j. mauricio lópez r. cenam time and frequency division statistical analysis of measurements (time...
TRANSCRIPT
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J. Mauricio López R.
CENAM Time and Frequency Division
Statistical Analysis of Measurements
(time domain)
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OUTLINE
1. Introduction
3. Allan Variance (AVAR)
4. AVAR uncertainty
5. Trazability
2. Time and Frequency Measurements
5. Time Deviation
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Introduction
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Real Clocks
There not exist the perfect clock, all the real clocks are unestable. The output frequency of a clock changes with time. A correct mathematical tool is needed to characterise the frequency instability of oscillators.
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)2sin()( tAtV
A: amplitude: frequencyt: time
+A
-A
Mathematical model for ideal frequency signal
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Quartz oscillators are a very common equipment to produce frequency signals but their frequency output could be more or less instable as function of many parameters
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V(t) = V0 sen(20t)
V(t) =[V0 + (t)] sen[20t + (t)]
tiempo
tiempo
Mathematical model for real (instable) frequency signals
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Precisión sin exactitud
Ni exactitud ni precisión
Exactitud sin precisión
Exacto y preciso
Tiempo TiempoTiempoTiempo
Estable de baja exactitud
Inestable de baja exactitud
Alta exactitud a largo tiempo e
inestable a corto tiempo
Alta estabilidad y alta exactitud
0
f fff
Exactitud y Estabilidad
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Standard deviation to specify frrequency satbility
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Time and Frequency Measurements
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Método de medición directa de frecuencia
MEDICIONES DE FRECUENCIA FIGURA 1 (04-2003F.DAT)
4999999.509
4999999.510
4999999.511
4999999.512
4999999.513
4999999.514
4999999.515
4999999.516
0 3600 7200 10800 14400 18000 21600 25200 28800 32400 36000
DATOS (10horasdemedición con =1s)
FR
EC
UE
NC
IA (
Hz)
Frecuencia Bajo
Calibración
Frecuencia Patrón para amarrar en
frecuencia al contador
Direct Frequency Measurement
Automatic data adquisition system
PC
Frequency Standard Device Under Test
Frequencymeter
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Método de medición de diferencia de fase
Pa tró n d e re fe re nc ia Pa tró n d e re fe re nc ia
Instrum e nto b a jo
c a lib ra c ió n
Instrum e nto b a jo
c a lib ra c ió n
Ad q u isic ió n d e d a to s a uto m a tiza d o
PC
Frecuencia Bajo
Calibración
Interfacede
Comunicación
Frecuencia Patrón para amarrar en
frecuencia al contador
Frecuencia Patrón para la calibración
Mediciones de Diferencia de Fase
Contador de Intervalos de
Tiempo
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4 - 1 6
F r e c u e n c i a e s t a b l e ( o s c i l a d o r i d e a l )
U n s t a b l e F r e q u e n c y ( R e a l O s c i l l a t o r )
T i m e
( t )
T i m e
( t )
V1
- 1T 1 T 2 T 3
1- 1
T 1 T 2 T 3
V ( t ) = V 0 s i n ( 2 0 t )
V ( t ) = [ V 0 + ( t ) ] s i n [ 2 0 t + ( t ) ]
( t ) = 2 0 t
( t ) = 2 0 t + ( t )
V ( t ) = s a l i d a d e l o s c i l a d o r , V 0 = A m p l i t u d n o m i n a l p i c o - a - p i c o ( t ) = a m p l i t u d d e r u i d o , 0 = f r e c u e n c i a n o m i n a l ( t ) = f a s e , a n d ( t ) = r u i d o d e f a s e
t d)t(d
21=
t d)t(d
21 = )t( 0
π
Φπ
frequency, ousInstantane
V
I n e s t a b i l i d a d e n f r e c u e n c i a ( r u i d o )
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Allan Variance
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La Varianza de Allan es la herramienta usada para el análisis en el dominio del tiempo de mediciones de Tiempo y Frecuencia siendo un estimador de la dispersión de las mediciones, determinando así, la estabilidad del oscilador bajo calibración.
