CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
A COMPLEX CORRELATION FILTER
II
FOR RADAR DIRECTION TRACKING
A thesis submitted in partial satisfaction of the
requirements for the degree of Master of Science in
Engineering
by
Dale Alan Goudey
/
June, 1978
The Thesis of Dale Alan Goudey is approved:
A. F. R~tcliff"JV
I. Hashimoto, Chairman
California State University, Northridge
ii
TABLE OF CONTENTS
Page
TABLE OF TABLES .
iv
TABLE OF FIGURES.
v
vi
ABSTRACT . . .
INTRODUCTION . • .
1
Chapter
. . .. .
1.
PROBLEM STATEMENT
2.
SIGNALS AVAILABLE FOR PROCESSING.
3.
MONOPULSE MEASUREMENTS.
3-1.
3-2.
4.
. ..
3
7
11
Synthesis . . . .
• . • •
Angle Extraction Techniques .
11
16
THE EFFECT OF MULTIPLE POINT TARGETS ON MONOPULSE
ANGLE MEASURES. . • . • . . . . . • • • • • • • .
20
5.
STATISTICAL ANALYSIS OF SIGNALS FRON POINT TARGETS.
28
6.
ANALYTICAL MODEL OF SIGNALS FROM EXTENDED TARGETS
38
7.
MAXIMUM LIKELIHOOD ESTIMATION .
..
52
SUMHARY . •
..
REFERENCES. •
53
APPENDICES
Appendix A;
Appendix B:
Statistics of arg[A* B] for a Poin-t Target
and Additive Gaussian Noise. • • . •
•
•
. •
fl
' 55
Statistics of 2Re{Dn/Sn} for a Point
Target and Additive Gaussian Noise •
59
Appendix C:
Maximum Likelihood Estimate of
68
Appendix D:
Statistics of the Maximum Likelihood
Estimate of ~- . . • . . • • • • • •
iii
~
••
. .....
.,
72
· TABLE OF TABLES
Page
3.1.
MEASURES OF DIRECTION TRACKING ERROR.
iv
19
TABLE OF FIGURES
Page
3.1.
ANTENNA CONFIGURATION FOR PARTIAL DIRECTION YiliASUREMENT . . . 14
3. 2.
ANTENNA CONFIGURATION FOR COMPLETE DIRECTION MEASUREMENT . . 15
4.1.
A* B FOR A TYPICAL TWO-POINT TARGET • • • • • •
. 26
4.2.
A* B FOR AN EQUAL MAGNITUDE TWO-POINT TARGET
• 27
4.3.
D/S FOR AN EQUAL MAGNITUDE TWO-POINT TARGET.
. 27
5.1.
STANDARD DEVIATION OF M2 VERSUS SNRS . , . •
33
5.2.
STAliDARD DEVIATION OF M , M , AND M VERSUS SNRA • • .
2
1
3
• 34
5. 3.
MEAN SQUARED ERROR OF M WITH ADDITIVE GAUSSIAN NOISE.
1
35
5.4.
MEAN SQUARED ERROR OF 11 WITH ADDITIVE GAUSSIAN NOISE. . .
2
36
5.5.
MEAN SQUARED ERROR OF M WITH ADDITIVE
3
7.1.
PROBABILITY DENSITY FUNCTION OF l1Ll FOR CHANGING N
7.2.
PROBABILITY DENSITY FUNCTION OF M:L
7.3.
STANDARD DEVIATION OF ML
7.4.
MEAN SQUARED ERROR OF ML . . . • • . • •
1
7. 5.
MEAN SQUARED ERROR OF ML
1
2
AND ML
2
1
GAUSSI~~
FOR CHANGING
VERSUS SNR
AS AN ESTIMATE OF ¢.. •
v
NOISE.
r
.
37
. 48
..
.
.
48
49
50
51
ABSTRACT
A COMPLEX CORRELATION FILTER
FOR RADAR DTRECTION TRACKING
by
Dale Alan Goudey
Master of Science in Engineering
The problem of processing measured data for use with the Kalman
filter is considered.
Specifically, the problem is that of tracking
an airborne target with measurements obtained by radar.
observation and filtering processes are investigated.
The direction
The problems
considered are direction observation noise due to multiple targets,
multipath reflections in the mainlobe, target angular extent, and
theiThal noise in the receiver.
It is assumed that a stable track of
the ensemble of radar reflector points is desired rather than resolving
and tracking the individual points.
A signal processing technique
which reduces the adverse effects of target angular extent and more
closely matches the observations to the Kalman filter observation
· model is presented.
vi
It is assumed that the radar system makes use of the monopulse
type of antenna system.
With the information available from this
type of antenna, a maximum likelihood estimate of the direction of
the return signal can be formulated which is in the form of a complex
correlation pre-filter.
The performance of the maximum likelihood
direction estimator is compared to that of the maximum likelihood
angle estimator which has been investigated in the literature.
Of key interest in this paper is the fact that the statistics
of the estimators discussed are obtained by direct mathematical
analysis rather than by a computer simulation of the noise processes.
This report also includes the first treatment of the statistics of
D/S (a common angle measurement) which properly presents the functional
relationship between the threshold level and the variance of the
measurement for the assumption of a point target and additive Gaussian
noise.
vii
INTRODUCTION
In most studies of direction measuremen.ts obtained by monopulse
radar systems the measurement is in the form of angle error.
Investigations of the statistics of the most common direction
measurement in monopulse systems, D/S, have been presented in the
literature.
Haximum likelihood estimates of. angle error have been
presented as well as studies of their statistics.
This paper is a study of a different type of direction measurement.
The measurement is in the form of the vector dot product of
a direction error vector with a measurement vector.
This quantity is
a direct measure of the residual of a Kalman filter whose states
include target position in cartesian coordinates, or target direction
cosines relative to the radar.
The measurement is thus easily
applied as an observation to the Kalman tracking filter.
A maximum likelihood estimate of the Kalman filter residual is
formulated.
A comparison is made bet"':J"een the maximum likelihood
angle estimators presented in the literature and the estimator
developed here.
Statistical analysis :is used to obtain expressions
for the mean squared error of the new estimator, as well as its
probability density function.
Chapter 1 of this paper defines tlte problem and provides the
motivation for the analysis which follows.
Chapter 2 describes the
relation$hips between the spatial radar signals and the signals
obtained at the antenna output ports.
'The relationships between
these signals and direction error observations are discussed in
1
2
·chapter 3.
Chapter 4 describes the effects of multiple point targets
on themeasurement of direction error, which results in noise that is
correlated with signal amplitude.
A statistical study of the direction measurements is made in
chapters 5, 6, and 7.
Chapter 5 investigates the observation
statistics for additive Gaussian noise and single targets with no
angular extent, as a function of the signal to noise power ratio.
Chapter 6 discusses a statistical model of targets with angular
extent.
A maximum likelihood estimate of direction error is presented
in chapter 7, along with its statistics.
The appendix contains the derivations of the analytic solutions
to the measurement statistics, which are new results.
Chapter 1
PROBLEM STATEMENT
The classical approach to radar direction tracking is to form
an angle observation which is a measure of" angle error.
The
observation formation process is designed such that the indicated
error is linear with respect to the actual angle error over the
desired region of operation.
The linear measurement allows the
application of classical control system techniques to perform the
filtering and antenna control required to keep the antenna mainlobe
positioned on the target.
Much work has been done in minimizing the effects of noise on
the angle measurement [refs. 1 and 2).
linearity of the angle error
The work is based on the
observation~
Though the assumption
of linearity is valid to an acceptable degree within the normal
operating region, it will be shown that it is not necessary to form
an observation linear in angle for optimal direction estimation.
Target tracking can be achieved by
obser~ing
and filtering
the target vector position or direction cosines (unit vector
direction~
Modern radar control systems use Kalman filter techniques to perform
target trajectory filtering and estimations [refs. 3 and 4).
The
observations for use in the Kalman filter need not be in the form
of angle measure, as with the classical approach.
Any measurement
or observation which can be accurately modeled as a linear combination
of the Kalman filter state variables and observation noise, or a
linear combination of the difference between the actual and estimated
3
4
Kalman filter state variables and noise, can be used by the Kalman
filter.
The Kalman filter is the optimal linear estimator based on
hypotheses of observation noise, target acceleration statistics,
and of target dynamics.
The Kalman filter may use the three target direction cosines and
their first and second derivatives as the system state variables
>
with which to filter and predict the target direction.
The estimated
target direction is then given by a unit vector, made up of the three
direction cosine estimates.
Alternatively, the Kalman filter state
variables may be the vector range, velocity, and acceleration of the
target.
In this implementation, the target dynamics upon which the
Kalman filter is based will be simple kinematics in a cartesian
reference frame [ref. 4].
The analysis which follows deals with the
direction cosine approach, though the results can be applied to the
vector range, velocity, acceleration formulation.·
Observations for the Kalman filter are modeled as a linear
combination of the state variables and noise, as shown in equation 1.1.
(1.1)
y=Hx+v
where
·H
observation matrix
x = system state
v
observation noise
Equation l. 2 shows a possible formulation of the state vector.
The state vector is made up of the target direction cosines and
their first and second derivatives.
co~ines
The set of three direction
make up a unit vector in cartesian coordinates, as shown in
5
"A
.
X
1:
=
(1.2)
+
A
+
A=
where
equation 1. 3.
A
N,E,D
(1.3)
North, East~ Down components
of the unit direction vector
An observation of target direction can be written in
the form of a linear combination of the components of the unit
direction vector and noise as shown in equation 1.4.
matrix H is, therefore, defined as in equation 1.5.
The observation
The observation
(1.4)
y
H
=
(~
hE
hD
0
0
0
0
0
(1.5)
0)
model can be written using vector dot product notation as shown in
equation 1.6, which provides a geometric interpretation of the scalar
+
y
+
h•A+v
(1.6)
+
observation of the vector quantity A.
The observation y is used by the discrete Kalman ftlter as shown
"
in equation 1.7, where K is the Kalman gain matrix, and x(k!k
1) is
the estimated state vector at time k given observations up to time
6
;Cklk) = ~Cklk- 1) + K[y(k) - ~(klk- l)J
k- 1.
Equation 1.7 can be written as shown in equation 1.8.
