1
Linea P opaga ion o Unce ain y
in P obe Posi ion Compensa ed Mul iline-TRL
Robin Schmid
d
d d
d
, Ma co Ga elli, Membe , IEEE,
And ea Fe e o, Fellow, IEEE, Michael Dieudonn´
e, Dominique Sch eu s, Fellow, IEEE
Abs ac —The quan ifica ion o unce ain y sou ces in on-
wa e S-pa ame e s measu emen s is c ucial o cla i y he quali y
o he ex ac ed models used in P ocess Design Ki s. In ac , a
clea unce ain y budge expands ou unde s anding o whe e
fi s ac ions can be aken o imp o e such measu emen s. In
his pape , we p opose a simplified app oach o unce ain y
quan ifica ion wi hou he equi emen o any wa eguide o
coaxial connec ion, e.g., wi hou emo ing he p obes. Then, we
de elop he unce ain y p opaga ion algo i hm applicable o hese
unce ain y sou ces.
Index Te ms—B oadband communica ions, coplana wa eg-
uides, ins umen a ion and measu emen echniques, on-wa e
calib a ion, plana ansmission lines,
I. INTRODUCTION
ON-WAFER S-pa ame e s measu emen s allow accu a e
cha ac e iza ion o de ices and elimina e he inaccu a e
and o en cumbe some p ocess o de-embedding es fix u es.
Howe e , some issues may a ise when using es p obes,
mainly due o con ac epea abili y, ins umen a ion d i , and
signal leakage in o pa asi ic modes [1]. Many o hese unde-
si able e ec s can be minimized as demons a ed in [2], [3],
[4], imp o ing he accu acy o hese measu emen s.
A good example can be ound in [5], whe e A z was able o
ob ain aceable on-wa e measu emen s wi h a comp ehensi e
unce ain y budge . Mo e pa icula ly, he unce ain ies o he
es fix u e we e e alua ed wi h Eu ame ’s guidelines [6],
which equi es p obe disconnec ion and calib a ion a he
connec o .
In a simila con ex , we we e able in [7] o significan ly
imp o e measu emen s su e ing om p obe misplacemen s
on con ac pads by quan i ying he modifica ion o e o boxes
wi h he p obe posi ion. Howe e , he unce ain y quan ifica-
ion pe o med in [7] was limi ed o calib a ion esiduals using
exp essions om [8], noise, and p obe posi ion unce ain y.
In he cu en pape , ou aim is o comple e he unce ain y
s udy om [7] and include es fix u e d i and subs a e p o-
Manusc ip ecei ed Ma ch 29, 2025; e ised May 23, 2025; accep ed June
8, 2025. The wo k was in pa suppo ed by he 23IND10 OnMic o p ojec .
The p ojec (23IND10 OnMic o) has ecei ed unding om he Eu opean
Pa ne ship on Me ology, co-financed om he Eu opean Union’s Ho izon
Eu ope Resea ch and Inno a ion P og amme and by he Pa icipa ing S a es.
(Co esponding au ho s: Robin Schmid )
Robin Schmid , And ea Fe e o, Ma co Ga elli, and Michael Dieudonn´
e
a e wi h Keysigh Technologies Inc. (e-mail: obin.schmid @keysigh .com)
Robin Schmid and Dominique Sch eu s a e wi h KU Leu en.
Colo e sions o one o mo e o he figu es in his pape a e a ailable
online a h p://ieeexplo e.ieee.o g.
Digi al Objec Iden ifie : 10.1109/TMTT.2025.3580938
cessing a ia ions, be o e p opaga ing hem in ou e sions o
he mul iline-TRL algo i hm de i ed om [9], [10] and in he
ex ac ed displacemen unc ions o he p obes. Addi ionally,
we p opose an app oach o noise quan ifica ion less sensi i e
o d i and a simplified me hod o d i quan ifica ion, bo h
possible when p obes a e s ill connec ed o he Vec o Ne wo k
Analyze . Recei e linea i y e o s we e howe e no consid-
e ed in he ame o his esea ch.
In he fi s pa o his pape , we cla i y how noise, d i , and
wa e inhomogenei y a e es ima ed. The second pa consis s
o Jacobian-based fi s -o de p opaga ion o unce ain y in
bo h he calib a ion algo i hm and he displacemen unc ion
used o p obe misplacemen compensa ion. Finally, we com-
pa e unce ain ies p opaga ed in he di e en implemen a ions
o he mul iline-TRL, including he mos ecen e sions om
[11], [12].
II. UNCERTAINTY QUANTIFICATION
A. Unce ain y Due o Noise
S-pa ame e measu emen s a e based on he a io o com-
plex numbe s cohe en ly measu ed by I/Q ecei e chains
ha ecei e a signal om di ec ional couple s. These ecei e
chains a e subjec o bo h he mal noise and oscilla o phase
noise, which a e esponsible o ace noise and noise floo in
S-pa ame e s. The modeling o hese fi s unce ain y sou ces
can be done based on [13], whe e mul iplica i e and addi i e
andom a iables a e in oduced in each ecei e o he VNA,
ollowing he model (1), wi h ˆ
xi he ac ual alue o he
eading, δnHi and δnLi he complex andom a iables o
high- and low-le el noise.
xi=ˆ
xi(1 + δnHi) + δnLi (1)
In [13], a simple expe imen is p oposed o es ima e he
con ibu ion o high- and low-le el noise o he measu emen s,
based on se e al ecei e measu emen s made a wo di e en
signal le els. Following his me hod, o a sys em o a leas
wo po s, no addi ional load connec ion is equi ed, as a low
signal le el in he ecei e s can be achie ed when he opposi e
po is ac i e.
This allows us o w i e he exp ession (2), whe e he
measu emen s o a single s anda d wi h a high eflec ion
coe ficien and a ansmission coe ficien close o he noise
floo a e su ficien o eco e he a iance o he di e en
ecei e noises. In on-wa e measu emen s, i is in gene al
possible o each hese equi emen s by lowe ing he chuck
and sepa a ing open p obes by se e al mm.
