Patent Publication Number: US-10312587-B1

Title: Designing an antenna array to meet specified performance criteria

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     N/A 
     BACKGROUND 
     Geolocation refers to techniques for determining the geographic location of an object. Various types of geolocation exist. The present invention is applicable to environments where the object to be geolocated emits a signal. In such environments, various measurements can be performed on the received signal to estimate the location of the emitting object. For example, a receiver can perform angle (or direction) of arrival techniques to estimate the angle between the emitting object and the boresight vector of the antenna&#39;s receiver. 
     Angle of arrival techniques are often performed by detecting phase differences at a number of antennas that receive the signal emitted by the object. To detect the angle of arrival, the detected phase differences are compared to calculated phase differences (based on steering vectors) for each of a number of possible angles to identify which calculated phase differences most closely correlate with the detected phase differences. The angle that corresponds to the most closely correlated phase differences can then be identified as the angle of arrival of the received signal. 
     Because phase is cyclical, there may be instances where multiple sets of calculated phase differences closely correlate with the detected phase differences. In such instances, the angle of arrival calculations may yield multiple possible angles of arrival such that there would be ambiguity as to what the actual angle of arrival was. For example, a calculated phase difference corresponding to an incorrect angle of arrival may have a higher correlation than the calculated phase difference corresponding to the correct angle of arrival. Such ambiguity may be worsened in multipath environments since the phase of the multipath signal will interfere with the phase of the direct path signal. 
     BRIEF SUMMARY 
     The present invention is generally directed to systems, methods and computer program products for designing an antenna array to meet specified performance criteria. A system can be configured to receive various performance criteria as inputs, and from these inputs, identify how antenna elements of an antenna array should be arranged so that the antenna array will meet the performance criteria. An iterative process can be performed to identify at least one arrangement of antenna elements that will best meet the performance criteria while also complying with specified structural constraints. 
     In one embodiment, the present invention is implemented by an antenna array design system as a method for identifying an arrangement of antenna elements that will meet required performance criteria based on structural constraints. The antenna array design system receives input that defines required performance criteria for an antenna array including an angle of arrival accuracy and a probability of ambiguous arrival. The antenna array design system also receives input that defines structural constraints for the antenna array including a number of antenna elements, a field of view, and a maximum distance between any two antenna elements in one dimension. The antenna array design system calculates, based on the structural constraints, a number of samples that are required to achieve the angle of arrival accuracy, and also calculates, based on the structural constraints and the number of samples, a minimum signal to noise ratio reduction parameter. The antenna array design system identifies, based on the structural constraints, a search space within which antenna elements of the antenna array will be positioned. The antenna array design system then selects a first arrangement of the number of antenna elements within the search space, calculates, for each of a number of angles within the field of view, a signal to noise ratio reduction parameter for the first arrangement, and determines that a minimum of the noise ratio reduction parameters calculated for the first arrangement exceeds the minimum signal to noise ratio reduction parameter. The antenna array design system also calculates, for each of a number of angles within the field of view, an angle of arrival error resulting from coherent multipath for the first arrangement, generates a sum of the angle of arrival errors for the first arrangement, and determines that the sum of the angle of arrival errors for the first arrangement is less than a corresponding sum of angle of arrival errors for one or more other arrangements of antenna elements. As a result, the antenna array design system generates output that identifies that the first arrangement meets the required performance criteria and provides better multipath resistance than the one or more other arrangements. 
     In another embodiment, the present invention is implemented as an antenna array design system that comprises one or more non-transitory computer readable media storing computer-executable instructions which when executed implement a method for identifying an arrangement of antenna elements that will meet required performance criteria based on structural constraints. The method comprises: receiving input that defines required performance criteria for an antenna array, the required performance criteria including an angle of arrival accuracy and a probability of ambiguous arrival; receiving input that defines structural constraints for the antenna array, the structural constraints including a number of antenna elements, a field of view, and a maximum distance between any two antenna elements in one dimension; calculating, based on the structural constraints, a number of samples that are required to achieve the angle of arrival accuracy; calculating, based on the structural constraints and the number of samples, a minimum signal to noise ratio reduction parameter; identifying, based on the structural constraints, a search space within which antenna elements of the antenna array will be positioned; and iteratively evaluating a plurality of possible arrangements of the antenna elements within the search space. The iterative evaluation includes, for each possible arrangement: calculating, for each of a number of angles within the field of view, a signal to noise ratio reduction parameter for the arrangement; determining whether a minimum of the noise ratio reduction parameters calculated for the arrangement exceeds the minimum signal to noise ratio reduction parameter; when the minimum of the noise ratio reduction parameters calculated for the arrangement exceeds the minimum signal to noise ratio reduction parameter, calculating, for each of a number of angles within the field of view, an angle of arrival error resulting from coherent multipath for the arrangement and generating a sum of the angle of arrival errors for the arrangement; and determining whether the sum of the angle of arrival errors for the arrangement is less than a corresponding sum of angle of arrival errors for previously evaluated arrangements, and if so, saving the arrangement as a best arrangement. 
     This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This Summary is not intended to identify key features or essential features of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Understanding that these drawings depict only typical embodiments of the invention and are not therefore to be considered limiting of its scope, the invention will be described and explained with additional specificity and detail through the use of the accompanying drawings in which: 
         FIG. 1  illustrates an example coordinate system that can be used to calculate various performance criteria for an antenna array; and 
         FIG. 2  illustrates a flow diagram of a process for designing an antenna array to meet specified performance criteria. 
     
