Note: Descriptions are shown in the official language in which they were submitted.
1 1'2 ~ ~9
Background of the Invention
This invention relates generally to airborne radar systems
utilizing the Doppler effect to derive navigational information
and particularly to signal processing techniques for improving
the accuracy of systems of such character.
As is known in the art,in any so-called Doppler navigation
radar system in an aircraft, components of velocity o~ the air-
craft may be derived with respect to orthogonal axes having a
known relationship to the surface of the earth. Because there
are three orthogonal components of velocity of interest, radar
echo signals in a plurality ~at least three) of radar beams
illuminating different portions of the underlying terrain must
be used. A Doppler navigation system, employing three non-
coplanar radar beams, which has found wide acceptance is the
so-called Janus system. Such a system has both forward and
rearward looking beams, and is usually comprised of two forward
looking beams and one rearward looking beam arranged in such a
manner as to form the ~reek letter ~. In such a system, the
difference between the Doppler shifts received from the rearward
looking beam and the forward looking beam on the same side of
the aircraft provides a measure of the "along heading" velocity
VH; while the difference between Doppler shifts received from
the two forward looking beams provides a measure of the "cross-
~- heading" or "drift" velocity VD. The vertical velocity component
is derived by combining the Doppler shifts obtained from the rear-
ward looking beam and the forward looking beam an the opposite
side of the aircraft.
~ ecause each Doppler radar beam has a finite beamwidth, the
return signal associated with any beam comes from different direc-
tions. Thus, return signals for each radar beam contain Doppler
3~'2~
frequencies within a frequency spectrum rather than a singleDoppler frequency. The Doppler frequency of interest, which is
proportional to the velocity component along any beam, is the
mean frequency of the Doppler frequency spectrum. Because of
the appreciable spectrum width, the instantaneous central fre-
quency of the Doppler signal is, however, subject to random
fluctuations about its mean value, giving rise to an error in
tracking the center of the beam and, therefore, in measuring
velocity. In addition, signal returns from isolated large
targets or the effect known as "sea bias~ can cause measurement
errors by apparently changing the center frequency of the
Doppler frequency spectrum.
As is known in the art, errors which result from some
apparent changes in the return signal spectrum may be reduced
by utilizing a monopulse radar having beams with antenna patterns
directed toward the earth fore and aft of the aircraft velocity
vector, as is described in an article entitled: "Null-Tracking
Doppler Navigation Radar" by P. G. Smith, IEEE Transactions on
Aerospace and Navigational Electronics, March 1963, Pages 50-55.
In such a system, the Doppler frequencies corresponding to the
null plane of the monopulse elevation difference channel are
tracked. The null plane is used because antenna gain is apparently
zero at the null, and the slope of the apparent antenna gain curve
on either side of the null plane is very steep. The accuracy of
null tracking depends upon the average slope of the difference
pattern in the ViCillity of the null. Thus, any known null tracking
system is useful only when the null plane of the elevation differ-
ence pattern is substantially orthogonal to the ground track of
the aircraft, i.e. when the azimuth difference null is essentially
parallel to an isodop, in order to produce a Doppler spectrum in
-2--
1~5~i~
the elevation difference channel which has a sharp null. As the
null plane deviates from parallelism with an isodop, degradation
of the difference pattern null occurs. This means that roll or
yaw angles in excess of 15 experienced by the aircraft may de-
grade the null to an unacceptable degree to the detriment of
system accuracy. In addition, because any known null tracking
system is restricted to use with beams which are substantially
in the ground track directlon, returns from sideward looking
beams are still subject to beam spreading with errors due to
apparent changes in the center frequency of the Doppler frequency
spectrum as described above.
As is also known in the art, tile degradation of system
accuracy as a function of drift angle may be reduced by using an
antenna which is azimuth-stabilized along the ground track. In
such a system, the difference between the Doppler shifts obtained
from two sideward looking beams on opposite sides of the ground
track is used to drive a servo which turns the antenna in azimuth
~.
until the difference is forced to zero. This condition occurs
-^, when the ground intersections of the two beams lie on the same
isodop, thereby aligning the antenna with the aircraft track.