The Allan Variance is a statistical tool used for the time domain analysis of time and frequency measurements to estimate the noise behind measurements and to determine the stability of an oscillator under calibration.
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22y 2
1σ iy 22
2
2y 2
1σ ix
Frecuencia Fase
iii yyy 1 iiii xxxx 122 2
Concepto de la Varianza de Allan
ii
i
xxy
1
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Varianza de Allan para Mediciones de Frecuencia
1N
1i
2i1i
0
2y yy
1N21
σ
donde:
Varianza de Allan
Número de datos espaciados 0
i-ésima medición de fase
2y
N
iy
Tiempo de observación = m0
m =2n cálculos posibles
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mN
iimimiy xxx
mN
2
1
222
2 222
1
donde:
Varianza de Allan
Número de datos espaciados 0
Tiempo de observación = m0
i-ésima medición de fase
2y
Nix
m =2n cálculos posibles
Varianza de Allan para Mediciones de Diferencia de Fase
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Barras de Incertidumbre
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Distribución 2
Para df < 100
2
2
2 )(y
ysdf
2ys
donde:Estimado de la Varianza de Allan
df Número de grados de libertad
Varianza de Allan verdadera2
y
Distribución X
2
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Barras de incertidumbre
2
2
2 )(y
ysdf
025,0975.0 2
22
2
2
dfsdfs y
yy
Tablas 2
Barra Inferior Barra Superior
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Tabla 2
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Para df > 100
Barras de incertidumbre
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Barras de incertidumbre
22 96,121
025,0 h
Para df > 100
22 96,121
975,0 h
12 dfh
donde:
Barra Inferior
Barra Superior
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Número de Grados de Libertad
NBS Technical note 679
White Phase
Flicker Phase
White Frquency
mN
mNNdf
221
4112
ln2
1lnexp
Nmn
Ndf
54
4222
132
2
mm
NN
mN
df
Flicker Frequency
Random Walk Frequency
234
5
19,43,2
22
2
mparamNm
Ndf
mparaNN
df
2
22
3
41312
N
mNmNm
Ndf
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4-25
Por debajo del ruido “fliker”, los cristales de cuarzo tipicamente tienen una dependencia -1 (white phase noise). Los patrones atómicos de frecuencia muestran una dependencia del tipo -1/2 (white frequency noise) para tiempos de promediación cercanos al tiempo de ataque del lazo de amarre, y -1 para tiempos menores del tiempo de ataque. Tipicamente los ’s para el ruido flicker son: 1 s para osciladores de cuarzo, 103s para relojes de rubidio y 105s para Cesio.
y()-1
-1
0
Tipo de ruido:
Whitephase
Flickerphase
Whitefreq.
Flickerfreq.
Randomwalk freq.
-12 12
Dependencia temporal de Dependencia temporal de yy(())
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Ejemplos de cálculo de varianza de Allan
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NIST Special Publication 1065
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NIST Special Publication 1065
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Incertidumbre de medición
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La pendiente del gráfico ofrece información sobre la diferencia fraccional de frecuencia “promedio” entre los
osciladores
mff
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La desviación de los datos experimentales respecto a la línea recta que mejor los representa ofrece información
sobre la incertidumbre en la desviación fraccional de frecuencia
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La desviación de los datos experimentales respecto a la línea recta que mejor los representa ofrece información
sobre la incertidumbre en la desviación fraccional de frecuencia
0
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La desviación de los datos experimentales respecto a la línea recta que mejor los representa ofrece información
sobre la incertidumbre en la desviación fraccional de frecuencia
1
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La desviación de los datos experimentales respecto a la línea recta que mejor los representa ofrece información
sobre la incertidumbre en la desviación fraccional de frecuencia
3
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La desviación de los datos experimentales respecto a la línea recta que mejor los representa ofrece información
sobre la incertidumbre en la desviación fraccional de frecuencia
)( y
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)(
ymff
0
0
fff
ff
Frecuencia del
oscilador
Frecuencia de
referencia
))(1(0 ymff
))(21(0 ymff
Factor de cobertura k=2
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......)( 2210
122
fhfhh
fh
fh
fhfS y
=-2 Random walk frequency=-1 Flicker frequency=0 White frequency=+1 Flicker phase=+2 White phase
y2 ()
= -1 Random walk frequency= -1 Flicker frequency= -1/2 White frequency= 0 Flicker phase = +1/2 White phase
Dominio de la frecuencia
Dominio del tiempo
0 1 2 3 4 5 6 7 8 -17
-15
-13
-11
-9
log
logy
White PM o flicker PM
White FM
Flicker FM Random walk FM
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103
100 101 102 103
10-1
100
101
102
=2 =1
=0
=-1
=-2
N (número de mediciones)
),2(
),(2
2
N
Random walk frequency
Flicker frequency
White frequency
Flicker phase
White phase
Varianza de Allan
Varianza estándar
=--1
Varianza de Allan vs Varianza Estándar
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Traceability
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NMI 2
Labs. Sec.
