Methods and apparatus for direction of arrival measurement and radio navigation aids

The disclosure relates to methods, algorithms and apparatus for direction-of arrival (DOA) measurement/computation based on long-baseline, phase-difference, paired-antenna interferometry and on DOA-computing array processing algorithms. Specifically, methods and algorithms based on direct, cyclically unambiguous estimation of the cosine of the DOA are described for resolving the cyclic ambiguities in long-baseline, phase-difference paired-antenna interferometers, and for steering the computations to the vicinities of the solutions in computation-intensive array processing algorithms, thereby reducing computation load and time. The invention enables the design of DOA-determination systems and radio navigation aids that combine desirable characteristics (such as high resolution and accuracy, simpilicity, low cost, self-calibration, etc.) of different methods of extracting/computing DOA data from the outputs of antenna elements that are positioned in diverse arrangements for realizing complementary apertures.

BACKGROUND OF THE INVENTION 
This disclosure relates to methods, algorithms and apparatus for enabling 
unambiguous, high-resolution measurement/computation of the direction of 
propagation of a traveling wavefront. 
It is well-known in the art of measuring direction of arrival (henceforth, 
DOA) of a traveling wave that the resolution, accuracy and immunity to 
multipath and multi-signal interference of the measurement are increased 
by increasing the physical aperture of the receiving sensor in emitter 
location relative to the receiver, and of transmitting illuminator in 
receiver location relative to the transmitter (as in radio navigation 
aids). Said physical aperture may be realized either by a "continuous" 
structure (a reflector or a lens) that collects incident energy over the 
extent of the aperture and focuses it onto a receiving "point" sensor, or 
beams it out of a "point" radiator; or by a spatially spread set of 
discrete (receiving or transmitting) units which in effect "sample" a 
physical area or volume. This invention relates to the latter method of 
realizing a physical aperture, and "aperture" is defined here as the 
maximum physical lineal separation between members of said discrete set. 
This invention relates to discrete aperture-sampling sets of antennas with 
which the direction of propagation of a traveling wavefront is ultimately 
derived either from the phase-shift differences between the outputs of the 
most widely separated pairs of antennas, or by employing the outputs of 
the various discrete antennas to set up an algebraic system which is then 
solved for characteristic indicators of said direction of propagation. In 
this disclosure we shall refer to the phase-shift difference method as 
paired-antenna interferometry (or PAI, for short), and to the algebraic 
system approach as the DOA-computing array processing algorithm (or APA, 
for short) approach. 
The baseline length of a paired-antenna, phase-difference measuring 
interferometer (PAI) is the key parameter for high-resolution and accuracy 
of direction-of-arrival (DOA) measurement by this means. The baseline 
length is in essence the aperture of a DOA sensor in which this sensing is 
based on the phase difference accumulated by the incident wavefront in 
transit from one end of the baseline to the other. If the path length 
traversed between these ends is equivalent to more than one wavelength of 
the incident wave, then the corresponding phase shift will include an 
integer multiple of 2.pi. rad that will not be revealed by a 
phase-difference detector. The phase-difference measurement is then said 
to be cyclically ambiguous. Cyclic ambiguity is resolved in prior art by 
means of other phase-difference measurements that are also subject to 
cyclic ambiguity that are performed in parallel between the outputs of 
additional pairs of antennas separated by judiciously chosen, 
progressively shorter baseline lengths; or, in cooperative situations that 
so avail, between the outputs of the same pair of antennas, on components 
of the same signal that differ appropriately in frequency. As a result, 
complexity and cost of an interferometer go up with baseline length, 
largely because of escalating costly provisions for resolving the cyclic 
ambiguities. The special design requirements and the added initial 
nonrecurring acquisition cost and later recurring calibration, operation 
and maintenance costs, of the prior art methods of cyclic ambiguity 
resolution, set severe limits on affordable or permissible aperture extent 
(interferometer baseline length). 
The technique disclosed herein provides a means for resolving the cyclic 
ambiguities of the phase difference between wavefronts of the same signal 
wave at the positions of a pair of antennas that are separated by a 
baseline length equal to an an arbitrary number of wavelengths of said 
signal wave by employing a method of directly obtaining a first estimate 
of the cosine of the direction of arrival (DOA) through a cyclically 
unambiguous, or non-PAI, measurement. In this disclosure, we introduce the 
concept of "hybrid interferometry", wherein one opts to employ 
longbaseline phase-difference-measurement interferometry only for the 
"fine" measurement of the DOA, and other means for the "first estimate" or 
"coarse" measurement, that resolves the cyclic ambiguity in the fine 
measurement. 