Ground track stabilization by such a method is slow and is imple-
mented only by relatively complex, heavy, expensive hardware~
Such a system is, in any event, not suitable to prevent degradation
~; in accuracy resulting from aircraft roll and yaw.
As is also known in the art, an antenna radome will diffract
a radar beam passing therethrough. This diffraction, which is
dependent on the beam pointing direction, causes an error to be
introduced in the determination of the angle associated with the
Doppler frequency of the return signal and therefore in the com-
putation of the aircraft velocity.
Summary of the_Invention
With this background of the invention in mQnd, it is an object of
this invention to provide an improved signal processing techni~ue for an air-
borne Doppler navigation radar system using a monopulse radar.
Another object of this invention is to provide a method of increas-
ing the accuracy of an airborne manopulse radar used in a Doppler navigation
system without any need for a complex mechanical drive for the antenna of such
a ra~r.
Still another object of this invention is to provide an airborne
m~nopulse D~ppler navigation radar ~hose performance is not affecbed by air-
craft roll, pitch, or drift angles.
Still another object of this invention is to provide an airborne
monopulse D~ppler navigation radar whose performance is not affected by radome
induced antenna beam point mg errors.
These and other Qbjects of the invention are generally attained by
providing an airborne monopulse radar in a Doppler navigation system and a pro-
cessor wherein a selected monopulse null plane may be rotated to make it appear
that ~uch selected null plane is parallel to the isodops in the portion of the
illuminated terrain, regardless of the orientati of the radar beam.
In a preferred ecbodiment of this invention, such rotation is
accomplished by weighting the difference channel signals by, respectively, the
sine and cosine of a so-called mon~pulse rotation angle. ~he m~nopulse angle
(B) is evaluated for each different orientation of the radar beam in accordance
with the attitude and velocity of the aircraft. m e attitude and velocity of
the aircraft in turn are determined i~ part by inertial measurements which are
continuously processed in an onboard computer.
In accordance with the present invention, there is provided an im-
proved method of operating an airborne Doppler navigation radar system utiliz-
ing an inertial platform and a monopulse radar having an antenna adapted to
produce a plurality of directive bea~s, the centerline of each one of such
beams being in a fixed angular relationship with respect to the radiatin~ face
J
g
of such antenna, for illuminating different portions of the terrain underlying
an aircraft carrying such system so that return signals in each one of such
beams may be processed in an azimuth difference and an elevation differenoe
channel to derive measurements:of the velocity vector of the aircraft with
respect to the earth, in particular, the components of the velocity vector
along said beams, such improved method comprising the steps of: (al aligning
the antenna so that the center of the portion of the terrain illuminated by a
first selected one of the directive beams is disp~sed at an angle with respect
to the aircraft flight path; (b) energizing the antenna to form, sequentially,
each one of the plurality of directive beams; and (c) pro oessing the returns
from ea~h sequentially formed beam as received by the antenna to derive the
weighting vzlues required to rotate the null of the elevation difference chan-
nel for the next group of sequentially formed radar beams by "N" degrees,
where N is the angular difference in degrees between the null through the
oe nterline of the beam which is tangent to a selected isodop.
.
~.~
- 4a -
"
Brief Description of the Drawings
The above-mentioned and other features of the invention
will become more apparent by reference to the following descript-
ion taken in connection with the accompanying drawings in which:
FIG. lA is a sketch, somewhat simplified, of an air-
craft employi~g a monopulse Doppler navigation radar to deter-
mine its velocity by directing a series of radar beams toward the
terrain beneath the aircraft;
FIG. lB is a sketch, greatly simplified, of the portion
lQ of the terrain illuminated by radar beam 22 of FIG. lA useful in
understanding how the radar on board the aircraft of FIG. lA
processes the return signals from such portion to determine air-
craft velocity;
FIG. lC is another sketch of the terrain illuminated by
radar beam 22 of F~G. lA illustrating portions of the terrain ~-
~: from which returns of constant Doppler shift are obtained;
, .,
FIG. lD is a sketch of a Doppler sensitivity curve
obtained by processing the returns from the portion of the terrain
; illustrated in FIG. lC;
'1
~ 20 FIGS. 2A, 2B and 2C are sketches of the portion of the
:~ terrain illuminated by radar beam 40 of FIG. lA which correspond,
respectively, to FIG6. lB, lC and lD;
FIG. 3 is a block diagram, somewhat simplified, of a
radar sy-stem incorporating our invention;
:: FIGS. 4A and 4B are sketches of a uniform sphere
surrounding the aixcraft of FIG. lA which are useful in under-
standing the contemplated signal processing technique;
FIGS. 5, 5A and 5B are vectox diagrams useful in
unde~standing the contemplated signal processing technique; and
3Q FIG 6 is a vector diagram illustrating the effect of
monopulse axis rotation according to our in~ention.
