A
GENERAL
ANALYSIS
OF
ERRORS
IN
NOISE
MEASUREMENT
SYSTEMS
#
L.
P adell,
A.
Come on,
A.
Rami ez*
ABSTRACT
Gene al
and
app oxima ed
exp essions
a e
p esen ed
which allow
he
iden i ica ion
and
quan i ica ion
o
e o s
in
noise
igu e
measu emen s
ha
use
he
Y- ac o
me hod.
This
o mula ion
is
a
gene aliza ion
o
ha
in
/2/
o
all
measu emen
si ua ions
(in
pa icula
i
is
no
necessa y
o
include
an
inpu
isola o ).
The
o mulas
a e
u ilized
o
de e mine
he
e o
limi s
on
he
noise
igu e
measu emen
o
a
mic owa e
low-noise
ampli ie .
INTRODUCTION
Causes
o
e o
in
noise
igu e
measu emen
pe sis
e en
in
mode n,
mi
c op ocesso -con olled
measu emen
sys ems.
E ec
o
e o s
is
especially
impo an
in
he
cha ac e iza ion
o
low-noise
de ices,
whe e
depa u es
o
indica ed
noise
igu e
alues
om
ac ual
ones
a e
mo e
signi ican .
Analysis
o
such
e o s
has
gene a ed
a
conside able-
amoun
o
li e a-
u e.
In
1975
Mamola
and
Sannino
p esen ed
a
gene alized
analysis
o
sou -
ce
misma ch
e ec s
on
noise
pa ame e
measu emen s/l/.
The
exp essions
de-
i ed
he e
allow
he
de e mina ion
o
he
co ec
alue
o
he
e ec i e
inpu
noise
empe a u e
o
he
wo-po
om
he
expe imen al
da a.
In
1981
S id
s udied
he
special
bu
impo an
case
in
which
an
ideal
(s12=0)
in-
pu
isola o
is
used
in
he
measu emen
/2/.
In
1984
Kuhn
s udied
he
in lu
ence
o
he
gain
a ia ion
wi h
inpu
impedance
on
measu emen
e o s
/3/.
In
he
p esen
wo k
gene al
and
app oxima ed
exp essions
a e
de i ed
which
allow
he
iden i ica ion
and
quan i ica ion
o
e o s
in
noise
igu e
measu
emen s.
This
o mula ion
gene alizes
he
one
in
/2/
o
all
measu emen
si-
ua ions
(namely,
i
is
no
necessa y
o
include
an
inpu
isola o )
and
al-
so
i
akes
in o
accoun
he
second
s age
co ec ion.
Non
con e ing
wo-
po s
a e
assumed.
As
a
p ac ical
example,
he
o mulas
a e
u ilized
o
de
e mine
he
e o
limi s
on
he
noise
igu e
measu emen
o
a
mic owa e
low
noise
ampli ie .
NOISE
MEASUREMENTS
IN
REAL
CONDITIONS
Figu e
I
shows
he
se -up
and
quan i ies
in ol ed
in
a
noise
igu e
measu emen
o
a
wo-po
de ice
using
a
wo-s a es
noise
sou ce
(**)
Wi hou
loss
o
gene ali y
a
'cold'
sou ce
empe a u e
T
=T
is
assumed
c
o
h oughou
he
pape .
The
noise
'igu e
shown
by
he
ins umen
(Fdmeas)
di-
e s
om
he
igh
Fd
alue
( he
alue
unde
nominal
condi ions
sO)
because
se e al
depa u es
om
an
'ideal'
measu emen
si ua ion
occu :
-
The
noise
sou ce
e lec ion
coe icien
is
no
he
nominal.
In
addi i-
on,
i
changes
be ween
he
wo
s a es
('cold'
and
'ho ').
This
causes
*E.T.S.E.Telecomunicacio.
Dep .
o
Teo ia
del
Senyal
i
Comunicacions
A.M.R.
G oup.
Ap.30002,
08080
Ba celona
-
Spain
+This
wo k
has
been
suppo ed
by
he
CAICYT
(Resea ch
P ojec
3346/83
and
TELEFONICA
(Con ac
e .
U.P.C.