© 2025. Pe sonal use o his ma e ial is pe mi ed. Howe e , pe mission o ep in / epublish his ma e ial o ad e ising o p omo ional
pu poses o o c ea ing new collec i e wo ks o esale o edis ibu ion o se e s o lis s, o o use any copy igh ed componen o his
wo k in o he wo ks mus be ob ained om he IEEE. Link o publishe e sion wi h DOI: 10.1109/TMTT.2025.3580938
2
xi,i
xi,j=xi,i 1
xi,j 1∗δnHxi
δnLxi wi h j6=i(2)
Since hese measu emen s may ake some ime, he ecei e
measu emen s xi,i o (2) can be subjec o ins umen a ion
d i , as i may be he case o olde e sions o equency
ex ende s, o o ex emely low high-le el noise (e.g., like
o he new PNA DDS sou ces [14]). The e o e, he o al
dis ibu ion o xi,i will also be a ec ed by d i , e ec i ely
inc easing he a iance o he dis ibu ion. In such case, local
di e ences p o e o be much mo e obus agains d i when
quan i ying noise. This in oduces equa ions (3) ha use he
dis ibu ion o local di e ences ins ead.
di (xi,i)
di (xi,j )=xi,i 1
xi,j 1∗√2δnHxi
√2δnLxi wi h j6=i(3)
In Fig. 1, he alidi y o he new equa ions was es ed by
ex ac ing noise le els wi h epea ed ecei e measu emen s
wi h a sho connec ed o bo h po s, while he RF powe
le el was slowly d i ing. We clea ly obse e ha d i s had
much less impac on his new ex ac ion me hodology as
he acquisi ion ime inc eases and he high-le el noise e m
educes wi h he IF bandwid h.
Fig. 1: Po 1 Recei e noise le els ex ac ed o a ious IF Bandwid h
(do ed: om [14], con inuous: This Pape )
Noise models ound in he li e a u e usually decompose
each S-pa ame e wi h a mul iplica i e pa and an addi i e
pa [6], [15]. Using he o mula ion (1), we can de i e, o
he fi s o de , he exp ession (4) whe e we easily iden i y he
composi ion o he mul iplica i e and addi i e pa s in any
ecei e a io, o ming he aw S-pa ame e s o he swi ch
e ms [16].
xi
yj
=ˆ
xi
ˆ
yj
(1+δnHxi −δnHyj −1
|ˆ
yj|δnLyj)+ 1
|ˆ
yj|δnLxi (4)
In (4), we unde s and ha he e en ual co ela ions o he
high-le el noise in he ecei e s mus also be aken in o
accoun i one wan s o e ie e he ac ual noise le els in
he a io measu emen s. In ac , he phase noise o he VNA
oscilla o s is mainly esponsible o he high-le el noise,
and i is consequen ly clea ha he high-le el noise will
be co ela ed. Meanwhile, we assume ha he noise floo
essen ially o igina es in he mal noise and should he e o e
be unco ela ed.
To es ima e such co ela ions, we can simply s udy he
p oduc (5).
xiy∗
j−xiy∗
j≃ˆ
xiˆ
yj
∗δnHxiδn∗
Hyi(5)
Then, using local di e ences in (6), i is also possible o
obse e c oss-co ela ions, while minimizing d i impac .
di (xi)di (y∗
j)≃2∗ˆ
xiˆ
yj
∗δnHxiδn∗
Hyi(6)
A e he ex ac ion o noise le els and co ela ions based
on epea ed measu emen s o a eflec s anda d, we could only
eliably de e mine he co ela ions o high-le el noises on he
same po , be ween he eflec and he e e ence ecei e s.
Fig. 2: S anda d De ia ion o noise p opaga ed o S-pa ame e s (30dB
a enua o )
In Fig. 2, we compa e he ac ual s anda d de ia ion o a
30dB a enua o ’s S-pa ame e s due o noise wi h he p edic ed
s anda d de ia ion using ou model. Noise e o was p opa-
ga ed o he fi s o de using (4) wi h and wi hou co ela ions,
espec i ely labeled ”Jacobian” and ”Jacobian no co ”), which
p o ed simila o he esul s ob ained using METAS UncLib
[17]. By aking in o accoun he c oss-co ela ions be ween
he ecei e chains, he model p edic s well he expec ed
noise a iances o eflec ion coe ficien s. Howe e , since we
neglec ed co ela ions in ansmission pa ame e s, he impac
o noise emains o e es ima ed.
B. Unce ain y Due o D i
D i e o s in on-wa e measu emen s o igina e om a -
ious sou ces, om he empe a u e-dependen beha io o
Vec o Ne wo k Analyze s’ ecei e s and equency ex ende s’
mixe s o s abili y o he p obe s a ion. Time-dependen d i
depends on many ac o s, such as oom- empe a u e s abili y,
he powe a which mul iplie chains a e d i en, and he ai flow
a ound he p obe s a ion. As a consequence, an addi ional
expe imen is equi ed o unde s and he magni ude o such
de ia ion unde condi ions simila o hose o he measu e-
men s.
A lowe equencies, co esponding o he use o coaxial
connec o s, i is possible o moni o d i s hanks o an elec-
onic calib a ion (e-cal) by pe o ming a epea able calib a ion
egula ly o e a pe iod o se e al hou s. By his means, i
is possible o ob ain he a iance o each o he e o e ms,
e en ually including hei co a iance by defining a mul i a ia e
no mal Wiene andom p ocess [18]. Howe e , as is he case
in on-wa e measu emen s, when no e-cals a e a ailable and
making a epea able connec ion can be an issue, measu emen
d i can only be assessed wi h he egula measu emen o
a unique s anda d. The desc ip ion o a possible me hod
can be ound in [6], whe e a e an ini ial calib a ion, he
measu emen o a simple flush- h ough s anda d ( h u) is used.
