Abstract:
A method and apparatus for determining and correcting for phased array mispointing errors, particularly those due to structural deformation, is disclosed. The method comprises the steps of receiving a signal from each of a plurality of signal sources at at least one receiving sensor disposed away from the phased array in a direction at least partially toward a receiver of a transmitted signal from the phased array, and determining the phased array pointing from the received signals. The apparatus comprises a receiving sensor for receiving a signal from each of a plurality of signal sources, the receiving sensor disposed away from the phased array in a direction at least partially toward a receiver of a transmitted signal from the phased array, and an array pointing computer for determining the direction of the phased array from the received signals.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates in general to methods of directing spacecraft payloads and in particular to a method and apparatus for determining and correcting for the pointing error of a phased array antenna on a spacecraft. 
     2. Description of the Related Art 
     Satellite systems are widely used to transmit information to many ground users. In satellite-based communication, it is desirable to transmit information to ground-based users in certain areas, but not the ground-based users in other areas. This is accomplished with the use of “spot beams” that concentrate the energy of the transmitted signal to a limited terrestrial area. To assure optimum reception by all ground-based users, to prevent interference among users in different areas, and to reduce the probability of unauthorized reception at ground stations not authorized to receive the transmitted spot beam, it is important that the spot beam be accurately directed to the proper terrestrial locations. Deviation of antenna pointing typically causes a drop of signal power for communications to and from the spacecraft and ground user in the satellite&#39;s services areas, thus degrading the communications services provided by the satellite. 
     Antenna pointing is usually controlled by a control system so that antenna communication beams will be accurately directed to the proper target(s). 
     Spot beam pointing accuracy can be limited by many factors. One of these factors is deformation of the structures supporting the phased array antenna on the spacecraft bus/body. Such errors can result from thermal gradients, launch environment effects, or other factors. Further, because sensors that are used to determine spacecraft pointing are usually placed at locations remote from the transmitting or receiving antennas and the components subject to structural deformation, such errors are typically unobservable by these sensors. 
     One technique for ameliorating this problem is to use an attitude sensor such as a star tracker, Earth sensor, or beacon sensor very close to or on the communication antenna itself. Unfortunately, this approach cannot be economically applied to satellites that have multiple communication antennas. Also, the use of beacon sensors can be unacceptably expensive because a terrestrial beacon station must be maintained for the on-board beacon sensor. This is especially the case for non-geosynchronous satellites because a single terrestrial beacon station will not be able to cover the entire orbit of the satellite and many stations are usually needed. What is needed is a system and method for compensating for these errors. The present invention satisfies that need. 
     SUMMARY OF THE INVENTION 
     To address the requirements described above, the present invention discloses a method and apparatus for determining pointing of a phased array. The method comprises the steps of receiving a signal from each of a plurality of signal sources at at least one receiving sensor disposed away from the phased array in a direction at least partially toward a receiver of a transmitted signal from the phased array, and determining the phased array pointing from the received signals. The apparatus comprises a receiving sensor for receiving a signal from each of a plurality of signal sources, the receiving sensor disposed away from the phased array in a direction at least partially toward a receiver of a transmitted signal from the phased array, and an array pointing computer for determining the direction of the phased array from the received signals. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Referring now to the drawings in which like reference numbers represent corresponding parts throughout: 
     FIG. 1 is a diagram illustrating a satellite or spacecraft; 
     FIG. 2 is a diagram depicting the functional architecture of a representative spacecraft control system; 
     FIGS. 3A-3C are diagrams depicting elements of a phased array pointing determination and correction device; 
     FIG. 4 is a diagram illustrating one implementation of the phased array pointing determination and correction device; 
     FIGS. 5A and 5B are flow charts illustrating exemplary process steps that can be used to practice the present invention; and 
