Patent Publication Number: US-2023155649-A1

Title: Array Shape Reconstruction for Distributed Systems

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of and priority under 35 U.S.C. § 119(e) to U.S. Provisional Patent Application Ser. No. 63/280,943 entitled “Methods for Global Shape Reconstruction for Distributed Systems” filed Nov. 18, 2021, and U.S. Provisional Patent Application Ser. No. 63/296,217 entitled “Array Shape Reconstruction Through Mutual Coupling” filed Jan. 4, 2022, which are incorporated herein by reference in their entirety for all purposes. 
    
    
     FIELD OF THE INVENTION 
     The present invention generally relates to reconfigurable phased arrays incorporating a constellation of antennas and methods for determining the current configuration of phased arrays. 
     BACKGROUND 
     Antenna arrays continue to play a growing role in a broad range of applications, from communications, sensing, ranging, power transfer, projections, and imaging. In addition to planar arrays, conformal arrays, which are rigidly fixed to a nonplanar surface, have long been used in low profile (aerodynamically and/or visually) radio frequency (RF) systems, mostly for streamlining purposes. In mechanically static arrays, unusual shape and orientation of array elements may be analyzed and accounted for during the design phase as these systems are used for a single, rigid shape. With an unusual but known shape array, pattern synthesis may be utilized using geometric analysis and various optimization algorithms, including genetic algorithms. While these techniques are effective, array shape may not be static and known. Emerging applications using flexible microwave and mm-wave antenna arrays in wearables and light-weight, deployable apertures in space change shape dynamically, may include capabilities beyond conventional static conformal arrays. Flexible phased arrays, which can constantly change shape during deployment and operation, may also suffer from irregular element position and orientation. However, their antenna positions may not be known in advance and accounted for during design. As such, it may be advantageous to have flexible phased arrays which may be able to determine their shape and dynamically update phases to control beam coherence and direction during operation. 
     SUMMARY OF THE INVENTION 
     Various embodiments are directed to a reconfigurable phased array including: a constellation of antennas configured to receive and transmit radiation, where each of the antennas senses incidental power from the transmitted radiation from the other antennas of the constellation of antennas; a computer system configured to: measure the incidental power sensed by the each of the antennas; perform a physical constraint mapping of the constellation of antennas; perform an array shape construction to determine the current position of each of the antennas based on the physical constraint mapping and the incidental power sensed by each of the antennas. 
     In various other embodiments, the computer system is further configured to utilize the determined current position to determine antenna excitation configurations of the antennas. 
     In various other embodiments, the constellation of antennas is positioned on a flexible substrate. 
     In still various other embodiments, the flexible substrate is bendable to reposition the constellation of antennas into an ideal position of all the elements. 
     In still various other embodiments, the computer system is further configured to control the position of the constellation of antennas into the ideal position of all the elements. 
     In still various other embodiments, the physical constraint mapping utilizes the possible mutual angle, antenna distance, and/or local curvature of the constellation of antennas. 
     In still various other embodiments, the physical constraint mapping includes modeling the incidental power sensed by the each of the antennas into a raw mutual coupling matrix S∈   N×N . 
     In still various other embodiments, the raw mutual coupling matrix is defined by: 
     
       
         
           
             
               
                 S 
                 mn 
               
               = 
               
                 
                   
                     
                       a 
                       m 
                     
                     ⁢ 
                     
                       
                         D 
                         m 
                       
                       ( 
                       
                         θ 
                         mn 
                       
                       ) 
                     
                     ⁢ 
                     
                       a 
                       j 
                     
                     ⁢ 
                     
                       
                         D 
                         n 
                       
                       ( 
                       
                         θ 
                         
                           n 
                           ⁢ 
                           m 
                         
                       
                       ) 
                     
                   
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                 
                 ⁢ 
                 
                   e 
                   
                     - 
                     
                       j 
                       ⁡ 
                       ( 
                       
                         
                           ϕ 
                           m 
                         
                         + 
                         
                           ϕ 
                           n 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             ; 
             and 
           
         
       
       
         
           
             