    
    
     DETAILED DESCRIPTION 
     In this specification, the term “performance criteria” will be used to generally refer to three different metrics of an antenna array: (1) “angle of arrival accuracy” or σ AoA ; (2) “probability of ambiguous arrival” of P ambig ; and (3) “multipath resistance” which will be represented as the error caused by coherent multipath or Δϕ. Although these terms will be mathematically defined below, they can generally be defined as follows: an antenna array&#39;s “angle of arrival accuracy” defines how accurate an angle of arrival system&#39;s estimated angle of arrival matches the actual angle of arrival of a received signal; an antenna array&#39;s “probability of ambiguous arrival” defines how frequently an angle of arrival system will select an incorrect angle; and an antenna array&#39;s “multipath resistance” defines the degree to which coherent multipath affects the accuracy of the antenna array. As will be described below, required performance criteria can be provided as input to an antenna array design system, and the system can identify an arrangement of antenna elements that will meet the performance criteria. In some embodiments, a signal to noise ratio of the link (SNR in  which represents the expected SNR of the signal when it is received at the antenna elements) may also be provided as a required performance criteria, while in other embodiments, the SNR of the link may be assumed. 
     The term “structural constraints” will be used to refer to any constraints on the physical configuration of the antenna array. For example, structural constraints may include a number of antenna elements in the antenna array (N AR ), a required field of view (AZ min , EL min , AZ max , EL max ), the maximum distance between any two antenna elements in one dimension (d max ), etc. Based on the specified structural constraints and the required performance criteria, an antenna array design system can identify a most suitable arrangement of antenna elements for an antenna array. 
       FIG. 1  provides an example of the antenna array geometry that will be used in this description. It should be understood, however, that the present invention should not be limited to any particular orientation of the coordinate system. Of importance is the fact that the measurements of an angle of arrival system are based on some reference orientation of the axes. 
     In  FIG. 1  and the following description, the relative position of an antenna element of the antenna array will be defined by the position vector P i  (P ix , P iy , P iz ) where the units of the position vector are assumed to be carrier cycles. For simplicity,  FIG. 1  shows only one antenna element, but of course, an antenna array will include multiple antenna elements that are positioned in a one, two or three dimensional arrangement. The vector U (Az, El) represents the vector that points in the direction defined by azimuth and elevation angles. In the depicted coordinate system, a point at 0° azimuth would lie on the x-z plane and a point at 0° elevation would lie on the y-z plane. As shown below, each element g i  of a steering vector G defines the antenna gain of a particular antenna element in the direction of the vector U (Az, El). This steering vector consists of a magnitude part (which for simplicity is assumed to be unity) and a phase part that is proportional to the dot product between U (Az, El) and the antenna element&#39;s position vector P i (P ix , P iy , P iz ).
 
 g   i   =e   −j2πU·P     i      (1)
 
where
 
 U ( Az,El )=( u   x   ,u   y   ,u   z )=(cos( El )cos( Az ),cos( El )sin( Az ),sin( El ))   (2)
 
