Physical and Statistical Properties of the Complex Monopulse Ratio ANDREI MONAKOV, Member, IEEE St. Petersburg State University of Aerospace Instrumentation Angular glint gives rise to random pointing errors, which can deteriorate the direction-of-arrival (DOA) estimation accuracy. In monopulse radars the glint manifests itself through a complex valued process known as the complex monopulse ratio (CMR). Physical and statistical properties of the glint are considered. It is proposed to use the Poynting vector formalism for the CMR physical interpretation. The complex probability density function (pdf) and correlation function of CMR are derived. Manuscript received September 15, 2011; revised February 12 and July 2, 2012; released for publication July 12, 2012. IEEE Log No. T-AES/49/2/944512. Refereeing of this contribution was handled by U. Nickel. Author’s address: Department of Radio Engineering, St. Petersburg State University of Aerospace Instrumentation, Bolshaya Morskaya, 67, Saint Petersburg, 190000, Russia, E-mail: (a [email protected]). 0018-9251/13/$26.00 c ° 2013 IEEE I. INTRODUCTION Angular glint is a phenomenon that manifests itself in observations of extended radar targets. The glint gives rise to random pointing errors, which can considerably deteriorate the direction-of-arrival (DOA) estimation accuracy. These errors are due to the interference of signals reflected by target scattering centers [1]. The glint aggravates the accuracy of all known radar systems independently of their antenna type, polarization, transmitted power, receiver sensitivity, signal bandwidth, etc. In monopulse radars the glint manifests itself through a complex valued process known as the complex monopulse ratio (CMR). Suggested by S. M. Sherman in his Ph.D. dissertation [2], CMR was originally based on the hardware ability of monopulse radars to produce so-called “complex indicated angles,” which correspond to the CMR’s real and imaginary components. Since that time CMR has been thoroughly investigated and proposed to improve quality of the DOA estimation of extended radar targets [3—5], unresolved targets and multipath [6—9], and radar target recognition [10]. CMR is formulated as ³ (t)= » (t)+ i´(t)= e ¢ (t) e § (t) (1) where » (t) = Re[³ (t)] is an in-phase component (IC), ´(t) = Im[³ (t)] is a quadrature-phase component (QC) of CMR, and e § (t) and e ¢ (t) are complex envelopes of signals in the sum and difference monopulse channels. The IC depends on the wave front tilt in an observation point. If a point-like target is observed, IC is proportional to the target off-axis angle, and it is usually used in monopulse radars for the DOA estimation. In case of an extended target, IC follows the glint fluctuations and can be used to estimate the glint. It is well known that the QC is nil when the target is point-like [2, 6, 7]. When the target is extended, QC manifests itself, and this fact can be used to detect extended targets and to improve the quality of the target angular position estimation [3—5]. Both components of CMR exist simultaneously with the glint and can be measured in practice. The physical nature of IC can be interpreted easily by means of any of the two concepts of the glint that are known now. In 1959 D. D. Howard explained angular glint as fluctuations of the normal to the target signal phase front [11]. In [12] J. E. Lindsay used the phase gradient formalism to represent the glint. Later, a new concept was suggested. In [13, 14] it is shown that there is a component of the time-averaged Poynting vector that corresponds to the power flow in orthogonal to the radial direction. Although these two concepts have been known for a long time, there is still some controversy in 960 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013
Transcript
Physical and Statistical
Properties of the Complex
Monopulse Ratio
ANDREI MONAKOV, Member, IEEE
St. Petersburg State University of
Aerospace Instrumentation
Angular glint gives rise to random pointing errors, which can
deteriorate the direction-of-arrival (DOA) estimation accuracy.
In monopulse radars the glint manifests itself through a complex
valued process known as the complex monopulse ratio (CMR).
Physical and statistical properties of the glint are considered. It
is proposed to use the Poynting vector formalism for the CMR
physical interpretation. The complex probability density function
(pdf) and correlation function of CMR are derived.
Manuscript received September 15, 2011; revised February 12 and
July 2, 2012; released for publication July 12, 2012.
IEEE Log No. T-AES/49/2/944512.
Refereeing of this contribution was handled by U. Nickel.
Author’s address: Department of Radio Engineering, St.