- 4a -
~125~9
The geometry involved with a Doppler navigation radar system
which gives rise to the problem to be solved is shown in FIG. lA.
Referring now to FIG. lA, an aircraft 10, equipped with a mono-
pulse radar 12 having an antenna 14 (which conveniently may be a
conventional phased array) fixed relative to the longitudinal axis
of the aircraft 10, is shown to be flying parallel to an assumed
flat terrain 16 at a constant altitude "h~. Three orthogonal
axes marked "x", "y", and "z" having an origin at the aircraft 10
are shown. The components of interest of the velocity of the air-
craft 10 are the components along such axes. The aircraft 10 here
is shown to be moving so that its ground track 19 is parallel to
the z-axis. The hyperbolas 20a, 20b, and 20c represent the loci
~e' of points on the terrain 16 from which echo signals having a
constant Doppler shift are received. Such loci are commonly
referred to in the art as "isodops".
- The antenna 14 is shown to direct a radar beam 22 toward the
terrain 16 in such a manner that the centerline 24 of beam 22 forms
an angle (A) with the velocity vector 18. In the absence of air-
craft yaw and drift, the aircraft heading vector 26 is coincident
with the velocity vector 18 and the radar beam 22 i5 symmetrlcal
about the ground track vector 19.
Referring now to FIG. lB, which is an enlargement of the area
illuminated by radar beam 22, the monopulse elevation null 28 of
the monopulse elevation difference pattern 30 is shown to be
parallel to isodop 20b while the azimuth null axis 32 (sometimes
herein referred to as the elevation error axis 32 for reasons to
be clear hereinafter) is orthogonal to the isodop 20b. In order
to gain a better understanding of the processing technique involved
in determining the desired Doppler shift, reference is made to
l~Z~i~19
FIG. lC wherein another representation of the area illuminated
by radar beam 22 is given. The vertical sections labeled fO,
fl f 1~ ... fn~ f n represent areas of constant Doppler shift,
each of which is centered on a corresponding Doppler frequency.
Doppler frequency fO corresponds to the isodop which is tangent
to the elevation null, here isodop 20b. The negative subscripts
represent negative Doppler shifts or decreased frequency with
respect to the center frequency fO, while the positive subscripts
represent positive Doppler shifts or increased frequency with
respect to center frequency fO. The x's 34 represent unit re-
flectors at each of the Doppler frequencies.
Referring back to FIG. lB, it may be seen that unit reflectors
- to the left of elevation null 28 are on the negative side of the
elevation difference pattern 30, while unit reflectors to the
right of elevation null 28 are on the positive side of the eleva-
tion difference.pattern 30. As is well known, the sign of the
signals out of a monopulse receiver are positive (+) for returns
coming from unit reflectors located on one side of the elevation
null 28, and negative (-) for returns coming from unit reflectors
on the opposite side of elevation null 28.
The Doppier navigation radar 12 processes the return signals
along the elevation error axis 32, plotting net signal strength
versus Doppler frequency. The result is an S-shaped sensitivity
curve 36, as shown in FIG. lD. The Doppler frequency of interest
is that which corresponds to the cross-over point 38 of the sensi-
tivity curve 36. The slope of the sensitivity curve 36 in the
region of the cross-over point 38, which is related to the depth
of the elevation difference pattern null, determines how accu-
rately the cross-over point may be tracked.