     In non-mathematical terms, for a signal emitted by an object positioned at a particular angle (Az, El) from the antenna array&#39;s reference coordinates, the value of g i  will define what the phase of the signal should be when it is received at the antenna element at position P i  relative to the phases of the signal when received at antenna elements at other positions. 
     The present invention is generally directed to techniques for identifying the position P i  at which each of a number of antenna elements should be placed to produce an antenna array that will meet specified performance criteria. Based on structural constraints, an antenna array design system can iterate through possible arrangements of antenna elements and, for each possible arrangement, evaluate whether the arrangement would produce an antenna array that meets specified performance criteria. By performing this iterative process, the antenna array design system can identify a particular arrangement that may best meet the specified performance criteria or at least identify a number of possible arrangements that will meet the specified performance criteria. In this way, an antenna array design suitable for a particular environment/purpose can be quickly and efficiently determined. 
       FIG. 2  provides a flowchart of a process  200  that an antenna array design system can perform to identify an arrangement of antenna elements that meets specified performance criteria and complies with structural constraints. The antenna array design system can be any computing device including a standard computer that executes software that is configured to implement process  200 . 
     Process  200  includes a first step  201  in which the structural constraints and required performance criteria are specified. As an example only, it may be desirable to design an antenna array to be used on an aircraft carrier to geolocate aircraft as they approach for landing. In such a case, the antenna array would need to be highly accurate (e.g., σ AoA =) 0.01°, have a very low probability of ambiguous arrival (P ambig =10 −10 ), and have a high multipath resistance (i.e., low values of Δϕ within the field of view). Also, since the aircraft to be geolocated will be relatively close to the antenna array, the SNR of the link may be relatively high (e.g., SNR in =10 dB). Further, the field of view for the antenna would need to encompass the area from which the aircraft approach for landing (e.g., AZ min =−20, AZ max =20°, °, EL min =−5°, EL max =5°). The number of antenna elements could be specified as 8 (e.g., because the angle of arrival system includes 8 analog/digital converters) and the maximum distance between any two antenna elements could be specified as 48λ (e.g., to comply with any maximum size limitations on the antenna array). Again, these values will be used for illustrative purposes only and should not be viewed as limiting the invention. 
     Process  200  includes steps  202   a - 202   c  in which a number of initial calculations are performed based on the inputs. In step  202   a , the minimum number of samples (N samp ) that are required to achieve the specified angle of arrival accuracy (σ AoA ) is calculated. A suitable equation for calculating angle of arrival accuracy is as follows: 
                     σ   AoA   2     =                  N   AR     ⁢       ∑   i     ⁢       ∂     g   i         ∂   Az           +       ∑   i     ⁢         ∂     g   i   *         ∂   Az       ⁢       ∑   i     ⁢     g   i                  2         N   samp     ⁢         SNR     i   ⁢           ⁢   n       ⁡     (                ∑   i     ⁢       g   i   *     ⁢       ∂     g   i         ∂   Az                2     +       N   AR     ⁢     real   ⁡     [       ∑   i     ⁢       g   i   *     ⁢         ∂   2     ⁢     g   i         ∂     Az   2             ]           )       2                 (   3   )               
Equation (3) requires knowledge of the positions of the antenna elements which, for purposes of the present invention, are not known. Therefore, it can be assumed that all antenna elements are arranged on a single axis which allows an upper bound on the angle of arrival accuracy to be defined as:
 
                     σ   AoA   2     ≤     2       N   samp     ⁢         SNR   in     ⁡     (     2   ⁢   π   ⁢           ⁢     d   max     ⁢     cos   ⁡     (   α   )         )       2                 (   4   )               
where SNR in , d max , and a are all provided as inputs in step  201 . The value of a can be set to the maximum angle (e.g., 20° azimuth) off boresight which is where the angle of arrival accuracy will be minimized.
 