Petersburg State University of Aerospace Instrumentation,
Bolshaya Morskaya, 67, Saint Petersburg, 190000, Russia, E-mail:
real and imaginary components. Since that time CMR
has been thoroughly investigated and proposed to
improve quality of the DOA estimation of extended
radar targets [3—5], unresolved targets and multipath
[6—9], and radar target recognition [10]. CMR is
formulated as
³(t) = »(t) + i´(t) =e¢(t)
e§(t)(1)
where »(t) = Re[³(t)] is an in-phase component (IC),´(t) = Im[³(t)] is a quadrature-phase component (QC)of CMR, and e§(t) and e¢(t) are complex envelopesof signals in the sum and difference monopulse
channels. The IC depends on the wave front tilt in an
observation point. If a point-like target is observed,
IC is proportional to the target off-axis angle, and
it is usually used in monopulse radars for the DOA
estimation. In case of an extended target, IC follows
the glint fluctuations and can be used to estimate
the glint. It is well known that the QC is nil when
the target is point-like [2, 6, 7]. When the target is
extended, QC manifests itself, and this fact can be
used to detect extended targets and to improve the
quality of the target angular position estimation [3—5].
Both components of CMR exist simultaneously
with the glint and can be measured in practice. The
physical nature of IC can be interpreted easily by
means of any of the two concepts of the glint that are
known now.
In 1959 D. D. Howard explained angular glint
as fluctuations of the normal to the target signal
phase front [11]. In [12] J. E. Lindsay used the
phase gradient formalism to represent the glint.
Later, a new concept was suggested. In [13, 14] it is
shown that there is a component of the time-averaged
Poynting vector that corresponds to the power flow in
orthogonal to the radial direction.
Although these two concepts have been known
for a long time, there is still some controversy in
960 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013
scientific publications about if they are identical
or not. The first publications on the glint [13, 14]
showed that both concepts produce similar results in
the case of simple electromagnetic target models. In
[15] it was claimed that the concept was equal only
when the geometrical optics approximation could be
used to evaluate the target scattered field. In [16] this
assertion was argued, and it was proved that, in the
case of the target model consisting of electric and
magnetic dipoles, the concepts produced the same
results for the glint. In [17] the authors returned to the
theme and showed that, in the case of more elaborate
target models, which took into account polarization
characteristics of the target and receiving antenna, the
phase gradient method (PGM) and the Poynting vector
method (PVM) were not equivalent.
The physical nature of QC is not so easy to
disclose. It is quite clear that QC cannot be explained
within the phase gradient concept. Indeed, estimation
of the wave front tilt, which is actually performed in
a monopulse radar by means of two squinted antenna
subapertures (amplitude comparison) or the wave front
observation in two spatially separated points (phase
comparison), cannot be used to indicate the presence
of an extended radar target: the phase front tilts of
electro magnetic (EM) fields produced by a point-like
target and an extended target are indistinguishable.
Thus, this concept can be used only to interpret
IC. Therefore, there is nothing to do but to try the
Poynting vector formalism to disclose the physical
nature of QC.
The presented paper is devoted to the analysis
of physical and statistical properties of CMR and
its real and imaginary parts. The proposed physical
interpretation of QC may solve the dilemma between
the two glint concepts in favor of PVM. In Section II
the complex Pointing vector formalized is introduced
to reveal the physical nature of CMR. Section III
is devoted to statistical analysis of CMR. Although
there are some publications where the probability
density function (pdf) of CMS and the correlation
functions of its real and imaginary parts are obtained
(see, e.g. [10], [18], [19]), the present publication
aims to provide a concise analysis of the statistical
characteristics of CRM. Finally, the paper ends with a
conclusion in Section IV.
II. POYNTING VECTOR FORMALISM AND PHYSICALPROPERTIES OF CMR
Let us suppose that an extended target can be
represented with a set of a large number of specular
points distributed over the target volume, i.e., we use
the multiple-point model [1, 13, 20]. For simplicity
we assume that each specular point is a source of
a linearly polarized plane wave, which polarization
vector lies in XOY-plane (Fig. 1), and the observationpoint O is the origin of the plane. This assumption
Fig. 1. Coordinate system and field vectors of nth specular point.