Labs. Ind.
NMI 1
Labs. Sec.
Labs. Ind.
BIPM
u1
u2
u´1
u´2
u´0 u0
Concepto de Trazabilidad
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NMI 2
Labs. Sec.
Labs. Ind.
NMI 1
Labs. Sec.
Labs. Ind.
BIPM
u1
u2
u´1
u´2
u´0 u0
Concepto de Trazabilidad
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NMI 2
Labs. Sec.
Labs. Ind.
NMI 1
Labs. Sec.
Labs. Ind.
BIPM
u1
u2
u´1
u´2
u´0 u0
CMC´sCMC´s
Concepto de Trazabilidad
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NMI 2
Labs. Sec.
Labs. Ind.
NMI 1
Labs. Sec.
Labs. Ind.
BIPM
u1
u2
u´1
u´2
u´0 u0
CMC´sCMC´s
CMC´s CMC´s
Concepto de Trazabilidad
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NMI
Labs. Sec.
DUT
BIPM
u1
u2
u0
Concepto de Trazabilidad
210 uuu
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NMI
Labs. Sec.
DUT
BIPM
Concepto de Trazabilidad
,0
ff y()0
,1
ff y()1
,2
ff y()2
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NMI
Labs. Sec.
DUT
BIPM
Concepto de Trazabilidad
,0
ff y()0
,1
ff y()1
,2
ff y()2
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NMI
Labs. Sec.
DUT
BIPM
Concepto de Trazabilidad
,0
ff y()0
,1
ff y()1
,2
ff y()2
,ff y()
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NMI
Labs. Sec.
DUT
BIPM
Concepto de Trazabilidad
,0
ff y()0
,1
ff y()1
,2
ff y()2
n
i iff
ff
1
n
ii
yy1
2 )()(
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Concepto de Trazabilidad
Frecuencia, f
BIPM
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Frecuencia, ff0
BIPM
NMI
Concepto de Trazabilidad
![Page 60: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/60.jpg)
Concepto de Trazabilidad
Frecuencia, ff0
BIPM
NMI
f0
![Page 61: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/61.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1
NMILab Acred.
BIPM
![Page 62: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/62.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1
BIPM
NMI
f1
Lab Acred.
![Page 63: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/63.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1 f2
BIPM
NMI DUTLab Acred.
![Page 64: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/64.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1 f2
BIPM
NMI DUT
f2
Lab Acred.
![Page 65: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/65.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1 f2
BIPM
NMI DUT
f
Lab Acred.
![Page 66: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/66.jpg)
Concepto de Trazabilidad
Frecuencia, ff0f1 f2
BIPM
NMI DUT
n
ii
ff1
Lab Acred.
![Page 67: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/67.jpg)
Concepto de Trazabilidad
y()0
y()1
y()2
n
ii
yy1
2 )()(
![Page 68: J. Mauricio López R. CENAM Time and Frequency Division Statistical Analysis of Measurements (time domain)](https://reader035.vdocuments.mx/reader035/viewer/2022062807/5665b4681a28abb57c914642/html5/thumbnails/68.jpg)
J. Mauricio López R.
CENAM Time and Frequency Division
Statistical Analysis of Measurements
(time domain)
THANK YOU!!