               
                 θ 
                 mn 
               
               = 
               
                 
                   cos 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                   
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     wherein a m  is the total fixed amplitude offset in antenna m, D m (θ) is the directivity of antenna m for an angle θ relative to broadside, {right arrow over (l mn )} is the vector pointing from the phase center of antenna i to the phase center of antenna n, ϕ m  is the total fixed phase offset in antenna m, and   is the unit vector normal to the surface of the array, that describes the orientation of antenna m. 
     In still various other embodiments, the physical constraint mapping further includes performing simulation and analysis to calibrate the constellation of antennas. 
     In still various other embodiments, the physical constraint mapping further includes performing a one-time calibration of the constellation of antennas in a predetermined configuration. 
     In still various other embodiments, the predetermined configuration includes a completely flat configuration. 
     In still various other embodiments, the completely flat configuration yields a coupling matrix defined by: 
     
       
         
           
             
               
                 S 
                 
                   m 
                   ⁢ 
                   n 
                 
                 flat 
               
               = 
               
                 
                   
                     
                       a 
                       m 
                     
                     ⁢ 
                     
                       
                         D 
                         m 
                       
                       ( 
                       
                         θ 
                         
                           m 
                           ⁢ 
                           n 
                         
                         flat 
                       
                       ) 
                     
                     ⁢ 
                     
                       a 
                       n 
                     
                     ⁢ 
                     
                       
                         D 
                         n 
                       
                       ( 
                       
                         θ 
                         
                           n 
                           ⁢ 
                           m 
                         
                         flat 
                       
                       ) 
                     
                   
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                         flat 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                 
                 ⁢ 
                 
                   e 
                   
                     - 
                     
                       j 
                       ⁡ 
                       ( 
                       
                         
                           ϕ 
                           m 
                         
                         + 
                         
                           ϕ 
                           n 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                                 flat 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
             where 
           
         
       
       
         
           
             
               
                 θ 
                 
                   m 
                   ⁢ 
                   n 
                 
                 flat 
               
               = 
               
                 π 
                 2 
               
             
             , 
           
         
       
     
     when m≠n, and the constellation of antennas are each identical which makes D n (θ)=D m (θ)=D(θ) for all m&#39;s and n&#39;s. 
     In still various other embodiments, the physical constraint mapping further includes simplifying the coupling matrix such that the coupling matrix is defined by: 
     
       
         
           
             
               S 
               mn 
               flat 
             
             ≅ 
             
               
                 
                   
                     a 
                     m 
                   
                   ⁢ 
                   
                     a 
                     n 
                   
                   ⁢ 
                   
                     
                       D 
                       ⁡ 
                       ( 
                       
                         π 
                         2 
                       
                       ) 
                     
                     2 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     - 
                     
                       j 
                       ⁡ 
                       ( 
                       
                         
                           ϕ 
                           m 
                         
                         + 
                         
                           ϕ 
                           n 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                                 flat 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, the physical constraint mapping further includes defining a normalized transfer function, H. 
     In still various other embodiments, the normalized transfer function is defined by: 
     
       
         
           
             
               H 
               mn 
             
             = 
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       θ 
                       mn 
                     
                     ) 
                   
                   ⁢ 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       θ 
                       nm 
                     
                     ) 
                   
                 
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       π 
                       2 
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     - 
                     
                       jk 
                       ⁡ 
                       ( 
                       
                         | 
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           - 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                                 flat 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, the physical constraint mapping further includes assuming a constant curvature of the constellation of antennas. 
     In still various other embodiments, assuming a constant curvature makes the normalized transfer function defined by: 
     
       
         
           
             
               H 
               mn 
             
             = 
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                           flat 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       k 
                       ⁡ 
                       ( 
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         - 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, the physical constraint mapping includes determining the polarity of the shape of the constellation of antennas. 
     In still various other embodiments, the polarity is defined by: 
     
       
         
           
             
               p 
               = 
               
                 ( 
                 
                   
                     ∑ 
                     
                       i 
                       ≠ 
                       j 
                     
                   
                   
