(
&'
%
2
(i+2n)τ
(i+n)τ
0
0
1 1# 1
1
2
mod σy (τ ) = E
y(t) dt −
y(t) dt
2 n i=0 τ (i+n)τ0
τ iτ0
with τ = nτ0 .
mod σy2 (τ ) = E
+∞
y(t) wM (t) dt
−∞
wavelet-like
variance
wM
energy
!"#

√1 t

−

2τ 2


 √ 1 (2t − 3)
2τ 2
=
*
1

√
−
(t − 3

2

2τ


0
E{wM } =
!
compare the energy
0<t<τ
τ < t < 2τ
2τ < t < 3τ
elsewhere
+∞
−∞
$2 %
2
wM
(t) dt
1
2τ
wM
0
τ
0
−1
2τ
2τ
1
=
2τ
1
E{wM } = E{wA }
2
this explains why the mod Allan variance is always lower than the Allan variance
time
3τ t
TABLE I
MPARISONS OF
A LLAN VARIANCE ( TRADITIONAL COUNTER ),
51A
TRIANGLE VARIANCE ( HIGH - RESOLUTION COUNTER ) AND MODIFIED
Spectra and variances
CE RESULTING FROM CHARACTERISTIC NOISE .
A
CUTOFF FREQUENCY, fH , IS INTRODUCED
H ERE , PM STANDS FOR PHASE MODULATED AND FM STANDS FOR FREQUENCY MOD
FOR THE A LLAN VARIANCE OF WHITE PHASE NOISE AND FLICKER PHASE NOISE TO AV
INFINITE RESULT.
W E ALSO IGNORE THE SMALL SINUSOIDAL TERM .)
Noise Type
Sy (f )
2)
Allan (⇤A
Modified Allan
Triangle
White PM
h2 f 2
3 fH
4 2
h2 ⌅ -2
2 (⌅ )
= ⇤A
h2 ⌅ -3
8
2 (⌅ )
= 2 f1 ⇥ ⇤A
h2 ⌅ -3
2 (⌅ )
= 3 f8 ⇥ ⇤A
Flicker PM
h1 f
White FM
h0
Flicker FM
h-1 f -1
Random Walk FM
h-2 f -2
Frequency Drift (ẏ = Dy )
00
-
1.038+3 ln(2 fH ⇥ )
4 2
2 (⌅ )
= ⇤A
1
h ⌅ -1
2 0
2 (⌅ )
= ⇤A
3
2
H
h1 ⌅ -2
2 ln(2) h-1
2 (⌅ )
= ⇤A
2
3
⇥ 2 h-2 ⌅
2 (⌅ )
= ⇤A
1 2 2
D ⌅
2 y
=
3 ln( 256
)
-2
27
h
⌅
1
2
8
3.37
2 (⌅ )
⇤
A
3.12+3 ln fH ⇥
1
h ⌅ -1
4 0
2 (⌅ )
= 0.50 ⇤A
3 311/16
2 ln(
) h-1
4
2 (⌅ )
= 0.67 ⇤A
11 2
⇥ h-2 ⌅
20
2 (⌅ )
= 0.82 ⇤A
1 2 2
D ⌅
2 y
2
2
H
6 ln( 27
)
16
h1 ⌅ -2
12.56
2 (⌅ )
= 3.12+3
⇤
A
ln fH ⇥
2
h0 ⌅ -1
2
3
2 (⌅ )
= 1.33 ⇤A
(24 ln(2) 27
ln(3)) h-1
2
2 (⌅ )
= 1.30 ⇤A
23
30
⇥ 2 h -2 ⌅
2 (⌅ )
= 1.15 ⇤A
1 2 2
D ⌅
2 y
is replaced TABLE
with II0 for consistency with the general literature
Amplitude
ORDER ERROR ,
,
IN THE
A LLAN ,
TRIANGLE AND MODIFIED
fH is the high cutoff frequency, needed for the noise power to2 be finite
VARIANCES CAUSED BY THE INCLUSION OF DEAD - TIME .
T HE
2 , IS THE ORIGINAL VARIANCE , ⇤ 2 ,
NCE WITH DEAD - TIME , ⇤⇥
d
2
ENTED BY : ⇤⇥ ⇥ (1 + ) ⇤ 2 . (*F OR THE MOST DIVERGENT
d
A LLAN VARIANCE HAS A COMPLICATED DEPENDENCE ON THE
F FREQUENCY, fH ; HOWEVER , FOR SIMPLICITY WE GIVE THE
S.T. Dawkins, J.J. McFerran, A.N. Luiten, IEEE
√
2τ
1
√
2τ
wA
HE
t
Trans. UFFC
54(5)
p.918–925,
May
2007
1
−√
Π estimator —> Allan variance
given a series of contiguous non-overlapped measures
measure series
ν0
ν1
ν2
ν3
1/τ
wΠ (t)
1/τ
wΠ (t− τ )
......
time
t
t
+1/( 2 τ)
wA(t)
−1/( 2 τ)
0
τ
t
2τ
the Allan variance is easily evaluated
! "
$
#
2
1
2
σy (τ ) = E
y k+1 − y k
2
......
t
52