Referring back to FIG. lA, it may be seen that when aircraft
--6--
~i25419
10 is subjected to a wind velocity vector 40', the aircraft must
be yawed (here at an angle ~) to maintain its flight path along
the velocity vector 18. Centerline 24 of radar beam 22 moves
with the aircraft heading vector 26 such that it is no longer in
the same vertical plane with the velocity vector 18, and there-
fore, the elevation null 28 is no longer parallel to isodop 20b.
The net result is that the elevation null is degraded and the
slope of the sensitivity curve 36 is diminished with a concomitant
loss of tracking accuracy. This effect may be more clearly under-
stood by referring to FIG. 2A where isodop 20b is shown crossingthe elevation null 28 at an angle ~; and to FIG. 2B where isodops
' fl~ f 1~ ... f ~ f , are shown crossing the elevation error
axis 32 at an angle ~. As the isodops are no longer parallel to
the elevation null 28, the unit reflectors 34 on any isodop fall
on both the positive and negative sides of the elevation differ-
ence pattern 30. The resulting sensitivity curve 42 shown in
FIG. 2C,which is obtained by plotting signal strength versus
` Doppler frequency along the elevation error axis 32, has a more
.~ ~ shallow slope than the sensitivity curve 36 of FIG. lD.
Referring now back to FIG. lA, it may be seen that a loss in
tracking accuracy will also result if the antenna 14 directs a
beam to one side of the aircraft 10, i.e. beam 40. Thus, there is
a loss in tracking accuracy associated with radar beams whose
centerlinés are not coplanar with aircraft velocity vector 18.
;
llZ5~9
.
Referring now to FIG. 3, a radar system having a monopulse
receiver for null tracking is shown. Such system includes a
microwave front end (not numbered), comprised of four antenna
elements, 46a-46d, connected in a conventional manner to form a
sum channel 50a,an elevation channel 50b (i.e. a difference
channel), and an azimuth channel 50c for receiver section 52.
It is noted that the sum channel is coupled, selectively, to
receiver section 52 and transmitter 54, in a conventional manner,
here by circulator 56.
Receiver section 52 is a conventional heterodyne receiver
and includes RF amplifiers 58a-58c, a local oscillator 60, mixers
62a-62c, and IF amplifiers 64a-64c, which are all of conventional
design and arranged to convert RF signals applied to receiver
section 52 into IF signals on lines 66a-66c. Such IF signals are
passed through range gates 67a-67c, which are o-~ conventional
design and are controlled by range gate generator 69, also of
conventional design. The azimuth and elevation difference signals,
on lines 66c, 66b, respectively, are directed to the monopuIse
rotation processor 68 wherein they are subsequently wei~hted by
the sine and cosine of the rotation angle B in rotation angle
multipliers 70a-70b (i.e. IF sine and cosine phase multipliers
which are of conventional design). The rotation angle B is the
angle through which the monopulse axes must be rotated to maintain
proper alignment to the generated isodops and thereby prevent mono-
pulse null fill-in.
The monopulse rotation angle B is calculated in rotation
angle computer 72, in a manner to be described hereinafter, from
data received from inertial platform 74 and motion compensator 75.
The aircraft attitude is determined from inertial platform 74
which is comprised of gyros and accelerometers. Analog signals
-8-
l~Z~
from inertial platform 74 are digitized by A/D converters 76a-76n
and are processed in rotation angle computer 72 along with velocity
information from motion compensator 75 to yield the required rota-
tion angle B.