                     ( 
                     
                       
                         sign 
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       - 
                       
                         E 
                         
                           m 
                           ⁢ 
                           n 
                         
                         1 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     where p=−1 indicates a convex shape and p=+1 indicates a concave shape. 
     In still various other embodiments, the physical constraint mapping further includes performing recursive wrap correction to calculate a corrected phase wrap matrix constraining utilizing the polarity. 
     In still various other embodiments, the physical constraint mapping further includes truncating a search space for Euclidean distance to a single one-wavelength range. 
     In still various other embodiments, truncating the search space includes solving the following equation: |{right arrow over (l mn )}|∈[R′ mn λ, (R′ mn +1)λ). 
     In still various other embodiments, truncating the search space results in an Euclidean distance of: 
     
       
         
           
             
               E 
               
                 m 
                 ⁢ 
                 n 
               
               3 
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         arg 
                         ⁢ 
                            
                         min 
                       
                       
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ∈ 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   mn 
                                   ′ 
                                 
                                 ⁢ 
                                 λ 
                               
                               , 
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       mn 
                                       ′ 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 λ 
                               
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           H 
                           mn 
                           meas 
                         
                         - 
                         
                           
                             H 
                             mn 
                           
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                   
                   ) 
                 
                 2 
               
               . 
             
           
         
       
     
     In still various other embodiments, the constellation of antennas is part of a global antenna system and the computer system is further configured to use the current position of each of the antennas in the constellation of antennas to stitch together with a position of other antennas in the global antenna system for a global reconstruction of the global antenna system. 
     Various other embodiments are directed to a method of determining a current configuration of a phased array, the method including: providing a constellation of antennas configured to receive and transmit radiation towards a far field target, wherein each of the antennas senses incidental power from the transmitted radiation from the other antennas of the constellation of antennas; measuring the incidental power sensed by the each of the antennas; performing a physical constraint mapping of the constellation of antennas; and performing an array shape construction to determine the ideal position of each of the antennas based on the physical constraint mapping and the incidental power sensed by each of the antennas. 
     In various other embodiments, the method further includes utilizing the determined current position to determine antenna excitation configurations of the antennas. 
     In various other embodiments, the constellation of antennas is positioned on a flexible substrate. 
     In still various other embodiments, the flexible substrate is bendable to reposition the constellation of antennas into the ideal position of all the elements. 
     In still various other embodiments, the method further includes controlling the position of the constellation of antennas into the ideal position of all the elements. 
     In still various other embodiments, the physical constraint mapping utilizes the possible mutual angle, antenna distance, and/or local curvature of the constellation of antennas. 
     In still various other embodiments, performing the physical constraint mapping includes modeling the incidental power sensed by the each of the antennas into a raw mutual coupling matrix S∈   N×N . 
     In still various other embodiments, the raw mutual coupling matrix is defined by: 
     
       
         
           
             
               
                 S 
                 mn 
               
               = 
               
                 
                   
                     
                       a 
                       m 
                     
                     ⁢ 
                     
                       
                         D 
                         m 
                       
                       ( 
                       
                         θ 
                         mn 
                       
                       ) 
                     
                     ⁢ 
                     
                       a 
                       j 
                     
                     ⁢ 
                     
                       
                         D 
                         n 
                       
                       ( 
                       
                         θ 
                         nm 
                       
                       ) 
                     
                   
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                 
                 ⁢ 
                 
                   e 
                   
                     - 
                     
                       j 
                       ⁡ 
                       ( 
                       
                         
                           
                             ϕ 
                             m 
                           
                           + 
                           
                             ϕ 
                             n 
                           
                           + 
                           k 
                         
                         | 
                         
                           
                             l 
                             
                               m 
                               ⁢ 
                               n 
                             
                           
                           ¯ 
                         
                         | 
                       
                       ) 
                     
                   
                 
               
             
             ; 
             and 
           
         
       
       
         
           
             