The consequence o d i can be hen exp essed as in [15] wi h
3
a pe u ba ion ma ix posi ioned a he calib a ion plane, using
unco ela ed mul iplica i e e ms in ansmission and addi i e
e ms in eflec ion.
On-wa e , a h u can only be ealized when p obes a e
placed on he wa e , which means ha hese measu emen s
will also be subjec o ib a ion o he p obe s a ion, he e o e
p og essi ely a ec ing he ini ial good con ac . Thus, we
p opose he use o he mo e eliable p obe open o d i
quan ifica ion, a e an ini ial p obe ip calib a ion. As we
only access he d i o he eflec ion coe ficien a each
po , i is no possible o unde s and exac ly which e o
e m is esponsible o he d i . The e o e, we chose o
use conse a i e es ima es o he d i o he ins umen as
desc ibed o po pby equa ions (7), (8) and (9), simila o
hose o a flush- h ough in [6]. Howe e , we should be awa e
ha such a eflec ion-based es ima e will no be sensi i e o LO
phase d i p esen in he ansmission coe ficien phase, which
can be acked based on he measu emen o any ecip ocal
s uc u e.
∆pp =max(|ℜ(Spp( j)−Spp( i))|,|ℑ(−)|)(7)
∆app =a g(Spp ( j)/Spp ( i))(8)
∆mpp =|Spp ( j)/Spp ( i)|−1(9)
Based on hese alues, i is possible o define se e al Wiene
andom p ocesses Wbased on a ime-dependen no mal
dis ibu ion Nas desc ibed by Eq. (10). A classical app oach
o he es ima ion o αuses local di e ences applied o he
p ocess s udied.
W(α, ∆ ) = N(0,Σ = α∆ )(10)
Ye , as seen in he p e ious pa , he eflec ion coe ficien is
subjec o ace noise and so will he local di e ences. To
educe he influence o high-le el noise, we fi s con ol e
he del a unc ions o size Nwi h a Keize window o size
M, ob aining
∆k. Finally, a be e es ima e o αis ound in
Eq. (11), which uses obse ed di e ences be ween any ime
iand j. Fo cla i y, he o mula ion desc ibes he p obabili y
ha he del a unc ion is unde a ce ain ime-dependen alue
desc ibed by he Wiene p ocess, independen ly o he s a ing
ime.
ˆαk=1
ΣiΣjwij
N−M
X
i=0
N−M−i
X
j=M
wij
∆k( i, j)·
∆k( i, j)
i+j− j
(11)
wij =1
N−M−j(12)
The d i e o is hen exp essed a he calib a ion plane
using he pe u ba ion ma ix (13), whe e δW ep esen s in-
dependen andom a iables ob ained om he Wiene p ocess
o (13) wi h αex ac ed on each o he del a unc ions defined
in (7), (8) and (9).
Dp( ) = δWpp1(1 + δWmpp1)ejδWapp1
(1 + δWmpp2)ejδWapp2δWpp2(13)
To e i y ha he di e en me hods a e indeed equi alen ,
we pe o med a d i s udy using an e-cal, whe e we apply
he di e en me hodologies men ioned abo e: using a pai o
sho s, a pai o opens, a h u line and finally, by di ec ly
acking he e o e ms h ough ecalib a ion. A ci cui boa d
was also in oduced a one o he po s o a ec he e o box
and be able o obse e a mo e significan d i . The esul s
a e plo ed in Fig. 3 o a ious e ifica ion s uc u es. Unde
hese condi ions, we clea ly obse e an o e lap o he cu es
o di e en ex ac ion me hodologies, which gi es us some
confidence in hei equi alence. Finally, ou es ima e ˆαgi es
sa is ac o y esul s, enclosing he obse ed d i e o wi hin
he 95% unce ain y bounds.
Fig. 3: Es ima ed d i 95% bounds (con inuous) agains ac ual e o
(dashed) on e-cal e ifica ion s anda ds
C. Wa e P ocessing Inaccu acy
The manu ac u ing o calib a ion s uc u es and de ices on-
wa e is also subjec o non-ideali ies, esponsible o pa o
he measu ed S-pa ame e s’ unce ain y. To unde s and hei
impac on calib a ion, we pe o med a dimensional cha ac e -
iza ion wi h a con ocal mic oscope on some o he calib a ion
s anda ds. The fi s esul s a e al eady desc ibed in Table I o
ou o iginal pape [7]. On he basis o hese measu emen s,
a co a iance ma ix ep esen ing he dimensional a ia ion
ac oss he en i e wa e was de e mined, which, wi h he CPW
model o [19], could be p opaga ed o he p opaga ion cons an
γand line impedance Z0o he ansmission line.
These we e hen used in o de o unde s and he unce ain y
ha a ec s he ansmission lines o he calib a ion ki and he
access lines ha connec p obes o he di e en s anda ds and
DUTs. Finally, o he sake o simplici y, we did no conside
he dimensional a ia ion o he pads and ape s.
Fig. 4: Measu ed p opaga ion cons an o ”L1” and ”L3” samples,
95% e o ba s indica e he po ion o e o o igina ing om geome y
a ia ion
As shown in Fig. 4, a significan sp ead can be obse ed in
he elec ical p ope ies o he ansmission line be ween he
di e en samples, whe e he p opaga ion loss o ”L1” sample
4
Fig. 5: Measu ed impedance o ”L1” and (cu en ) ”L3” samples,
95% e o ba s indica e he geome y a ia ion on model fi ed wi h
me hod o [24]
is mo e han wice he loss o he cu en ly s udied ”L3”
sample, o he exac same dimensions. In ac , he .39µm
gold deposi ed on he InP wa e o L1sample was measu ed
wi h an a e age conduc i i y o 11.1 MS/m, while L3sample
was measu ed a 21.0MS/m. This indica es ha he leak in
he eac o obse ed du ing ab ica ion had a significan impac
on he p ope ies o conduc o s. A mo e de ailed desc ip ion
o he ab ica ion p ocedu e can be ound in [20].