(1. 7)
The
A
"'
x(ktk)·
....
x(klk-
+
1)
+ K[h •
+
+
(A - A)
+ v}
expression in the brackets is the filter residual.
(1.8)
The residual can
be obtained from the signals provided by a monopulse antenna.
is no need to form the observation as shown in equation 1.1.
There
The
method of obtaining the residual observation in such a manner that
minimizes the noise contribution of v is the topic of investigation.
The statistics of v are also investigated.
Chapter 2
SIGNALS AVAILABLE FOR PROCESSING
The radar signals observed from distant point targets can be
· written in the form shown in equation 2 .1.
j w. [ t
N
s(t,;)·
= Re{ L
ai(t) e
1
+
The signal is a function
+
+
(A • • r) I c]
1
(2 .1)
}
i=l
where
number of targets
N
wi
+
A.
1
c
frequency of reflected wave from target i
= unit
vector pointing in the direction of target i
velocity of wave propagation
=
a. (t)
complex envelope of return from target i
1
+
of time t and positio.n r.
It is assumed that within the region of
space for which the signal is observed (the physical dimensions of
the antenna), the amplitude of each component wave is not a function
of position.
The polarization of the component waves are not shown.
It is assumed that the polarization of each wave is constant within
+
the region o£ space for which s ( t, r) is observed.
\<lith the additional
:assumption that the polar response of the antenna is constant over its
surface, the effect of wave polarization is a constant \vhich can be
included in the factor a(t).
For a planar array antenna, the signal becomes:
N
s(t,x,y).
L
Re{
i=l
+
'rhe vectors e
+
X
and e
y
j t•} • [ t
a.(t) e
1
1
+
+
+..
+
xl c + e y I c) ]
A • (e
x
Y
}
(2.2)
are orthogonal unit vectors in the plane of the
7
8
array.
Introducing the antenna weighting function, I(x,y), the
equation becomes:
-+
jw.[t +A •
N
s(t,x,y) = Re{I(x,y)
I
a.(t) e
i=l
1
1
,-+
~xe
-+
+ ye )]
~ x
c Y }
(2.3)
If the array employs a time gradient technique to point the beam off
the mechanical boresight of the array, the antenna weighting function
can be modeled as shown in equation 2.4.
w
I(x,y)
!I(x,y)l
+
- .o~
J- H
e
c
•
[-+
xe
x
+ ye-+]
y
(2.4)
1\
-+
where
A
w
0
unit vector in the direction of the beam
transmitted frequency
If there is only one point target contained in s(t,x,y), or if
the signal of a target is separable from that of others by appropriate
filtering, then the signal can be used to determine the direction,
range, and range rate of the target.
The range rate is indicated by
the doppler frequency shift, while the range is indicated by a time
shift.
To see how the information on target direction can be extracted,
it is instructive to
form of
2.5.
s(t,x,y)~
inv~stigate
the three dimensional Fourier trans-
The general Fourier transform is shown in equation
Equations 2.6 and 2.7 are simplifications of equations 2.2 and
-jxw
co
F(w )
X
s.(x)
= ·
J
-oo
f(x) e
x dx
(2.5)
. .
(2. 6) .
9
-jxw
I(x)
=
jr(x)l e
X
(2.7)
2.4 where wT is the target spatial frequency, and wx is the estimated
spatial frequency, as defined in equations 2.8 and 2.9.
The spatial
Fourier transform of s(x) I(x), shown in equation 2.10 is seen to be
w
wT = - (A
c
. -+e )
w -+
= - (A
. -+e )
-+
w
X
c
X
X
-+
21f -+
= - (A • e )
=
21f
"
-+
-+
(A • e )
X
J:oo
s(x)ji(x)l e
jwT
= Joo aji(x)l e
-oo
(2.9)
X
A.
-jwx
S(w ) =
(2 .8)
X
A.
X
X
(2.10)
dx
-jwx
X
e
dx
-+
the response of the antenna to a target at A with the beam pointed in
...
-+
the A direction. Equation 2.10 is the equation for the far field
pattern of the antenna, given in texts on radar antenna characteristics.
The nomenclature of equations 2.8 and 2.9 have been introduced to
clarify the problem in the context of direction observations applied
to the Kalman filter.
The measurement of wT or of (wT - wx) will
provide an observation suited to the Kalman filter as discussed
earlier.
Scanning the mainlobe by beam steering techniques to locate or
track the target is equivalent to examining the spatial frequency
content of the return signal in the
pl~n~
of the antenna.
The set
of space-time frequencies at which the three dimensional Fourier
10
transform is a maximum is the observed location of the target in the
space-time frequency domain, providing target direction and range rate
information.
Equation 2.8 provides the relationship between spatial
frequency, time frequency, and direction.
Chapter 3
MONOPULSE MEASUREMENTS
3-1.
Synthesis
Antenna mainlobe scanning techniques are appropriate for
initial location and resolution of targets.
Once a target is
resolved, however, it is desirable to obtain measurements of wT or
w'f - w without resort to scanning.
-
X
This is accomplished by monopulse
angle measurement techniques [ref. 5].
The term monopulse arose due
to the ability to obtain an angle measurement on a single pulse,
since no sweeping of the mainlobe is involved.
The residual, used by the Kalman filter as shown in equation 1.8,
21T -+
=- e
A
-+
"'
-+
X
is related to wT and wx as shown in equation 3.1.
estimator is desired.
(3.1)
• (A - A}
A spatial frequency
The observed signal is given by equation 2.6.
The function s(x) can be used to determine the residual (wT - wx) as
shown in equation 3.2, where arg[u] denotes the phase angle of the
(3.2)
arg[s(z) s*(z - d)] = (wT- wx) d
complex quantity u and s* is the complex conjugate of s.
The residual
is indicated by the phase angle of the complex spatial autocorrelation
function of s(x).
The autocorrelation function provides valid direction information
only when a single spatial frequency i~ present.
A spatial frequency
filter must be used to exclude any undesired spatial frequencies
11
.12
before the spatial c_orrelation function. :is computed.
A(x)
= Joo
s(z) h(x -
Equation 3. 3 is
z) dz
(3.3)
-oo
the convolution integral which illustrates the method of spatial
frequency filtering.
The function h(x} 10 the spatial impulse response
of the filter, is the weighting function
the spatial filtering.
~f
the antenna which performs
The spatial freqmaency response of the filter
is given by the Fourier transform of h(x)..
the signal at the output of the antenna,
The real part of A(x) is
~hich
will undergo signal
processing to transform the time frequer.He_y down to the desired value
and synthesize the real and imaginary ccmmponents of the complex
quantity.
To provide the two signals necessary for forming the correlation
indicated in equation 3. 2, two weightinr.; functions satisfying
equation 3. 4 are required.,
The outputs
f,;)f
these trtJo antennas are then
used as shown in equation 3. 5 to obtain 1tihe residual observation.
arg[A* B] = (w
T
process is best illustrated by figure
~ uJJ )
:x
3~1-,
d
The
(3,5)
vJhich shows the configur-
ation of the antenna required to satisfy ·equation 3. 4 along the x-axis
for all directions of the return
identical.
signaL~
The two antennas are
Both are symmetric about the.ir own centers.
Figure 3.2
shows a possible arrangement .£or obtain..iflilg two orthogonal residuals,
as desired to completely determine the t:.arget ., s direction.
13
The technique just described is called phase comparison monopulse.
The target direction is indicated by the phase difference between the
signals of two spatially separated antennas.
Denoting the signals
from figu-re .3.1 as A and B, the measurement of the residual, equation
3.1, is obtained from the phase difference between A and B as shown by
equation 3.6.
The measured phase difference is used as an observation
A
cj>
=
arg[A* B]
=
Z'IT d
A
~
(~
•
X
-
~)
(3.6)
A
+
+
+
= h • (A - A)
by the Kalman filter, with equation 3.7 used as the row of the
+
h
= 2'IT
A
d +
e
X
observation matrix H of equation 1.5 for the computation of the
Kalman gain matrix.
(3.7)
14
FIGURE 3.1.
ANTENNA CONFIGURATION FOR PARTIAL DIRECTION MEASURE}1ENT
15
FIGURE 3.2.
ANTE~~A
CONFIGURATION FOR COMPLETE DIRECTION MEASUREMENT
3-2.
Angle extraction techniques
A measurement of
¢ can be obtained from the A, B signal pair
as discussed previously.
not in this form.
Often, however, the available signals are
In systems where angle measure is required, the
antenna is usually designed to produce the two signals, S and D,
which produce a measure of angle error by the ratio D/S.
is the sum signal.
The sum signal is produced by an antenna weighting
function that is even about the antenna center.
gain pattern is also even about the beam axis.
difference signal.
The S signal
The resulting antenna
The D signal is the
The difference signal is produced by an antenna
weighting function that is odd about the antenna center.
resulting antenna gain pattern is odd about the beam axis.
The
The antenna
is designed so that a target near the boresight of the antenna will
produce the S and D signals exactly in phase or 180 degrees out of
phase with one another depending on the direction of the error.
The
quantity D/S is then a real quantity which for small errors is
proportional to angle error.
It will be shm..rn later that D/S is a
complex quantity for targets that cannot be modeled as a single point.
There is a functional relationship bet;;..reen the S, D pair and the
A, B pair of signals.
An antenna weighting function that is even
about the center of the antenna can be produced by adding the A and B
signals together, producing the sum signal.
An antenna weighting
function that is odd about the center of the antenna can be produced
by subtracting the B signal from the A signal, producing the difference
signal.
The relationships are shown :in equation 3. 8 and 3. 9.
The
factor j in equation 3.9 is required to rotate the difference signal
16
17
S
=A+
D
= j(A-
(3.8)
B
(3. 9)
:B)
to bring it into phase with the sum signal.
Using equations 3.8 and 3.9, the angle measure D/S can be
written in terms of A and B, as shown in equation 3.10.
D
s
=
Assuming
Im{A* B}
t11A+BII
(3.10)
2
a point target and no signal processing errors or noise, the
magnitudes of A and B will be equal, and the angle measure given by
D/S is related to
~
as shown in equation 3.11.
The antenna can be
D/S = tan ¢/2
(3.11)
designed to produce a measurement from D/S that is linear in angle
(amplitude comparison monopulse), or a measurement from A* B that is
linear in phase (phase comparison monopulse), as in equation 3.6.