permits all further calculations to be performed
in two-dimensional (2D) space. Restrictions that
arise with the assumption are not principal, and the
generalization for three-dimensional (3D) space
is quite obvious. Then, the plane wave of the nthspecular point can be characterized with three vectors
(kn,en,hn), n= 1,2, : : : , where kn = xsin®n+ ycos®n isa unit wave vector; en =¡xen cos®n+ yen sin®n, hn =Z¡10 (kn£ en) are electric and magnetic components; ®nis DOA of the wave; Z¡10 is the intrinsic impedance
of free space; and x, y are unit vectors of the X andY coordinate axes. Then, the electric and magneticcomponents of the total EM field in the observation
point O are E=Pn en and H=
Pnhn, and the
complex Poynting vector (CPV) (see, e.g., [21]) is
P=1
2[E£H¤] = 1
2Z0
Xm,n
em£ (kn£ e¤n)
=1
2Z0
Xm,n
[(em ¢ e¤n)kn¡ (em ¢ kn)e¤n]: (2)
It is worth noting that the application of CPV for
the analysis of CMR is a principal difference of the
presented paper from previous articles [16, 15, 17].
It immediately follows from (2) that CPV has two
components
Px =1
2Z0
Xn
en sin®nXn
e¤n ¼1
2Z0DS¤ (3)
Py =1
2Z0
Xn
en cos®nXn
e¤n ¼1
2Z0jSj2 (4)
whereD =
Xn
®nen
S =Xn
en
(5)
and the axial approximations sin®n ¼ ®n, cos®n ¼ 1are used. These approximations are valid when the
target specular points are close to the Y-axis, which isthe case when the target range is much larger than its
linear dimensions.
As follows from (3) and (4), Py is real, and,therefore, it corresponds to an active power flow in
the radial direction. The component Px has real and
MONAKOV: PHYSICAL AND STATISTICAL PROPERTIES OF THE COMPLEX MONOPULSE RATIO 961
imaginary parts. The real part Re[Px] correspondsto an active power flow in orthogonal to the radial
direction, but the non-zero imaginary part Im[Px]testifies that there is a reactive power flow in the
orthogonal direction. In the case of a point-like target,
Im[Px] = 0, and the reactive flow disappears. The samephenomenon, as mentioned above, takes place with
QC ´.Let us consider a monopulse radar observing our
target. Suppose that the antenna is linearly polarized
with the orientation p= x and that the sum and
difference antenna patterns are f§(®) and f¢(®),respectively. Then, signals in the radar channels are
e§ =Xn
f§(®n)(p ¢ en) =Xn
cos(®n)f§(®n)en
e¢ =Xn
f¢(®n)(p ¢ en) =Xn
cos(®n)f¢(®n)en:
(6)
Let us expand the antenna patterns into the Taylor
series in the vicinity of ®= 0 and, due to the axialapproximation, neglect the infinitesimals of the second
and higher orders
f§(®)¼ f§(0)f¢(®)¼ ¹f§(0)®
(7)
where ¹= f 0¢(0)=f§(0) is the difference pattern slope.Then, substituting (7) into (6), we can express signals
in the sum and difference channels via signals S andD defined in (5)
e§ ¼ f§(0)Xn
en = f§(0)S
e¢ ¼ ¹f§(0)Xn
®nen = ¹f§(0)D:
(8)
Hence, the Poynting vector components can be
estimated as
Px =e¢e
¤§
2¹Z0f2§ (0)
=1
2¹Z0f2§ (0)
fRe[e¢e¤§] + iRe[e¢(ei¼=2e§)¤]g
Py =1
2Z0f2§ (0)
je§ j2:
(9)
As follows from (9) it is necessary to have two
phase detectors to estimate the real and imaginary
components of Px and one amplitude detector toestimate Py. The difference signal e¢ is appliedto the phase detectors directly, but the sum signal
e§ , before being connected to the phase detector,which corresponds to the estimator of the imaginary
component of Px, should be 90± phase shifted.
Actually, this hardware is used in monopulse radars
that are capable of estimating the CMR.