               
                 θ 
                 mn 
               
               = 
               
                 
                   cos 
                   
                     - 
                     1 
                   
                 
                 ( 
                 
                   
                     · 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                   
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     where a m  is the total fixed amplitude offset in antenna m, D m (θ) is the directivity of antenna m for an angle θ relative to broadside, {right arrow over (l mn )} is the vector pointing from the phase center of antenna i to the phase center of antenna n, ϕ m  is the total fixed phase offset in antenna m, and   is the unit vector normal to the surface of the array, that describes the orientation of antenna m. 
     In still various other embodiments, performing the physical constraint mapping further includes performing a one-time calibration of the constellation of antennas in a completely flat configuration. 
     In still various other embodiments, the completely flat configuration yields a coupling matrix defined by: 
     
       
         
           
             
               S 
               
                 m 
                 ⁢ 
                 n 
               
               flat 
             
             = 
             
               
                 
                   
                     a 
                     m 
                   
                   ⁢ 
                   
                     
                       D 
                       m 
                     
                     ( 
                     
                       θ 
                       
                         m 
                         ⁢ 
                         n 
                       
                       flat 
                     
                     ) 
                   
                   ⁢ 
                   
                     a 
                     n 
                   
                   ⁢ 
                   
                     
                       D 
                       n 
                     
                     ( 
                     
                       θ 
                       
                         n 
                         ⁢ 
                         m 
                       
                       flat 
                     
                     ) 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 e 
                 
                   - 
                   
                     j 
                     ⁡ 
                     ( 
                     
                       
                         ϕ 
                         m 
                       
                       + 
                       
                         ϕ 
                         n 
                       
                       + 
                       
                         k 
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               where 
               ⁢ 
                   
               
                 θ 
                 
                   m 
                   ⁢ 
                   n 
                 
                 flat 
               
             
             = 
             
               
                 π 
                 2 
               
               . 
             
           
         
       
     
     when m≠n, and the constellation of antennas are each identical which makes D n (θ)=D m (θ)=D(θ) for all m&#39;s and n&#39;s. 
     In still various other embodiments, performing the physical constraint mapping further includes simplifying the coupling matrix such that the coupling matrix is defined by: 
     
       
         
           
             
               S 
               
                 m 
                 ⁢ 
                 n 
               
               flat 
             
             ≅ 
             
               
                 
                   
                     a 
                     m 
                   
                   ⁢ 
                   
                     a 
                     n 
                   
                   ⁢ 
                   
                     
                       D 
                       ⁡ 
                       ( 
                       
                         π 
                         2 
                       
                       ) 
                     
                     2 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     - 
                     
                       j 
                       ⁡ 
                       ( 
                       
                         
                           ϕ 
                           m 
                         
                         + 
                         
                           ϕ 
                           n 
                         
                         + 
                         
                           k 
                           ⁢ 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                                 flat 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, performing the physical constraint mapping further includes defining a normalized transfer function, H. 
     In still various other embodiments, the normalized transfer function is defined by: 
     
       
         
           
             
               H 
               mn 
             
             = 
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       θ 
                       
                         m 
                         ⁢ 
                         n 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       θ 
                       
                         n 
                         ⁢ 
                         m 
                       
                     
                     ) 
                   
                 
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       π 
                       2 
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       k 
                       ⁡ 
                       ( 
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         - 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, performing the physical constraint mapping further includes assuming a constant curvature of the constellation of antennas. 
     In still various other embodiments, assuming a constant curvature makes the normalized transfer function defined by: 
     
       
         
           
             
               H 
               mn 
             
             = 
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     D 
                     ⁡ 
                     ( 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                           flat 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     ) 
                   
                   2 
                 
               
               ⁢ 
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                       flat 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
               
               ⁢ 
               
                 
                   e 
                   
                     
                       - 
                       j 
                     
                     ⁢ 
                     
                       k 
                       ⁡ 
                       ( 
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         - 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       ) 
                     
                   
                 
                 . 
               