     FIGS. 6A and 6B are diagrams depicting further embodiments of the present invention. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     In the following description, reference is made to the accompanying drawings which form a part hereof, and which is shown, by way of illustration, several embodiments of the present invention. It is understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the present invention. 
     FIG. 1 illustrates a three-axis stabilized satellite or spacecraft  100 . The spacecraft  100  is either situated in a stationary (geostationary or geosynchronous) orbit about the Earth, or in a mid-Earth (MEO) or low-Earth (LEO) orbit. The satellite  100  has a main body or spacecraft bus  102 , a pair of solar panels  104 , a pair of high gain narrow beam antennas  106 , and a telemetry and command omni-directional antenna  108  which is aimed at a control ground station. The satellite  100  may also include one or more sensors  110  to measure the attitude of the satellite  100 . These sensors may include sun sensors, earth sensors, and star sensors. Since the solar panels are often referred to by the designations “North” and “South”, the solar panels in FIG. 1 are referred to by the numerals  104 N and  104 S for the “North” and “South” solar panels, respectively. 
     The three axes of the spacecraft  100  are shown in FIG.  1 . The pitch axis P lies along the plane of the solar panels  140 N and  140 S. The roll axis R and yaw axis Y are perpendicular to the pitch axis P and lie in the directions and planes shown. The antenna  108  points to the Earth along the yaw axis Y. 
     The spacecraft  100  includes a phased array antenna  112  mounted on the spacecraft bus  102  or a supporting structure. The phased array antenna  112  can be used to transmit signals with wide angle or spot beams as desired. The spacecraft  100  also includes a boom  116  or other appendage, having a receiving sensor  114  such as a receiving horn mounted on the boom so that it&#39;s sensitive axis is directed substantially at the planar array. The boom-mounted calibration sensor sometimes used with phased array antennas can be used as the receiving horn  114  and boom, thus allowing the calibration system to be used to perform on-orbit pointing correction. As will be discussed in greater detail below, the boom  116  and receiving horn  114  permit the phased array pointing error to be accurately determined and compensated for. 
     FIG. 2 is a diagram depicting the functional architecture of a representative attitude control system. The spacecraft  100  includes a processor subsystem  274 , which includes a spacecraft control processor (SCP)  202  and a communication processor (CP)  276 . 
     The SCP  202  implements control of the spacecraft  100 . The SCP performs a number of functions which may include post ejection sequencing, transfer orbit processing, acquisition control, stationkeeping control, normal mode control, mechanisms control, fault protection, and spacecraft systems support, among others. The post ejection sequencing could include initializing to assent mode and thruster active nutation control (TANC). The transfer orbit processing could include attitude data processing, thruster pulse firing, perigee assist maneuvers, and liquid apogee motor (LAM) thruster firing. The acquisition control could include idle mode sequencing, sun search/acquisition, and Earth search/acquisition. The stationkeeping control could include auto mode sequencing, gyro calibration, stationkeeping attitude control and transition to normal. The normal mode control could include attitude estimation, attitude and solar array steering, momentum bias control, magnetic torquing, and thruster momentum dumping (H-dumping). The mechanisms mode control could include solar panel control and reflector positioning control. The spacecraft control systems support could include tracking and command processing, battery charge management and pressure transducer processing. 
     Input to the spacecraft control processor  202  may come from any combination of a number of spacecraft components and subsystems, such as a transfer orbit sun sensor  204 , an acquisition sun sensor  206 , an inertial reference unit  208 , a transfer orbit Earth sensor  210 , an operational orbit Earth sensor  212 , a normal mode wide angle sun sensor  214 , a magnetometer  216 , and one or more star sensors  218 . 
     The SCP  202  generates control signal commands  220  which are directed to a command decoder unit  222 . The command decoder unit operates the load shedding and battery charging systems  224 . The command decoder unit also sends signals to the magnetic torque control unit (MTCU)  226  and the torque coil  228 . 
     The SCP  202  also sends control commands  230  to the thruster valve driver unit  232  which in turn controls the liquid apogee motor (LAM) thrusters  234  and the attitude control thrusters  236 . 
     Wheel torque commands  262  are generated by the SCP  202  and are communicated to the wheel speed electronics  238  and  240 . These effect changes in the wheel speeds for wheels in momentum wheel assemblies  242  and  244 , respectively. The speed of the wheels is also measured and fed back to the SCP  202  by feedback control signal  264 . 