The
design goals are different, but an antenna designed for one mode of
operation can be used in the other mode of operation •.vith only slight
errors involved.
In any case, equation 3.6 or the assumption of
linearity of D/S are approximations only.
A list of five possible measures of angle or of
table 3.1.
~
is given in
The measures are written in terms of the A, B pair and
the S, D pair, using the relationships in equations 3.8 and 3.9.
relationship of each measure.with
~is
The
shown, with the assumption that
18
the magnitudes of A and B are equal.
The statistics of some of these
measures are investigated in chapters 5, 6, and 7.
The phase comparison direction measurement developed in the
. preceeding section does not provide. a measurement that is linear with
respect to angle error, as is usually desired for angle tracking.
It
can be sho~~. however, that there is little difference between the
two measures when the tracking error is small (much less than a
beamwidth).
Equation 3.6 can be rewritten in terms of angle error as
The angles n and n are measured from the
shown in equation 3.12.
<P
=
21T d
I'
(3.12)
[sin n - sin n]
A
- 21T
- -d
A
"
cos(n + n)[2 sin(n - n)]
2
2
+
antenna normal, in the plane of e and the antenna normal.
From
equation 3.12 it is evident that the phase error of a phase comparison
monopulse system can for small angle errors be approximated as shown
in equation 3.13, where
e
is the angle error and a is the angle at
which the beam is pointed from the antenna normal.
cos a.
(3 .13)
~-
NAME
f(A,B)
f(S,D)
Ml
arg[A* B]
arg[S* S - D* D + 2j Re{S* D}]
I
Im{A* B}
M2
z IIA+
M3
1:14
Ms
~
Bll
2
~
5
Im{A* B}
2 Re{S* D}
II A* BII
,[(S* S ~ D* D) + 4(Ra{S~ D}) 2] 1/ 2
Im{A* B}
2
2
!AII + IIBII l
n
2 Re{S* D}
¢
2 tan ¢/2
sin
<P
sin ¢
S* S + D* D
Im{A* B}
2 Re{S* D}
Re{A* B}
S* S - D* D
TABLE 3.1.
L__
2 Re{ D}
RELATION TO ¢
WHEN !!All= IIBII
tan
<P
MEASURES OF DIRECTION TRACKING ERROR
------~~~
I-"
\0
Chapter 4
THE EFFECT OF }UJLTIPLE POINT TARGETS ON MONOPULSE ANGLE MEASURES
The previous section considered the return signal of single point
targets.
The assumption of a single point target from which the
discriminant is.formed is often justifiable in multiple target cases.
where the targets can be resolved in angle, range, or range rate.
There are ·situations, however, where returns from two or more points
will be encountered which cannot be separated by doppler filters,
range gates, and the antenna beamshape.
In these situations, which
can arise from a single extended target at close range, multipath
reflections, or multiple independent targets, the direction indication
from any of the previously discussed techniques may not provide valid
direction information [ref. 6, pp. 13-22].
To illustrate the effects of multiple point targets on monopulse
angle measurements, it is easiest to investigate the response of the
monopulse measure to a two point target.
For illustration, measures
11 and N of table 3.1 are investigated.
2
1
Assuming that the two points lie in directions corresponding to
phase angles
<P
1
and <P , the signals due to each separate target are as
2
shown in equations 4.1 through 4.4.
The factors a
. <Pl
Bl = a 1 eJ
eji/J
A2 = a2
a2
and a
2
are the
(4 .1)
Al = al
B2
1
ej(I/J + 4>2)
20
(4.2)
(4.3)
(4.4)
21
amplitudes 1 and 1/J is. the relative phase of the two point returns,
The antenna gain in the direction of each target is assumed to be
included in the factors a
and a •
2
1
Using these assumptions, the
angle error will be indicated by the signals as shown in equations 4.5
and 4.6.
The resulting A* B and D/S signals are given in equations
A= A + A
2
1
(4.5)
B + B
1
2
(4.6)
B
4.7 and 4.8.
=
The D/S signal becomes a complex quantity, and is
referred to as the Complex Indicated Angle [ref. 7].
(4. 7)
A* B
<1>1
D
al sin
2+
.(1/1 +
al cos
If the directions
<1>
2
2 +
o~
.(1/J
eJ
a2
+
-
2
a2 eJ
s
<1>1
<1>2
<I>
-
2
2
<I>
1)
<1>1
<1>2
sin2
)
(4.8)
<1>2
cos
2
the two point targets are such that <1>
1
and
are not equal, then the directions indicated by A* B and D/S will
be a function of the relative phase 1/J between the two point targets.
As the relative phase changes, the D/S quantity will trace out a circle
[ref. 7] in the complex plane, while A* B will trace out a line.
The-measures of direction tracking error shown in table 3.1 will
all be affected by the relative phase of the two points of the two
22
point target.
glint.
The resulting angle measurement noise is referred to as
The effects of glint can be reduced if several measurements of
the signal pair A, B (or S, D) can be made with different phase
From equation 4.7 it is evident that if
relationships~-·
1
B.
A~
1
1
is averaged over sufficiently many ~., uniformiy distributed over 2TI,
1
the result will be a weighted measurement of $
equation 4.9.
I
A~
i=l
1
the greatest magnitude.
~
and
2
and
~
2
1
-
Na
as shown in
A~
1
B.1 with
2 dJ.$1
2 .$2
+ Na eJ
2
1
(4.9)
This will produce a weighted measurement of
Equation 4.10 is valid only
B.}
1
1
B.
as shown in equation 4.10.
Max{A~
when $
and $
An alternate scheme would be to use the
N
~l
1
(4.10)
1
2
are between ±TI, and the difference between
~l
and ¢
2
is less than TI.
Similar methods have been suggested using the D and S signals.
Using these signals it has been shown that good results are obtained
when the quantities S* D and S* S are averaged over
several~.
1
's.
'The quotient (S* D)/(S* S) is then a reliable measure of target
direction [ref. 8].
Another method is to use the D/S quantity that
exhibits the largest magnitude of S [ref. 9].
While the two point target is being observed by the radar, the
phase difference
~
between the two return signals may be changing due
to the relative movement of the two points aiong the radar's line
of sight.
Neglecting doppler effects, the phase difference is given·
23
by' equation 4.11, where D is the depth of the target, and A. is the
2D.
l
=--
(4.11)
A.i
wavelength o£ the radar signal.
The factor of two is required to
account for the two-way travel of the radar signal.
The quantity A* B, when filtered, will produce the quantity shown
in equation 4.9 if the relativemotion of the target points is
great enough with respect to the filter time constant.
A* B is not filtered, but its phase angle
<P
If the signal
is extracted and filtered,
the resultant filtered direction observation will not produce a proper
measure of direction.
This is due to the effects of the phase
difference term in equation 4.7.
and typical values of a , a ,
2
1
<P
1
A plot of A* B for all values of lJ!
, and $
2
is given in figure 4.1.
The error on the indicated target direction is correlated with the
amplitude of the quantity A* B.
The quantity A* B must be filtered
while the phase difference varies over-Zn before the phase angle
<P
is extracted.
Observations of the target with different phase differences lJ!.
l
between the two points can be obtained by the relative movement of
the target points.
Since this movement is not guaranteed and cannot
be physically induced when desired, another technique should be used.
Assuming the actual difference between the distanees to the two
target points remains constant, the phase differe;nce between the two
points can be changed by changing the wavelength of the transmitted
radar signal.
A change in vmvelength will change the relationship
26,
between
~
and
targe~
direction, as indicated by equation 3.6.
However, if the depth of the target is much greater than ten wavelengths, then a slight change in A. will produce a great change in 1jJ
while affecting the relationship between
~
and target direction very
little.
It should be recognized that the relationships given in table 3.1
between ¢ and the measures M and M are not correct for the two
2
4
point target case.
This is due to the fact that the magnitudes of
A and B are not necessarily equal when more than one point target
is being observed, as was assumed when the relationships bet,veen ¢
and M or M of table 3.1 were derived.
2
4
Since in a two point target
situation measures M and M are not measuring the same quantity
2
1
~.
the two measures can be expected to behave quite differently in this
situation.
The classic two point target situation occurs when a target flies
closely over a body of water.
If the target is observed from a point
near the surface of the water, specular reflectioBB will produce an
image of the target belovr the surface of the water.
Assuming that
the antenna mainlobe is directed midway between the target and its
image, the A* B and D/S signals are produced as shD<vn in figures 4.2
and 4.3, respectively.
Figure 4. 3 shows how tracking in such a sit1s:ation can be unstable,
since the indicated angle will be a function the phase difference 1/J
between the. two points, which will vary as the aurget moves,
This
phenomenon has been noted in ,actual tracking syst;ems [ref. 6,
pp. 285-286].
Figure 4.2 indicates that a proper1y designed system
I
'
25
using the A* B signal and M may provide more stable tracking.
1
The
phase a11gle of A* B remains constant, though care must be taken not
to be misled by the possible ambiguity
of±~, when A* B =-a ej~.
It is seen here that the phase angle of the complex correlation
function A* B maylprovide a more stable measure of target direction
that the measure D/S when two point targets of equal amplitude are
encountered.
In chapter 7 this same measure is shown to be the
maximum likelihood estimate of
number of point reflectors.
~
when the target is made up of a large
26
Im
1
Re
-1
1
a
-1
FIGURE 4.1.
1
=
1
a2
.5
$1
rr/4
q, 2
= rr/2
A* B FOR A TYPICAL TWO-POINT TARGET
27
a·
1
~
~
1
2
=
a
2
=
1
= rr/4
= 0
FIGURE 4.2.
A* B FOR AN EQUAL MAGNITUDE TWO-POINT TARGET
D/S
2
1
Re
q+
-t---c---J--+--t-----+-~4---+--T--c-
1
2
-1
1
-2
~1
~
FIGURE 4. 3.
=
'Jf/4
2 = 0
D/ S FOR AN EQUAL :t<IAGNITUDE TWO-POINT TARGET
Chapter 5
STATISTICAL ANALYSIS OF SIGNALS FROM POINT TARGETS
For target tracking using a Kalman filter, the variance of the
observation must be known.