     Importantly, equation (4) does not require knowledge of the antenna element arrangement and can therefore be used in process  200  to search for a suitable arrangement of antenna elements. Given the inputs received in step  201 , equation (4) can be solved for N samp —the only unknown. Solving for N samp  yields the number of samples that need to be taken to obtain the required angle of arrival accuracy. 
     In step  202   b , the parameter SNR out  can be calculated. SNR out  defines the SNR of the link at the worst possible angle and is directly related to the probability of ambiguous arrival. In the present example, this angle would be ±20° azimuth. In more detail, an incorrect (or ambiguous) angle of arrival will be detected in cases where the correlation between the received signal and an incorrect steering vector is larger than the correlation between the received signal and the correct steering vector. Such a scenario can be represented as: 
                       ∑   k     ⁢              ∑   l     ⁢       x   i     ⁢     g   i   *              2       &gt;       ∑   k     ⁢              ∑   l     ⁢       x   i     ⁢     g     0   ⁢   i     *              2               (   5   )               
where g 0i  are the elements of the steering vector corresponding to the true angle of arrival and g i  are the elements of the incorrect steering vector. From equation (5), a metric T(α) can be defined as follows:
 
                     T   ⁡     (   ∝   )       =           ∑   k     ⁢              ∑   l     ⁢       x   i     ⁢     g   i   *              2       -       ∑   k     ⁢              ∑   l     ⁢       x   i     ⁢     g     0   ⁢   i     *              2         &gt;   0             (   6   )               
Due to the central limit theorem, the metric T(α) can be approximated as Gaussian such that the probability of ambiguous arrival (P ambig ) can be calculated as the probability that T(α)&gt;0 as follows:
 
                     P   ambig     =       ∑     m   ≠     m   0         ⁢     P   ⁡     (       T   ⁡     (     α   m     )       &gt;   0     )                 (   7   )                 P   ambig     =       ∑     m   ≠     m   0         ⁢     Q   ⁡     (         SNR   out     ⁡     (     α   m     )         )                 (   8   )               
where SNR out (α m ) represents the SNR of T(α) for a signal arriving from the direction α i  based on the assumption that SNR out &gt;10 dB and Q( ) is the Q-function.
 
     Even though P ambig  depends on SNR out  from all possible angles, it can be approximated by assuming that the worst case angle (α wc ) will produce a SNR out  that is much smaller then at the other angles such that: 
                     P   ambig     =         ∑     m   ≠     m   0         ⁢     Q   ⁡     (         SNR   out     ⁡     (     α   m     )         )         ≈     Q   ⁡     (         SNR   out     ⁡     (     α   wc     )         )                 (   9   )               
Equation (9) can be evaluated to identify the value of SNR out  that will cause P ambig  to remain below the specified level even at the worst case angle.
 
     With SNR out  determined in step  202   b , step  202   c  can be performed to calculate ΔSNR min  which defines the largest SNR reduction that an antenna element could have at any angle within the defined field of view. A suitable equation for calculating ΔSNR min  is as follows: 
                       SNR   out     ⁡     (     a   i     )       =         N   samp     ⁢       SNR   in     ⁡     (       N   AR   2     -              ∑   i     ⁢       g     0   ⁢   i       ⁢     g   i   *              2       )           4   ⁢           ⁢     N   AR                 (   10   )               SNR out (α m )= N   samp SNR in ΔSNR(α m )   (11)
 