             
           
         
       
     
     In still various other embodiments, performing the physical constraint mapping includes determining the polarity of the shape of the constellation of antennas. 
     In still various other embodiments, the polarity is defined by: 
     
       
         
           
             
               p 
               = 
               
                 ( 
                 
                   
                     ∑ 
                     
                       i 
                       ≠ 
                       j 
                     
                   
                   
                     ( 
                     
                       
                         sign 
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                               flat 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       - 
                       
                         E 
                         
                           m 
                           ⁢ 
                           n 
                         
                         1 
                       
                     
                     ) 
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     where p=−1 indicates a convex shape and p=+1 indicates a concave shape. 
     In still various other embodiments, performing the physical constraint mapping further includes performing recursive wrap correction to calculate a corrected phase wrap matrix constraining utilizing the polarity. 
     In still various other embodiments, performing the physical constraint mapping further includes truncating a search space for Euclidean distance to a single one-wavelength range. 
     In still various other embodiments, truncating the search space includes solving the following equation: |{right arrow over (l mn )}|∈[R′ mn λ, (R′ mn +1)λ). 
     In still various other embodiments, truncating the search space results in an Euclidean distance of: 
     
       
         
           
             
               E 
               
                 m 
                 ⁢ 
                 n 
               
               3 
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         arg 
                         ⁢ 
                            
                         min 
                       
                       
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ∈ 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   mn 
                                   ′ 
                                 
                                 ⁢ 
                                 λ 
                               
                               , 
                               
                                 
                                   ( 
                                   
                                     
                                       R 
                                       mn 
                                       ′ 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 λ 
                               
                             
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           H 
                           mn 
                           meas 
                         
                         - 
                         
                           
                             H 
                             mn 
                           
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                   
                   ) 
                 
                 2 
               
               . 
             
           
         
       
     
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee. 
       The description and claims will be more fully understood with reference to the following figures and data graphs, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention. 
         FIG.  1    is an example of an EDM produced by a decentralized flexible phased array in accordance with an embodiment of the invention. 
         FIG.  2    is an illustration of the shape geometry at the intersection of two element clusters for Angle Stitching in accordance with an embodiment of the invention. 
         FIG.  3    is an illustration of the shape geometry at the intersection of two element clusters for Tangent Vector Stitching in accordance with an embodiment of the invention. 
         FIG.  4    is an image of an example of a flexible phased array. 
         FIG.  5    illustrates a block diagram of a modular framework for mutual coupling-based shape reconstruction in accordance with an embodiment of the invention. 
         FIG.  6 A  is an image of an antenna pair coupling set-up. 
         FIG.  6 B  is a plot of a coupling power measurement compared with two different propagation models 
         FIG.  6 C  is a plot of a coupling phase measurement. 
         FIG.  7 A  is an image of a ground plane backed dipole array. 
         FIG.  7 B  is an unwrapped coupling phase measurement plot. 
         FIG.  8    is a visualization of forbidden and allowed regions of the θ mn ⊗|{right arrow over (l mn )}| vector space for physically restricted phase arrays. 
         FIG.  9    shows a 2.5 GHz flexible phased array with 0.6λ antenna spacing and dipoles antennas that extend 0.25λ normal to the local ground plane in accordance with an embodiment of the invention. 
         FIG.  10    is a visualization of the Spiral Match algorithm in accordance with an embodiment of the invention. 
         FIG.  11    is a measurement setup for coupling measurements of connectorized arrays in accordance with an embodiment of the invention. 
         FIG.  12 A  illustrates an example folded dipole antenna with design dimensions. 
         FIG.  12 B  is a plot of measured and simulated folded dipole input matching and adjacent element coupling. 
         FIGS.  13 A and  13 B  illustrate shape reconstruction results for passive 2.5 GHz folded dipole phased arrays. 
         FIG.  14 A  is an image of a patch antenna with design dimensions. 
         FIG.  14 B  is a plot of a measured and simulated patch antenna input matching and adjacent element coupling. 
         FIGS.  15 A and  15 B  are shape reconstruction results for passive 2.5 GHz patch antenna phased arrays. 
         FIG.  16 A  is an integrated circuit-based flexible phased array with 8 elements bent with front shown. 
         FIG.  16 B  is the back of the integrated circuit-based flexible phased array of  FIG.  16 A  with feed network, chips, interposers, and circuitry shown. 
         FIG.  17    is a schematic block diagram of a an RFIC element circuit. 
         FIG.  18 A  is an image of an RFIC die. 
         FIG.  18 B  is an image of an interposer with RFIC. 
         FIG.  18 C  is an image of 3 interposers mounted on flexible printed circuit board. 
         FIG.  19 A  illustrates a radiator mounted perpendicular to the surface of the ground plane which avoids the need for a thick, likely rigid, layer of substrate. 
         FIG.  19 B  is the measured and simulated far-field patterns with bending. 
         FIG.  19 C  is the measured and simulated far-field patterns without bending. 
         FIG.  20    is shape reconstruction results for integrated 10 GHz phased arrays with dipole antennas for various radii of curvature. 
         FIG.  21    illustrates an example operating procedure for performing the calibration of the shape of the antenna array. 
     
