analysis and graphical representation of s-parameter data for...
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© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 1
Analysis and graphicalrepresentation of S-parameter data
for Euromet 555 measurementcomparison exercise
Topics :– E555: what is it?– How data was provided.
• Problems.
– Analysis• Outlier elimination• Comparison Reference Value and Confidence Region• Monte Carlo
– Draft A representation
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 2
E555: What is it?
• Measurement comparison exercise– European Metrology Collaboration
Project 555– Supplementary Comparison
Euromet.EM.RF-S16
• Measurement of Scattering Parameters
• 12 Participants: BNM-LNE/LAMA,Nmi VSL, INMS NRC, NIST, METAS,CSIR-NML, EPSG/MTA, IEN, SQF,NPL, INTA, UKAS No“0147“
• Status: Measurements done. Draft A in preparation.
• Devices: 2.4 mm system– Attenuator: 3, 20, 40 dB– Load: male, female– Missmatch: male, female
• Frequencies: 50 MHz - 50 GHz
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 3
Measurements performed
• Full Data Set: 63 frequencies– 50, 100, 500 MHz– 1, 2, 3, .. , 39, 40 GHz– 40.5, 41.0, 41.5, .. , 49.5, 50.0 GHz
Euromet 555Visualisation Tool
Further Analysis:Outlier eliminationComparison Reference Value (CRV)95% Confidence regionresults shown in draft A
• Annex Data (7 frequencies) withuncertainties:
– 0.05, 1, 10, 20, 30, 40, 50 GHz
• Required Format:– magnitude and phase (r, φ)
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 4
Problems with (r,φ) data
Small r:
• r-u(r) < 0
0.03
0.02
0.01
0.00
-0.01
-0.02
S11
mag
nitu
de
LA
B 01
LA
B 02
LA
B 03
LA
B 04
LA
B 05
LA
B 06
LA
B 07
LA
B 08
LA
B 09
LA
B 10
LA
B 11
LA
B 12
Load 407, f = 0.05 G Hz
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 5
Problems with (r,φ) data
Small r:
• r-u(r)<0
• u(φ) overestimated
• Large variations inu(r) and u(φ)
0.030
0.020
0.010
0.000
S11
mag
LAB
01
LAB
02
LAB
03
LAB
04
LAB
05
LAB
06
LAB
07
LAB
08
LAB
09
LAB
10
LAB
11
LAB
12
300
200
100
0ph
ase
(deg
)
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
3 dB, f = 0.050 GHz
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 6
Overview of analysis of annex data
• Ignore uncertainties in r and φ• Transform data from (r,φ) to complex representation (x,y)• Eliminate outliers using robust statistics• Calculate mean for real (x) and imagenary (y) component to determine
comparison reference value (CRV)• Calculate standard uncertainties and correlation for CRV• Determine 95% confidence region in x-y for CRV• Determine 95% confidence interval for r and φ using monte carlo
method
References:N.M. Ridler and M.J. Salter, Metrologia 39, 2002, 295-302M.J. Salter, Anamet Report 033, April 2001
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 7
Transformation
• Data Points: 12..1),( =Φ ir ii
• Transformation:– Real component:– Imagenary Component:
),(),( iiii yxr �Φ
iii
iii
ry
rx
Φ⋅=
Φ⋅=
sin
cos
0.030
0.020
0.010
0.000
S11
mag
LAB
01
LAB
02
LAB
03
LAB
04
LAB
05
LAB
06
LAB
07
LAB
08
LAB
09
LAB
10
LAB
11
LAB
12
300
200
100
0
phas
e (d
eg)
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
3 dB, f = 0.050 GHz
a
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 8
Analysis II: Outlier Elimination
),( mm yx• Determine spatial median:
• Median of absolute deviation (MAD):
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
( ) ( ){ }NiyxyxmedMAD mmii ,,1;,, �=−=
• Outlier criterion:
( ) ( ) MADyxyx mmii ⋅>− 75.3,,
• Use robust estimator
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 9
Analysis III: Comparison Reference Value (CRV)
• CRV:
• Standard uncertainties in CRV:
∑=
=N
iix
Nx
1
1 ∑=
=N
iiy
Ny
1
1
( )∑=
−−⋅
=N
ii xx
NNxu
1
2
)1(1
)(
( )∑=
−−⋅
=N
ii yy
NNyu
1
2
)1(1
)(
-0.004
-0.002
0.000
0.002
0.004
Im[S11]
-0.004 -0.002 0.000 0.002 0.004Re[S11]
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