Let us now consider the angular errors that appear
in the case of radar observation of the target. It
follows from (9) that the apparent signal source
location corresponds to
» =Re[Px]
Py=Re[DS¤]jSj2 =
Re[e¢e¤§]
¹je§ j2: (10)
This expression (up to the a-priori known parameter
¹¡1) coincides with IC »(t) in (1) and is extra proofthat the Poynting vector formalism can be used to
evaluate the target glint. Similarly, we can write
´ =Im[Px]
Py=Im[DS¤]jSj2 =
Im[e¢e¤§]
¹je§ j2(11)
and this equation shows the best correlation with QC
´(t) in (1). Therefore, QC ´(t) does not correspondto any direction of the active power flow, but it
designates the reactive power flow of the field. This
is the actual physical nature of QC. It is necessary to
note that normalization of the estimate Im[Px] to Py in(11) is performed due to the automatic gain control
(AGC) loop that is an essential part of any monopulse
radar and which input is connected to the amplitude
detector producing the signal je§ j2 » Py. From the
physical point of view, this normalization does
not have any meaning, but, as is shown further on,
it can be justified via analysis of QC statistical
properties.
Thus, the Poynting vector formalism permits not
only an accurate evaluation of the target glint but also
an explanation of the physical nature of CMR and its
components. This nature can be traced in statistical
properties of CMR.
III. STATISTICAL PROPERTIES OF CMR
A. Complex Probability Density Function
Suppose that the target specular points produce
statistically independent random signals en, n= 1,2, : : :and that the number of points is large enough to
consider signals S and D jointly Gaussian distributed.
Then, the joint complex pdf of S and D is
f(S,D) =1
¼2 detRexpf¡(S¤D¤)R¡1(SD)Tg (12)
where (¢)T designates the matrix transposition, and thecorrelation matrix R can be written as
R= Ps
μ1 ®t
®t (¯2t +®2t )
¶(13)
where Ps =P
n Pn is a mean power of the sum signal
S, ®t =Pn ®nPn=
Pn Pn is an angular position of
the target center of reflections, ¯t = [Pn(®n¡®t)2Pn
=Pn Pn]
1=2 is an effective angular extent of the
target, and ®n and Pn, n= 0,§1,§2, : : : are angularpositions and signal mean powers of the scatterers.
962 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013
Then, after simple calculations, given in Appendix I,
the complex pdf of CMR can be derived as
f³(z) =¯2t
¼(¯2t + jz¡®tj2)2: (14)
Marginal pdfs of IC and QC can be easily computed
from (14)
f»(x) =¯2t
2(¯2t +(x¡®t)2)3=2(15)
f (y) =¯2t
2(¯2t + y2)3=2
: (16)
Both density functions depend on the effective angular
extent ¯t, but comparison of (15) and (16) shows theprinciple difference of IC and QC: the IC pdf depends
on the target angular position, but the QC pdf does
not have this property. This fact proves the conclusion
made in the previous section: QC does not correspond
to any direction of the active power flow.
In conclusion let us note that density (14) has a
remarkable feature: its first absolute moment
Ej´j=Zjyjf (y)dy = ¯t (17)
is equal to the effective angular extent. Hence, QC can
be used to estimate ¯t via simple temporal averaging.
Because of this result the normalization of Im[Px]
to Py , which does not have any physical meaning, isabsolutely justified.
B. Correlation Function and Power Spectral Density
In Appendix II it is shown that the CMR
correlation function is
R³(¿) =_a1 _a3¡ _a22_a21
ln1
1¡j _a1j2+_a22_a21
j _a1j21¡ j _a1j2
(18)
where
_a1(¿) =Xn
Pn _rn(¿).X
n
Pn
_a2(¿) =Xn
®nPn _rn(¿).X
n
Pn
_a3(¿) =Xn
®2nPn _rn(¿).X
n
Pn:
(19)
_a2(0) = ®t = 0, _a3(0) = ¯2t , and _rn(¿) is the correlation
coefficient of a signal en from the nth specular point.The glint correlation function was derived in [13]
and is equal to
R»»(¿) =a3a1 cos(°3¡ °1)¡ a22 cos2(°2¡ °1)
a21ln
1p1¡ a21
+a22 cos
2(°2¡ °1)1¡ a21
(20)
where ak = j _akj, °k = arg _ak, and k = 1,2,3. SinceR³(¿) = R»»(¿) +R´´(¿)¡ 2iR»´(¿ ), it is possible to findthe QC correlation function R´´(¿) and the reciprocalcorrelation function R»´(¿) from (18):
R´´(¿) =a3a1 cos(°3¡ °1)¡ a22 cos2(°2¡ °1)
a21ln
1p1¡ a21
¡ a22 sin
2(°2¡ °1)1¡ a21
(21)
R»´(¿) =a3a1 sin(°3¡ °1)¡ a22 sin2(°2¡ °1)
a21ln
1p1¡ a21
+a22 sin2(°2¡ °1)
1¡ a21: (22)
Let us consider the power spectral densities that
correspond to the derived correlation functions.