The binary representations of the sine and cosine of the
rotation angle B are translated to suitable analog signals in DlA
converters ~8a-78b and are sent to the monopulse rotation processor
68 on lines 80a-80b, respectively. The rotated monopulse data are
processed in combiner 84 in such a manner as to form new monopulse
10 axes Xl and yl on lines 86a-86b which may be represented, respec-
tively, by the following equations:
xl = Az cos B + El sin B (1)
yl =-A sin B + El cos B (2)
where:
A = the azimuth difference channel signal
El = the elevation difference channel signal
For the purpose of clarity, the remainder of the radar system
:~ in FIG. 3 is shown to process only the data on line 86a, correspon-ding to the yl axis. The signal processing for the Xl axis data on
20 line 86b is identical to that for the yl axis data described here-
inafter.
rlixers 88a-88b, in conjunction with local oscillator 90,
; translate each of the IF signals on lines 86a and 92, corresponding
to the yl axis and the sum channel data respectively, to a suitable
video frequency so that they may be digitized by A/D converters
94a-94b. Before being fed to mixer 88b, the sum channel data on
line 66c is attenuated in ~ariable attenuator 98 in order to com-
pensate for the insertion loss experienced by the difference
channel data in passing through the rotation processor 68 and the
30 combiner 84. The digital data from A/D converters 94a-94b is then
applied to the Doppler filter bank Fast Fourier Transform (FFT)
g
i9
signal processors 96a-96b which perform Doppler processing,
signal to noise enhancement, and thresholding for each Doppler
filter. The FFT is performed on a number of range gates for each
coherent transmitter frequency to produce corresponding sets of
frequency spectra which represent the resolvable target Dopplers
obtained during the radar dwell time interval.
The rotated difference channel data and the sum channel data
~` are passed to a processor, here designated normalization processor
100, to normalize the former in a conventional manner. Such pro-
cessing is performed independently for each Doppler filter in the
filter bank. The result of the normalization is a reduction in
the effect of scintillation in radar return signals. The normal-
ized data represent a plot of monopulse error angle versus Doppler
frequency (i.e. a discriminator curve similar to that shown in
FIG. lD). Such data are then passed through processor 102 wherein
only data from filters below a preset threshold are accepted for
processing. The accepted data in the vicinity of the difference
pattern null then undergo a "least squares" curve fitting process
from which the zero gain intercept is extracted. The zero gain
intercept is the Doppler frequency estimate corresponding to the
monopulse null. This process is repeated for each range gate,
obtaining independent measurements of the Doppler frequency at
each monopulse null. The average of such measurements yields the
final estimate of the desired null Doppler frequency. Such
average is passed to motion compensator processor 75 wherein it is
converted to velocity coordinates and is subsequently used to
compute the track and climb angles, in a manner to be described in
greater detail hereinafter. The revised estimates of the track
and climb angles are sent to the rotation angle computer wherein
they are used to update the rotation angle B.
-10 -
5~
The data from processor 102 represents Doppler frequency data
measured in beam coordinates. In motion compensator processor 75 J
such data is first converted to velocity data and is then trans-
formed into reference space coordinates by means of multiplying
the data by a matrix comprising the direction cosines of the beam
pointing angles. This transformation may be expressed as:
V = [DCI V ~3)
1 0 V V
REFERENCE BEAM
;` ~
The radar derived velocity data is then combined with the
velocity data from inertial platform 74 to correct for the long
term errors in the velocity data from inertial platform 74. The
data is combined as shown in Equation 4.
~Vx = Vx ~radar) - Vx (platform) + Nx
~Vy = Vy ~radar) - Vy (platform) ~ N
~V e V (radar) - V (platform) ~ N
The ~V terms represent the errors between the radar derived velo-
city data and the platform derived velocity data, and may be seen
to include a noise term (N). The noise term is eliminated and the
error data is smoothed by submitting the data to a filtering or
smoothing process. Such a process, which is described in an
article entitled, "Optimizing the Dynamic Parameters of a Track-
While-Scan System" by J Sklansky, RCA Review, June 1957, Pages
163-185, may be represented as shown in Equations 5 and 6. For
the sake of clarity only, the x component of the velocity vector
li;~S~l~
V is shown in Equations 5 and 6, it being understood that the y
and z components of V undergo an identical process.
(ll) = aVx(n) + ~QVx(n) ~ ~x(n)] (5)
~x(n) = ~Vx(n) + ~/T[~Vx(n) - ~Vx(n)] (6)
The following definitions apply to the terms of Equations 5 and
~ Vx(n) is the smoothed value of velocity error for the Nth radar
dwell
.rX(n) is the predicted value (from the N-l radar dwell) of the
velocity error for the Nth radar dwell
Vx(n) is the rate of change of the smoothed value of velocity
error for the Nth radar dwell
is a constant 0<~1
~ is a constant given approximately by ~
; T is the duration of the Nth radar dwell
The velocity error predictions for the N+l radar dwell are given
by
~x(n~ Vx(n) + T~Vx(n)
/~ _
(n+l) = ~Vx(n) ~8)
where
Vx(n+l) is the predicted rate of change of the velocity error
for the N+l radar dwell
The velocity error predictions for the N+l ràdar dwell are
combined with the platform derived velocity data to calculate new
values for the climb (C~ and track (T) angles. Referring now to
11~5~9
FIGS. SA and 5B, it may be seen that the aircraft track angle T
involves rotation of the aircraft ground velocity vector, V
about the Y-axis. The following definitions therefore apply:
cos T = Vz/Vgs sin T = ~x/vgs
where
Vgs = ~x + Vz ~9)
The aircraft climb angle C involves rotation of the aircraft
velocity vector V about the X-axis and therefore the following
definitions apply:
sin C = -Vx/V cos C = Vg/V (10)
where
V = ~x + Vy + Vz
The updated values for the climb ~C) and track (T) angles are used
to calculate the value of the monopulse rotation angle (B) for the
N+l radar dwell.
The process for computing the rotation angle B, through which
the monopulse difference channel data are rotated to form the new
monopulse axes Xl and yl~ will now be described. As prevlously
mentioned, in order to maintain the depth of the monopulse differ-
ence pattern null so as to identify a frequency or velocity with a
given beam direction, it is desirable that the data in the two
monopulse difference channels be processed along an axis normal to
an isodop. This is accomplished by rotating the two receiver
channels through an angle B so as to effectively form two new mono-
pulse receiver channels Xl and yl where:
[ 1 ]= (B)~ ] (11)
-13-
19
The rotation angle (B) must be found for each of the radar
beams and will, in general, be different for each beam because
the aircraft attitude and velocity vectors will affect each beam
differently, The rotation angle (B) is formed in rotation angle
computer 72 from inputs received from inertial platform 74 and
motion compensator processor 75. The rotation process involves
the transformation of the radar beam coordinates in "velocity
space" to corresponding coordinates in "antenna space".
Before proceeding, it should be noted that W. H. Von Aulock,
in an article entitled "Properties of Phased Arrays," Proceedings
of IRE, Oct. 1960, Pages 171S-1727, has shown that in discussing
the scanning properties of phased array antennas. it is advanta-
geous to use contour maps of the antenna pattern in so-called
"T-space" which is a projection of a unit sphere of constant range
on the plane of the array.
Referring now to FIG. 4A, a unit range sphere 200 is shown
surrounding an aircraft 202 which is equipped with a Doppler navi-
gation radar ~not shown~. The intersection of a Doppler cone 204
(which is herein defined as a conical contour of constant Doppler
frequency) with the unit sphere 200 is shown to form a circle 206.
The intersection of the Doppler cone 204 with a ground plane 208
is shown to form a hyperbola 210, and the intersection o the unit
sphere 200 with the ground plane 208 is shown to form a circle 212.
Referring now to FIG. 4B, the aircraft 202 is shown to be
flying in the XZ plane with its velocity vector V coinciding with
the Z axis. The aircraft 202 is shown to direct a Doppler cone
204 in the direction of the Z-axis. The Doppler cone 204 inter-
sects the unit sphere 200 at points marked, respectively, 214a,
214b. As previously mentioned, the intersection of the Doppler
cone 204 with the unit sphere 200 forms a circle 206
li ~ 5~1 9
which is projected into "T-space" or "antenna space" as a circular
isodop 218 The circular isodop 218 in "T-space" or "antenna
space" is shown to lie in $he XY plane. The angle G, which is
shown to be one-half the Doppler cone angle, is defined as the
angle between the velocity vector V, which here is coincident
with the Z-axis, and the radius of the unit sphere 216, which here
is defined as being equal to unity. The angle G is here also
defined as the predetermined beam pointing angle. The vertical
distance then, from the Z-axis, to the intersection points 214a,
214b, is seen to be equal to sine G. As the intersection of the
Doppler cone 204 with the constant range sphere 200 is projected
into "antenna space" or "T-space" as a circular isodop 218, the
radius of the circular isodop 218 may also be seen to be equal to
the sine G. It may be seen therefore, that specifying a particular
beam pointing angle G will also define a particular circular isodop
in "antenna space" or "T-space". Any point T on the selected
circular isodop may now be defined by means of the following
expressions:
X = sin G sin t ~12
Y = sin G cos t ~13~
where t ls defined as the angle between the radius of the isodop
(sine G) and the Y-axis.