    
    
     DETAILED DESCRIPTION 
     Various embodiments of the invention relate to a reconfigurable phased array and a method for determining the ideal configuration of a phased array. Certain disclosed embodiments include a reconfigurable phased array including a constellation of antennas configured to receive and retransmit radiation towards a far field target. The constellation of antennas may be a passive or active phased array. The active phased arrays can be driven by bulk components or with integrated circuits (ICs). The shape of arrays driven by ICs can be reconstructed using the IC as the central node performing the mutual coupling measurements. If the IC has both RF and computing functionality, all reconstruction can be programmed onto a single die. Shape reconstruction can be used with many antenna variants. To implement a system with any of these, the directivity pattern and the applicable, specific electromagnetic behavior may be changed into the coupling model. Examples of antenna variants include: Flexible polyimide antenna(s); Flexible metal antenna(s); Dipole antenna(s); Flexible dipole antenna(s); Patch antenna(s); Flexible air-gap patch antenna(s); Circularly polarized antenna(s); Helical antenna(s); and/or Yagi-Uda antenna(s). The antennas may or may not include a ground plane. Further, the antennas may be mounted on a stretchable substrate or a foldable substrate which may be utilized to reposition each element into different configurations. 
     Each of the antennas senses incidental power from the transmitted radiation from the other antennas of the constellation of antennas. This incidental power may be referred to as mutual coupling. The reconfigurable phased array further includes a computer system configured to: measure the incidental power sensed by the each of the antennas; perform a physical constraint mapping of the constellation of antennas; perform an array shape construction to determine the ideal position of all the elements based on the physical constraint mapping of the constellation of antennas and the incidental power sensed by each of the antennas. The physical constraint mapping may be performed by a Spiral Match algorithm which is further discussed below. 
     As flexible arrays and high-dimensional antenna systems continue to develop in maturity and in utility, it may be advantageous to be able to calibrate their shape. These systems enable myriad applications, such as imaging, sensing, wireless power transfer, communication, etc. These applications may benefit from including a process that the array itself or some other device has some sense of the array shape so that complex operations (e.g. beam steering, wavefront shaping, computational imaging, etc.) can be performed. Various previous approaches rely on communication between each element in the array. In large, distributed systems, where two elements might be driven by different RFICs or by different clock lines, it is not always possible to perform shape calibration. Disclosed herein are methods for performing global shape reconstruction for distributed systems by stitching together many local shape measurements. These methods may enable a myriad new applications not previously addressed, including but not limited to shape calibration for very large, distributed array systems, flexible electronics, distributed arrays in a large network, or disconnected transceivers in an imaging, sensing, or communications system. 
     The disclosed shape reconstruction method utilizes a mutual coupling measurement to generate a physical constraint matrix that is transformed into shape information for utilization in beam steering, sensing, imaging, or a host of other applications. The underlying circuitry is equipped to take this coupling measurement. In some embodiments, the underlying circuitry is a radio frequency integrated circuit (RFIC) which contains all of the necessary functionality for normal operation, including Tx and Rx paths for full tranceiving functionality. When the chip receives a command to perform a mutual coupling measurement, it sets one (antenna) output to the transmitting path and another output to the receiving path. The outgoing (Tx) and incoming (Rx) waves are mixed down to baseband where they can be ported off-chip—perhaps to an application-specific integrated circuit (ASIC) or microcontroller analog-to-digital converter (ADC)—where they can be used for shape reconstruction. 
     This process prevents the measurement of mutual coupling between antennas that are driven by separate RFICs because the receiving and transmitting paths cannot be mixed. This sets a cap on the scope of a mutual coupling matrix for a large system that may be comprised of multiple RFICs. Such a system will be limited to producing a purely block-diagonal mutual coupling matrix which will, in turn, produce a purely block-diagonal physical constraint matrix. This makes it impossible to reconstruct the shape globally, and limits to only reconstructing the shape within the scope of a single RFIC. An illustration of this problem is given with the example of a purely block-diagonal matrix in  FIG.  1   .  FIG.  1    is an example of an EDM produced by a decentralized flexible phased array in accordance with an embodiment of the invention. The decentralized flexible phased array includes 3 RFICs responsible for 3 antennas which each can only measure mutual coupling between themselves. 
     Large array systems rely on the distribution of RF control functionality (e.g. power generation, amplification, phase shifting, etc.) to multiple RFICs because of limitations of centralizing control over many outputs in a single RFIC. For example:
         Power concerns: Generating the necessary power for hundreds of transceivers with reasonable SNR on a small area may concentrate tremendous heat that could degrade performance.   Line losses: A centralized chip may include routing to all the outputs, including comparatively longer high-frequency RF lines which introduce high resistive losses and reduces output power.   Chip size: A chip with hundreds of outputs presents a geometric challenge. The chip may have to be designed as a long strip to maximize the perimeter-to-area ratio so that the necessary output lines can be routed. Moreover, a routing problem may occur that may mitigate design freedom regarding antenna spacing and position.
 