In Fig. 5, he line impedance ex ac ion using [21] and [22]
did no ma ch he me hod o [23]. This can be explained by he
p esence o a con inuous g adien o conduc i i y wi hin he
me al s ips, which, simila o su ace oughness [24], a ec s
local cu en densi ies and abno mally inc eases conduc o in-
duc ance ou side o wha is expec ed om a uni o m ma e ial.
P ope modeling o he e ec howe e equi es in o ma ion
on he conduc i i y p ofile [25]. We he e o e chose o model
he unce ain y as a a ia ion a ound he alue ex ac ed using
[21].
III. UNCERTAINTY PROPAGATION
Fo fi s -o de p opaga ion o unce ain y, we compu ed
a Jacobian ma ix a ound he solu ion o each s ep o he
mul iline-TRL algo i hm, whe e unce ain y sou ces a e con-
side ed as pe u ba ions ei he loca ed a he calib a ion plane
o a he measu emen plane.
A fi s , we de i ed he equa ions o he p opaga ion con-
s an om he NIST’s implemen a ion o [9] and [26]. Then,
he e o e ms a e de e mined based on he gene al algo i hm
o [10]. Finally, he displacemen unc ion ha we in oduced
in [7] is included in he comple e calib a ion scheme. A
summa y o he p opaga ion o unce ain y is p esen ed in Fig.
6, whe e unce ain y sou ces and measu emen s a e p esen ed
a he inpu o each o he calib a ion s eps.
A. P opaga ion Cons an
To compu e he unce ain y o he p opaga ion cons an , we
fi s in oduce he di e en pa ame e s ha a ec he accu acy
o each line’s measu emen . As in [13], we apply noise di ec ly
a he ecei e le el in Eq. (14) wi h he ma ix δBmand δAm
he b- and a-wa e ma ix pe u ba ed by noise, e.g. δBm=
Bm+Nb,δAm=Am+Nawi h Naand Nb he ma ices
con aining noise e o s applied on each ecei e as in oduced
in Eq. (1).
Fig. 6: P ocedu e o unce ain y p opaga ion
The epea abili y o p obe con ac s is in oduced a he
calib a ion plane le el in Eq. (15) wi h he cascade ma ices
R1and R2desc ibed in [7] and s2 he unc ion ha con e s
S- o T-pa ame e s. The influence o wa e p ocessing on he
p opaga ion cons an will be conside ed la e when compu ing
e o e ms.
Mi=s2 (S aw) = s2 (δBmδA−1
m)(14)
Ni=R1(x1i)·e−γli0
0eγli·R2(x2i)(15)
In [26], a bes common line is ound and, o ming a pai
wi h each o he o he (N−1) lines, we ob ain (N−1)
eigen alues om which an op imal weigh ed a e age is ound.
In he o iginal algo i hm, he op imal weigh ing was ound on
he basis o unce ain y exp essed by a pe u ba ion ma ix
in oduced a he calib a ion plane. I was al eady demon-
s a ed he e ha a he fi s o de , he de e mined p opaga ion
cons an is only sensi i e o changes in he ansmission
coe ficien . In ou condi ions, since epea abili y is also ex-
p essed a he calib a ion plane, only he ansmission o ou
displacemen ma ix is o impo ance.
Mk=Mij =MiM−1
j(16)
Nk=Nij =NiN−1
j(17)
We he e o e define he p oblem o N−1eigen alues
based on he measu emen s o each pai o lines Miand
Mj, which con ain noise e o , and he modeled lines Ni
and Nj, which con ain epea abili y e o . As measu ed Mij
and modeled Nij exp essed in (16) and (17) bo h sha e he
same eigen alues, we can exp ess (18), whe e λMdesigna e
he eigen alue a e he co ec oo choice.
Λ(Mk) = Λ(Nk) = λMk =1
2(λ1+1
λ2
)(18)
Gi en ou no a ion o (19), we e o mula ed he fi s -
o de exp essions ob ained by Ma ks in [26] by including he
andom pa o he ansmission coe ficien o he displacemen
ma ix loca ed a po p o each line i,δS pi
21, gi ing (20).
gNk = (lj−li)γk=ln(λNk)∼ˆ
gNk +δgNk + ∆gNk (19)
5
δgNk =δS 1j
21 +δS 2j
21 −δS 1i
21 −δS 2i
21 (20)
This o mula ion explici ly gi es he unce ain y o mula ed
a he calib a ion plane. We can he e o e w i e he Jacobian
ma ix o (21) linking gNk o all eal and imagina y e ms
o he le and igh epea abili y ma ix. We did no include
he ecip oci y di ec ly he e o con enience, as hey a e aken
ca e o by he co ela ions o he epea abili y e ms.
JN(32×2) =1
20−I2−I20... I2I20T(21)
δgNk =JN·δxLNi
δxLNj (22)
Also, knowing he posi ion o he p obes o each ouch-
down, i is possible o co ec each o he gNk by sub ac ing
he bias in oduced in Eq. (23), whe e ∆S pi
21 designa es
he sys ema ic pa o he ansmission coe ficien o he
displacemen ma ix.
∆gNk = ∆S 1j
21 + ∆S 2j
21 −∆S 1i
21 −∆S 2i
21 (23)
Fo he second s ep, i is now necessa y o accoun o
unce ain y exp essed a he measu emen plane, e.g., on he
eigen alues o Mij. Fo his pu pose, we nume ically compu e
he Jacobian ma ix linking gMk o he pe u ba ions accoun ed
o in (14), gi ing he Jacobian ma ix JMk used in (24), whe e
δxLMi
consis s o a ec o o 16 e ms ep esen ing he impac
o noise on he aw S-pa ame e s o he line Li.