It is therefore an object of this invention to provide an alternative 
method and means for resolving long interferometer baseline 
phase-difference cyclic ambiguities at significant reduction in costs and 
complexity in comparison with said prior art methods, and thus eliminate 
the ambiguity resolution requirement of a long-baseline PAI as a primary 
factor limiting affordable baseline length. 
The invention also applies to DOA-computing array processimg algorithms (or 
APA's) such as those known in the art by the descriptive labels of 
beamforming, maximum-likelihood, MUSIC (for multiple signal 
classification) and ESPRIT (for estimation of signal parameters via 
rotational invariance techniques). Characteristically, all said algorithms 
involve complex and lengthy computations that inherently start with a 
search-and-plot procedure to reveal the peaks of a measure or an indicator 
of signal presence versus DOA. In all of said APA's, the computation load 
and time would be significantly reduced if additional information is 
provided to point the way to solutions in the form of at least coarse 
first estimates of DOA's of some or all incident signals picked up by the 
antennas in the array. 
It is therefore another object of this invention to provide a method for 
significantly reducing the computation load and time of DOA-computing 
APA's by employing a method of directly obtaining first estimates of the 
cosines of the DOA's of some or all of a number of incident signals, and 
hence restricting the required computations only to the refinement of 
those estimates. 
It is yet a further object of this invention to provide methods and 
algorithms for estimatimg the cosine of the direction of arrival of a 
wavefront that are not subject to, or require/involve resolution of, 
cyclic ambiguity. 
These and other objects and features of this invention will become apparent 
from the claims, and from the following description when read in 
conjunction with the accompanying drawings.

DETAILED DESCRIPTION 
The hybrid interferometry concept is illustrated in FIG. 1. In this 
concept, a nonambiguous, coarse DOA measurement is performed in some 
manner other than phase-difference interferometry, with accuracy 
sufficient to completely resolve the ambiguity of a fine measurement based 
on an extended baseline defined only by two widely spaced antennas. For 
purposes of economy, the coarse measurement is carried out in terms of a 
direction-dependent parameter not subject to cyclic ambiguity, by means of 
a moderate-to-small-aperture sensor; i.e., one with an aperture much 
smaller than the highly cyclically ambiguous long-baseline, 
phase-difference interferometer. 
For a wavefront cos .omega..sub.c t with a wavelength, 
.lambda.=c/(.omega..sub.c /2.pi.), the difference, T.sub..phi., in times 
of arrival at two antennas spaced L apart causes a phase shift difference 
between the pick-ups of the signal by these antennas, expressible as 
##EQU1## 
where L=Distance separating the two antennas, the so-called baseline 
length 
.phi.=Radial angle of incidence of the wavefront relative to the 
orientation of the line connecting the antennas 
c=The speed of propagation 
For (L/.lambda.) cos .phi.&gt;1, the right-hand-side of Eq. 
(1) is expressible as 
EQU 2.pi.(L/.lambda.) cos .phi.=2.pi.K+.delta. (2) 
where K=an integer, and 0.ltoreq..delta.&lt;2.pi.. Inasmuch as cycles of a 
sinewave are indistinguishable one from the other, the component 2.pi.K is 
not detectable as a phase difference between two replicas of the same 
sinewave, and the only output of a phase-difference detector will be 
.delta.. Such a measurement is therefore said to be "cyclically 
ambiguous", meaning of course ambiguous in the number K of full 2.pi.'s 
that must be added to .delta. in order to account fully for the effect of 
the difference, T.sub..phi., between the times of arrival of the wavefront 
at the two separated antennas. 
In practice, the two components of phase difference in Eq. (2) are 
determined separately: 2.pi.K by a coarse measurement (i.e., not as fine a 
measurement as for the full baseline, but) fine enough to provide for the 
ambiguity resolution (henceforth, AR); and .delta. by a "fine" measurement 
to bring out the baseline instrumental resolution (henceforth, BIR) and 
accuracy in the determination of the baseline end-to-end phase difference, 
and, hence, the direction cosine, cos .phi.. 