     The spacecraft control processor also sends jackscrew drive signals  266  to the momentum wheel assemblies  243  and  244 . These signals control the operation of the jackscrews individually and thus the amount of tilt of the momentum wheels. The position of the jackscrews is then fed back through command signal  268  to the spacecraft control processor. The signals  268  are also sent to the telemetry encoder unit  258  and in turn to the ground station  260 . 
     The SCP  202  communicates with the telemetry encoder unit  258 , which receives the signals from various spacecraft components and subsystems indicating current operating conditions, and then relays them to the ground station  260 . The telemetry encoder unit  258  also sends ground commands to the SCP  202  that executes various ground command spacecraft maneuvers and operations. 
     The wheel drive electronics  238 ,  240  receive signals from the SCP  202  and control the rotational speed of the momentum wheels. The jackscrew drive signals  266  adjust the orientation of the angular momentum vector of the momentum wheels. This accommodates varying degrees of attitude steering agility and accommodates movement of the spacecraft as required. 
     The use of reaction wheels or equivalent internal torquers to control a 3-axes stabilized spacecraft allows inversion about yaw of the attitude at will. In this sense, the canting of the momentum wheel is entirely equivalent to the use of reaction wheels. Other spacecraft employ external torquers, chemical or electric thrusters, magnetic torquers, solar pressure, etc. to control spacecraft attitude. 
     The CP  276  and SCP  202  may include or have access to one or more memories  270 , including, for example, a random access memory (RAM). Generally, the CP and SCP  202  operates under control of an operating system  272  stored in the memory  270 , and interfaces with the other system components to accept inputs and generate outputs, including commands. Applications running in the CP  276  and SCP  202  access and manipulate data stored in the memory  270 . The spacecraft  100  may also comprise an external communication device such as a satellite link for communicating with other computers at, for example, a ground station. If necessary, operation instructions for new applications can be uploaded from ground stations. The CP  276  and SCP  202  can also be implemented in a single processor, or with different processors having separate memories. 
     In one embodiment, instructions implementing the operating system  272 , application programs, and other modules are tangibly embodied in a computer-readable medium, e.g., data storage device, which could include a RAM, EEPROM, or other memory device. Further, the operating system  272  and the computer program are comprised of instructions which, when read and executed by the SCP  202 , causes the spacecraft processor  202  to perform the steps necessary to implement and/or use the present invention. Computer program and/or operating instructions may also be tangibly embodied in memory  270  and/or data communications devices (e.g. other devices in the spacecraft  10  or on the ground), thereby making a computer program product or article of manufacture according to the invention. As such, the terms “program storage device,” “article of manufacture” and “computer program product” as used herein are intended to encompass a computer program accessible from any computer readable device or media. 
     FIG. 3A is a diagram showing elements of the phased array pointing device  300 . The phased array pointing device  300  comprises a boom or appendage  116  extending from the spacecraft bus  102 . A receiving sensor  114  such as a radio frequency (RF) horn is attached to the boom  116  at the end of the boom  116  opposite the boom&#39;s attachment to the spacecraft bus  102 . The receiving sensor  114  is disposed away from the phased array  112  on the surface of the spacecraft bus  102 , and in a direction at least partially toward a receiver of a signal transmitted from the phased array  112  (in a direction away from the spacecraft bus  102 ). 
     The phased array pointing device  300  also includes a plurality of signal sources  302 A- 302 D (hereinafter alternatively referred to as signal source(s)  302 . Although four signal sources  302  are shown (up signal source  302 A, down signal source  302 C, left signal source  302 D and right signal source  302 B), the present invention can be implemented with a fewer or greater number of signal sources  302 . In the illustrated embodiment, the signal sources  302  are RF horns disposed about the periphery and at the center of each side of the phased array  112 , and together span a two-dimensional plane coincident with the phased array  112 . 
     In the illustrated embodiment, the signal sources  302  form four transmitting beams that form a directional pyramid  122 . The transmitted beams are received by the receiving sensor  114  along a null vector  120  a short distance away. 
     The four signal sources  302  have the location, line of sight separations, and beam widths described in Table 1 below: 
     
