Abstract:
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.

Description:
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 &#34;continuous&#34; structure (a reflector or a lens) that collects incident energy over the extent of the aperture and focuses it onto a receiving &#34;point&#34; sensor, or beams it out of a &#34;point&#34; radiator; or by a spatially spread set of discrete (receiving or transmitting) units which in effect &#34;sample&#34; a physical area or volume. This invention relates to the latter method of realizing a physical aperture, and &#34;aperture&#34; 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 2π 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 &#34;hybrid interferometry&#34;, wherein one opts to employ longbaseline phase-difference-measurement interferometry only for the &#34;fine&#34; measurement of the DOA, and other means for the &#34;first estimate&#34; or &#34;coarse&#34; 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&#39;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&#39;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&#39;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&#39;s by employing a method of directly obtaining first estimates of the cosines of the DOA&#39;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. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the drawings: 
     FIG. 1 is a diagram of a long-baseline interferometer employing an ambiguity resolving subsensor according to the invention. 
     FIG. 2 is a block diagram embodiment of an algorithm for deriving an ambiguity resolving estimate of the direction cosine from the outputs of spaced antennas according to the invention. 
     FIG. 3 is a block diagram of an embodiment of the invention for deriving the value of the cosine of the radial direction of arrival of a signal wavefront from the detected sinusoidal Doppler frequency modulation induced by simulating antenna around the perimeter of a circle. 
     FIG. 4. is a block diagram of yet another embodiment for deriving determinants of the radial direction cosine from induced sinusoidal Doppler frequency modulation according to this invention. 
     FIG. 5 is an illustrative diagram of an embodiment of the invention for deriving the radial direction cosine from the output of abrupt switching between two spaced antennas. 
    
    
     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 ω c  t with a wavelength, λ=c/(ω c  /2π), the difference, T.sub.φ, 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 
     φ=Radial angle of incidence of the wavefront relative to the orientation of the line connecting the antennas 
     c=The speed of propagation 
     For (L/λ) cos φ&gt;1, the right-hand-side of Eq. 
     (1) is expressible as 
     
         2π(L/λ) cos φ=2πK+δ                 (2) 
    
     where K=an integer, and 0≦δ&lt;2π. Inasmuch as cycles of a sinewave are indistinguishable one from the other, the component 2πK 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 δ. Such a measurement is therefore said to be &#34;cyclically ambiguous&#34;, meaning of course ambiguous in the number K of full 2π&#39;s that must be added to δ in order to account fully for the effect of the difference, T.sub.φ, 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: 2πK 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 δ by a &#34;fine&#34; 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 φ. 
     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 &#34;fine&#34; 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/λ 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 λ&#39;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 
     
         K=(L/λ) cos φ+δ/2π                     (3) 
    
     Since δ is measured with high resolution, and L/λ is, a priori, known or measurable with high precision, we need only measure cos φ to within a resolution, ε cos  φ, such that 
     
         (L/λ)|ε.sub.cos φ |&lt;1/2(4) 
    
     Accordingly, the measurement of cos φ need be good only to within a peak error of 
     
         |ε.sub.cos φ |.sub.peak =1/(2L/λ)(5) 
    
     A cos φ measurement of such coarseness should therefore be sufficient to resolve the cyclic phase-difference ambiguities of a baseline of length 
     
         L/λ&lt;(1/2)/|ε.sub.cos φ |.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 
     
         σ.sub.K.sup.2 =(L/λ).sup.2 σ.sub.cos φ.sup.2 +σ.sub.δ.sup.2 /(2π).sup.2                 (7) 
    
     The distinction between coarse and fine measurements allows us to attribute an uncertainty ε K  in K entirely to the coarse measurement of cos φ, 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 φ to additive gaussian noise with a peak factor p, the probability that the error ε K  in computing K from the measurement of cos φ will exceed a peak of pσ K  is given by ##EQU2## where erf(. . . ) is the error function. Since ε K  must not exceed 1/2 if the ambiguity is to be resolved correctly, we set pσ K  =1/2, which then enables us to express the probability that the coarse measurement of cos φ will not correctly resolve the ambiguity for the long baseline as ##EQU3## 
     Expressions for cos φ will next be determined for a number of candidate methods for providing AR estimates of cos φ. 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 φ: As an Adcock directional sensor, or for inducing cos φ-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 θ 1  relative to North is, in response to a wavefront described by cos ω c  t at the center of the circle, 
     
         e.sub.θ1 (t)=2E.sub.s cos α sin {(πD/λ) cos α cos (.θ.sub.1 -θ)} sin ω.sub.c t        (10) 
    
     where 
     α=Elevation angle of arrival above the plane of the circle 
     θ=Azimuth angle of arrival relative to North 
     
         cos φ=cos α cos θ                          (11) 
    