δgMk =JMk ·δxLMi
δxLMj (24)
Combining he di e en equa ions esul s in Eq. (25).
gk≡gMk +δgMk +δgNk −∆gNk (25)
As he e is no co ela ion be ween he a iances o he
measu ed and modeled p opaga ion cons an s, as sugges ed
by Ha ab in [12], we can di ec ly sum hese wo con ibu ions
o he o al unce ain y in (26), wi h (27) and (28).
Vgk =VMgk +VNgk (26)
VMgk =JMk VMLi0
0VMLjJT
Mk (27)
VNgk =JNk VNLi0
0VNLjJT
Nk (28)
wi h
VMLi=δxLMi δxT
LMi
=Vn16×16 (29)
VNLi=δxLNi δxT
LNi
=V 18×80
0V 28×8(30)
The p opaga ion cons an is hen es ima ed on he basis o
he weigh ed a e age o he di e en esul s gk. In (31), we
ei e a e he clea o mula ion om [9], whe e he op imal
solu ion is calcula ed based on he ec o o line leng h
di e ences L and he co a iance ma ix V is ob ained based
on unco ela ed noise wi h simila a iance in oduced on he
eal and imagina y pa o he ansmission coe ficien o he
lines. Howe e , his fi s o mula ion is no compa ible wi h
he co a iance ma ix ha we compu ed so a , as we a e
manipula ing a co a iance ma ix o complex alues.
γ=LTV−1G
LTV−1L=C1×N−1·G(31)
A con enien way o ep esen a complex numbe z, as
in oduced in Appendix C o [13], is he 2x2 ma ix (32).
z=ℜ(z)−ℑ(z)
ℑ(z)ℜ(z)(32)
By subs i u ing his ma ix o each e m o he Land G
ec o s, in oducing he L′and G′ma ices, i is he e o e
possible o ecompu e wi h (33) he bes linea unbiased
es ima o , hough his ime including he comple e co a iance
ma ix VG.
ˆγ=C′
op ·G′= (L′TV−1
GL′)−1·L′V−1
GG′(33)
The co a iance ma ix o he p opaga ion cons an gamma
can hen simply be ob ained wi h (34), by subs i u ing he
ep esen a ion o (32) o each elemen o he o iginal C,
ob aining C’, o by using he upda ed e sion C′
op .
VγM=C′·
Vg1... Vg1/gN−1
... ... ...
VgN−1/g1
... VgN−1
·C′T(34)
Wi hin his ma ix, since a common line is used o de e mine
he co a iance ma ix VGo he ec o G′, i is necessa y
o accoun o he co ela ion be ween he di e en e ms,
in oducing he ma ices Vgk/gm
. We can again spli he calcu-
la ion since he modeled and measu ed con ibu ions a e no
co ela ed, gi ing (35).
Vgk/gm
=JMk ·VMkm ·JT
Mn +JNk ·VNkm ·JT
Nn (35)
Wi h VMkm and VNkm compu ed as in (36) by placing he
pa ame e s o he common line Lisys ema ically fi s .
VMkm =VMLi 0
0 0(36)
B. E o Coe ficien s
Now ha unce ain y is exp essed o he p opaga ion con-
s an , we apply a simila me hodology wi hin he con ex o
[10], hough his ime using he comple e co a iance o he
lines desc ibed in (37), wi h he ma ix V he co a iance o
he ansmission line pa ame e s ob ained in sec ion II-C.
VLi =
Vn16×16 0 0 0
0V 18×80 0
0 0 V 28×80
0 0 0 V 4×4
(37)
Unce ain y con ibu ions a e in oduced on bo h sides o
he calib a ion equa ions (38) coming om [10], [7] whe e
E con ains he fi s six e o e ms, and k he symme y
ac o de e mined du ing he nex algo i hm s ep o sec ion
III-C. Repea abili y and opology a e applied o he modeled
esponse o he s anda ds con ained in he ma ix N, while
6
noise e ms a e applied a he measu emen plane, hus on
e ms appea ing in N,Gand H.
NE =G+kH (38)
Be o e including he displacemen e ms, he ideal esponse
o each line (39) is subjec o he p opaga ion cons an e o ,
composed o he sum o a measu ed pa δγM, ob ained
in sec ion III-A, and he p ocess- ela ed a ia ion δγTo
sec ion II-C. Finally, we also in oduce δΓLn = (δZLn −
ZL)/(δZLn +ZL) ha desc ibes he line misma ch caused
by changes in he c oss sec ion o he line, and R he
unc ion cascading he displacemen ma ices o S-pa ame e s
equi alen o Eq. (15).
SLn = R δΓLn e−(γ+δγM+δγT)ln
e−(γ+δγM+δγT)lnδΓLn (39)
The co a iance o he p opaga ion cons an Vγis hen
included in he comple e co a iance ma ix (40) nex o hose
o each line s anda d VLi
.
Vγ+L=
VγVL1/γ . VLN/γ
Vγ/L1
VL1.0
. . . .
Vγ/LN
0. VLN
(40)
Since each line measu emen is eused in his sec ion, he e
is an exis ing co ela ion Vγ/Ln
be ween he mean p opaga ion
cons an and he modeled line ansmission pa ame e s. These
can be calcula ed using (41), whe e we in oduce he co ela-
ion be ween he ex ac ed gkand he pa ame e s o each line
in (42). We hen de e mined he exp ession a he measu emen
plane XMand a he calib a ion plane in XNin (43).
VLn/γ2×52 =VT
γ/Ln
=δγδxT
Ln
=C′
δg1δxT
Ln
...
δgN−1δxT
Ln
(41)
δgkδxT
Li =XM2×16 XN2×16 02×4(42)
XM2×32 =δgMkδxT
LMn
=JMk "δxLMiδxT
LMn
δxLMj δxT
LMn#(43)
Finally, we calcula e he Jacobian ma ix JElinking he
pa ame e s conside ed wi h he ec o s Xand Y, solu ions
o he leas squa es p oblem o (38) as desc ibed in [10] and
[7]. We he e o e ob ain he co a iance ma ix o he e ms o
he Xand Y ec o s in (44).