619-TC/368)
(**)
See
also
Glossa y
o
e ms
below
924
G m,
Ted,
Tem ,
Te ,
G d
also
O
change.
-
The e
exis
unce ain ies
in
he
knowledge
o
sou ce
ho
empe a u e
(Thd
s.
Th).
-
The
noise
me e
also
con ibu es
o
he
o al
ead
noise
powe s.
To
ac
coun
o
his
e ec
a
second
s age
co ec ion
using
he
F iis'
o mu
la
is
commonly
done
using
calib a ion
da a.
-
The e
is
ins umen a ion
unce ain y.
Ins umen a ion
unce ain y
is
no
conside ed
in
he
ollowing
de i a-
ion
bu
i
is
added
in
he
inal
calcula ions.
Fo
each
noise
sou ce
s a e
(1
and
2)
he
ollowing
powe s
a e
ead
by
he
ins umen
du ing
measu e-
men
(N1,N2)
and
du ing
calib a ion
(Nlc,N2c):
N1
k(T
+
T
(P
)*B.G
(P
)
I
c
e
SC
SC
N2
k(Th
+
T
(P
))B13G
(P
2h~a
+
e ( shV
.G
(PsQ
Ni
IC
k(T
em
sc'
msc1'~,
N
(k(Tc
+
Te(Fs))
.B.G m(sh)
2c
ha
em
sh
m
sh
Noise
igu es
(Fmeas)
a e
calcula ed
by
he
ins umen
using
he
exp e
ssion
/5/,/6/:
Y(I
-
T
ITO)
+
(T
h/T
-
1)
meas
y
-
I
whe e
Y
is
he
Y- ac o
ac ually
measu ed
by
he
ins umen ,
de ined
by:
Y
=
N2/N1
(du ing
measu emen )
(2a)
Y
=
N2c/N
1c
(du ing
calib a ion)
(2b)
Subs i u ing
(2a)
in
(1)
yields
F
(P
SC
)
m
(T
h-T
)
meas
M
(T
-T
)+
T
(N
F
(P1
)
-HMF
(P
(3)(*
h
ha
o o
h
sh
c
SC
This
undamen al
exp ession
ela es
he
displayed
(measu ed)
noise
i
gu e
o
a
chain
( o med
by
he
wo-po
unde
measu emen
and
hemeasu e-
men
sys em),
o
pa ame e s
o
he
measu emen
se -up
and
also
iden i ies
he
causes
o
e o .
Fmmeas
can
be
ob ained
in
a
simila
manne
by
subs i u ing
(2b)
in
(1).
The
ins umen
calcula es
he
de ice
gain
om
/5/:
N
-
N
N2
1I
Gdm~eas
N
-N
(4)
2c
Ic
and
hen
i
deembeds
Fdmeas
using
he
F iis'
o mula:
F
-1
F
F
-
mmeas
(5)
dmeas
meas
Gdmeas
EXPRESSION
OF
NOISE
MEASUREMENT
ERROR
Subs i u ing
(1),
(2),
(3)
and
(4)
in
(5)
he
ollowing
exp ession
e
la ing
he
displayed
noise
igu e
o
he
wo-po
unde
measu emen
o
(*)
See
Glossa y
o
e ms
below
925
he
pa ame e s
o
he
measu emen
sys em
is
de i ed:
M
h(T
-T)+T
M
mF
(h
-m(
sc)
mc
h
F
(
)m
(ThT
)
+
mh
ha
o
mhm
s
SCch0Gad
(O)N/M1
Fdmeas
Mh(T
-
T')
+
T'(hF ( sh)
-
M
cF
(
(6)
(S)
We
a e
in e es ed
in
he
expec ed
limi s
o
he
e o
AF
(*)
Exp e-
11
d~*
ssions
o
he
e o
limi s
can
be
ob ained
by
di e en ia ing
he
F iis'
o mula
F
(
)
-i
F
=0
-
0
mo
(*)
d(O)
F (O)
Gd
(0)
yielding
=
3F
@FI+Fm oo
Fd
=
-
G
(O)
iFm
+
G
(0)
Gad
(7)
ad
~~~~ad
Subs i u ing
o
F meas,
Gdmeas
in
(7)
we
ob ain,
wi h
he
lowes
o de
o
app oxima ion
(which
is
su icien
i
e o s
a e
no
exceedingly
high)
F
( sc
)Mc(ThTo)
Fm( P
)Mmc
(Th
To)
F
M
N
F
SC
____F
______
Mc
h
meas
+
h
_0
imeas-I
(8)
d
F
(F
-)V(Th-T
)
meas
F
(
)M
h(Th-T)
G
+
M
G
kOJl'$ThaTo)
measFm
00
m
h
dmeas
Mmh
o
dmeas
o
which
he
addi ional
app oxima ions
mh
(T
ha.