Of all the variables in Eq. (2), only K cannot be measured directly, and 
therefore must be inferred from measurements of the other quantities. 
Since K is a discrete integer, its value is quantized, changing in quantum 
steps of unity. Accordingly, measurements leading to it need only be 
"fine" enough to yield a number within .+-.1/2 of the actual K. 
The traditional method of performing the measurement leading to K is to 
exploit the fact that the ratio L/.lambda. can be arranged, a priori, to 
have a set of values judiciously chosen to provide a progression of 
reduced (and hence less precise) actual (if different values of L are 
used) or virtual/electical (if different frequencies, or .lambda.'s, are 
simultaneously received) baseline lengths. This enables a number of 
different phase-difference measurements to be performed in parallel on 
either (i) the outputs of 3 or more antennas paired to provide a set of 
exactly known baseline lengths, the shorter ones resolving the ambiguities 
of the longer ones, leading in the end to the AR of the longest baseline; 
or (ii) on sinewaves of different frequencies originating in the same 
source and picked up by the same baseline pair of antennas, or by 3 or 
more antennas arranged for a judicious selection of baseline lengths. 
The alternative to the traditional method, offered by the hybrid 
interferometry approach, follows from solving Eq. (2) for K, yielding 
EQU K=(L/.lambda.) cos .phi.+.delta./2.pi. (3) 
Since .delta. is measured with high resolution, and L/.lambda. is, a 
priori, known or measurable with high precision, we need only measure cos 
.phi. to within a resolution, .epsilon..sub.cos .phi., such that 
EQU (L/.lambda.).vertline..epsilon..sub.cos .phi. .vertline.&lt;1/2(4) 
Accordingly, the measurement of cos .phi. need be good only to within a 
peak error of 
EQU .vertline..epsilon..sub.cos .phi. .vertline..sub.peak =1/(2L/.lambda.)(5) 
A cos .phi. measurement of such coarseness should therefore be sufficient 
to resolve the cyclic phase-difference ambiguities of a baseline of length 
EQU L/.lambda.&lt;(1/2)/.vertline..epsilon..sub.cos .phi. .vertline..sub.peak(6) 
If the error in computing K form the substitution of results of 
measurements in Eq. (3) is attributed to random errors in the 
measurements, then the mean squared error in computing K from Eq. (3) is 
given by 
EQU .sigma..sub.K.sup.2 =(L/.lambda.).sup.2 .sigma..sub.cos .phi..sup.2 
+.sigma..sub..delta..sup.2 /(2.pi.).sup.2 (7) 
The distinction between coarse and fine measurements allows us to attribute 
an uncertainty .epsilon..sub.K in K entirely to the coarse measurement of 
cos .phi., in which case we drop the second term on the right-hand-side of 
Eq. (7). If further we attribute the error in measuring cos .phi. to 
additive gaussian noise with a peak factor p, the probability that the 
error .epsilon..sub.K in computing K from the measurement of cos .phi. 
will exceed a peak of p.sigma..sub.K is given by 
##EQU2## 
where erf(. . . ) is the error function. Since .epsilon..sub.K must not 
exceed 1/2 if the ambiguity is to be resolved correctly, we set 
p.sigma..sub.K =1/2, which then enables us to express the probability that 
the coarse measurement of cos .phi. will not correctly resolve the 
ambiguity for the long baseline as 
##EQU3## 
Expressions for cos .phi. will next be determined for a number of candidate 
methods for providing AR estimates of cos .phi.. The techniques considered 
are all based on measurables that are intrinsically free of cyclic 
ambiguities. 
ADCOCK-SENSOR-BASED AMBIGUITY RESOLVERS 
Antennas arranged with uniform spacing around the perimeter of a circle can 
be employed in at least two ways that we consider here for the AR coarse 
measurement of cos .phi.: As an Adcock directional sensor, or for inducing 
cos .phi.-dependent sinusoidal FM. In this Section, we consider the 
Adcock-based techniques. 