Suppose that the target consists of two scatterers,
which have equal mean signal powers, identical signal
correlation coefficients, and symmetrical angular
positions: P1 = P2, r1(¿ ) = r2(¿ ) = exp[¡¼(¢f¿)2],®1 =¡®2 = 0:5¢®, where ¢f is a signal spectrumwidth and ¢®= (®1¡®2) is an angular extend ofthe target. The scatterers are supposed to be complex,
independent objects [1, p. 13]. The model is not of a
theoretical interest only. It can be used to represent
a pair of aircrafts observed in the same resolution
volume.
In Fig. 2 the results of the computations are
presented for two scenarios: 1) the scatterers do not
move relative to each other and 2) the scatterers
move with different velocities, and their spectra are
shifted relatively to each other for ¢F = 200 Hz.In both situations the spectrum width ¢f is set to20 Hz. For the first scenario it is not difficult to show
that S³(f) = 2S»»(f) = 2S´´(f). Therefore, the CMRspectral density S³(f) and densities of its componentsS»» and S´´ have similar shape, and particularly, theydo not have any sidelobes. For the second scenario
the situation is much more complicated. S»»(f) andS´´(f) have absolutely different bearing. IC keeps itsmain component in the vicinity of zero-frequency, and
only its sidelobes, which are situated at harmonics
of the difference frequency ¢F and are not verypronounced, testify that there is some relative motion
of the scatterers. Otherwise, the QC spectral density
S´´(f) does not have any zero-frequency component,and it is concentrated in the vicinity of the difference
frequency ¢F and its second harmonic. This fact wasnoted in [10] and can be used for the classification
of extended radar targets. The CMR spectrum S³(f)is just an envelope of the IC and QC densities, and
its first sidelobe is comparable with the value of the
mainlobe.
Let us find out what process corresponds to the
CMR low-frequency component of the spectrogram in
MONAKOV: PHYSICAL AND STATISTICAL PROPERTIES OF THE COMPLEX MONOPULSE RATIO 963
Fig. 2. Spectral power densities S³ (f), S»(f), S´´(f), Sh³i(f) for two-point target. (a) Points move with similar velocities ¢f = 0 Hz.(b) Points move with different velocities ¢f = 200 Hz.
Fig. 2. CMR of our two-point target is
³(t) =®1e1(t)+®2e2(t)e
i¢−t
e1(t)+ e2(t)ei¢−t
=®1 +®22
+®1¡®22
1¡ ½(t)ei¢−t
1+ ½(t)ei¢−t(23)
where ¢− = 2¼¢F and ½(t) = e2(t)=e1(t) is the ratioof complex amplitudes. If the rate of amplitude
fluctuations of e1(t) and e2(t) is much less than thedifference frequency (¢FÀ¢f), it is possible torewrite (23) as a Fourier series
³(t) =®1 +®22
+®1¡®22
8>>>><>>>>:+1+2
1Xn=1
[¡½(t)]nein¢−t, j½(t)j< 1
¡1¡ 21Xn=1
·¡ 1
½(t)
¸nein¢−t, j½(t)j> 1
: (24)
Hence, in CMR there is a low-frequency component
h³(t)i=½®1, je1(t)j> je2(t)j®2, je1(t)j< je2(t)j
(25)
that corresponds to evolutions of the angular position
of the “brightest” scatterer within the target. This
“telegraph wave” component of the glint was first
reported in [13], [7] and studied afterwards in [1],
[22], [23].
In Appendix III the correlation function of h³(t)i isderived as
Rh³i(¿) =¢®2
4
h1¡
p1¡ r2
i: (26)
Spectral density corresponding to the correlation
function Rh³i(¿) is shown in Fig. 2 with the dash-dotline. The figure proves that the low-frequency
component of CMR corresponds to the telegraph wave
(25). This result again testifies that only IC contains
information on the absolute angular positions of radar
targets.