VE24×24 =JEVγ+LJT
E(44)
C. Symme y Fac o k
To finally ob ain he co a iance o all he e o e ms, we
again sepa a e he e ms ac ing a he measu emen plane
and a he calib a ion plane. In addi ion o he ideal eflec
esponse, we conside he misma ch and change o p opaga ion
cons an o he ansmission line sec ions connec ing he eflec
s anda ds o pads o leng h lo s, using he local p ocess co a i-
ance desc ibed in sec ion II-C. As he exac eflec beha io
ΓRis app oxima ely known a e calib a ion, we in oduce he
a io (45) ha defines he a ia ion o he eflec ion coe ficien
wi h espec o changes in his access line. Fo each po ,
δΓ1,δΓ2 ep esen s he line o se misma ch and δγ1,δγ2 he
change in p opaga ion cons an .
∆Γ=Γb
Γa
=δΓ2+ΓRe−δγ2lo s (1 + δΓ2e−δγ2lo s +o(δΓ2
2))
δΓ1+ΓRe−δγ1lo s (1 + δΓ1e−δγ1lo s +o(δΓ2
1))
(45)
The ideal esponse o he eflec s is hen modified in (46) by
conside ing p obe con ac epea abili y and asymme y, which
also modifies he ma ices ∆G,∆Hand N′in (22), (23) and
(24) o [7].
C11 =La11 +L2
a12Γa[1 + La22Γa+o(LkiiLkjj)]
C22 =Lb11 +L2
b12Γa·∆Γ[1 + Lb22Γa·∆Γ+o(LkiiLkjj)]
(46)
Noise is added o he scheme simila ly o he lines in he
p e ious sec ion. We he e o e ob ain he co a iance ma ix o
each o he eflec s anda ds VRi56×56
, as an addi ional p ocessing
accu acy co a iance ma ix o shape P4×4was included o
conside he change in p opaga ion cha ac e is ics o he le
and igh o se s.
Finally, a Jacobian ma ix is again compu ed acing back
he sensi i i y o he final e o e ms o he p e iously com-
pu ed X and Y ec o s and o he unce ain y sou ces consid-
e ed in he eflec s anda ds. Wi h (47) he comple e co a iance
con aining noise, d i , p ocess a ia ion, and epea abili y o
each o he eflec s, we finally ob ain he co a iance o he
e o e ms wi h (48).
VE+R=
VE0.0
0VR1.0
. . . .
0 0 . VRM
(47)
VK14×14 =JKVE+RJT
K(48)
D. Displacemen Func ion
In [7], we demons a ed he ex ac ion o a displacemen
unc ion linking he a ia ion o S-pa ame e s o he ansi-
ions o he posi ion o he p obes. This displacemen unc ion
is also subjec o he unce ain y sou ces p e iously discussed,
essen ially educing possible esul enhancemen . We now
p opaga e unce ain y in he pa ame e s o his unc ion. Fo
a po 2 p obe, hese pa ame e s a e fi ed on he basis o he
ma ix (49), wi h T′
aw designa ing he swi ch e ms co ec ed
measu emen s o any 2-po ansmi ing de ice when he
p obe is placed on i s pads a a posi ion pi.
R2(xi) = L−1T2T′−1
aw0T′
awiT−1
2L(49)
Because he ex ac ion me hod is insensi i e o he exac
S-pa ame e s o he on-wa e de ice, and he pad geome y
a ia ion was no aken in o accoun , p ocess a ia ion was
no conside ed. Then, we chose he mic oscope esolu ion o
σx=.5µm as he inde e minacy o he posi ion o he p obe.
V =
Vp0.0
. . .
0. Vpn
(50)
7
The comple e co a iance o he inpu pa ame e s can he e-
o e be w i en as in (50), whe e Vpi designa e he co a iance
o he measu emen aken o each posi ion o he p obe (51).
Vpi =Vn16×16 0
0σ2
x(51)
These inpu pa ame e s p opaga e in o he fi ing pa ame e s
βo unc ion using (52).
Vβ=JβV JT
β(52)
Finally, we ob ain he co a iance ma ix V o he displace-
men unc ion in (53) conside ing i s pa ame e s βand hei
co a iance Vβ, he measu ed posi ion xo he p obe and i s
inde e minacy σx.
V 8×8= (β, x, Vβ, σx)(53)
E. Co ec ion
The las s ep o ob ain he unce ain y o he DUT S-
pa ame e s is he p opaga ion du ing co ec ion o sys ema ic
e o s, which can be equi alen ly desc ibed in he classical
8- e ms e o model [16], o mo e easily in [13]. Using he
la e , we ob ain Eq. (54) ha accoun s o noise, d i , and
displacemen unc ion. Finally, δK,δM,δL and δH desc ibe
he e o e ms wi h co a iance, Rand Da e unc ions
cascading epea abili y and d i ma ices a he calib a ion
plane.
D( R(SDUT )) = (δKδBm−δMδAm)(δLδBm−δHδAm)−1
(54)
Finally, we p opaga e hese unce ain ies on he basis o a
8×62 Jacobian ma ix and he comple e 62 ×62 co a iance
ma ix o he unce ain ies conside ed.
IV. RESULTS AND DISCUSSION
The di e en algo i hms we e implemen ed in Py hon wi h
he sciki - lib a y [27]. A fi s , we alida ed he app oach
by compa ing he esul s wi h o he a ailable me hods [12].
Then we applied he algo i hm o he measu emen s om [7],
whe e a mo e comple e unce ain y e alua ion was pe o med.
Finally, since we compu ed he Jacobians du ing unce ain y
p opaga ion, comple e unce ain y budge s a e quickly ob-
ained.