Thus, successful global shape reconstruction of a large, distributed system includes methods that go beyond those presented previously.
       

     Assumptions may be made about the curvature. Firstly, it is known because of the design of the array, the spacing between adjacent RFIC blocks. Thus, the problem of stitching multiple blocks together into a global shape converges to solving for the angles between adjacent blocks. Secondly, it may be assumed that the surface of the array is continuous (free of kinks or folds). It is known that—in some distance limit—the curvature is constant. If the spacing between adjacent RFIC “blocks” is small relative to the bend constraints placed by the array components, it can be assumed that the distant limit is on the order of the element spacing and, thus, that the curvature between blocks is constant. These constraints can be leveraged in a number of ways to turn locally reconstructed shapes into a global shape reconstruction. 
     Local shape reconstruction is defined as knowledge of the position of elements in an array cluster relative to other elements in that same array cluster. Relative position can be imagined as a vector from one element in the cluster to another element in the cluster: 
         r   i,j   k   =r   j   k   −r   i   k   (1)
 
     where r i,j   k  is the relative position vector and r j   k  is the position of element j in cluster k. j∈{1, 2, . . . , N k } where N k  is the number of elements in cluster k. Local shape reconstruction provides the relative position vector, r i,j   k , but not the absolute position vector, r j   k . Local shape reconstruction, and by proxy the tuple of relative position vectors that constitute a result, {r i,j+1   k |i∈{1, 2, . . . , N k −1}}, may be invariant under translation and rotation. Thus, performing global shape reconstruction between two clusters implies solving for two degrees of freedom: the position of the first element and the direction of the first relative position vector in the second block. A global shape may still retain the outstanding two degrees of freedom from the first cluster. 
     First, a 1D array may be provided with the potential to bend into shapes in two dimensions. This array may be composed of K clusters of N elements each, where—for example—element r N   k  is adjacent to element r 1   k+1 . A “cluster” may be defined as a grouping of elements driven by the same RFIC; local shape reconstruction gives us the shape of a cluster. Successful global shape reconstruction may be defined as knowledge of the shape of multiple clusters together. Said otherwise, it is the elimination of the degrees of freedom for all but the first cluster. Thus, the position of the first element and direction of the first relative position vector are defined relative to the position of elements in the previous cluster: 
         r   k,k+1   =r   1   k+1   −r   N   k   (2)
 
         r   k,k+1   =p   k,k+1 (cos(θ k,k+1 ) {circumflex over (x)} +sin(θ k,k+1 ) ŷ )  (3)
 
         r   1,2   k+1   =p   1,2   k+1 (cos(θ 1,2   k+1 ) {circumflex over (x)} +sin(θ 1,2   k+1 ) ŷ )  (4)
 