The observation variance is used in
computing the optimal Kalman filter gains.
It is, therefore, of some
interest to investigate the statistics of the noise on the observation
~
of the direction error.
The observation
~
used by the Kalman filter will be corrupted
by antenna positioning errors, beam pointing errors, atmospheric
refraction, target angular extent, signal processing errors, and
thermal noise in the receiving electronics.
Only Gaussian receiver
noise will be considered here.
The signal obtained from the radar signal processor due to a
point target can be modeled as shown in equation. 5 .1.
The additive
(5.1a)
A
B = a ej~ + n
(5 .lb)
B
noises nA and nB are independent, zero mean complex Gaussian random
variables.
With these assumptions, the mean and variance of the
measures of
~
sho\-m in table 3.1 can be determined analytically.
Equations 5. 2 through 5. 5 sho·w the first and second moments
about zero of the observed
are derived in appendix A.
~-
from M of table 3.lft
1
The equations
Using these equations the variance and
mean square error of the observed
28
~
can be computed as shown in
29
co
S(k,SNRA)(-l)k+l sin k<P
E[Ml] = 2 .
k=l
k
(5.2)
co
2
1T
2]
cos k<j>
E•M
S(k,SNRA) (-l)k
l 1 =-+ 4
k2
3
k=l
(5.3)
I
I
1r
SNRA
S(k, SNR) =
-SNRA
e
4
2
= _ _a_ _
SNR
SNRA
SNRA 2
[Ik-l (-.-) + Ik+l (--)]
-2- 2
-2- 2
a
(5. 4)
2
(5.5)
A
equations 5.6 and 5.7, respectively.
The mean square error is the
term desired for use in computing the Kalman gain matrix.
(5. 6)
(5.7)
Equations 5.8 through 5.17 show the statistics of M as derived
2
in appendix B.
It is assumed that no observation is made when the
magnitude of S is less than the threshold T.
E[M
E [ (M2)
2
2
jlsl
> T] = 2 K E[D]
This requirement is
(5.8)
E[S]
jl sI .>
(5. 9)
T]
sl
K = SNR -S S
0
(5.10)
30
.00
sk
=
I
SNRi
i
I
s
i=O (i + k)! j=O
(5 .11)
.'
J.
(5.12)
00
p =
I
SNRi
i
t
s
(5.13)
i=O (i + 1)! (i + 1) j=O
00
I
(-x)
J• I•
i
(exponential integral) (5.14)
i=l i • i!
y = 0.57721566 . . . (Euler's Constant)
(5 .15)
(5.16)
(5 .17)
imposed due to the form of the measure M , with S in the denominator.
2
The variance of D/S is unbounded if the restriction is not made..
When
the restriction is made the variance is bounded anrl well behaved as
shown in figure 5.1, which shows the standard dev-iation of M for
2
various threshold levels as a function of SNR.
Equations 5.18 through 5.20 show the statistics of measure M
3
of table 3 .1.
Equations 5.16 through 5.18 were derived in reference 10.
31
-SNRA
=
E. SNR
4
E[M;] =
l
e
A
SNR
[I (~) +
0 2
(1 - (1 - 2y)
2
. SNRA
r 1 (--)] 2
sin
(5.18)
<P
2
(5.19)
cos 2</>)
2
y
1
= -=--
-SNR
(1 - e
A)
(5. 20)
2 SNRA
The variance of measures M , M , and M are shown in figure 5.2
3
2
1
as a function of SNRA for
<P
= 0.
It is seen that they exhibit
slightly different behavior at low SNRA.
behave nearly identically.
At high SNRA' however, they
They all approach equation 5.21
1
VAR(M) - - -
asymptotically as SNRA increases.
. (5.21)
Equation 5.21 is an accurate
approximation of observation variance for SNRA above 10 dB, as seen
by figure 5.2.
If SNRA can be estimated from measurements of signal
and noise power, equation 5.21 can be used to compute the observation
:variance as required for use by the Kalll'.an filter.
Additional noise
terms must be added due to antenna positioning errors and other
sources of error.
From figure 5.2 it may appear that M is the poorest of the
1
three measures of </>.
The lesser error of M and M at
2
3
<P ""
0 is
obtained only with a sacrifice in performance for other values of <f>,
·.· Figure 5. 3 shows the mean square error of measure M arising from the
1
32
Gaussian noise.
The curves were obtained from equations 5.2 through
5.5, and equation 5.7.
Figure 5.4 shows the mean square error of
measure M when used as an estimate of
2
~.
These curves were obtained
from equations 5.8 through 5.17, with TNR equal to one.
errors increase as
~
increases.
Note that the
Figure 5.5 is a similar graph of M
3
obtained from equation 5.18 through 5.20.
Here the tradeoff of
performance at zero error for poorer performance at greater tracking
error is more pronounced.
1.0
.8
TNR
= 1.
<P
=0
,6
STMl)ARD
DEVIATION
(radians)
.4
TNR
~
= 10.
~
..__
•2
0
2
4
6
8
SNR
FIGURE 5.1,
s
10
(dB)
12
14
16
18
STANDARD DEVIATION OF M VERSUS SNR
2
8
w
w
'
I
1.0
.8
.6
STANDA...liD
DEVIATION
(radians)
•4
.2
-2
0
2
4
6
8
10
12
14
SNRA (dB)
FIGURE 5.2.
STANDAP~
DEVIATION OF M1 , M2 , AND M3 VERSUS SNRA
w
.j:-..
~
35
.1.3
1.2
1.1
1.0
.9
.8
MEAN OF • 7
SQUARED
ERROR
CradianJ). 6
.5
.4
6 dB
.3
8 dB
.2
10 dB
12 dB
.1
~--------------·------·-----·~~~---------------14 dB
0
0
'IT
'IT
31T
'IT
S'IT
7'IT
'IT
9'IT
16
8
16
4
16
16
2
16
$ (radians)
FIGURE 5 . 3.
MEAN SQUARED ERROR OF M
1
WITH ADDITIVE GAUSSIAN NOISE
36
1.3
1.2
1.1
1.0
.9
.8
.7
MEAN OF
SQUARED
ERROR
•6
(radians 2 )
.5
.4
.3
.2
.1
'""""'--...,------....----...-----------or------· .. ' .
1f
1f
31f
1f
51f
31f
71f
1f
91f
16
8
16
4
16
8
16
2
16
cp
FIGURE 5.4.
(radians)
MEAN SQUARED ERROR OF M
2
WITH ADDITIVE GAUSSIAN NOISE
37
1.2
L1
1.0
.9
.8
•7
MEAN OF
SQUARED
.6
ERROR
2
(radians )
.5
• 4.3
4 dB
.2
.1
'If
37f
'If
8
16
4
16
77f
'If
97f
16
2
16
cp (radian:s)
FIGURE 5.5.
MEAN SQUARED ERROR OF H
3
WITH ADDITIVE GAUSSIAN NOISE
Chapter 6
ANALYTIC MODEL OF SIGNALS FROM EXTENDED TARGETS
In chapter 4 it was shown that the angular extent of a target
has a significant effect on direction measurements.
The depth and
angular extent of the target, together with the target angular motion,
produces both amplitude and angular variations (scintillation) when
conventional monopulse measures are used,
The Swerling I and II radar target models are analytic models
of the radar return signal [ref. 18].
The models exhibit properties
similar to the amplitude scintillation encountered in actual radar
systems, and are useful in obtaining analytic expressions for
detection probabilities.
The models can be extended to include the
effects of angle scintillation.
Such a model was proposed by
Sam Thaler, and is described in detail in reference 8.
The model
will be described here and applied to the question of error
statistics on the measurement of
~-
Assume that the target is made up of many points, each of which
vary in radar cross section and range (over at least a wavelength)
in a random manner with time.
The signal A from a monopulse antenna
system will be the sum of the contributions from each point.
If
each contribution is a similarly distributed complex·random variable
with uniformly distributed phase, then the sum will be approximately
zero mean complex Gaussian distributed.
If it is assumed that
enough d . me elapses between observations. (but not bet,veen individual
pulses of an observation) that amplitudes are statistically
38
39
t~e
uncorrelated, then the signal described is
signal
~eceived
from a radar target.
Swerling I model of the
The magnitude of the signal at
any point in time is Rayleigh distributed, while the power is
exponentially distributed.
A similar argument can be made for the B signal of the monopulse
system.
Since the A and B signals are both modeled as zero mean
complex Gaussian, it.would simplify matters to assume that over the
aspect angles of the target encountered, the A and B signals are
jointly Gaussian.
In such a case, the covariance of A and.B can
~
be written as shown in equation 6.1, where
is due to the direction
(6.1)
E[A* A]
E[B* B]
(J
2
(6.2)
of the target centroid and r is a real quantity between zero and one
related to the effects of target angular extent on the decorrelation
of the signals A and B.
The assumptions made in references 2, 8, and 16 were related to
the S and D signals, uhich were also assumed correlated complex
Gaussian with zero mean.
This assumption is identical to the same
assumptions on A and B due to the relationships between the two
signal pairs, as given in equations 3.10 and 3.11.
In reference 16,
it·is shown that the statistic r of equation 6.1 is related to
target angular extent
small cf>.
ew
approximately as shown in equation 6.3 for
It is assumed that the target. is uniformly distributed
over the width
<1>
w
in units defined by equation 3.13.
40
<1>2
--w
4
r
12
=
(6.3)
4>2
+ ·w
-
4
12
The joint Gaussian model allows Gaussian receiver noise to be
introduced in the statistics of the return signals with no change
in the form of the distribution of the signals.
The sum of two
Gaussian signals, the target signal and the noise signal, will also
be Gaussian.
For example, if the target has no angular extent, then
the target signal statistics are as shown in equations 6.4 and 6.5.
•,;.
E[A* B]
= eJ"'
E[A* A]
=
a
2
T
(6.4)
E[B* B]
(6.5)
If independent zero mean Gaussian noises of variance cr
2
n
are added to
the A and B signals, then the signals are still jointly Gaussian with
statistics as shmvn in equations 6.6 and 6.7.
The phase angle, 4>,
2
r
crT
=
2
crT +
(J
2
2
= crT
+
(J
2
(6.6)
D.
(J
2
n
of the complex correlation coefficient is unchanged.