     D=Diameter of circle 
     λ=Wavelength of incident wave 
     E 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/λ, we can synthesize two resultant signals described by 
     
         e.sub.NS (t)≃(n/2)E.sub.s (πD/λ){ cos.sup.2 α cos θ} sin ω.sub.c t,                         (12) 
    
     Corresponding to a North-South diameter and 
     
         e.sub.EW (t)≃(n/2)E.sub.s (πD/λ){cos.sup.2 α sin θ} sin ω.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 
     
         e.sub.o (t)=E.sub.s cos α cos ω.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 π/2 rad to obtain 
     
         e.sub.o,π/2 (t)=E.sub.s cos α sin ω.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 φ 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 NS  (t) in FIG. 2, is shifted in frequency by a fixed amount, denoted ω 1 , sufficient to make signals at ω c  and ω c  +ω 1  separable compoletely by an ordinary filter. The frequency-shifted signal is then added to the other signal, with the signal corresponding to e o  (t) at least a few times stronger than that corresponding to e 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 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 NS  (t) to deliver a corresponding frequency-reference signal with a constant amplitude independent of DOA. This latter signal is phase-shifted π/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 θ; from which ##EQU6## Substitution from Eqs. (18) and (19) into Eq. (11) yields cos φ. (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 φ are derived by assuming errors in the measured quantities in Eq. (15). The results are 
     
         ε.sub.cos φ ≃{(n/2)(πD/λ)}.sup.-1 (ε.sub.NS /A.sub.o)-(ε.sub.o /A.sub.o) cos φ(20) 
    
     
         |ε.sub.cos φ |≦{(n/2)(πD/λ)}.sup.-1 |ε.sub.NS |/A.sub.o+|ε.sub.o |/A.sub.o                                        (21) 
    
     and, for random errors, 
     
         σ.sub.cos φ, max.sup.2 ={σ.sub.NS.sup.2 /(nπD/2λ).sup.2 +σ.sub.o.sup.2 }/A.sub.o.sup.2(22) 
    
     where 
     A o  =Amplitude of output of Center Antenna 
     ε o  =Error in measurement of A o   
     ε NS  =Error in measurement of amplitude of e NS  (t) 
     and σ o  and σ 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 φ. 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 ω c  t will be transformed by the rotation of the receiving antenna into an exponent-modulated signal at the receiver input, described by 
     
         e.sub.rec (t)=E.sub.s cos α cos {ω.sub.c t+ψ(t)}(23) 
    
     where E s  is an amplitude-level factor, 
     
         ψ(t)=(2πr/λ) cos α cos (ω.sub.m t-θ)(24) 
    
     θ is measured relative to the orientation of a reference diameter, and α above the plane, of the circle. An FM demodulator delivers ##EQU7## where κ d  is a proportionality constant. This shows that ##EQU8## where  ○X  denotes convolution; h lp  (t) is the unit-impulse response of a lowpass filter that passes 0 Hz and rejects all frequencies at and above ω m  rad/s, and has a DC response given by H lp  (jO); and 
     
         G={κ.sub.d (ω.sub.m r/λ)H.sub.lp (jO)}/2(27) 
    
     The operations expressed in Eq. (26) can be implemented as shown in FIG. 3. FIG. 4 shows how α and θ can be extracted from e out  (t). 
     Expressions for errors in the determination of cos φ are derived by assuming errors in in the measured quantities in Eq. (26). The results are 
     
         |ε .sub.cos φ |.sub.max ≃|ε.sub.θ |.sub.max +|ε.sub.Am |.sub.max /A.sub.m   (8) 
    
     and, for random errors, 
     
         σ.sub.cos φ,max.sup.2 =σ.sub.74 .sup.2 +σ.sub.Am.sup.2 /A.sub.m.sup.2 =σ.sub.Am.sup.2 /(A.sub.m.sup.2 /2)                                                       (29) 
    
     where ##EQU9## ε Am  =Error in determination of A m  ε.sub.θ =Error in the phase of detected tone 
     σ Am  and σ.sub.θ are rms values of random errors 
     