       
         
               
               
               
               
             
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 LOS Angular 
                   
                 Location 
               
               
                   
                 Separation 
                   
                 Separation 
               
               
                   
                 from Beacon 
                   
                 from Beacon 
               
               
                   
                 Null Vector 
                   
                 Null Vector 
               
               
                   
                 122 
                 Beamwidth 
                 122 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                 Up Signal Source 302A 
                 φ EL   
                 ψ 
                 d AZ   
               
               
                 Down Signal Source 302C 
                 −φ EL    
                 ψ 
                 −d AZ    
               
               
                 Left Signal Source 302D 
                 φ AZ   
                 ψ 
                 p EL   
               
               
                 Right Signal Source 302B 
                 −φ AZ    
                 ψ 
                 −p EL    
               
               
                   
               
             
          
         
       
     
     FIGS. 3B and 3C are diagrams showing selected elements of the phased array pointing determination and correction device  300  from perspective “A” shown in FIG. 3A, and FIG. 3C is a diagram showing elements of the phased array pointing device  300  from perspective “B” shown in FIG.  3 A. 
     FIG. 4 is a diagram illustrating an embodiment of further elements of the phased array pointing device  300 . The array pointing device  300  includes an array pointing computer  402  communicatively coupled to the receiving sensor  114  and the phased array  112 . The receiving sensor  114  is communicatively coupled to a receiver  402 , which detects and demodulates the signals sensed by the receiving sensor  114 . The received signals are provided to a signal selector  406 , which separates the signals received from each of the signal sources  302 , so that the signal from each can be appropriately analyzed. As each signal may be distinguishable from the others by transmitting one at a time, or at different frequencies, or with different codes, the functionality of the signal selector  406  may be intermingled with that of the receiver  404 . The output of the signal selector  404  is provided to a signal magnitude computer  408  which determines the magnitude of the signals received at the receiving sensor  114 , and a phase detector  410 , which determines the phase of each of the receiving signals. The phase information is provided to a distance computer  414 , which computes a distance between each of the signal sources  302  and the receiving sensor  114 . The output of the signal magnitude computer  408  is provided to the deviation angle computer  412 . The output of the deviation angle computer  412  and distance computer  414  are provided to an array pointing correction computer  416 , which generates a phased array pointing error. The pointing error is combined with the phased array pointing command to compensate for the computed errors, and provided to the phased array  112 . 
     FIGS. 5A and 5B are flow charts illustrating exemplary process steps that can be used to practice the present invention. Referring first to FIG. 5A, a plurality of signals are transmitted from the signal sources  302  in the direction of the receiving horn  114 , as shown in block  502 . In one embodiment, the boresight of the horns used to transmit the plurality of signals are directed away from the receiving horn  114  and cross each other between the signal sources  302  and the receiving horn  114  at focus point  118 . 
     The plurality of signals are received by the receiving horn  114  and the receiver  404 , as shown in block  504 . In the illustrated embodiment, the receiving horn  114  is disposed away from the phased array  112  in the direction that the phased array  112  ordinarily transmits signals. This is shown in block  504 . The received signals are then distinguished from one another, either by the time that they were received, the modulation frequency of the transmitted signal or by a signal code. This is shown in block  506 , and in the embodiment illustrated in FIG. 4, this is performed by the signal selector  406 . The phased array pointing (either the error between the indicated direction and the measured direction or the actual pointing direction) is determined from the received signals, as shown in bock  508 , and a phased array pointing correction is computed from the phased array pointing, as shown in block  510 . 