A. Compa ison Be ween Di e en Implemen a ions
To e i y he alidi y o ou app oach, we compa e he e-
sul s ob ained wi h ou app oach wi h a second implemen a ion
ha uses au oma ic di e en ia ion wi h METAS unclib [15],
and wi h he e sion in oduced in [12]. In Fig. 7 we plo he
di e ence in he p opaga ion cons an be ween he common
line pai ing me hod om he Mul iCal implemen a ion o [26]
and he me hod in oduced in [11]. Addi ionally, he s anda d
de ia ions ob ained a e fi s -o de p opaga ion o he noise
e o a e sensi i ely he same o all h ee e sions. This shows
ha in e ms o fi s -o de momen and hence o a small e o ,
he e is no di e ence be ween he wo app oaches.
Fig. 7: S anda d De ia ion o P opaga ion Cons an due o Noise and
Di e ence o Es ima es Ob ained wi h TUG and NIST Algo i hms
The mo e in e es ing pa conce ns he compu a ion o e o
e ms using he gene al o mula ion o Sil onen [10] (TCX
“ ansmi ing ci cui - any ci cui - unknown ci cui ”) a he
ha using eigen ec o -based me hods, as in [26] and [12]. In
[7] we al eady obse ed ha he esul s ob ained wi h he TCX
algo i hm we e close o he esul s ob ained om he op imal
mTRL o [8], whe e hey al eady demons a ed a educed
unce ain y compa ed o he Mul iCal implemen a ion.
Fig. 8: Calib a ed 288um Line beha io (con inuous) and s anda d
de ia ion due o noise only (do ed)
In Fig. 8, we compa e he esul s ob ained wi h he algo-
i hm o [12] wi h ou e sion o he mul iline-TRL algo i hm.
Again, ou esul s we e also es ed agains au oma ic di e -
en ia ion. The esul s confi m a sligh educ ion in s anda d
de ia ion o he eflec ion coe ficien s when using he TCX
app oach ins ead o eigen ec o -based me hods o [28] and
[7].
As expec ed, he e is a mo e significan educ ion in un-
ce ain y o he ansmission coe ficien , since he Th u line
was no conside ed ideal in ou app oach. This is di e en o
o he e sions o he mul iline-TRL, as hey a e all en o cing
he calib a ion plane o be placed exac ly in he middle o he
Th u. I has he ad an age o clea ly speci ying he calib a ion
plane in e e ence o he p obe posi ion bu a he cos o
ansmission measu emen accu acy. This is especially ue in
on-wa e measu emen s, whe e pa asi ic modes pa icipa e in
he deg ada ion o he ansmission o sho lines [29].
B. Displacemen Func ion
Nex , we analyze he accu acy o he ex ac ed displacemen
unc ion. As desc ibed in sec ion III-D, we only conside ed
posi ion inde e minacy and noise influence on he pa ame e s
o he in e pola ion unc ion. In he plo Fig. 9, he o al un-
ce ain y bounds o he displacemen unc ion’s S-pa ame e s
o a p obe displacemen o 8µm we e ob ained o each o
he ollowing scena ii: when e o e ms a e ob ained by (i.)
applying he nonlinea op imiza ion me hod (om l) o (ii.) he
TCX app oach, and when in e pola ion is based on (a.) he
polynomial me hod o (b.) he lossless me hod.
8
Fig. 9: Displacemen Func ion S-pa ame e s (x= 8µm) wi h 95%
e o ba s plo ed o ii.a. and ii.b.
The esul s fi s show ha he e is no significan di e -
ence be ween he esul s ob ained wi h nonlinea op imiza ion
(om l) and he gene al algo i hm ( cx) bo h in alue and
in 1s -o de momen s. A no able di e ence be ween he wo
in e pola ion me hods is ha a bias appea s o he eflec ion
coe ficien angle a low equency. This is ela ed o he well-
known change in beha io o coplana wa eguide lines a low
equencies when he skin dep h app oaches he conduc o
hickness. Al hough, since his phenomenon occu s a ela-
i ely low equencies, he co esponding magni ude o he
eflec ion coe ficien in oduced by a displaced p obe is qui e
low, and hus has a limi ed impac on compensa ed esul s.
A mo e impo an di e ence be ween he wo in e pola ion
app oaches is he magni ude o he ansmission coe ficien ,
which is clea ly mo e unce ain o he polynomial app oach,
as he eflec ion coe ficien in he lossless app oach is mainly
dependen on eflec ion coe ficien s, i.e. on 1−|s11|2.
Fig. 10: Unce ain y Budge on unc ion (ex ac ed om TCX)
(con inuous: polynomial me hod, dashed: lossless me hod)
Finally, in he budge o unce ain y con ibu ions in bo h
polynomial and lossless unc ions Fig. 10, we sepa a ed he
impac on he unc ion pa ame e s o he p obe posi ion
a iance caused by he image p ocessing, labeled ”posi ion”,
om i s di ec impac du ing compensa ion, labeled ”inde-
e minacy”. Conce ning he polynomial app oach, we can
obse e ha he a ia ion seen on he ansmission magni ude
a high equency is indeed no coming om noise, bu is mo e
likely due o non-local e ec s ha we e no aken in o accoun .
Addi ionally, he use o simplified models only ma ginally
imp o es he ansmission phase because he posi ion inde e -
minacy emains he same, e en hough he unc ion pa ame e s
imp o e.
C. Unce ain y o he P opaga ion Cons an
We now compu e he p opaga ion cons an and i s unce -
ain y bounds es ima ed on he measu emen s om [7], his
ime wi h a di e en app oach ha is cla ified by Eq. (19). The
esul s shown in Fig. 11 fi s show a educ ion in he e ec i e
pe mi i i y in e band discon inui ies a e compensa ion.