A vector equation relating a beam in "velocity space" to a
beam which has been selected a priori in "antenna space" is:
Yv I = [C] [T] [A]t y ¦ (14)
where
llZS419
[A] = the transpose of the attitude matrix and is com-
prised of the roll, pitch, and heading matrices
~T] = the track matrix
~ C~ = the climb matrix
The subscript "v" represents the beam coordinates in velocity
space, and the subscript "a" represents the beam coordinates in
antenna space.
The transformation process will now be described with refer-
ence to PIG. 5. The two-dimensional vector V in FIG. 5 has com-
ponents x and y in one coordinate system. The same vector hascomponents xl, yl, in a coordinate system which is rotated from
the first coordinate system through an angle ~. The components
xl, yl can be expressed in terms of the components x, y by:
xl = x cos ~ ~ y sin ~
y = y cos ~ - x sin ~ (15)
or in the matrix notation by:
~ xl 1 ~ cos ~ sin ~ 1 ~x 1
L yl ~ -L -sin ~ cos ~ ~ ly ~ ~16)
The two-element column matrices in Equation~l~represent the
vector V of FIG. 5 in two different coordinate systems. The
second-order matrix represents the rotational transformation from
.~ the coordinate system x, y to the coordinate system xl, yl,
A vector in three dimensional space is expressed in terms of
its three mutually orthogonal components x, y, z. Since this
triad has three degrees of rotational freedom, the relationship
between it and another orthogonal triad xl, yl, zl can be
-16-
il~54~9
expressed as an angular rotation about each of the three axes in
some particular succession. Thus, as mentioned hereinabove, the
attitude matrix is comprised of the roll, pitch, and heading
matrices and is formed by successively rotating about the Z, X,
and Y axes through the roll, pitch, and heading angles, respec-
tively. Such transformation relates a beam in "antenna space" to
the inertial platform onboard the aircraft, which is herein
defined as reference space. This transformation may be expressed
as:
a- cos R sin R O 1 0 0 cos H O sin H XR-
a = -sin R cos R O O cos P sin P O 1 0 R ~17)
a O O 1 O -sin P cos P -sin H O cos H R
Equation tl7) is inverted to yield:
YR ¦ ~ [A] ¦Ya ~18)
where [A] is defined as the transpose of the attitude matrlx lA].