IV. CONCLUSION
In spite of the long history of investigation, thereis still a discussion in scientific publications on thenature of the glint–a phenomenon that arises whilean extended radar target is observed. There are twoconcepts that can describe the glint: the PGM andPoynting vector formalism. Both methods permitthe errors induced by the glint for all known DOAestimators to be accurately predicted. The main idea
964 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013
of the presented article is that the dilemma can be
resolved if abilities to explain the physical nature of
the CMR of the vying methods are determined. The
PGM cannot explain the existence of the imaginary
part of CMR because the estimation of the target
EM field in two spatially separated points (phase
comparison monopulse) or two squinted antenna
subapertures (amplitude comparison monopulse),
which is actually performed in a monopulse radar, can
be used to estimate only one geometrical parameter
of the field, namely the phase front gradient. In
the presented paper it is shown that the Poynting
vector formalism can be used to disclose the physical
nature of the in-phase and quadrature-phase CMR
components. While the IC corresponds to the
direction of the active power flow in the scattered
field, the QC is determined by the reactive power
flow. Due to its nature the QC is nil if the target is
point-like because, in this case, only the active
power flow exists. Otherwise, if the target is
extended, the QC accompanies the glint generated
by the target.
The pdf and correlation function of CMR are
derived. The latter permits a full spectral analysis
of CMR and its components to be conducted. This
analysis shows that the power spectral densities of the
CMR components are affected by the target type quite
differently. The IC weakly depends on the internal
motion of the target’s specular points. At the same
time the spectral density of QC changes its bearing
completely when this motion is present. This fact
can be used for the classification of extended radar
targets.
APPENDIX I. COMPUTATION OF THE CMRPROBABILITY DENSITY FUNCTION
Let us consider the joint complex Gaussian pdf of
S and D (see (12))
f(S,D) =1
¼2 detRexpf¡(S¤D¤)R¡1(SD)Tg (27)
where the correlation matrix R is given in (13).
Polar transformation of the variables in (27) yields
the joint pdf
f(½s,½d,'s,'d)
=2½s½d(¼Ps¯t)
2exp
½¡ 1
Ps¯2t
[(®2t +¯2t )½
2s + ½
2d
¡2®t½s½d cos('d¡'s)]¾(28)
where ½s = jSj, ½d = jDj, 's = argS, 'd = argD.After introduction of new variables à = 'd¡'s,x= ½d=½s cos('d ¡'s) and x= ½d=½s sin('d¡'s) and
integration of f(½s,x,y,Ã) over Ã, we have
f(½s,x,y)=2½3s
¼(Ps¯t)2exp
½¡ ½2sPs¯
2t
[(®2t +¯2t )+x
2+y2¡2®tx]¾:
(29)
Then, after the final integration of (29) over ½s, the
joint pdf of x and y is
f(x,y) =¯2t
¼[¯2t +(x¡®t)2 + y2]2: (30)
Hence, the complex pdf of CMR can be written as
f³(z) =¯2t
¼(¯2t + jz¡®tj2)2: (31)
Density function (31) is a particular case of more
general results reported in [18], [19], [24].
APPENDIX II. COMPUTATION OF THE CMRCORRELATION FUNCTION
Let us suppose that the signals S(t) and D(t) are
complex Gaussian random processes and that the
target angular position ®t = 0. Then the correlation
matrix of these signals is
R= Ps
0BBBBB@1 _a1(¿) 0 _a2(¿)
_a¤1(¿) 1 _a¤2(¿) 0
0 _a2(¿) _a3(0) _a3(¿)
_a¤2(¿) 0 _a¤3(¿) _a3(0)
1CCCCCA (32)
where _a1(¿), _a2(¿), and _a3(¿) are given in (19). Then,
the joint complex pdf of the vector z= (SS¿DD¿ )T is
f³(z) =1
¼4 detRexpf¡zHR¡1zg (33)
where S = S(t), S¿ = S(t+ ¿), D =D(t), D¿ =D(t+ ¿),
and H designates the conjugate matrix transposition.
The characteristic function of pdf (33) is
³(r) = expf¡ 14rHRrg (34)
where r= (pp¿qq¿ )T.