     FIG. 5B is a flow chart describing exemplary process steps that can be used to determine the phased array pointing from the received signals. In block  512 , a magnitude of each of the received signals is determined. In the embodiment illustrated in FIGS. 3A-3C, there are four signal sources, including an up signal source  302 A, a down signal source  302 C, a left signal source  302 D, and a right signal source  302 B. 
     Next, an azimuth and elevation deviation angle is computed from the magnitude of each of the received signals, as shown in block  514 . This can be accomplished as according to equation (1) below.                        EL   meas     =            α                       Mag   up     -     Mag   down           Mag   up     +     Mag   down             ,                 AZ   meas     =            β            Mag   left     -     Mag   right           Mag   left     +     Mag   right                         Equation                   (   1   )                                  
     wherein Mag up  is a magnitude of the received signal from the up signal source  302 A, Mag down  is a magnitude of the received signal from the down signal source  302 C, Mag left  is a magnitude of the received signal from the left signal source  302 D, Mag right  is a magnitude of the received signal from the right signal source  302 B, α is a first scale factor, and β is a second scale factor. 
     The phase of each of the received signals is also computed, as shown in block  516 . A distance is computed between the signal sources  302  and the receiving horn  114 , as shown in block  518 . This can be accomplished according to equations (2a)-(2d) below:                D   up     =         phase   up       2      π            λ   up               Equation                   (     2      a     )                   D   down     =         phase   down       2      π            λ   down               Equation                   (     2      b     )                   D   left     =         phase   left       2      π            λ   left               Equation                   (     2      c     )                   D   right     =         phase   right       2      π            λ   right               Equation                   (     2      d     )                                  
     wherein D up , D down , D left   , and D   right  are measured distances from the up, down, left, and right signal sources ( 302 A,  302 C,  302 D and  302 B) to the receiving sensor, respectively, and λ is wavelength of the radio frequency (RF) signal. 
     Next, as shown in block  520 , a pointing error of the phased array  112  is determined from the distance between the signal sources  302  and the receiving horn, and the azimuth and elevation deviation angles. This can be accomplished a variety of ways. For the four signal source embodiment disclosed in FIGS. 3A-3C this can be accomplished as follows:                  [           Δ                   θ   array_x                 Δ                   θ   array_y             ]     =                  I   xy          (       ∇     M   T            ∇   M       )         -   1            ∇     M   T       *     [           Δ                 EL               Δ                 AZ               Δ                   D   up                 Δ                   D   down                 Δ                   D   left                 Δ                   D   right             ]         ,     
            I   xy     =            [         1       0       0       0       0       0           0       1       0       0       0       0         ]               Equation                   (   3   )                                  
     wherein the array pointing error is αθ array     —     x  is the angular error in one direction and Δθ array     —     y  is the angular error in a direction orthogonal from the first angular error ΔEL and ΔAZ are the difference between the elevation and azimuth deviation angles EL meas  and AZ meas  described above and the nominal pointing angle (ΔEL=EL meas −EL 0 , and ΔAZ=AZ meas −AZ 0 ), ΔD up , ΔD down , ΔD left , and ΔD right  describe the difference between the distances from each of the signal sources and the receiving horn  114  D up , D down , D left , and D right  and the nominal (measured distance, not accounting for spacecraft bus deformation, e.g. ΔD up =D up −D up     0   , ΔD down =D down −D down     0   , ΔD left =D left −D left     0   , and ΔD right =D right −D right     0   ). 
     The gradient ∇M is computed from a sensitivity matrix ∇F as described below.                ∇   F     =       [           I   EL           T     center_receive      _EL                 I   AZ           T     center_receive      _AZ                   v   up_receive          S   up_center             v   up_receive                 v   down_receive          S   down_center             v   down_receive                 v   left_receive          S   left_center             v   left_receive                 v   right_receive          S   right_center             v   right_receive           ]          
     [                      C   Null_SC         0           0         C   Null_SC                      ]             Equation                   (   3   )                                  
     wherein 
     