Overlapped Λ estimator —> MVAR
by feeding a series of Λ-estimates of frequency in the formula of the Allan variance
! "
$
#
2
1
2
σy (τ ) = E
y k+1 − y k
2
as they were Π-estimates
ν0
ν1
ν2
1/τ
wΛ(t)
1/τ
wΛ (t− τ )
.....
ν3
t
time
t
+1/( 2 τ)
wM (t)
.....
t
−1/( 2 τ) t
0
τ
2τ
3τ
one gets
exactly
the
modified
Allan
variance!
!
mod σy2 (τ ) = E
with τ = nτ0 .
"
1 1
2 n
n−1
#$
i=0
1
τ
%
(i+2n)τ0
(i+n)τ0
1
y(t) dt −
τ
%
(i+n)τ0
iτ0
y(t) dt
&'2 (
53
Joining contiguous values to increase τ
graphical proof
w(1)
mod Allan
τ=τB
t
m=2
t
τ=2τB
w(2)
t
m=4
t
τ=4τB
w(3)
t
m=8
w(4)
t
τ=8τB
converges to Allan
t
m = 1!
m = 2!
m = 4!
m ≥ 8!
!
!
mod Allan
this is not what we expected
...
the variance converges to the
(non modified) Allan variance
There is a mistake in one of my articles: I believed that in the case of the
Agilent counters the contiguous measures were overlapped. They are not.
54
Non-overlapped Λ estimator —> TrVAR
55
by feeding a series of Λ-estimates of frequency in the formula of the Allan variance
! "
$
#
2
1
2
σy (τ ) = E
y k+1 − y k
2
as they were Π-estimates
1/τ
wΛ(t)
time t
1/τ
wΛ(t–2τ)
time t
+1/(√2 τ)
wTr(t)
time t
–1/(√2 τ)
τ
2τ
3τ
4τ
one gets the triangular variance!
S.T. Dawkins, J.J. McFerran, A.N. Luiten, IEEE Trans. UFFC 54(5) p.918–925, May 2007
56
E. Rubiola
Experimental methods in AM-PM noise metrology
— book project —
© Roberto Bergonzo
Front cover: The Wind Machines
Artist view of the AM and PM noise
Courtesy of Roberto Bergonzo, http://robertobergonzo.com
Conclusions
•
•
•
•
•
Review of general techniques
The trigger may not what it seems – however, in unusual conditions
Sophisticated interpolation techniques
The thermometer-code interpolator is simple with modern FPGAs
The Λ (triangular) estimator provides higher resolution than the Π
(rectangular) estimator, but can be used with periodic phenomena only
• Mistakes are around the corner if the counter inside is not understood
• Some of the reported ideas are suitable to education laboratories and
classroom works (I used a bicycle odometer and milestones to
demonstrate the Λ estimator)
Thanks to J. Dick (JPL, retired), V. Giordano (Femto-ST), C. Greenhall (JPL), D. Howe
(NIST) M. Oxborrow (NPL), F. Vernotte (Observatory of Besancon) for discussions & more
To know more:
1 - http://rubiola.org, slides and articles
2 - http://arxiv.org, document arXiv:physics/0503022v1
3 - Rev. of Sci. Instrum. vol. 76 no. 5, art.no. 054703, May 2005.
home page http://rubiola.org
57

home page http://rubiola.org E. Rubiola Outline

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