~quation ~18) relates a beam in "antenna space" to a beam in
"reference space". Transforming a beam from "reference space" to
"velocity space" involves rotating that beam successively through
the aircraft track and climb angles. This transformation process
is given by:
1 0 0 cos T O sin T ~XR~
v = O cos C sin C O 1 0 R ~19)
v O -sin C cos C -sin T O cos T R
112~ 9
Finally, combining equations (18) and ~19) results in:
Y ¦= [C] ~T~ ~A]t lly (1~)
Equation ~14)may be further simplified as follows:
IY ¦ = [D]t ¦ (20)
v BE~I a BEAM
Where [D]t is a composite matrix given by:
[D]t = [C] ~T] [A] (21)
;
Expanding Equation (20) results in:
X (beam) = dl A (beam) + dl Y ~beam) + dl3 Z ~beam)
~ Y ~beam) = dl X ~beam) + dl22 Y (beam) ~ dl3 Za tbeam) ~22)
: where dll, dl2, etc. are elements of the composite transformation
; matrix ¦D] , and
Z ~beam) = [1 x2 (beam) y2 (beam)] 1/2 ~23)
Inverting vector Equation ~20) yields:
-18-
Ya = [Dl ¦YV (24)
The procedure now is to map an isodop from velocity space into
~antenna space," obtain the slope of the isodop in 'antenna spacel'
and then evaluate this slope at the coordinates of a predetermined
beam pointing angle. In "velocity space", the locus of points of
equal Doppler (i.e. an isodop) is given by the following expres-
sion:
Xv~ sin G sin t
Yv = sin G cos t ~25)
_Zv cos G
where the angles G and t have been previously defined with reference
to FlG. 4b. The isodop of Equation (25) maps into "antenna space'~
X , Y , Z where:
20-xa- i sin G sin t
a = [D] sin G cos t t26)
LZa cos G
Expanding Equation ~26) results in the following:
Xa dll sin G sin 12 13 ~27)
a 21 sin G sin t + d22 sin G cos t + d23 cos G ~28)
Z = (1 - Xa a) (29)
-19 -
5419
where dll, dl2, etc. are elements of the transformation matrix L~-
Taking the first derivative of Equations ~27) and (28~ yields:
dY
a = d sin G cos t -d sin G sin t ~30
dX
a = dll sin G cos t -dl2 sin G sin t (31)
: Dividing Equation ~30) by (31) yields:
dYa d sin G cos t -d sin G sin t
dXa dll sin G cos t -dl2 sin G sin t ~32)
which further reduces to:
dY d Y -d X
a 21 v 22 v
: = (33)
dXa dll Yv -dl2 Xv
Equation (33) represents the slope of the isodop in antenna space~
This slope should now be evaluated at the coordinates of the beam
by means of combining equations (33) and (22~ to yield:
.
-20-
~ZS~l9
!
dY ~d dl X + d dl Y + d dl z
a 21 21 a21 22 a 21 23 a
(34)
dX (d dl X + dl Y + d dl z
a 11 21 a22 a 11 23 a
22 11 ad22 dl2 Ya + d22 d
(d dl X + d dl Y + d dl
12 11 a12 12 a 12 13 a
Simplifying Equation (34): yields:
dY (d dl -d dl )X +(d dl -d dl )Y +(d dl -d dl )Z
a 21 21 22 11 a 21 22 Z2 12 a 21 23 22 13 a
dX (d ldll-dl2dll)Xa+(dlldl22-dl2dl2)Ya (dll 23 12 13 a
where Xa, Ya, Za are the coordinates of the predetermined beam.
Referring now to FIG. 6, the pertinent angles and slopes are shown.
The tangent of the Angle B between the lines labeled respectively
X, Xl is given by:
`:
M - M
tan B -2 1 (3~)
: 1 + M2 M
where:
M2 is the slope of X
Ml is the slope of X
Since the slope of X - Ml = O and the slope of Xl = M2 = M, then it
30 follows that:
21-
tan B = M = - 1
dYa/dXa
evaluated at the beam since Xl is normal to the isodop.
Also, it follows that:
cos B = ~
~2 + 1
sin B = M
~M2 + 1
Thus, the rotated axes Xl, yl of FIG. 6 may be related to the
unrotated axes X, Y by means of the following expression:
[~ ] [ cos B sin Bl [X] (38)
yl -sin B cos B
or
xl = X cos B + Y sin B ~39)
yl = -X sin B + Y cos B (40~
Referring back now to FIG. 3, the inputs to the monopulse rotation
processor 68 correspond to the sin B, cos B terms in Equations ~39)
and (40). The X, Y terms represent the azimuth and elevation
signals on lines 66c and 66b, respectively, and the Xl, yl terms
are the rotated azimuth and elevation signals shown on lines 86b
and 86a.
ll~S~i9
Having described a preferred embodiment of this invcntion,
it is now evident that other embodiments incorporating its
concepts may be used. It is felt, therefore, that this invention
should not be restricted to its disclosed embodiment but rather
should be limited only by the spiTit and scope of the appended
claims.
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