1-dim 68% confidence intervalsfor normal distribution
• Correlation coefficient:
( )( )( ) ( )yuxu
yyxxNNyxr
i
N
ii
⋅
−−−⋅=
∑=1)1(
1
),(
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 10
Two-dimensional normal distribution: Formalism
• 1-dim Gaussian: ( )Mean:Variance,:�
2exp)( 2
2
µσ
µ
−−∝
xxf
• 2-dim Gaussian: ( )( ) ( )( ) ( )
nCorrelatio:�
2
121
exp),( 2
2
2
2
2
−−
−−−
−−
−∝y
y
yx
yx
x
xyyxx
yxfσ
µσσ
µµρσ
µρ
– Contours of equal density:
( ) ( )( ) ( )const
yyxxconstyxf
y
y
yx
yx
x
x =−
−−−
−−
⇔= 2
2
2
2 2),(
σµ
σσµµρ
σµ
Ellipse equation:
circle,0y x, toparallelaxes0
⇔==⇔=
yx σσρρ
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 11
Two-dimensional normal distribution: Example
• S11 3dB 0.05GHz: ( )( )
( ) 4573.0,
0001.0,0002.0
0002.0,0024.0
−==
==−==
==−==
yxr
yuy
xux
yy
xx
ρ
σµ
σµ
1.5x10-3
1.0
0.5
0.0
-0.5
Im[S
11]
-3.0x10-3
-2.5 -2.0 -1.5 -1.0Re[S11]
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 12
Analysis IV: 95% Confidence Region
• Ellipse equation:(3dB f=0.05GHz)
),21,2(21
2 222
2
clnnFn
nk −+⋅
−+⋅
= υυ
υ
• Coverage factor k:– k determines confidence level– confidence level (cl) depends on:
• dimension n• degrees of freedom ν
S11 3dB f=0.05GHz:n=2, ν=N-1=10, cl=0.95
k=3.08
( ) ( )( ) ( ) 282210440.0966.1949.4 kxxxxyyyy ⋅=−⋅+−−⋅+−⋅ −
F distribution
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
0.0015
0.0010
0.0005
0.0000
-0.0005
Im[S11]
-0.0030 -0.0025 -0.0020 -0.0015 -0.0010Re[S11]
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 13
Analysis V: 95% Confidence Region for r and φ
Monte Carlo Method:Create 2-dim gaussian random distribution
22 yxr +=
1.5x10-3
1.0
0.5
0.0
-0.5
Im[S
11]
-3.0x10-3
-2.5 -2.0 -1.5 -1.0Re[S11]
4x10-3
3
2
1
0
coun
ts
3.2x10-3
3.02.82.62.42.22.01.81.6magnitude
8x10-3
6
4
2
0
coun
ts
185180175170phase
=Φ −
xy1tan
3dB f=0.05GHz:
46.00001.0)(0002.0)(
0002.00024.0
−===
=−=
ryuxu
yx
3.22.2
0004.00004.0 )()(
deg7.1750024.0+−
+− =Φ=
=Φ=
uru
r
95% of areabounded
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 14
Analysis V: 95% Confidence Region for r and φ
• Another example
22 yxr +=
=Φ −
xy1tan
S11 Load 446, f=1GHz:
17.000021.0)(00038.0)(
00054.000038.0
===
==
ryuxu
yx
5.762.39
0007.00004.0 )()(
deg4.550007.0+−
+− =Φ=
=Φ=
uru
r
-2x10-3
-1
0
1
2
Im[S
11]
-2x10-3
-1 0 1 2Re[S11]
7x10-3
6
5
4
3
2
1
0
coun
ts
2.0x10-3
1.51.00.5magnitude
6x10-3
5
4
3
2
1
0
coun
ts
150100500-50phase
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 15
Representation in draft A: Graphics
0.030
0.020
0.010
0.000
S11
mag
LAB
01
LAB
02
LAB
03
LAB
04
LAB
05
LAB
06
LAB
07
LAB
08
LAB
09
LAB
10
LAB
11
LAB
12
-200
0
200
phas
e (d
eg)
0.006
0.004
0.002
0.000
-0.002
-0.004
Im[S11]
0.0060.0040.0020.000-0.002-0.004Re[S11]
Device: Load 446, f: 1.000 GHz
© metas, M. Zeier ANAMET 20th Meeting September 2003 slide 16
Representation in draft A: TableLo ad 4 4 61 . 0 0 0 GHz
S 1 1 mag u(mag) phase u(phase) Re Im Flag(deg) (deg)
LAB 0 1 0 .0019 0 .0114 147 .84 180 .00 -0 .0016 0 .0010 1
LAB 0 2 0 .0008 0 .0177 -168.17 180 .00 -0 .0008 -0 .0002 1
LAB 0 3 0 .0024 0 .0075 18.00 180 .00 0 .0023 0 .0007 1
LAB 0 4 0 .0004 0 .0114 -39 .28 180 .00 0 .0003 -0 .0003 1
LAB 0 5 0 .0008 0 .0081 159 .10 11.69 -0 .0007 0 .0003 1
LAB 0 6 0 .0022 0 .0013 25.96 43.82 0 .0020 0 .0010 1
LAB 0 7 0 .0020 0 .0060 56.00 180 .00 0 .0011 0 .0017 1
LAB 0 8 0 .0008 0 .0080 45.77 180 .00 0 .0005 0 .0005 1
LAB 0 9 0 .0023 0 .0111 12.13 174 .86 0 .0022 0 .0005 1
LAB 1 0 0 .0012 0 .0165 143 .13 180 .00 -0 .0009 0 .0007 1
LAB 1 1 0 .0014 0 .0057 92.16 180 .00 -0 .0001 0 .0014 1LAB 1 2 0 .0009 0 .0300 -79 .78 180 .00 0 .0002 -0 .0009 1
S 1 1 mag u(mag) phase u(phase) Re u(Re) Im u(Im)
CRV 0 .0007 +0.0007 55.35 +76.53 0 .0004 0 .0004 0 .0005 0 .0002
-0 .0004 -39 .24
S 1 1 Re Im OutB
M e d ian 0 .0004 0 .0005 0 .0051
Co rre lat io n Re[S11] Im[S11]M at rix Re[S11] 1 .0000 0 .1660
Im[S11] 1 .0000