     where N AR  represents the number of antenna elements, N samp  is the result of step  201  described above, SNR in  is an input (or assumed) and ΔSNR(α m ) represents the SNR reduction. In other words, the smaller the value of ΔSNR, the smaller the value of SNR out  will be. 
     Equation (11) can be evaluated for all angles of interest (e.g., from −20° to) 20° to identify the minimum for ΔSNR(α m ). ΔSNR min  can be set to this identified minimum in step  202   c . ΔSNR min  therefore defines the smallest value of ΔSNR(α m ) that will still yield an acceptable probability of ambiguous arrival. 
     With N samp , SNR out , and ΔSNR min  calculated, process  200  can proceed to step  203  in which the search space is defined. The search space can be any volume within a three dimensional space that will cause the antenna array to comply with the structural constraints. For example, the search space may be selected to provide the specified field of view and to comply with the maximum distance parameter d max . In many cases, the antenna array may only need to provide a narrow field of view such that a two dimensional arrangement of antenna elements would be sufficient. In such cases, the search space can be defined as a two dimensional area (i.e., a plane) to minimize the processing required. Although the antenna elements could be placed anywhere in the search space, further constraints on the search space may be defined to allow an exhaustive search to be performed in a reasonable amount of time. For example, the search space could be limited to positions on the y and z axes. As part of defining the search space, the antenna array design system may also identify each possible antenna element position within the search space. These possible positions can be constrained by spacing requirements (e.g., a half wavelength (λ/2) between each possible position) such that the search space is divided into a grid of possible positions. In short, the result of step  203  is a search space comprised of a number of possible positions for antenna elements. 
     In step  204 , the iterative portion of process  200  is commenced. In particular, step  204  entails the selection of a new arrangement of antenna elements within the search space. In this context, “new” refers to an arrangement that has yet to be evaluated. Each arrangement can include the number of antenna elements, N AR , which was specified in step  201 , and each of the antenna elements can be positioned in one of the possible positions in the search space. In some embodiments, additional spacing constraints may be applied to ensure that two antenna elements are not positioned too closely together. For example, a minimum spacing of 4 wavelengths between adjacent antenna elements could be defined. 
     With a new arrangement selected and in step  205 , the antenna array design system can calculate ΔSNR i  for every possible angle within the field of view. This can be accomplished using equation (11) as defined above. Accordingly, the result of step  205  is a listing of the SNR reductions at every possible angle that will exist with the selected arrangement. 
     In step  206 , the minimum value of ΔSNR i  that was calculated in step  205  can be determined and compared with the value of ΔSNR min  which was calculated in step  202   c . Since ΔSNR min  represents the minimum value that will still provide the required probability of ambiguous arrival P ambig , as long as the minimum value of ΔSNR i  is greater than ΔSNR min , the probability of ambiguous arrival of the selected arrangement will be lower than the required probability P ambig . In other words, step  206  tests whether the selected arrangement of antenna elements will provide a probability of ambiguous arrival that is below (i.e., better than) the specified threshold P ambig . 
     If the minimum value of ΔSNR i  is less than ΔSNR min , it will be known that the selected arrangement is not a suitable arrangement (because the probability of ambiguous arrival would be too high) and process  200  can transition to step  211 . In step  211 , it will be determined whether there are additional arrangements within the search space that have not been evaluated, and if so, process  200  will transition back to step  204  for the next iteration. In contrast, if the minimum value of ΔSNR i  is greater than ΔSNR min  (i.e., the SNR reduction of the selected arrangement is not as bad as the worst case scenario that will still meet the required P ambig ), process  200  can transition to step  207  to allow the selected arrangement to be further evaluated. 
     In step  207 , the effects of coherent multipath on the selected arrangement in various directions are estimated. For simplicity, the effects of multipath on the azimuth angle will be described. It is noted, however, that a similar analysis can be performed with respect to the elevation angle. This multipath analysis employs the first and second derivatives of g i  (equation (1)) with respect to the azimuth angle which can be defined as follows: 
     
       
         
           
             
               
                 
                   
                     
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     As mentioned above, an angle of arrival system determines the angle of arrival by correlating the received signal x i  at each antenna element with conjugated elements of the steering vector g i  for a particular angle of arrival, takes the magnitude squared of the result, and compares the results for every angle of interest. The angle that produces the largest correlation C is selected as the angle of arrival. This correlation can be represented as: 
                   C   =       ∑   k     ⁢              ∑   l     ⁢       x   i     ⁢     g   i   *              2               (   14   )               
where the summation on k represents averaging over multiple samples. However, because the multipath effects are not affected by averaging over multiple samples, it can be assumed that k=1 yielding:
 
                   C   =       (       ∑   i     ⁢       x   i     ⁢     g   i   *         )     ⁢     (       ∑   i     ⁢       g   i     ⁢     x   i   *         )               (   15   )               
which can be expressed in matrix format as:
 
 C=GX   H   XG   H    (16)
 
where X and G are the vectors:
 
 X =( x   1   ,x   2   , . . . x   N )  G =( g   1   ,g   2   , . . . g   N )
 
and where (⋅) H  represents the conjugate transpose of a matrix and N represents the number of antenna element (N AR ).
 