0)
+
TO(&hF (sh)
McF (Psc))
14h(Tha
To)
M
(T
-T
)+T
(M
F
(P
)
-M
F
(P
))
14
(T
-
T
)
mh
ha
o
o
mhm
sh
mc
m
sc
mh
ha
o
ha
a e
ul illed
in
mos
p ac ical
cases
(noise
sou ce
ENR
>
5dB),
ha e
been
made.
Fo
low
noise
ampli ie s,
designed
o
ha e
a
noise
igu e
nea
he
mi-
nimun,
i
is
easonable
o
assume
ha
sligh
de ia ions
o
he
sou ce
im-
pedance
om
he
nominal
alue
do
no
p oduce
la ge
a ia ions
o
he
ac-
ual
noise
igu e
/3/,
so
ha
exp ession
(8)
can
be
app oxima ed
by
AF
N(M
(Th-TO)
-OIF
in(Mc
ThTO)
Fmmeas_
I4h
N
F
na-I1
(*)
d
Mh(Tha-TO)
meas(Mm
(Tha°TO)
Gdmeas
Mh
N
Gdmeas
Exp ession
(9)
indica es
ha
o
low-noise
ampli ie
measu emen
he
p e alen
causes
o
e o
a e
he
sou ce
impedance
a ia ion
due
o
he
noise
sou ce
swi ching
and
he
ENR
unce ain y,
and
i
is
hough
o
be
use ul
in
a
R&D
labo a o y
en i onmen .
I
also
shows
ha
including
a
well
cha ac e ized
isola o
a
he
noise
sou ce
ou pu
esul s
in
lowe
e o
limi s,
in
ag eemen
wi h
/4/.
Exp ession
(9)
should
no
be
used
o
wo-
po s
ha ing
an
inpu
isola o ;
in
ha
case
F
(
SC)/F (0)
is
known
/2/
F
(
)
I1
-
s7
ji2
F (0)
I
-
SC12
(assuming
ideal
isola o ,
s21=O)
(*)See
Glossa y
o
e ms
below
926
and
(8)
leads
o
less
pessimis ic
limi s
han
(9),
acco ding
o
/2/.
No e
ha
(9)
makes
only
use
o
F meas,
FxmeasesGdmeas
whjchcan
be
di ec ly
ead
du ing
calib a ion
and
measu emen .
PRACTICAL
EXAMPLE
Noise- igu e
measu emen
o
a
low-noise
na owband
mic owa e
ampli 'ie
a e
conside ed.
The
ampli ie
is
used
in
he
2nd
IF
sec ion
(2.62
GHz)
o
20-30
GHz
ea h
s a ions
ecei e s
and
i
was
designed
o
mee
e y
low
noi
se
eque imen s
(a ound
1
dB
NF).
I
has
wo
s ages
wi h
NEC's
NE75083
MES-
FET's.
The
measu emen
sys em
is
based
upon
a
HP
8970
B
noise
igu e
me e
and
HP
346B
noise
sou ces.
Fou
measu emen s
we e
made.
The
'i s
wo
used
a
noise
sou ce
wi h
ENR
=
15
dB.
The
hi d
and
ou h
used
a
noise
sou ce
wi h
ENR
=
5
dB.
The
second
and
ou h
added
an
inpu
isola o .
Expe imen-
al
esul s
o
Fdmeas
we e:
(a)
-
Fdmeas
=
1.17
dB
(ENR
15
dB,
no
inpu
isola o )
(b)
-
Fdmeas
=
1.28
dB
(ENR
15
dB,
wi h
inpu
isola o )
(c)
-
Fdmeas
=
1.07
dB
(ENR
5
dB,
no
inpu
isola o )
(d)
-
Fdmeas
=
1.35
dB
(ENR
5
dB,
wi h
inpu
isola o )
The
e o
limi s
we e
hen
calcula ed
using
(9)
o
(a)
and
(c)
and(8)
o
(b)
and
(d)
in
he
ollowing
manne
(see
Table
I)-
I'scI,dT'shl
and
ENR
unce ain y
we e
known
om
noise
sou ce
manu ac u e
s
da a.
Ilns umen a-
ion
unce ain y
we e
also
known
om
noise
igu e
me e
manu ac u e 's
da
a.
I'i|
was
measu ed.
Thus
i
was
possible
o
calcula e
Mc/Mh
unce ain y.
In
a
simila
way,
Mmc/Mmh,
Mh/Mmh
and
N/Mo
could
ha e
been
calcula ed
om
mj
(manu ac u e 's
da a)
and
j ol
(measu ed).
Howe e ,
as
Gdmeas
is
high
(29
dB),
he
con ibu ion
o
he
second
and
hi d
e ms
in
(9)
is
negligi-
ble.
Combining
Mc/Mh
unce ain y
wi h
ENR
and
ins umen a ion
unce ain ies
(in
bo h
wo s
case
and
RSS
manne )
he
e o
limi s
we e
ob ained.
Figu e
2
shows
he
measu ed
noise
'igu es
and
he
compu ed
e o
limi s
o
Fd.
Using
ansis o
manu ac u e 's
da a,
he
app oxima ion
F (Tsc)/F (0)
Z
1
was
calcula ed
o
a ec
he
es ima ion
o
he
e o
limi s
by
less
han
±
0.07
dB
in
case
(a)
and
less
han
±
0.02
in
case
(c).
CONCLUSIONS
A
gene al
exp ession
ela ing
he
noise
igu e
o
a
wo-po
displayed
by
a
noise
measu emen
sys em
ha
uses
a
Y- ac o
me hod,
o
he
a ious
pa ame e s
appea ing
in
a
eal
measu emen
si ua ion
in
p esen ed.
Simpli-
ied
exp essions
a e
de i ed
o
es ima e
e o
limi s
in
a
noise
igu e
measu emen
o
low-noise
ampli ie s
e en
hose
no
including
isola o s
a
hei
inpu s.
These
exp essions
only
equi e
measu ed
da a
o
Fm,
Gd,| iJi
and
j oj
and
manu ac u e 's
da a
o
he
es
o
he
pa ame e s.
Calcula-
ions
show
ha
e o s
can
be
as
high
as
±
0.74
dB
when
measu ing
a
I
dB
NF
ampli ie
wi h
no
inpu
isola o
and
ha
app osima ion
F ( sc)/F (O)
%
I
is
jus i ied
o
ampli ie s
wi h
noise
igu e
nea
he
minimun.
In
his
case,
he
measu emen
e o s
a e
domina ed
by
misma ches.
927
GLOSSARY
OF
TERMS
Fdmeas
Measu ed
Noise
Figu e
o
he
De ice
Fmmeas
Measu ed
Noise
Figu e
o
he
Noise
Me e
F meas:
Measu ed
Noise
Figu e
o
he
Chain
Gdmeas:
Measu ed
Gain
o
he
De ice
AFd
=
Fdmeas
-
Fd(O)
Fd
De ice
Noise
Figu e
Fm
Noise
Figu e
o
he
Noise
Me e
F
Noise
Figu e
o
he
Chain
Gad
:
De ice
A ailable
Gain
G
:
Chain
T ansduce
Gain
G m
:
Me e
T ansduce
Gain
Te
:
E ec i e
Inpu
Noise
Tempe a u e
o
he
Chain
Tem
:
E ec i e
Inpu
Noise
Tempe a u e
o
he
Me e
Th
:
Noise
Sou ce
Tempe a u e
(Calib a ion
Value)
Tha
:
Noise
Sou ce
Tempe a u e
(Ac ual
Value)
sc,h
:
Noise
Sou ce
Re lec ion
Coe icien
(Cold/Ho )
Fm
:
Inpu
Re lec ion
Coa icien
o
he
Noise
Me e
Fi
:
Two-Po
Inpu
Re lec ion
Coe icien
(du ing
Measu emen )
110
:
Two-Po
Ou pu
Re lec ion
Coe 'icien
(du ing
Measu emen )
'00
-
Io
when
s'=0
(1
-
I1
(
12)-
II 212)
( s'
Ii)
=
Misma ch
Coe icien
ji
-
I'1 2j2
Mc
=
( sc,
i)
Mh
=
(Fsh-,
I;)
Mmc
=
( sc
i
m)
Mmh
=
( sh,
m)
N
=
( ooI
Fm)
MO
=
(O,I'i)
k
Bol zmann's
cons an
B
Sys em
Bandwi h
(much
na owe
han
ha
o
he
Two-Po )
REFERENCE
S
/1/
G.
Mamola,
M.
Sannino
"Sou ce
misma ch
e ec s
on
measu emen s
o
linea
wo-po
noise
em-
pe a u es"
IEEE
ans.
on
I.M.,
ol
IM-24
n.3,
sep .
1975,
pp.
239-242
/2/
E.
S id
"Noise
measu emen
o
low-noise
GaAs
FET
ampli ie s"
Mic owa e
Sys ems
News,
no .
1981,
pp.
62-70
/3/
N.J.
Kuhn
''Cu ing
a
sub le
bu
signi ican
cause
o
noise
igu e
e o "'
Mic owa e
Jou nal,
june
1984,
pp.
85-98
/4/
E.
S id
''No'ise
measu emen
checklis
elimina es
cos ly
e o s'
Mic owa e
Sys ems
News,
dec.
1981,
pp.
88-107
/5/
"Noise
igu e
measu emen s.
P inciples
and
applica ions"
Hewle -Packa d
semina ,
1982
/6/
"Fundamen als
o
RF
and
Mic owa e
Noise
Figu e
Measu emen s"
Hewle -Packa d
Applica ion
No e
57-1,
July
1983
928

il
measu ed
a,c
b,d
Mc/Mh
unce ain y(dB)
a,c
b,d
(.M!
(W.C.)
15
dB
0.03
0.04
±0.34
0+.02
0.56
0.061
0.26
5
dB
0.01
0.01
±0.11 ±0.006
ENR
unce ain y:
±0.3
dB
(W.C.),
±0.1
dB
(RSS)
Ins umen a ion
unce ain y:
±0.1
dB
TABLE
I
-
Pa ame e s
in e eining
in
AFd
Mm( s
F m)
CALIBRATION
G m( s, m)
TemC s)
Gpd( i
, m)
L
_
_
_
,
_
F
(
s)
I
+Te
( s)
/To
Fmc
s)u1+Tem( s)/To
G ( 3, i)
FIGURE
1
Te ( s)
-
Block
Diag.
1
.
q::
0)
-o
03
.
_-
,
C-,-
I
a
CD
0)
:D
-
1a
5
4
AMPLIFIER
INCLUDINC
3
AN
INPUT
ISOLATOR
B
(IslI
=-0.061)
(EXPRESSION
(3))
I
9
~
~~~~~Ii
LIMITS
DUE
TO
i
1
TRUMNTATION
ANDD-
. ERUNCERTAINTY:
----WORST
CASE
1-T--------I
1-
'-
'1-RSS
-1
MEASUREMENT
WITH
A
'
15
DB
ENR
NOISE
SOURCE
WORST
CASE
MEASUREMENT
WITH
BB
':
5
DB
EIIR
IIOISE
SOURCE
RSS
-
BjA/
MEASUREMENT
WITH
15
DB
ENR
NOISE
SORCE
WORST
CASE
MEASUREMENT
WITH
EBBS
/5
DB
ECR
NOISE
SOURCE
,
j-
-7-------
AMPLIFIER
NOT
INCLUDING
INPUT
ISOLATOR
(i ,1FJ=0.56
.
B'
/:
.
(EXPRESSION
(4))
.
|
|
I
A
.5
I
Unce ain y
(dB)
FIGURE
2
-
E o
Limi s
929
Noise
Sou ce
ENR(dB)
I scI
I shl
Nl
c,
N2
C
Ni
,N2
-
-
-
i
-
-
s
I
I I
I
I
I
I
I_
-
a