(6. 7)
41
Using this model, the statistics of the measures of
obtained.
~
can be
In reference 15 it is shown that the first and second
moments of the measure M of table 3.1 is as shown here in equations
1
6.8 through 6.10.
In reference 16 it was shown that the mean and
variance of M is as shown in equations 6.11 through 6.13.
2
A
threshold is required to keep the variance bounded, as was the case
for the point target model in additive Gaussian noise.
00
E[M ]
2
1
I
S(k,r)
(-l)k+l sin kp
k=l
(6.8)
k
k
2
(-1) cos kp
S(k,r)
E[Mi] =..:!!..__+ 4
k2
3
k=l
00
I
r
k
r 2 c.!. + 1)
2
S(k,r) =
F c.!.,
2
k!
E[M ]
2
VAR(M )
2
E (x)
1
(6.9)
E[2D/S]
=fro 1
X Z
2
k+
'
1 •
~,
r2)
(1 + cos _j_) tan $/2
.!+ cos tP
r
2
2(1·- r)
(1 + r cos
k
~)
e
2
2
2
T /2a
T2
El (-~)
(6.10)
(6.11)
(6.12)
2a
e-z dz (exponential integral)
(6.13)
Chapter 7
MAXIMUM LIKELIHOOD ESTIMATION
An optimal estimator of a parameter can be formulated based on
the following information.
The statistics of the observations
conditioned on the parameter to be estimated must be known.
The
criterion by which the estimator is judged to be optimal must be
defined.
Depending on the criterion chosen, the a priori statistics
of the quantity to be estimated may be required.
The signals A and B from a monopulse antenna require nonlinear
processing to form quantities related to target direction.
It is
difficult to formulate a nonlinear estimator that properly exploits
a prior statistics.
For this reason the usual approach to estimation
in this type of situation is to first form an optimal estimate of a
parameter linear with respect to the quantity desired using an
optimality criterion that makes no use of a priori statistics.
This
quantity is then applied to a linear filter which makes proper use of
a priori statistics.
This is the approach taken here in estimating target direction.
Directior1 estimation is performed by two filters.
The first filter
is based on the statistics of the received signals using the maximum
likelihood criterion.
The second filter (a Kalman filter) is based
on the statistics of the linearized observation, the statistical
model of the system state, and the minimum mean square error
criterion.
The resulting estimate has. nq pretention of overall
42
43
optimality for any single criterion, hut the optimality of each stage
gives one confidence in the approach.
The estimator of the quantity
based on the observed A and B
<P
signals can be formulated using maximum likelihood techniques.
The statistics of A and B are assumed to be those developed in
chapter 6.
Gaussian.
The signals A and B are zero mean, jointly complex
~~l'
The maximum likelihood estimate,
equation 7 .1.
of
<P
is as shown in
It is assumed that N independent samples of the
N
MLl =
arg[
<P
I
signal pair are available.
2
i
J_
The independent samples can be obtained
by frequency agility [refs. 8, 9].
estimates of r and a
(7.1)
A* B.]
i=l
Joint maximum likelihood
of equations 6.1 and 6.2 can also be formulated.
They are shown here in equations 7.2 and 7.3.
These maximum likelihood
estimates are derived in appendix C.
"2
a
1
N
I
(A~
J_
2 i=l
r
1
=
~
a
A.
J_
+
(7. 2)
B* B.)
i
l
N
l.
i=l
(7.3)
A*i B.J_
The statistics of the maximum likelihood estimator given by
equation 7.1 is given by equations 6.8 through
where N
=
1.
6~~0
for the case
In appendix D it is shown that the }iirobability
density function of the maximum likelihood estimat:'e ML
1
of
<P
is
44
'
· given by equation 7. 4 for any N.
The f:irst and second moments
f(<j>)
(7. 4)
1
+ r
2
m
2 "'
cos (<I> - ¢)
I
{1- r
2
2 "'
.
cos (¢ - <j>)]- 1
i.=l
X
i=l
Tr
(2N
- 2k
+ l)
k=l
2N -.
2"~
"
-1.
+ r cos(p - ¢) sin
{'o
(r cos(<j> - <j>))
[1- r2 cos2(; - <j>)]N+l/2
x
N-1
Tr (2N
k=l
of ML
1
- 2k
+ 1)
2N - 2k
about zero are given by equatious 7.5 through 7,7.
The
function f(x) is the Gamma function, while the function F(a,b;c;z)
is the hypergeometric function as defin.e.d in reference 11.
These
equations are also derived in appendix lD.
B(k,N,r) (-l)k+l sin ~!.
z l:
k
k=l
E[}1L~]
2
= _:!!__
3
00
+
(7 .5)
4
I
k=l
S(k,N,r)
(-l)k cos k¢
--~--
k
-
(7.6)
'
45
k
2 N
k
S(k,N,r) = r (1 - r _) r (N + k) f(l + k) F(N + k 1 + -; k + 1; r2)
'
2
2
2
2
(N- 1)! k!
=
rk f(N + k) f(l + ~)
2
2
F(l + ~ - N, k. k + 1; r 2 )
'
2
2·
(N - 1)! k!
(7. 7)
Figure 7.1 shows the probability density function of ML
r
=
0.5 and N
=
1, 2, 4, 8, and 16.
density function of ML
1
for N
=
1
for
Figure 7.2 shows the probability
2 and r = .5, · .75, and .9.
The maximum likelihood estimate of
~
is closely related to the
maximum likelihood estimate of angle error which is developed in
references 2, 8, and 16, and elsewhere.
The maximum likelihood
estimate of angle error is derived from the assumption that signals
S and D are zero mean, jointly Gaussian random variables.
same model discussed in chapter 6.
This is the
The statistics of S and D can be
written in tenns of the statistics of A and B, as shown in equations
7.8 through 7.10.
Equations 3.8 and 3.9 were used to arrive at
these relationships.
E[S* D]
2r cr
E(S* S] = 2cr
2
2
sin
+ 2r
(7.8)
~
cr
2
cos
~
(7.9)
E[D* D]
(7.10)
In reference 16 the joint maximum likelihood estimates of F
and c
2
are derived, where F and c
and 7.12.
2
are defined in equations 7.11
The reference relates the statistics F and c
2
to target
angle centroid and radius of gyration (a measure of angular extent).
46
F =
E[S* D]
(1 + cos cp) tan rp/2
l+ cos rp
r
E[S* S]
c
2
=
E[D* D]
2
1 - r
- F* F =
(7.11)
(7.12)
(1 + r cos rp)2
E[S* S]
Their maximum_ likelihood estimates are shown in equations 7.13 and
From the foirn of equation 7.13, it is evident that the
7.14.
N
I
lML
2
2
s~
].
i=l
F
D.].
(7.13)
N
I
s~
].
i=l
s.
].
N
"2
c
I
=
D~
_].
i=l
D.
-- ]_
- F*
(7 .14)
F
N
I
s~
]_
i=1
s.
].
quantity F is related to the measure M of table 3.1.
2
F, a maximum
likelihood estin1ate ofF in equation 7.11, can be used to estimate rp.
The probability density function and first and second moments
of the maximum likelihood estimate of F were derived in references 16
and 17.
The results are given in equation 7.15 and 7.16 in terms
E[ML ]
VAR[ML ]
2
2r sin cp
E[2F]
2
(7.15)
1 + r cos cp
=
1
2(
N
)
1
1 - r
(1 +
2
r cos ¢)2
(7.16)
47
2
Using these equations the maximum likelihood
ofr, <P, and cr •
measures ML
and ML
1
2
of
<P
can be compared.
Figure 7.3 shows the standard deviation of measures ML
for
= 0 and N = 4 as a function of SNR.
<P
1
and ML
2
The statistic r is related
to the effective SNR by equation 7.17, which is obtained from
SNR
r = ---1 + SNR
equation 6.6.
to ML
SNR.
1
(7.17)
From the figure it would appear that ML
2
is superior
for SNR less than 12 dB, and both are equivalent for greater
As was the case in chapter 5, the measures should also be
compared at values of <P other than zero.
Figures 7.4 and 7,5 show the mean square error of the estimators
ML
1
and ML
2
as a function of
equivalent rand
<P
w
<P
for various values of SNR.
The
parameters of equations 7.17 and 6.3 are shown.
Here it is seen that at high SNR, ML
is superior.
1
the benefits of the lower error of ML
1
At lower SlnR,
for large <P diminish.
The
appropriate estimator may thus depend on the tracking errors and noise
levels expected to be encountered.
.5
.4
r
= 0.5
.3
f(cp-<fl)
.2
.1
-IT
FIGURE 7.1.
'IT
0
2
'IT
'IT
2
PROBABILITY DENSITY FUNCTION OF
~~
1
FOR CHANGING N
1.2
1.0
.8
.6
.4
.2
-'IT
FIGURE 7.2.
'IT
2
0
'IT
2
PROBABILITY DENSITY FUNCTION OF ML
1
FOR CHANGING r
i.O
.8
STANDARD
DEVIATION
(radians)
....
• 6 .•
.4
I
ML1
N
'"
~
=4
.2
2
4
6
8
14
12
10
16
SNR (dB)
FIGURE 7. 3.
STA...lli!DARD DEVIATION OF ML
1
AND ML
2
VERSUS SNR
,.,..
\,0
50
.30
.25
SNR = 4 dB
(r = 0.715)
.20
MEAN OF
SQUARED
ERROR
2
(radians )
.15
6 dB
(r = 0.8)
.10
8 dB
(r = 0.94)
.05
16 dB
1------_(.:.:r:_=_0:...:•..:..9_7LJ.....;.':......:.1>.:..w_=_1T;_/4.;.:.)_ _ _ _ _ _ _ _ _ _ _ _
1f
1f
31f
1f
51f
16
8
16
4
16
cp
FIGURE 7. 4.
31f
-8
(radians)
MEAN SQUARED ERROR OF ML
1
71f
1f
9n
16
2
16
51
.30
.25
.20
.15
SNR
MEAN OF
SQUARED
ERROR
(r
=
4 dB
= 0.715)
2
(radians )
.10
8 dB
0.94
-L-----(r =
.os
l
16 dB
(r =
0.975,
. :::
1T
1T
31T
16
8
16
1T
5Tf
31T
4
16
8
-
<I>
FIGURE 7 • 5 •
---
1f
91T
16
2
16
(radians)
MEAN SQUARED ERROR OF JIU.