       
           I   EL =[100 ], I   AZ =[010], 
       
     
     C Null     —     SC  is a direction matrix describing a transformation from a spacecraft body reference frame to a null vector  120  (extending from the center of the phase array  112  to the receiving horn  114 ) reference frame; 
     S up     —     center  is a skew symmetric position vector matrix describing a vector from the center of the phase array  112  to the up signal source  302 A; 
     S down     —     center  is a skew symmetric position vector matrix describing a vector from the center of the phase array  112  to the down signal source  302 C; 
     S left     —     center  is a skew symmetric position vector matrix describing a vector from the center of the phase array  112  to the left signal source  302 D; 
     S right     —     center  is a skew symmetric position vector matrix describing a vector from the center of the phase array  112  to the right signal source  302 B.                  T     center_receive      _EL       =     [         0         1     d   center_receive           0         ]       ,           Equation                   (   4   )                     T     center_receive      _AZ       =     [           1     d   center_receive           0       0         ]       ,              and           Equation                   (   5   )                   v   i     =         ⌊           x   i_receive           y   i_receive           z   i_receive           ⌋       d   i_receive       .             Equation                   (   6   )                                  
     and wherein 
     i={up, down, left, right} 
     d center     —     receive  is a distance from a center of the phased array to the receiving sensor; 
     d i     —     receive  is a distance from a vector from the i th  signal source to the receiving sensor, and 
     x i     —     receive , y i     —     receive , z i     —     receive  are x, y, and z components of the vector from the i th  signal source to the receiving sensor. 
     Using the foregoing relationships, the gradient ∇M is computed as: ∇M=∇F(:,[1,2,4,5,6]) (all of the rows and the first, second, fourth, fifth, and sixth columns of a sensitivity gradient matrix ∇F ). The use of a subset of the columns of the sensitivity gradient matrix ∇F assures appropriate numerical conditions and that the appropriate parameters can be computed. 
     Further, the error in the pointing error estimate can be determined as:                  [           E     θ   x                 E     θ   y             ]     =             I   xy          (       ∇     M   T            ∇   M       )         -   1       *     (       ∇     M   T            ∇   N       )     *   Δ                   θ   array_z       +           I   xy          (       ∇     M   T            ∇   M       )         -   1            [           n   el               n   az               n   d_up               n   d_down               n   d_left               n   d_right           ]           ,           Equation                   (   7   )                                  
     wherein ∇N=∇F(:,3) (all of the rows and the third column of ∇F), E 74      x    is the error in the pointing error estimate in a first direction, E θ     y    is an error in the pointing error estimate in a second direction orthogonal to the first direction, n el , n az , n d     —     up , n d     —     down , n d     —     left , and n d     —     right  represent noise in the measurement of the deviation angles and the distances from the up, down, left and right signal sources  302  to the receiving sensor  114 . 
     The foregoing is ultimately derived from the relationship:                [           Δ                 EZ               Δ                 AZ               Δ                   D   up                 Δ                   D   down                 Δ                   D   left                 Δ                   D   right             ]     =         ∇   F     *     [           Δθ   array_x               Δ                   θ   array_y                 Δ                   θ   array_z                 Δ                   x     array_to      _receive                   Δ                   y     array_to      _receive                   Δ                   z     array_to      _receive               ]       +     [           n   el               n   az               n   d_up               n   d_down               n   d_left               n   d_right           ]               Equation                   (   8   )                                  
     wherein the terms Δθ array     —     x , Δθ array     —     y , and Δθ array     —     z  represent the angular deformation in spacecraft body frame of the structures supporting the phase array  112  on the spacecraft bus  102  and Δx array     —     to     —     receive . Δy array     —     to     —     receive , and Δz array     —     to     —     receive  represent the translational deformation of the structures supporting the phase array  112  on the spacecraft bus  102 . 
     As shown in FIG. 4, the pointing error determined in block  520  can be added or subtracted from the phased array beam pointing commands, thus compensating for phased array beam pointing errors and increasing the angular accuracy of beams generated by the phased array  112 . 
     FIG. 6A is a diagram of another embodiment of the present invention, in which elements of the phased array  112  itself are used for the signal sources  302  instead of separate RF horns. Such beams can be formed using appropriate portions  602 A- 602 D of the phased array. 
     FIG. 6B is a diagram of another embodiment of the present invention, in which signal sources  302 A- 302 D are used to generate signals used to determine the distance from the signal sources  302 A- 302 D to the receiving sensor  114 , but in which the portions  602 A- 602 D of the phased array  112  are used to generate signals used to determine azimuth and elevation deviation angles. In this embodiment, the parameters described in Table 1 are represented as described in Tables 2A and 2B below: 
     
       
         
               
             
               
               
               
             
               
               
               
             
           
               
                 TABLE 2A 
               
             
             
               
                   
               
               
                 PHASE ARRAY ELEMENT-FORMED BEAMS 
               
             
          
           
               
                   
                 LOS Angular Separation 
                   
               
               
                   
                 from Beacon Null Vector 
               
               
                   
                 122 
                 Beamwidth 
               
               
                   
                   
               
             
          
           
               
                 Up Signal Source 602A 
                 φ EL   
                 ψ 
               
               
                 Down Signal Source 602C 
                 −φ EL    
                 ψ 
               
               
                 Left Signal Source 602D 
                 φ AZ   
                 ψ 
               
               
                 Right Signal Source 602B 
                 −φ AZ    
                 ψ 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 2B 
               
             
             
               
                   
               
               
                 DISTANCE-MEASUREMENT HORNS 
               
             
          
           
               
                   
                 Location Separation from 
               
               
                   
                 Beacon Null Vector 122 
               
               
                   
                   
               
             
          
           
               
                   
                 Up Signal Source 302A 
                 d AZ   
               
               
                   
                 Down Signal Source 302C 
                 −d AZ    
               
               
                   
                 Left Signal Source 302D 
                 p EL   
               
               
                   
                 Right Signal Source 302B 
                 −p EL    
               
               
                   
                   
               
             
          
         
       
     
     Although described with respect to a phased array  112  used to transmit signals, the foregoing invention can also be applied to a phased array used to receive signals as well. In this embodiment, a receiving beacon pyramid is formed on the phased array by the signals transmitted to the phased array  112  by a transmitting horn disposed on the boom  116  and nominally along the null vector of the receiving pyramid. 
     Conclusion 
     This concludes the description of the preferred embodiments of the present invention. The foregoing description of the preferred embodiment of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto. The above specification, examples and data provide a complete description of the manufacture and use of the composition of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended.