Lawrence Berkeley National Laboratory
Recent Work
Title
GENERALIZED ANALYSIS OF PHASE-SENSITIVE DETECTION CIRCUIT OPERATING CHARACTERISTICS AT THE SIGNAL DETECTION
IN THE PRESENCE OF NOISE
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https://escholarship.org/uc/item/36h8w44j
Author
Leskovar, Branko.
Publication Date
1971-05-01
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University of California
Submitted
to
the
International
Symposium
on
Electrical
Network
Theory
London,
England,
September
6-10,
1971
" '
r
, J
; ,
, I
, ,
. ,
,
,~
.;..
J
B.
Leskovar
UCRL-20252
Summary
Preprint
C.
2.
...
~..
'
.....
GENERALIZED
ANALYSIS
OF
PHASE-SENSITIVE
DETECTION
CIRCUIT
OPERATING
CHARACTERISTICS
AT
THE
SIGNAL
DETECTION
IN
THE
PRESENCE
OF
NOISE
Branko
Leskovar
May
1971
AEC
Contract
No.
W
-7405-eng-48
,
TWO-WEEK
LOAN
COpy
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a
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be
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copy,
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Tech.
Info.
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Ext.
5545
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as
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of
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.
_0'
~;
,
I
-1-
GENERALIZED
ANALYSIS
OF
PHAsE-SENSITIVE
DETECI'ION
CIRaJIT
OPERATING
rnARACTERISTICS
AT
TI-IE
SIGNAL
DETECI'ION
INlHE
PRESENCE
OF
NOISE
'
Branko Leskovar
Lawrence
Radiation
Laboratory
,
university
of,California
Berkeley,
California
94720
Jtme
1971
UCRL-20252
SUJmlIary
. . . (1.,
2)
Previous
analysis
of
phase-sensitive
detectIon
CIrcuIts
have
been
based
on
the
assumption
that
the
additive
noise
is
non-
exis
tent
in
the
detector
reference
channeL
However,
in
contemporary
experimental
research
instnunentation
and
phase-lock
systems,
a
sig-
nificant
amount
of
noise
can be
present
in
the
reference
ch~e1.
(3)
Such
case
will
be
disOlssed,
asstDlling
that
the
input
signal
and
the
reference
wave
are
in
the
presence
of
independent,
stationary,
and
additive
Gaussian
noise.
The
generalized
detection
circuit
tmder
consideration
is
shown
in,
.
,.
Fig.
1.
The
circuit
input
consists
ofa
signal
s
(t)
superimposed on
a
broad-band
noise
n:(t).
After
filtering,
the
sum
si(t)=s(t)+ns(t)
is
applied
to
the
balanced
phase-sensitive
detector
•
The
detector
,.
reference
input
consists
of
a
reference
wave r
(t)
superimposed on a
broad-band
noise
n*
(t).
After
filtering,
the
sum
r
i
(t)=r(t)+nr(t)
is
.'
r
applied
to
the
detector
and
differential
circuit.
The
generalized
and
nonnalized
fonn
of
the
detector
operating
characteristl~s,~~
SJ'\i
CN
;+N;)1/2
=
(11/2)1/2~h(xs,lJr,ljI,n),
where
So
represents
the
the
output
signal;
lij
is
the
detection
efficiency;
N
r
and Ns
are
:nns
-2-
values
of
tile
narrow-band
noise
in
the
reference
channel
and
signal
channel,
respectively;
x =
SIN
is
the
input
signal-to-noise
ratio;
. s s
II
= R/N
is
the
reference
wave-to-noise
ratio;
n = N
IN
is
the
signal
. r r
..
. . s
·r
channel
noise-to-reference
channel
noise
ratio;lFlCxs,lJr,ljI,n)
denotes
a
confluent
hypergeometric
ftmction
of
the
fonnlFl(~1/2;1;Xs,1-Ir,l/I,n);
tJ,.
signifies
the
difference
of
the
hypergeometric
fWlctions;
1jJ
is
the
phase
angle
between
the
input
signal
and,
reference
wave.
Detector
,..
characteristics
nonlinearity
NA(X
s
)
relating
to
the
input
signal-to-
noise
ratio
is
described
by a
generalized
equation
The
tenns
lFlCxs,lJr,n)
and
lFl(llr,n)
denote
confluent
hypergeometric
functions
of
thefonn
lFl
C
-1/2;
l;xs
,1-I
r
,'nl
'aDdlFl
(1/2;2
;llr,n)
,
'respectively.
N*
(x )
asa
ftmction
of
x
is
calculated
and
plotted
in
Figs.'
2-3.
Ass
. 0
, " .
.
From
the
curves
it
can
be
Seen
thatNA(X~)
increases
with
inCreasing
,.
.
Je
s
'
for
given
1-Ir,and
n.
Also,NA(x
S
)
decreases
when
the
ratio
of
11
/x
increases.
particularly
when
1-1"'::'1.
Furthennore,
when
the
r s •
'ratio
n
is
large,
the
minimization
of
N; (x
s
)
requires
a
very
high
value
of
the
ratio
llr/xs.·
Foi
a
fixed
ration
and
phase
angle
1jJ,
.
,.
the
proper
choice
of
l1r
and
Xs
minimizes,
NA
(x
s
)
to
an
acceptable
amount.
This
is
important
in
a wide dynamic
range,
wide-band
detection
application
using
solid
state
circuit
componen.ts where
the
ratio
II
is
often
close
in
value
to
the
ratio
x
S
'
resulting
ina
r . .
.'
large
nonlinearity.
Curves iri
Fig.
3 show
that
for
n = 1
andxs
1,
the
in~rease
of
the
lly1xs
ratio
by one
order
of
magnitude
will
.
decr~ase
the'
nonlineari
ty
by· almos t
three
orders
of
magnitude.
"
..
Furthennore,
NA(x
s
)
is
reduced
even more
for
larger
ama:mts
of
xs.
-3-
For U
r
=
10,
increasia~
U
r
by one
order
of
magnitude
decreases
the
>lon1inearity by
moretium
four
'orders
of
magnitude.
,
"',.
Detector
characteristics
nonlineari
ties
~
(ljI)
and
Ne(l/I)
relating
,to
the
phase
angle
1/1
for
operating
points
I/I
B
= (2m+l)'II'/2 and
l/I
C
=
2m'11',
where m =
0,1,2,
•••
, 'are
described
by
equations
'
,
,.
'-1
~
(1/1)
= 1- [(n+n )
IFi[x
s
,IJ
r
,ljI
,n)
]/[utx
s
(1f/2-l/I)lF
l
(xs,ur,n)
] and
,.
,.
,=(ljI)
1-
[AIF1(x
s
,JJr·jl/l"n) /
[A
IFI
(x
s
,ur,n)]
Hypergeanetric
functions
II
(xs
'U
r
,ljI
;nJ
and
II
(x
s
' u
r
,n)
are
of
the
,,.
,
same form
as
in
the
expression
for
NA(I/I);
A
signifies
the,
differen~
" ,
,.
of
functions
of
the
fonn
IFI
(xs,ur,n).
The
nonlinearity
NB(ljI)
depends
strongly
upon
1/1,
x
s
' U
r
'
and
n,
as
shawn
in
Fig.
4..
To
obtain
the
,
,.
IIDSt
infonnatim
about
NB(I/I)
behavior,
the
nonlirlearity
is
calrulated
-3
and
plotted
as
a
function
of
xs'
The
rurvesshaw
that·
for
n
...
10
..
and
U
r
=
1,
the
nmlinearity
NB(x
s
)
is
practically
independent
of
,.
Xs
over
a wide dynamic
range.
en
the
cmtrary,
NB
(x
s
)
has
a
strong
~ndenceon
1/1,
n,and
U
r
•
From
the
runes
it
follows
that
the
proper
choice
Of )ir
for
a
given
n and
Xs
minimi;es
the
nonlinearity
•
.
~y
other
value
of
u.can
be
considered
as
nonoptimum.
" r .
,.
The
curves
for
the
nonlinearity
NC(x
s
)
in
Fig.
5 show
that
particular
values
of
J.lTminimizethe
nonlinearity.
For
this
purpose
,.
it
is
important
to
av.oid the
regions
where
NC
(x
s
) Curves have
maximums
•
.
~lysis
and
the
curves
show
that
the
presence
of
the
reference
channel
. ,
noise
has a dominant
role
in
the
detector
optin'lUn
operating
conditions.
,.
'"
Frama
comparison
of
the
nonlinearity
of
NB(x
S
)
and
NC(x
s
)
curves
i
tcan
be
seen
that
nonoptimum
values
of
u
r
' x
s
' and n have
far
more
'"
'"
influence
on
the
behavior
of
NC(x
s
)
than
on
~
(x
s
) •
In
general,
the
-4-
'"
nonoptimLim
II
r
,
x
s
' and n
values
maximally
increase
~B
(x
s
) approximate
ly
'"
one-half
order
of
magnitude
in
the
worst
case.
However,
NC(x
s
) 'nOn-
'"
optimulli
values
of
U
r
' x
s
'
and
n
increase
NC(x
s
) two
orders
of
magnitude. Consequently,
the
phase
angle
~
=
(2In+1)'II'/2
should
be
chosen
as
the
operating
point
wherever
possible.
ACKN<J\ILElXiMENT
The
author
wOuld
like
to
thank
Ruth
L.
Hinkins
for
canputer
,
programning.