To derive the CMR correlation function, let us use
the method of characteristic functions, whose essence
is in the representation of the joint pdf as the inverse
Fourier transformation of the joint characteristic
function and can be followed from the chain
R³(¿ ) =
ZD
S
D¤¿S¤¿f³(z)dz
=1
(2¼)8
Zdr³(r)
ZdDdD¤¿DD
¤¿ eiRe(Dq¤+D¤¿ q¿ )
£ZdSdS¤¿SS¤¿
eiRe(Sp¤+S¤¿ p¿ ): (35)
MONAKOV: PHYSICAL AND STATISTICAL PROPERTIES OF THE COMPLEX MONOPULSE RATIO 965
The last integrals in (35) should be treated as
generalized functions (see, e.g., [25], [26]), and it is
possible to show that
1
(2¼)2
ZDeiRe(Dq
¤)dD1
(2¼)2
ZD¤¿ e
iRe(D¤¿ q¿ )dD¤¿
=¡4±(q)±0(q¤)±0(q¿ )±(q¤¿ ) (36)
1
(2¼)2
ZeiRe(Sp
¤) dS
S
1
(2¼)2
ZeiRe(S
¤¿ p¿ )dS¤¿S¤¿
=¡ 1
(2¼)2p¤
jpj2p¿jp¿ j2
(37)
where ±(¢) and ±0(¢) are the Dirac ±-function and itsfirst derivative.
Substitution of (36) and (37) into (35) and
integration finally yields
R³(¿) =_a1 _a3¡ _a22_a21
ln1
1¡j _a1j2+_a22_a21
j _a1j21¡ j _a1j2
: (38)
APPENDIX III. CORRELATION FUNCTION OF THELOW-FREQUENCY COMPONENT
To derive the correlation function of low-frequency
component (25), let us return to (24). Analysis of the
equation shows that temporal averaging of ³(t) overthe period of ¢− yields the low-frequency component
h³(t)i. Hence, its correlation function can be derivedvia similar averaging of R³(¿).From (38) it immediately follows that, for the
two-point symmetrical target
R³(¿) =¢®2
4
264 1
cos2¢−¿
2
ln1
1¡ r2(¿)cos2 ¢−¿
2
¡r2(¿ )sin2
¢−¿
2
1¡ r2(¿)cos2 ¢−¿
2
375 : (39)
Expansion of the terms in brackets into the Taylor
series yields
R³(¿ ) =¢®2
4
(¡1+
1Xn=0
r2(n+1)(¿)
n+1cos2n
¢−¿
2+ [1¡ r2(¿)]
1Xn=0
r2n cos2n¢−¿
2
): (40)
The averaging of R³(¿) produces
Rh³i(¿ ) =1
2¼
Z ¼
¡¼R³(¿ )d(¢−¿ )
=¢®2
4
(¡1+4
1Xn=0
(2n)!
n!(n+1)!
·r2
4
¸(n+1)+ [1¡ r2(¿ )]
1Xn=0
(2n)!
(n!)2
·r2
4
¸n): (41)
Summation of the series in (41) gives the desired
correlation function
Rh³i(¿) =¢®2
4
h1¡
p1¡ r2
i: (42)
In general form the correlation function of the
low-frequency component is found in [23], and it is
equal to
Rh³i(¿ ) =·°¢®
(1+ °2)
¸2(1¡ (1+ °2)[(1¡ r21 )+ °2(1¡ r22 )]p
[(1+ °2)2¡ (r1¡ °2r2)2][(1+ °2)2¡ (r1 + °2r2)2]
)(43)
where °2 = P2=P1. Equation (42) is a particular case of
(43) for the symmetrical two-point target. It is worth
noting that (42) can be derived from the IC correlation
function R»»(¿) via the same averaging. It means that,
for the corresponding spectra, S³(!)! Sh³i(!) andS»»(!)! Sh³i(!) when ¢−!1.
REFERENCES
[1] Ostrovityanov, R. V. and Basalov, F. A.
Statistical Theory of Extended Radar Targets.
Norwood, MA: Artech House, 1985.
[2] Sherman, S. M.
Complex indicated angles in monopulse radar.
Ph.D. dissertation, University of Pennsylvania,
Philadelphia, Dec. 1965.
[3] Dzhavadov, G. G.
Informative properties of radar estimations of complex