2
AS
.fu~
-
7rr
ESTHIATE OF ¢
SUMMARY
A new direction error estimator for use in target tracking with a
Kalman filter has been developed.
A statistical analysis shows that
this estimator does have advantages over the maximum likelihood angle
error estimator when both are used to estimate
~.
Since the degree of
improvement depends on tracking error, the benefits of using the new
estimator will depend on the tracking errors encountered by the closed
loop tracking system.
The new estimator has less mean squared error
at greater levels of tracking error.
The actual characteristics of
the monopulse radar antenna can also play a role in making one
estimator desirable over the other.
The key contributions of this paper are the analytic expressions
for the statistics of the tracking error measurements.
The statistics
of measurement M with the assumption of additive Gaussian noise is
2
of particular interest since M is a popular tracking error measurP.2
ment.
Previous attempts have failed to show the functional relation-
ship between the threshold placed on the sum signal and the. statistics
of the measurement.
The relationship is given by equations 5,8
through 5.17 and illustrated in figure 5.1.
The appendix contains the derivations of the new results
presented in this paper.
52
REFERENCES
1.
J. W. McGinn, Jr., "Thermal Noise. in Amplitude Comparison
Monopulse Systems," IEEE Trans. Aerospace and Electronic Systems,
Vol. AES-2, pp. 550-556, September 1966.
2.
E. Mosca, "Angle Estimation in Amplitude Comparison Monopulse
Systems,tt IEEE Trans. Aerospace and Electronic Systems,
Vol. AES-5, pp. 205-212, March 1969.
3.
R. A. Singer and K. W. Behnke, "Real-Time Tracking Filter
Evaluation and Selection for Tactical Applications," IEEE
Trans. Aerospace and Electronic Systems, Vol. AES- 7, pp. 100-110,
January 1971.
4.
R. A. Singer, "Estimating Optimal Tracking Filter Perfonnance
for Hanned Maneuvering Targets," IEEE Trans. Aerospace and
Electronic Systems, Vol. AES-6, pp. 473-483, July 1970.
5.
D. K. Barton, "Radars, Volume 1, Honopulse Radar," Artech House,
Inc., Dedham, Hassachusettes, 1974.
6.
D. K. Barton, "Radars, Volume 4, Radar Resolution and Multipath
Effects/' Artecb House, Inc., Dedham, Massachusettes, 1975.
7.
S.M. Sherman, "Complex Indicated Angles in Monopulse Radar,"
Ph.D. Dissertation, University of Pennsylvania, December 1965.
8.
S. Thaler, "The Accuracy of Radar Angle Measurements,"
TIC 2764/63, Hughes Aircraft Co., Culver City, California,
August 1973.
9.
J. H. Loomis, III, "Frequency-Agility Processing to Reduce
Glint Pointing Error," IEEE Trans. Aerospace and Electronic
Systems, Vol. AES-10, pp. 811-820, November 1974.
10.
H. S. Nussbaum and H. J. Krizek, "Expressions for the Mean and
Variance of an Angle Discriminant," TIC 2241.24/117,
Hughes Aircraft Co., Culver City, California~ September 1976.
11.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions, Dover Publications, Inc., New York, 1965.
12.
I. S. Gradshteyn and I. W. Ryzhik, Table of Integrals Series
and Products, Academic Press, New York, 1965.
13.
A. Papoulis, Probability, Random Variables, ~nd Stochastic
Processes, McGraw Hill Book Company, San Francisco, 1965.
53
14.
M. R. Spiegel, Mathematical Handbook of Formulas and Tables,
McGraw Hill Book Company, San Francisco, 1968.
15.
A. J. Rainal, "Monopulse Radars Excited by Gaussian Signals,"
IEEE Trans. Aerospace and Electronic Systems, Vol. AES-2,
pp. 337-345, Nay 1966.
16.
S. Thaler, "The Mathematical Properties of Selected Monopulse
Techniques," TIC 2241/67, Hughes Aircraft Co., Culver City,
California~ August 1976.
17.
I. Kanter, "Multiple Gaussian Targets: The Track-on-Jam
Problem," IEEE Trans. Aerospace and Electronic Systems,
Vol. AES-13, pp. 620-623, November 1977.
18.
P. Swerling, "Probability of Detection for Fluctuating Targets,"
IEEE Trans. on Information Theory, Vol. IT-6, pp. 269-308,
April 1960.
APPENDIX A
Statistics of arg[A* B] for a Point Target
and Additive Gaussian Noise
The signal model is given in equations Al. through A6.
A and B
are complex quantities with magnitude c and relative phase angle
<f> .
. SA
A
n
A+ nA
= aeJ
(Al)
bej (<!> + SB)
B = B+ ~
n
2 "<!>
A* B = c eJ
[A[
E[n! nA]
(A3)
[Bj = c
(A4)
E[n~ nB]
E[nA] = E[nB]
(A2)
2o
2
(A5)
(A6)
0
The quantities nA and nB are independent complex Gaussian random
1
. b ~es
• h zero mean an d var1ance
.
2a 2 .
var1a
w1t
The sum of the signals
. and noise will have a relative phase that is perturbed by noise, as
represented by phase angles SA and eB.
The statistics of the phase angle between two complex quantities
A
n
and B
n
is most easily derived using a Fourier series expansion
of the phase angle.
The measured phase difference between two
complex quantities will be between
plu~
phase differences that may be greater.
55
~rid
minus pi, despite actual
This is easy to represent as
56
a Fourier series, as shown in equation A7.
Equation AS gives the
00
(-l)k+l sin kp
k
k=l
I
<PM
=2
1>2
= ..:!!:_ + 4
2
M
(A7)
00
3
(-l)k cos k<P
k2
k=l
I
(AS)
square of equation A7 [ref. 14, equations 23.9 and 23.14].
From
these equations, the first and second moments of arg[A* B ] about
n
zero are given by equations A9 and AlO.
n
Both statistics are related
to the expected value of cos k8 as shown in equations All and Al2,
co
E[arg[A* B ]]
n n
2
I
(-1)
k=l
2
2
E[(arg[A* B ]) ] =..:!!:_+ 4
n n
3
k+l 1
.
- E[s1n k(<f> + eB- eA)]
k
00
I
k=l
(-l)k
~ E[cos k(<f> + eB - eA)J
(A9)
(AlO)
k
(All)
2
sin k<P E [cos k8]
57
(Al2)
2
= cos k~ E [cos
where
e
k8]
is the phase angle of the bivariate GausRian random variable.
The joint probability density function of the magL'litude and phase
angle of a bivariate Gaussian random variable is given by equation
14-62 of reference 13, shown here in equation Al3.
The expected
value of cos k8 is given by equation Al4.
2 + 2
[_ b
c
2
2cr
f(b,8)
+ cb cos
cr
6]
2
(Al3)
E[cos k8]
(Al4)
Equation 9.6.19 of reference 11, shown here in equation Al5, can
be used to evaluate the integral of equation Al4· with respect to 8.
e
Equation A16 results.
z cos
e
cos(n8) de
=
I
n
(AlS)
(z)
Equation Al6 can be evaluat:<ed using equation
E[cos k8]
6.631.7 of reference 12, shown here in equation A17.
(Al6)
Using this
58
2
"".
J0
x e -a.x
J (Sx) dx
= ;;
B- e-
132
v
/Sa [Iv
(£) - I
- - -1 8a
8a 3 / 2
2
2
(£)
~ + l 8a
2
l
(Al7)
2
equation along with equation Al8, equation A19 results, where
I
S}TR
A
=
2
2
c /2cr •
(x)
n
= j-n
J
n
(jx)
(AlB)
The desired statistic is then given by equation A20.
E[cos k8]
(Al9)
SNRA
- -2- /n SNRA
,_k
e
= (j)-'-
SNRA
[I
2
1r
2
E (cos k8]
SNRA
= --
-SNRA
e
SNRA
[Ik-1 (--) + I
2
4
SNRA
(- - - ) - I
(---)]
k+l
2
k-1
2
2
2
2
SNR
(-~) ]2
k+l 2
(A20)
2
Combining equations A9 through Al2 and A20 gives the desired result,
equations A21 through A23.
00
E[arg[A* B ]]
n n
E[ (arg[A* B ])
n
n
2
J
2
(-l)k+1 S(k,SNRA) sin
k=l
k
I
2
= -. + 4
.
7r
3
cO
I
k=1
(-l)k S(k,SNRA)
k~
cos~
k2
1r SNRA
-SNRA
SNRA.
SNR
2
S(k,SNR) -- - - = e
[Ik_ (-.- -.) + Ik+l ( ? A)]
1
2
4
2
2
-
(A21)
(A22)
(A23)
APPENDIX B
Statistics of 2R {D /S } for a Point Target
e n n
and Additive Gaussian Noise
The signal mode is given in equations Bl through B4.
The
sn = s + ns
(Bl)
.en
D + nD = ~eJ
D
n
E[nfi ~]
E[n§ n 8 ]
2cr
(B2)
2
(B3)
SNRS = !s!2/2cr2
quantities n
variables.
~
and
8
(B4)
are independent complex Gaussian random
Since S and D have the same phase, for simplicity they are
What is desired is the statistics of 2R {D /S }
assumed to be real.
e
The probability density function of S
given that Is I > T.
n
s
= !s
n
I.
a2 + A2
a
f
as, 8 s
(a , e
s
s
)
= -
[- __:::_s~8
2rrcr 2
for a > 0, -rr <
es
n
cos
cr
<
2
(BS)
rr
which a < T are excluded, the resulting random variable S
59
for
+--=-s-
2cr2
e
If values of S
Aa
n
is given
n
by equation B5, where A= Is! and a
n
e
is
60
.e
distributed as shown in equations B6 and B7, where S =a eJ •
e
1
f (a, 8) = - fa
K
for a > T,
f
1
1 .e
Let U = - = -- eJ
Se
a
2R {D /S } for
e
n
n
Isn I
a ,8
s s
e
(a,-8)
(B6)
s' s
-~
e
<
< ~
(a ,e ) de
s
s
s
da
(B7)
s
Then the first and second moments of
T are given by equations B8 and B9.