In an Adcock directional sensor, the outputs of diametrically opposite 
pairs of antennas are first subtracted one from the other. The result for 
the pair separated by the diameter at azimuth angle .theta..sub.1 relative 
to North is, in response to a wavefront described by cos .omega..sub.c t 
at the center of the circle, 
EQU e.sub..theta.1 (t)=2E.sub.s cos .alpha. sin {(.pi.D/.lambda.) cos .alpha. 
cos (..theta..sub.1 -.theta.)} sin .omega..sub.c t (10) 
where 
.alpha.=Elevation angle of arrival above the plane of the circle 
.theta.=Azimuth angle of arrival relative to North 
EQU cos .phi.=cos .alpha. cos .theta. (11) 
D=Diameter of circle 
.lambda.=Wavelength of incident wave 
E.sub.s =Amplitude level factor 
It can be shown {1} that if the differenced outputs of a sufficient number 
of diametrically opposite pairs of antennas are combined in a prescribed 
way, then, except for a sequence of forbidden discrete values of 
D/.lambda., we can synthesize two resultant signals described by 
EQU e.sub.NS (t).perspectiveto.(n/2)E.sub.s (.pi.D/.lambda.){ cos.sup.2 .alpha. 
cos .theta.} sin .omega..sub.c t, (12) 
Corresponding to a North-South diameter and 
EQU e.sub.EW (t).perspectiveto.(n/2)E.sub.s (.pi.D/.lambda.){cos.sup.2 .alpha. 
sin .theta.} sin .omega..sub.c t, (13) 
Corresponding to an East-West diameter where n=(even) number of antennas 
around the perimeter of the circle. If, further, an antenna is placed at 
the center of the circle, then its output will be 
EQU e.sub.o (t)=E.sub.s cos .alpha. cos .omega..sub.c t (14) 
Inspection of the above equations shows that if the amplitudes in Eqs. (12) 
and (14) are first detected, then 
##EQU4## 
Alternatively, we may first phase-shift the output of the center antenna 
.pi./2 rad to obtain 
EQU e.sub.o,.pi./2 (t)=E.sub.s cos .alpha. sin .omega..sub.c t (16) 
From Eqs. (12) and (16), we have 
##EQU5## 
The division of hte predetected outputs can be performed computationally 
(digitally) at a very low IF. 
In a third alternative, cos .phi. can be extracted by the analog structure 
shown in FIG. 2. With reference to this figure, the ratio of amplitudes in 
Eq. (15) is obtained by means of an amplitude limiter. First, one of the 
two signals, e.sub.NS (t) in FIG. 2, is shifted in frequency by a fixed 
amount, denoted .omega..sub.1, sufficient to make signals at .omega..sub.c 
and .omega..sub.c +.omega..sub.1 separable compoletely by an ordinary 
filter. The frequency-shifted signal is then added to the other signal, 
with the signal corresponding to e.sub.o (t) at least a few times stronger 
than that corresponding to e.sub.NS (t). Amplitude-limiting the sum then 
effects the division of amplitudes required in Eq. (15) by yielding in the 
output of the limiter a signal component centered at the frequency of the 
input to the adder corresponding to e.sub.NS (t), whose amplitude is the 
desired ratio of amplitudes. A second amplitude limiter in the upper 
parallel branch operates on the signal corresponding to e.sub.NS (t) to 
deliver a corresponding frequency-reference signal with a constant 
amplitude independent of DOA. This latter signal is phase-shifted .pi./2 
rad, and then used to coherent-product demodulate the amplitude of the 
signal out of the lower branch, yielding a voltage proportional to cos 
(DOA) out of the lowpass filter. The structure in FIG. 2 embodies an 
algorithm that can also be implemented digitally/computationally. 
For a fourth alternative, note that the ratio of the amplitude in Eq. (13) 
to that in Eq. (12) is tan .theta.; from which 
##EQU6## 
Substitution from Eqs. (18) and (19) into Eq. (11) yields cos .phi.. (Note 
that the general expression that results from this substitution actually 
reduces to the expression in Eq. (15).) 
Expressions for errors in the determination of cos .phi. are derived by 
assuming errors in the measured quantities in Eq. (15). The results are 
EQU .epsilon..sub.cos .phi. .perspectiveto.{(n/2)(.pi.D/.lambda.)}.sup.-1 
(.epsilon..sub.NS /A.sub.o)-(.epsilon..sub.o /A.sub.o) cos .phi.(20) 
EQU .vertline..epsilon..sub.cos .phi. 