Fig. 11: P opaga ion cons an wi h 95% bounds
Al hough he eigen alue and op imal app oaches a e gi ing
simila esul s, we obse e mo e significan ipples in he
op imal app oach. This signifies ha he op imal app oach is
mo e p one o unce ain y han he eigen alue one. In addi ion,
in he op imal e sion, esidual-based unce ain y quan ifica-
ion using (55) seems o unde es ima e he unce ain y o he
e ec i e pe mi i i y, showing he limi s o such a gene al
o mula ion.
co (β)∼P| ij
k|2
n−m∗(JTJ)−1(55)
Finally, since we did no accoun o p obe- o-p obe
c oss alk, pa asi ic p obe e ec s, and he andom pa o p obe
con ac epea abili y, he ob ained budge is s ill incomple e.
This is qui e isible in he eal pa o he p opaga ion cons an ,
whe e he eigen alue app oach unde es ima es unce ain y.
This is pa icula ly isible in he WR3.4 band, whe e one o
he p obes was damaged and was esponsible o significan
c oss alk and pa asi ic mode injec ion. Consequen ly, only
he gene al esidual-based es ima ion was able o cap u e he
e ec .
D. Unce ain y o Calib a ed Measu emen s
The es o he p oposed p opaga ion algo i hm was e -
ified wi h nume ous e ifica ion s anda ds al eady desc ibed
in [7]. To emain concise, we ocus on one o he o se
load s anda ds, and compa e unce ain y p opaga ed h ough
he TCX and he op imal mul iline-TRL (om l) app oaches.
The calib a ed measu emen s wi h and wi hou p obe posi ion
compensa ion a e shown in Fig. 12, whe e he 95% confidence
in e als include noise, p obe placemen , and geome y. D i
was only aken in o accoun in he 0-220GHz equency band
since he expe imen s we e no ye p ope ly se up a he ime
o he measu emen s.
We ealize ha compa ed o he alues ob ained in [7]
o he op imal calib a ion, he s anda d de ia ion sligh ly
9
Fig. 12: Calib a ed o se load wi h 95% bounds ( op o bo om plo s:
mag uncompensa ed; mag compensa ed; angle uncompensa ed; angle
compensa ed)
inc eased, explained by he displacemen unc ion, which now
accoun s o he unce ain y in i s pa ame e s and includes
addi ional sou ces o he han he inde e minacy o he p obe
posi ion. The e is a ela i e ag eemen be ween he wo ap-
p oaches apa om he ollowing issues: he unde es ima ion
o unce ain y in ansmission magni ude o simila easons
han o he p opaga ion cons an ’s eal pa ; and a mo e
significan a iance a low equency o he TCX app oach,
since we accoun ed o p ocess a ia ion. Finally, his is a
passi e s anda d, and hence he di e ence be ween he o wa d
and e e se ansmission coe ficien ’s angles is a ibu able o
a ela i e change o he LO signal phase be ween po 1 and
po 2. This could ha e been caused by he mo emen s o he
cables connec ing he equency ex ende s o d i s ha we e
no aken in o accoun in he WR3.4 and WR2.2 bands.
Fig. 13: Unce ain y budge o S-pa ame e s’ magni ude ( op: un-
compensa ed, bo om: compensa ed)
In Fig. 13, we finally e ie e he unce ain y budge o he
o se load o ansmission and eflec ion magni udes, bo h o
he op imal and TCX e sions. Clea ly, he ag eemen be ween
he wo me hods in he uncompensa ed case comes om he
displacemen unc ion, labeled ” unc”. A e compensa ion,
mos o he e o in eflec ion a lowe equency is a ibu able
o p ocess a ia ion, labeled ” opo”. In ansmission, he
op imal algo i hm esiduals, labeled ” esiduals”, indica e ha
o he unce ain y sou ces a e p esen and no accoun ed o ,
like p obe c oss alk o example. Also, i is noise and d i ha
a e esponsible o ipples seen a he edge o each wa eguide
band, when he VNA po powe is educed and e o - e ms
de e io a e.
Finally, we unde s and ha each me hodology has i s
s eng hs, bu a comple e unce ain y s udy is only possible
i all pa asi ic modes a e a oided, o modeled and quan ified.
In ha con ex , a pa ially damaged p obe o bad cable man-
agemen can lead o an unp edic ed measu emen e o . This
makes quan ifica ion o unce ain y in on-wa e measu emen s
a pa icula ly di ficul ask, especially on a dense wa e , o en
encoun e ed in he indus y.
Fig. 14: Ex ac ed capaci ance o an open s anda d wi h 95% bounds
(g een: po 2 uncompensa ed, blue: po 2 compensa ed)
Meanwhile, he me hodology we p opose pe mi s a mo e
accu a e cha ac e iza ion o small de ices. In ac , as demon-
s a ed by Fig. 14, he e is a clea educ ion in he sp ead o
ex ac ed lumped elemen s ac oss equencies.
V. CONCLUSION
In on-wa e S-pa ame e measu emen s, comp ehensi e un-
ce ain y budge s a e o impo ance o help unde s and he
main sou ces o unce ain ies. In he case whe e calcula ed
bounds a e ob iously no ep esen a i e o he e o obse ed
on e ifica ion s uc u es, i can help iden i y ha he sou ce
o e o is no o igina ed in es fix u es o p obe con ac
epea abili y. Thus, i is mos likely an issue linked o he
design o he calib a ion se , a oo dense p obe en i onmen , o
ha p obes a e injec ing unwan ed p opaga ion modes. Such a
comp ehensi e s udy is necessa y in he ealiza ion o p ima y
calib a ion, and hus in on-wa e measu emen s, i is absolu ely
c i ical o ob aining eliable da a. I has been seen mo e
han once ha a po ion o yield loss is caused by un eliable
measu emen da a.
Fo ha pu pose, we enhanced and e ified a ew me hod-
ologies ound in he li e a u e o pe mi he in si u quan ifica-
ion o es fix u e noise and d i , e.g., when on-wa e p obes
a e s ill moun ed on he equency ex ende s, simpli ying he
ex ac ion o such quan i ies. We hen p oposed an app oach
o he quan ifica ion o unce ain y in displacemen unc ions
used in he compensa ed algo i hms o [7]. Al hough i is
possible o unde s and he e ec o p obe misplacemen s