>
2E[D
E[2Re{Dn/Sn}llsnl > T]
n
The
J E[U]
(B8)
2
E [ ( 2R {D Is } ) 11 s I > T]
e n n
n
(B9)
2
2
= 4E[(Re{D }) ] E[(Re{U}) ]
n
statistics of D are trivial since D is Gaussian distributed.
n
n
required statistics are given in equations BlO through Bl2.
E(D ]
n
E[ (R {D } ) 2 ]
e n
2
E[(I {D }) ] =
.
m n
The
The
D
(ElO)
2
D2 + cr·
(Bl1)
IJ
2
(Bl2)
statistics of U are given by equations Bl3 through BlS, which can
be evaluated using the joint density function of a and
e.
61
1
E[U]
E[- cos 6]
a
(Bl3)
2
2
1
E[(R{U}) ] = E[z
cos 6]
e
a
(Bl4)
2
. 2 e]
E[(I {U}) ] = E[ 21 s1n
m
a
(BlS)
Equation Bl3 is evaluated as follows:
E[ U]
-1
= El-
cos 6 ]
(Bl6)
a
1 fro f1T 1
=·
- cos e f(a,e) de da
K
T
-11
a
e
Aa cos 6
-SNR
1
= ---,.,
e
s
a
2
e de
cos
da
21rKa£
Using equation AlS, this becomes:
E(U] = ___1__
2 e
-SNR
s
fro
2
2
Te-a /2a [21T I (Aa)] da
("817)
l a2
21TKG
Expanding r (x) using equation Bl8, equation B17 becomes:
1
(x/ 2 )1+2i
i==O i! (i
E[U]
1
= -2
Ka
-SNR
e
s
I
(SNRS)2i+1
i =0 i !
+
fooT
a
(i
+
1) !
(Bl8)
1)!
2i+l
e
2
2
-a /2a
da
(Bl9)
62
<X>
I
x
n
e
-px
dx
e
n
k
,
I
-up
n~~ _ 1 } _
(B20)
k=O k! lln-k+l
u
Using equation 3.351.2 of reference .12, shown here in equation B20,
equation Bl9 becomes:
E[U]
1
=--e
-(SNR
+ TNR)
s
00
(SNR )Zi+l
.I
s
i=O
2Ki
+ 1)!
(i
i
( 2 crz) i+1
I
j=O
(TNR)j
(B21)
• I
J.
Rearranging terms gives:
1
SNR
A
I
-(SNR + TNR)
s
s
E[U] = - - - e
I
~NRi
TNRj
i=O (i + 1) ! j=O
K
(B22)
j !
Equation B8 has now been evaluated, as shown in equation B23.
E[2R {D Js
e
n
n
}I Is n I
>
D SNRs
T]
= 2--- e
S
K
(B23)
-(SNR + TNR)
s
i
00
I
i
I
SNR
i =0 ( i
+
1) ! j =0
• I
J.
Equation Bl4 can be evaluated as follows:
i
E[2 cos
a
1
=-
K
2
oo
(B24)
6]
1
a
2--2e
T a 21TG
Aa cos 8
2
a
J
cos
2
s de
da
Aa cos 8
a
2
1
1
[-+-cos 28] d8 da
2
2
63
-SNR
1
=-·2
e
s
K1Tcr
expand r
00
I
i=O
E[
2 cos 2
1
and r
0
(x/2)2i
• I
1..
2
I
=
• I
1..
(x/2)2(i+l)
i =0 i ! ( i + 2) !
(B25)
e]
a
1
=--e
2
2Kcr
-SNR
s
(A2/2cri2!)i+l(l/2cr2)i+l fooT
+
i=O
(i
+
a
2i+l
2)t
= -1-2 e -SNRs [fooT al e-a2/2cr2
da
oo
+ L'
i=l
2Ka
,oo
a
X
2i-l
e
-a 2/2 cr 2
00
da +
JT
X
I:
I
i=O
a
2i+l
2
2
-a /2cr
e.
da
(SNR ) i
------"s'---- ( 1
)i
_ I 2 ,.,2
v
i! i!
(SNR )i+l(l/2cr 2 )i+l
s
. I
1..
(;i + 2)!
J
The integrals in equation B25 can be evaluated using equation B20,
as shown in equation B26.
The remaining integral in equation B26
is the exponential integral [ref. 11, equations 5.1.1 and 5.1.11] as
64
1
E[2 cos
2
-SNR
1
e] = - - e
2Ko
a
uoo _l
s
2
e-a 2 /2o 2 da
(B26)
T a
(S"t-!"'R )i+l
co
i TNRj
1 -TNR
s
+-e
2
. I
2
i=O (i + 1) i! j=O J.
2
l:
. 1
+ -e
-TNR
E (TNR)
1
s
I
i
I
i=O i! (i + 2)! j=O
2
shown in equation B27.
(SNR )i+l
i! TNRj
• I
J
J.
Equation Bl5 is essentially solved, as seen
=
I
1
co
- e
-z
dz
2
2
T /2a z
= -y
L
- in TNR -
n=l
by comparing equations B24 and B28.
(B27)
n n!
The only difference is the change
of sign within the integral.
1
E[2
a
·
Sl.U
2
e1_
(B28)
Aa cos 8
0
2
65
Aa cos e
o·
1 - e -SNRs fooT la e -a2 /2(/ Io1T e
=K7Tcr
2
1
1
[ - - - cos 2e] de da
2
2
2
The value of K, as expressed in equation B7, can be evaluated
as follows:
e
Aa cos
a
K =
1
2
-SNR
=- e
a
-(SNR
=
e
2
de da
00
s
I
a
2i+l
(B29)
e
2
2
-a /2a
da
i=O
s
+ TNR)
00
I
SNRi
i=O . i!
i
I
j=O
TNRj
j!
The presentation of the results is greatly simplified by introclueing the following function representations:
co
Bk
00
p
i
SNRi
I ----- I
i=O (i + k)! j=O
I
2
l• .I
(B30)
• I
J.
i
SNRi
i=O (i + 1)
TNRj
I
j=O
TNRj
. I
J.
(B31)
66
The results are:
E[ZR {D /S}
e n n
I
Isn I
.
D 81
S
= 2--- SNR
> T]
13
s
(B32)
0
E[ 1 cos 2 6] = _l_ eTNR ~[E (TNR) + SNR e-TNR(P + S )]
2
2
a
4cr2
SO 1
s
(B33)
E[ 1 sin 2 6] = --1-- eTNR Jl[E (TNR) + SNR e-TNR(P- S )]
2
2
a
4cr2
SO 1
s
(B34)
Using equations B9, B14, B15, B33, and B34, one obtains:
E[(2Re{D /S })
n
n
2
1
Isn I
>
(B35)
T]
+ 4cr 2 --1--~[eTt{R E (TNR) + SNR (P4i
=
1
s0
5
s2)]
~[eTNR E (TNR) + SNR (P)]
s
s
1
0
2
2
+ .Q_ __S_[eTNR E (TNR) + SNR (P + S )]
s2 cr2So
=
8
1
y(SNR ,TNR) + (2 D)Z a(SNR ,TNR)
s
s
s
2
67
where
y(SNR -,TNR)
s
= ~[eTNR
13
E (TNR)
1
+
SNR (P)]
(B36)
. s
0
SNR
8
a(SNR ,TNR) = - - [eTNR E (TNR)
1
s
213
0
+
SNR (P
s
+
13 )]
2
(B37)
APPENDIX C
Maximum Likelihood Estimate of
~
Assume that N independent samples of the signal pair A., B.
:L
are available.
:L
A. and B. are jointly complex Gaussian distributed
:L
:L
with statics as given in equations Cl through C3, where r is a real
= E[B* B]
E[A* A]
(J
2
(Cl)
E[A* B]
(C2)
E[A] = E[B] = 0
number between zero and one.
(C3)
The joint density function of each pair
A., B. is given in equations C4 and CS.
:L
The joint maximum likelihood
:L
-l/2(A. B.)
c
f(A,B)
:L
(J
:L
(C4)
2
p
(CS)
• ,!,
re-J'Ya
2
estimate of a , r, and
~
2
(J
2
is sought.
The maximum likelihood estimate of the parameters r, ¢, and a
2
of the signal pair is the set of parameters which, when used in the
probability density function of the signals, maximizes the function
at the values of the signals observed.
rhe likelihood function, as
it is called, is given by equation C6 and C7.
68
The estimates are
69
".
I
N
"2
L({A.,B.} r,cp,e)
1
1
C
"-1~~~
B*
-1/2(A. B.) p
--lT ----:-e:
112
1
1
(C6)
i
i=l 1:P1
"2
~
jr@"2
a
re.
(J
p =
(C7)
"-"cjl"2
~:z
re J a
G
chosen to maximize equation C6 by differentiating equation C6 with
.....
.I\,
•
-"'2
respect to r, cp, and a , and equating tfhe three resulting equations
with zero.
The system of equations is <then solved, producing the
maximum likelihood estimates of r, cp, amd a
2
in terms of the signals
observed.
The problem is most easily approached by using the natural
logarithm of the likelihood function, rather than using the likelihood
function directly.
The log-likelihood: function can be written:
B*
"-l~~li
Constant
p
i
On.E~
The constant need not be retained.
"2
"·v2
reJ
a
a
"-·<V2
I
"2
re J a
r~
N
-Q.n
(A. B.)
i=l
a
seeks to maximize:
l
1
-1
. cp "2
"
"2
reJ a
a
A~
1
I
L" _.; .
"2
: re J a 2
B~
a
l
l
= -Q,n(l
-
"2
r
)
-
"2
Una
1
.~2(1
-
N
I
"2
r ) i=l
(A~~
l
A. +
l
B~
1
B.)
1
(C8)
70
+ ~___:::.2 . : :. .r---::-- Re { e- j
"2
"2
o (1 - r )
"
N
~ (A-If A. +
1.
l.
i=l
<1>
L
B~
B.)}
]_
l.
N
A~ B.], since this will
i=l l. 1.
maximize the real part of the quantity in the brackets.
By inspection it is evident that
<1>
= arg[ ~
Maximum likelihood estimates of o
2
and r remain to be derived.
aF
ao"2
=-
(C9)
2.