.vertline..ltoreq.{(n/2)(.pi.D/.lambda.)}.sup.-1 
.vertline..epsilon..sub.NS .vertline./A.sub.o+.vertline..epsilon..sub.o 
.vertline./A.sub.o (21) 
and, for random errors, 
EQU .sigma..sub.cos .phi., max.sup.2 ={.sigma..sub.NS.sup.2 
/(n.pi.D/2.lambda.).sup.2 +.sigma..sub.o.sup.2 }/A.sub.o.sup.2(22) 
where 
A.sub.o =Amplitude of output of Center Antenna 
.epsilon..sub.o =Error in measurement of A.sub.o 
.epsilon..sub.NS =Error in measurement of amplitude of e.sub.NS (t) 
and .sigma..sub.o and .sigma..sub.NS are the rms values of random errors. 
AR BASED ON INDUCED SINUSOIDAL DOPPLER FM 
A basis for DOA measurement is provided not only by phase change accured 
through the motion of the wavefront through the sensor aperture, but also 
by the rate of change of phase (the Doppler frequency shift) induced by 
moving a receiving antenna through the successive positions of the 
wavefront across the sensor aperture. Any motion of a receiving antenna 
relative to the source of radiation induces a Doppler frequency shift that 
depends in particular on the DOA of the incident.wavefront. In situ motion 
of a receiving antenna can be controlled so that it results in 
Doppler-shift modulation that can be detected unambiguously to provide cos 
.phi.. Two types of motion are of interest here: Circular motion and 
rectilinear motion. In this Section, we consider circular-motion-induced 
Doppler(CID). 
Consider an antenna in cicular motion. An incident signal described by cos 
.omega..sub.c t will be transformed by the rotation of the receiving 
antenna into an exponent-modulated signal at the receiver input, described 
by 
EQU e.sub.rec (t)=E.sub.s cos .alpha. cos {.omega..sub.c t+.psi.(t)}(23) 
where E.sub.s is an amplitude-level factor, 
EQU .psi.(t)=(2.pi.r/.lambda.) cos .alpha. cos (.omega..sub.m t-.theta.)(24) 
.theta. is measured relative to the orientation of a reference diameter, 
and .alpha. above the plane, of the circle. An FM demodulator delivers 
##EQU7## 
where .kappa..sub.d is a proportionality constant. This shows that 
##EQU8## 
where .circle.X denotes convolution; h.sub.lp (t) is the unit-impulse 
response of a lowpass filter that passes 0 Hz and rejects all frequencies 
at and above .omega..sub.m rad/s, and has a DC response given by H.sub.lp 
(jO); and 
EQU G={.kappa..sub.d (.omega..sub.m r/.lambda.)H.sub.lp (jO)}/2(27) 
The operations expressed in Eq. (26) can be implemented as shown in FIG. 3. 
FIG. 4 shows how .alpha. and .theta. can be extracted from e.sub.out (t). 
Expressions for errors in the determination of cos .phi. are derived by 
assuming errors in in the measured quantities in Eq. (26). The results are 
EQU .vertline..epsilon. .sub.cos .phi. .vertline..sub.max 
.perspectiveto..vertline..epsilon..sub..theta. .vertline..sub.max 
+.vertline..epsilon..sub.Am .vertline..sub.max /A.sub.m (8) 
and, for random errors, 
EQU .sigma..sub.cos .phi.,max.sup.2 =.sigma..sub.74 .sup.2 
+.sigma..sub.Am.sup.2 /A.sub.m.sup.2 =.sigma..sub.Am.sup.2 /(A.sub.m.sup.2 
/2) (29) 
where 
##EQU9## 
.epsilon..sub.Am =Error in determination of A.sub.m .epsilon..sub..theta. 
=Error in the phase of detected tone 
.sigma..sub.Am and .sigma..sub..theta. are rms values of random errors 
EQU .sigma..sub.Am.sup.2 =N.sub.o f.sub.m.sup.2 .beta..sub.n /P.sub.s(30) 
EQU .sigma..sub..theta..sup.2 =1/{2(A.sub.m.sup.2 /2)/.sigma..sub.Am.sup.2 
}=.sigma..sub.Am.sup.2 /A.sub.m.sup.2 (31) 
N.sub.o =Pre-FM-demodulation (i.e., IF) noise power spectral density, in 
watts/Hz or Joules 
f.sub.m =.omega..sub.m /2.pi. 