         σ.sub.Am.sup.2 =N.sub.o f.sub.m.sup.2 β.sub.n /P.sub.s(30) 
    
     
         σ.sub.θ.sup.2 =1/{2(A.sub.m.sup.2 /2)/σ.sub.Am.sup.2 }=σ.sub.Am.sup.2 /A.sub.m.sup.2                     (31) 
    
     N o  =Pre-FM-demodulation (i.e., IF) noise power spectral density, in watts/Hz or Joules 
     f m  =ω m  /2π 
     β n  =Effective noise bandwidth of an output bandpass filter centered at f m  Hz, in Hz 
     P s  =Pre-FM-demodulation (i.e., IF) average signal power, in watts 
     
         P.sub.s /(N.sub.o B.sub.nIF)≧10 dB 
    
     B nIF  =Pre-FM-demodulation (i.e., IF) effective noise bandwidth, in Hz 
     and the max&#39;s in Eqs. (28) and (29) are for α=0 and θ=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 λ/3 or less, along the intended &#34;path of motion&#34; (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 φ with the normal to the incident wavefront. Let the motion be at constant speed, in one direction, repeating f 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 ω c  t will be presented to the receiver input with a frequency shift given by {2,3} ##EQU10## In this case, 
     
         ε.sub.cos φ =ε.sub.f /(f.sub.m D/λ)(34) 
    
     and 
     
         σ.sub.cos φ =σ.sub.f /(f.sub.m D/λ) (35) 
    
     where ε f  is the error in counting the induced frequency shift, and σ 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 jω c  t is incident at a radial angle φ 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(ω c  t-Δψ), and at x=-d/2 by exp j(ω c  t+Δψ), where Δψ=(πd/λ) cos φ. 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 
     
         e.sub.rec (t)=e.sup.jψ(t) e.sup.jω.sbsp.c.sup.t  (36) 
    
     where ψ(t)=+Δψ or -Δψ. In this way, binary PSK may be applied to the signal in accordance with some code. Note that 
     
         e.sup.jψ(t) =cos Δψ±j sin Δψ    (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 Δψ, and an orthogonally phased phase-reversal modulated component of amplitude proportional to sin Δψ. 
     Now, let the antenna hopping be determined by a coded binary sequence c(t) of rectangular pulses, each of unit height, duration T 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, Δψ, 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 Δψ≃Δψ=(πd/λ) cos φ for d≦λ/4. The code modulation in Eq. (38) provides a means for enhancing the detected quadrature component. An IQ phase-lock loop automatically delivers sin Δψ≃Δψ=(πd/λ) cos φ, for d/λ≦λ/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., &#34;New Developments in Direction-of-Arrival Measurement Based on Adcock Antenna Clusters&#34;, NAECON&#39;89 Conference Proc.; May, 1989. 
     2. Baghdady, E. J., &#34;IDFM: A Novel Technique for Tracking, Navigation Aid and Flight Traffic Surveillance&#34;, Proceedings of the 1975 IEEE National Aerospace &amp; Electronics Conference, NAECON&#39;75, May, 1975. 
     3. Baghdady, E. J., &#34;Frequency Modulation by Synthetic Doppler: Theory and Some Novel Applications&#34;, Proceedings of the 1987 National Aerospace &amp; Aeronautical Electronics Conference, NAECON&#39;87, May 1987; pp. 310-316. 
     4. Baghdady, E. J., &#34;Directional Signal Modulation by Means of Switched Spaced Antennas&#34;, Proceedings of the 1987 IEEE Military Communications Conference, MILCOM&#39;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 2π 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 &#34;direction of propagation&#34; relative to a receiving sensor. 
     Radial direction, or angle, of arrival (RDOA) is the angle, φ, (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, θ, (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, α, (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 &#34;beamforming&#34;, &#34;maximum likelihood&#34;, &#34;MUSIC&#34;, &#34;ESPRIT&#34;. 
     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.