+
"2
0
"2 2
N
1
N
I A~
"2 (
(o ) (1 - r ) i=l
A. +
l.
I
i=1
l.
3F
(ClO)
ar
==
2r
---;_2
1 - r
"''
2r
N
I
"2
oL(l - r ) i=l
(A~ A. + B~1 B.)
+
1
l.
l.
"2
N
A~ B.j
) 2j
· i=l l. l.
------,..-2=2--=-,..-2--(1- r ) o
+
(1
II
r
I
0
Equation C9 provides:
"2
0
N
1
2(1 -
"2
r
)
<I
"
(A~A.+B~B.)-2r!l
]_
i=l
l.
l
]_
N
I
i=l
(Cll)
A"":B.jj)
l.
l
Substituting into equation ClO yields:
(A~ A.
l.
Zr
I
(l _ ;2)2 i=l
(Ai; A.
1.
1.
]_
+ B~
"
B.) -
l.
+ B* B.) + (l +
1
1.
(l _
l.
2r
N
II i=l
I
I
;z) zl!
;z>z 1=1
A-!; B
l.
A* B
i
.II)
(Cl2)
]_
.11
1
=
o
71
Rearranging yields: ·
N
211 I
r
= N
I
i=l
A~ B.j t
1
(A-t: A.
i=l
1
1
1
(C13)
+ B*i B.)
1
Substituting this back into equation Cll yields:
~2
= 1:
¥
2 i=l
(A~
1
A.1 +
B~
1
B.)
1
(Cl4)
APPENDIX D
Statistics of the Maximum Likelihood Estimate of
From appendix C, the maximum likelihood estimate of
shown here in equation Dl.
<f>
<f>
is as
Each pair of signals A., B. are complex
.•
.
~
~
N
arg[
L
A* B.]·
i
i=l
(Dl)
1.
jointly Gaussian random variables with statistics as shown in
Each B. can be written as shown in equation D4,
equations D2 and D3.
~
2
E[Bi; B.)
~
J
E[A1; A.]
~
J
( cro
for i
f
for i :/: j
., 2
reJ'~'cr
E[A1; B.]
~
J
j
(D2)
for i = j
=
(D3)
0 for i :/: j
(D4)
B.
~
where each T. is a complex Gaussian random variable independent of A.,
~
~
2
with zero mean and variance a •
Using vector notation, as shmvn in
' equations DS ·through D7, equation Dl can be written as shown in
(DS)
B
(D6)
=
72
73
Tl
T
T2
(D7)
=
TN
equation DB.
The vector inrter product AtT deserves special attention.
arg[A
t
~]
t
'd>
.I
arg[A (reJ 'A+ /1- r
2
_!)]
(D8)
Vectors A and T are independent zero mean complex Gaussian random
variables of dimension N.
The probability density function of T
is the same in all directions in 2N-dimensional space (including real
and imaginary components).
Because of this, the statistics of the
inner product of thj_s vector with a second vector is a function only
of the magnitude of the second vector, not on its direction.
If the
magnitude of A is known (the norm of A over 2N dimensions), then the
statistics of AtT are determined by the statistics of T.
The
distribution of AtT given the magnitude of A is zero mean, complex
2 2
Gaussian with variance given by IAI o .
Assuming that the magnitude of A is R, a real quantity, the
maximum likelihood estimate conditioned on R is as shown in
equation D9.
The quantity U is a complex Gaussian random variable
(D9)
with statistics identical to a single complex component of the vector
T.
2
The quantity R is Chi-square distributed with 2N degrees of
-·I '-t'
freedom.
The
squar~
of the magnitude of a vector is the sum of the
squares of its orthogonal components.
Since each of the 2N components
2
of A is Gaussian distributed, R is the sum of the squares of 2N
independent Gaussian random variables and is hence Chi-square
. distributed.
To determine the probability density function of
"~. it is
easiest to work with the quantity:
·~
s
;i-
2
r
+ -----U = b
eJ
r
e
j(~
+
8)
(DlO)
R.
Note that the phase angle of this quantity is identical to equation D9.
The joint density function of b and 8 in equation DlO conditioned on R
is expressed in equation Dll (from equation Al3).
r
2 2
br R
---
2
2n(l - r )
e
2 2
R
----2-[(b cos 8 - 1)
2(1 - r )
2
The probability
+ (b sin
2
den,sity function of R is given in equation Dl2.
2
e) ]
From these equations,
2
2
1
2N-2 -R /2
2
f (R ) = - - - = - - R
e
for R > 0
(N - 1)! 2N
the joint probability density function f(b,8) can be computed as
follows:
(Dl2)
75
2
f (b, e ,R )
(Dl3)
R2
1
·
= {3-----N
2N
R
e
-
:z< 1 + a)
(N - 1)! 2
where
a =
and
s=
2
r
[(b cos
2
1 - r
br
e-
1)
2
+ {h sin
(D14)
2
--'----2
21f(1 - r )
(D15)
(D16)
R2
S
- -2 ( 1 + a)
ooo _ _
1 __ .1:_ R2N
J
S(
N
(N - 1)! 2 .
1
)N+1
Joo
1 +a
D
~ (.
1
1
+
0
(z2)
e
1
.1:_
2
2
2N-2 e -z /2 dz
(N- 1)! 2N
2
) N+1 E [ z 2 ] ' wh ere z is Ch.i-square wit h
a
2N degrees of freedom
= 2NS(--1-)N+1
1 + a
Substituting equations Dl4 and D15 into equation D16 and rearranging
terms yields: ·
76
(Dl7)
From reference 14, equations 14.125, 14.139, and 14.140, the following
equations are obtained:
I
dx
(x2 + a2)n
-
X
n~l
I
1
2n - 1 i=l a
2i
2
2
(x + a )
i-1
l
f
n-1
k=l
(
2n- 2k - 1)
2n - 2k - 2
(Dl8)
X
-1(-)
tan
a
ln-2f (2n - 2k - 1)
+
(2n - 2) a2n-l k-1 2n - 2k - 2
-1
(Dl9)
Using these equations, f(e) is obtained as follows:
roo
f(8)
J0
=
f (b, e) db
(D20)
B_(l- r 2 )N Iooo _ _ _ _ _ _ _b_ _ _ _ _.
db
2
2
1
1r
/
[ (b - cos e)
+
- cos e]N+l
2
r.
=
2
B_(1 - r )N Joo
2
1f
r
. -cos
e
[z
2
z + cos e
N+l dz
1
2
+--- cos 8]
2
r
77
1 + r
X
2
cos
2
N
1
I
e
i=l (1 - r 2 cos 2 e)i
i-1
lnl
(2N - 2K + 1)
k=l
2N - 2k
+ r cos
. -1,(r
e Sln
e)
cos
2
2
(1 - r
cos e)N+l/Z
x
N-1
-IT
k=l
(2N - 2k + 1)
2N - 2k
The probability density function of
by substituting ¢ ~
~
follows from equation D20
~
for e on the right side of the equation and
for e on the left side.
The first and second moments of
~
are most easily obtained
from the probability density function of:
(D21)
Defining a new pair of random variables:
(D22)
b
2 2
_ - - - -- e - rR )
0 [ (b cos
2
2
+ (b sin e) ]
2(1 - r ) R"'"
Note that this equation and equation Dll are -different.
Continuing,
I
. 78
(D23)
b r cos e
2
2
2
- [R2 (1:_ +
r
) + _!__(
b
2 )]
- r
2
2
= ----=b~e_1 _
_ _ _ _ R2N-4 e
2
2(1- r )
R 2(1- r )
(N- 1)!
2N2~(1-
r
2
)
From reference 12, equation 3. 4 71.9:
(D24)
Using the substitutions
X
= R2
2N - 4
v-1==-_.:
2
~
b2
= ----::--
2
2(1 - r )
1
y =--2(1 - / )
one obtains:
f (b' 8)
(D25)
'
79
The moments of <P are given by equations D26 and D27 for reasons
k+l
00
A
I (-l)
k=l
k
2
E(<P]
00
I
= 2
E[sin k(<P + e) 1
(-l)k+l
k=l
k
(X)
(-l)k+l
I
= 2
k=l
2
E[~2] =..'!!:._+ 4
3
2
=..'!!:._+ 4
3
2
=..'!!:._+ 4
3
(D26)
E[sin k<P cos k8 + cos k<P sin k8]
sin k<P E[cos k8]
k
00
(-l)k
E[cos k(<P +e)]
2
k=1 k
I
(D27)
00
(-1)k
E[cos k<P cos ke - sin k<P sin k8]
2
k=l k
I
00
(-1)k cos k~ E[cos k8]
k2
k=l
I
similar to those given in appendix A.
Note that both statistics are
determined by E[cos k8], which can be obtained from equations D25 and
Al5 as follows:
E[cos k8]
2
=
J: J:
f(b,e) cos ke de db
( b
br
K
) I ( - - - ) db
2
D (N- 1)! 2N(l- r ) -~-l 1- r 2 k 1- r 2
oo
J
2bN
(D28)
80
From reference 12, equation 6.576.5:
A+~+ v) r(1-
bv r(l-
J
""O
A-
~ + v)
x-A K (ax) I (bx) dx = ------=2'----------'2~--J.l
v
2A+1 T{v + 1) a -A+v+l
X
1-A
F(1:- A+~+ v
j.t+V
2
2
(D29)
b2
v + 1; - )
2
a
Using the substitutions
-A
~
=N
= N- 1
1
a=-=--2
1- r
b
r
,
.l.
v
-
r
2
= k
one obtains:
(D30)
E[cos k6]
rk(l - r 2 )N
(N - 1)! k!
r (N
k
+ -=.)
2
,
r ( 1 + ~) F (N
2
Using equation 9.131.1 of reference
is:.
12~
k
+ -,
2
k
2
1 + -; k + 1; r )
2
an alternate form equation D30
81
rk f(N
E[cos k6]
+
+ ~)
k) f(l
2
2
--------~--------~
(N-l)!k!
k
F(l + - 2
N~
k
2
-; k + 1; r )
(D31)
2
This equation is a generalization of equation 15 of reference 15, in
which N was assumed to be 1.
Equations D26, D27, and D30 or D31
combine to give the desired result.