.beta..sub.n =Effective noise bandwidth of an output bandpass filter 
centered at f.sub.m Hz, in Hz 
P.sub.s =Pre-FM-demodulation (i.e., IF) average signal power, in watts 
EQU P.sub.s /(N.sub.o B.sub.nIF).gtoreq.10 dB 
B.sub.nIF =Pre-FM-demodulation (i.e., IF) effective noise bandwidth, in Hz 
and the max's in Eqs. (28) and (29) are for .alpha.=0 and .theta.=0. 
In situ antenna motion at speeds that are not mechanically feasible can be 
effectively and very accurately synthesized by electronically commutating 
the receiver input among the outputs of fixed antennas arranged, with 
uniform spacing of .lambda./3 or less, along the intended "path of motion" 
(e.g., the circle in above analysis). Such use of multiple antennas is 
strictly to designate sample positions along the path of motion, to be 
tapped consecutively by the commutator for connection to the receiver 
input, with time difference between successive samples equal to the time 
it would actually take a moving antenna to traverse the distance between 
the consecutive positions. The commutation process is then a process of 
sampling the phase-shift due to propagation along the path of motion in 
order to synthesize from those samples, by interpolation, the motion of 
one hypothetical antenna on the basis of the well-established principles 
of the sampling theorem of signal theory. This use of an array of antennas 
is quite different from the ways in which discrete antennas are used in 
the other RDF techniques, wherein the individuality and spatial 
discreteness of each antenna in the array is essential to the RDF 
mechanism. 
AR BASED ON RECTILINEAR-MOTION-INDUCED DOPPLER 
Let the receiving antenna execute a repetitive motion along a straight line 
of motion (LOM) of length D that makes an angle .phi. with the normal to 
the incident wavefront. Let the motion be at constant speed, in one 
direction, repeating f.sub.m times per second, with abrupt flyback from 
the last position to the starting position on the LOM. During the 
constant-speed traversals of the LOM, an incident signal described by cos 
.omega..sub.c t will be presented to the receiver input with a frequency 
shift given by {2,3} 
##EQU10## 
In this case, 
EQU .epsilon..sub.cos .phi. =.epsilon..sub.f /(f.sub.m D/.lambda.)(34) 
and 
EQU .sigma..sub.cos .phi. =.sigma..sub.f /(f.sub.m D/.lambda.) (35) 
where .epsilon..sub.f is the error in counting the induced frequency shift, 
and .sigma..sub.f is the rms value of random errors in the frequency 
count. 
AR BASED ON ANTENNA SWITCHING {4} 
A simple antenna hopping arrangement is shown in FIG. 5. Two antennas are 
positioned at x=.+-.d/2. A plane wavefront representing an unmodulated 
carrier exp j.omega..sub.c t is incident at a radial angle .phi. relative 
to the line connecting the two antennas. The phase as sensed at the 
midpoint, x=0, is taken as reference. The signal as sensed at x=d/2 is 
then represented by exp j(.omega..sub.c t-.DELTA..psi.), and at x=-d/2 by 
exp j(.omega..sub.c t+.DELTA..psi.), where .DELTA..psi.=(.pi.d/.lambda.) 
cos .phi.. Thus, if the receiver input is switched from the output of one 
of the antennas to that of the other, the received signal is represented 
by 
EQU e.sub.rec (t)=e.sup.j.psi.(t) e.sup.j.omega..sbsp.c.sup.t (36) 
where .psi.(t)=+.DELTA..psi. or -.DELTA..psi.. In this way, binary PSK may 
be applied to the signal in accordance with some code. Note that 
EQU e.sup.j.psi.(t) =cos .DELTA..psi..+-.j sin .DELTA..psi. (37) 
which, as illustrated in FIG. 5b), shows that the received signal can be 
decomposed into a carrier reference component of amplitude proportional to 
cos .DELTA..psi., and an orthogonally phased phase-reversal modulated 
component of amplitude proportional to sin .DELTA..psi.. 
Now, let the antenna hopping be determined by a coded binary sequence c(t) 
of rectangular pulses, each of unit height, duration T.sub.b sec and 
positive or negative polarity. The received signal is then represented by 
##EQU11## 
From Eqs. (36) and (37), we observe that the induced phase step, 
.DELTA..psi., can be extracted by first separating the quadrature 
component of the signal, cribed by Eq. (37) in the receiver, and then 
taking the ratio of their detected amplitudes to obtain sin 
.DELTA..psi..perspectiveto..DELTA..psi.=(.pi.d/.lambda.) cos .phi. for 
d.ltoreq..lambda./4. The code modulation in Eq. (38) provides a means for 
enhancing the detected quadrature component. An IQ phase-lock loop 
automatically delivers sin 
.DELTA..psi..perspectiveto..DELTA..psi.=(.pi.d/.lambda.) cos .phi., for 
d/.lambda..ltoreq..lambda./4. 