     With C calculated for each relevant angle, the angle corresponding to the largest value of C is selected as the estimated angle of arrival. This can be represented mathematically as: 
     
       
         
           
             
               
                 
                   
                     
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     Based on the assumption that the error caused by coherent multipath is very small, equation (17) can be approximated as: 
                         ∂     C   ⁡     (   Az   )           ∂   Az       ≅         ∂     C   ⁡     (     Az   =     Az   0       )           ∂   Az       +           ∂   2     ⁢     C   ⁡     (     Az   =     Az   0       )           ∂     Az   2         ⁢   Δ   ⁢           ⁢   ϕ         =   0           (   18   )               
where Az 0  represents the true angle of arrival and Δϕ represents the angle of arrival error caused by coherent multipath. Solving equation (18) for Δϕ yields:
 
     
       
         
           
             
               
                 
                   
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     The first derivative of C can be calculated from equation (16) as: 
                       ∂   C       ∂   Az       =         ∂     (       GX   H     ⁢     XG   H       )         ∂   Az       =           ∂   G       ∂   Az       ⁢     X   H     ⁢     XG   H       +       GX   H     ⁢   X   ⁢       ∂     G   H         ∂   Az                     (   20   )               
If the matrix multiplications in equation (20) are reordered and the entire second term is conjugated, it yields:
 
     
       
         
           
             
               
                 
                   
                     
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     The second derivative of C can be calculated in a similar manner to yield: 
     
       
         
           
             
               
                 
                   
                     
                       
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     Therefore, to calculate the angle of arrival error resulting from coherent multipath (Δϕ) in step  207 , equations (21) and (22) can be substituted into equation (19). Also, to calculate the first and second derivatives defined in equations (21) and (22), equations (1), (12) and (13), which define the elements of G, ∂G/∂Az and ∂ 2 G/∂Az 2  respectively, can be employed. 
     For reasons that are beyond the scope of this disclosure, the worst multipath error results when the relative phase between the direct path and the multipath is 90°. Therefore, in step  207 , the multipath effects need only be evaluated for 90° relative phase. In this way, the amount of processing required in step  207  can be minimized. The results of step  207  will therefore be an angle of arrival error (Δϕ i ) for every angle of interest in the field of view. 
     In step  208 , the angle of arrival error (Δϕ i ) due to multipath for all of the angles of interest can be summed to produce ΣΔϕ i  pertaining to the currently selected arrangement of antenna elements. This sum, ΣΔϕ i , can then be compared to the lowest identified sum, ΣΔϕ min , from previous iterations. In other words, the antenna array design system can identify whether the selected arrangement of antenna elements exhibits better multipath resistance than all previously evaluated arrangements (or, more appropriately, better than all previously evaluated arrangements that were determined to meet the required P ambig ). 
     If the selected arrangement does not exhibit better multipath resistance (i.e., if ΣΔϕ i  exceeds ΣΔϕ min ), process  200  can transition to step  211  to perform the next iteration. In contrast, if the selected arrangement exhibits better multipath resistance (i.e., if ΣΔϕ i  is less than ΣΔϕ min , the value of ΣΔ min  can be set to the value of ΣΔϕ i  in step  209  (to thereby preserve the new best value for subsequent iterations) and the selected arrangement of antenna elements (i.e., the positions of each antenna element) can be saved as the best arrangement in step  210 . Process  200  can then transition to step  211 . 
     Finally, after each possible arrangement of antenna elements has been evaluated, the determination in step  211  will be positive causing process  200  to transition to step  212  where the best arrangement identified during the iterative process is output. This best arrangement will be the arrangement that provides a probability of ambiguous arrival below the specified level and that exhibits the lowest sum of multipath errors across the angles of interest. 
     Although process  200  is described as outputting a single best arrangement, in some embodiments, more than one best arrangement could be identified. For example, in step  209  and  210 , the antenna array design system could identify and save the five arrangements that exhibit the lowest values for ΣΔϕ i . In this way, the antenna array design system could identify a number of possible arrangements that would meet the specified performance criteria. 
     In summary, the present invention provides a process by which an antenna array design system can identify at least one arrangement of antenna elements from among many possible arrangements that will best meet specified performance criteria based on specified structural constraints. The present invention therefore allows an antenna array to be more quickly and efficiently designed for a particular application. 
     The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive. The scope of the invention is, therefore, indicated by the appended claims rather than by the foregoing description.