AR BY MULTIPLE ANTENNA PATTERN INTERSECTIONS 
Other candidate methods for effecting the AR funtion include a variety of 
techniques based on steerable intersections and nulls of patterns of 
multiple antennas, such as (amplitude- and/or phase-comparison) monopulse, 
and homer-type DOA tracking systems. 
REFERENCES 
1. Baghdady, E. J., "New Developments in Direction-of-Arrival Measurement 
Based on Adcock Antenna Clusters", NAECON'89 Conference Proc.; May, 1989. 
2. Baghdady, E. J., "IDFM: A Novel Technique for Tracking, Navigation Aid 
and Flight Traffic Surveillance", Proceedings of the 1975 IEEE National 
Aerospace & Electronics Conference, NAECON'75, May, 1975. 
3. Baghdady, E. J., "Frequency Modulation by Synthetic Doppler: Theory and 
Some Novel Applications", Proceedings of the 1987 National Aerospace & 
Aeronautical Electronics Conference, NAECON'87, May 1987; pp. 310-316. 
4. Baghdady, E. J., "Directional Signal Modulation by Means of Switched 
Spaced Antennas", Proceedings of the 1987 IEEE Military Communications 
Conference, MILCOM'87, Oct. 1987; pp. 938-942. 
While there has been described what is at present considered to be 
representative embodiments of the invention, it will be obvious to those 
skilled in the art that various changes and modifications may be made 
therein without departing from the invention, and it is aimed in the 
appended claims to cover all such changes and modifications as fall within 
the true spirit and scope of the invention. 
For clarity in the statements of the appended claims, the following 
definitions of terms are provided: 
Cyclic ambiguity of a phase difference measurement means failure to show if 
said phase difference includes some integer multiple of 2.pi. radians, and 
what said integer is. 
Cyclic ambiguity ratio is the first term on the right-hand-side of Eq. (3). 
Direction of propagation is the angle formed between a line perpendicular 
to the plane of the wavefront and a reference line on the sensor platform. 
Direction of arrival (DOA) is used here synonymously with "direction of 
propagation" relative to a receiving sensor. 
Radial direction, or angle, of arrival (RDOA) is the angle, .phi., (see for 
example FIG. 5a)) between a line perpendicular to the plane of the 
wavefront and a reference line on the sensor platform. 
Azimuth direction, or angle, of arrival (ADOA) is the angle, .theta., (see 
for example FIG. 5(a)) between the projection, on a reference horizontal 
plane at the location of the sensor, of a line perpendicular to the plane 
of the wavefront and a reference North-South line on said horizontal 
plane. 
Elevation direction, or angle, of arrival (EDOA) is the angle, .alpha., 
(see for example FIG. 5(a)) between a line perpendicular to the plane of 
the wavefront and its projection on said reference horizontal plane. 
DOA-computing array processing algorithm (APA) is a computational procedure 
based on a formulation of a system of equations derived from starting 
expressions for the outputs of antennas arranged in an array for sensing 
the directions of arrival of signal waves. Examples in the art include 
algorithms known as "beamforming", "maximum likelihood", "MUSIC", 
"ESPRIT". 
Mutually coherent signal waves are different signal waves that have 
commensurable frequencies and t=0 (or initial) phases whose ratios are the 
same as the ratios of the corresponding frequencies. 
Commensurable frequencies are frequencies whose ratios are integers or 
quotients of integers. 
Diametrically spaced, positioned at the ends of the same diameter of a 
circle. 
Resultant N-S signal is the signal expressed by Eq. (12). 
Resultant E-W signal is the signal expressed by Eq. (13). 
Quadrature product demodulator is one that multiplies the signal by a 
synchronous carrier in quadrature-phase with the carrier reference of the 
signal, and low-pass filters the product. 
Designated characteristics includes instantaneous values of the output 
(i.e., of its phasor projection on the axis of reals), or of its 
instantaneous envelope, its instantaneous phase or phase difference from 
some reference or from another output, or its instantaneous frequency.