     Where p k,k+1  is the known, constant pitch between clusters k and k+1, p 1,2   k+1  is the known, constant pitch between elements 1 and 2 within cluster k+1, θ k,k+1  is the direction between the last element in cluster k and the first element in cluster k+1, and θ 1,2   k+1  is the direction between the first two elements in cluster k+1. Thus, the two degrees of freedom—position and direction—of the second cluster are fixed, defined relative to the first cluster by θ k,k+1  and θ 1,2   k+1 . An intersection of these two clusters and the relevant variables are shown in  FIG.  2   .  FIG.  2    is an illustration of the shape geometry at the intersection of two element clusters for Angle Stitching in accordance with an embodiment of the invention. 
     The problem of solving for global shape reconstruction may be defined as solving for the K−1 θ k,k+1 &#39;s and the K−1 θ 1,2   k+1 &#39;s. 
     To do so, smoothness may be maximized using the assumption of constant curvature at the block intersections. Smoothness may be defined as continuity of the first derivative. The angle difference, Δθ i   k , may be used as a metric for smoothness: 
       Δθ i   k =Δθ i,i+1   k −Δθ i−1,i   k   (5)
 
     Smoothness of shape implies that Δθ i   k  is constant in a small neighborhood about r i   k  (Δθ i−1   k ≈Δθ i   k ≈Δθ i+1   k ). Direction change at the edges can be defined as: 
       Δθ N   k =θ k,k+1 −Δθ N-1,N   k   (6)
 
       Δθ 1   k+1 =θ 1,2   k+1 −θ k,k+1   (7)
 
     To maximize the smoothness of the resulting stitched shape, it may be desired that these two direction changes to be similar to their neighbors: Δθ N-1   k  and Δθ 2   k+1 . Thus, linearly interpolating may be used to estimate for the edge direction changes: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       θ 
                       N 
                       k 
                     
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           θ 
                           
                             N 
                             - 
                             1 
                           
                           k 
                         
                       
                       + 
                       
                         Δ 
                         ⁢ 
                         
                           θ 
                           2 
                           
                             k 
                             + 
                             1 
                           
                         
                       
                     
                     3 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       θ 
                       1 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                           θ 
                           
                             N 
                             - 
                             1 
                           
                           k 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                           θ 
                           2 
                           
                             k 
                             + 
                             1 
                           
                         
                       
                     
                     3 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     These may be used to compute for the two unknown directions: 
       θ k,k+1 =θ N-1,N   k +Δθ N   k   (10)
 
       θ 1,2   k+1 =θ k,k+1 +Δθ 1   k+1   (11)
 
     Having solved for the two unknown directions, the two degrees of freedom for cluster k+1 have been eliminated and, in doing so, successfully “stitched” clusters k and k+1 together. The positions of the first two elements in the k+1 cluster can be computed using (3) and (4). This process may be continued for each k∈{2, 3, . . . , K}. 
     Antenna position may be used to obtain information about the shape of the array. The tangent vector, corresponding to a single element, i, may be given by: 
         t   i   k   =r   i+1   k   −r   i−1   k   (12)
 
     This formulation may be used to help stitch two clusters, k and k+1, each with N k  and N k+1  elements, together. To the first order, (2) sets the following conditions: 
         t   N     k     k   =r   1   k+1   −r   N     k-1     k   (13)
 
         t   1   k+1   =r   2   k+1   −r   N     k     k   (14)
 
     which relate the location of one cluster with respect to the next. As can be seen, two equations may completely define the two degrees of freedom for the k+1 cluster. This geometry is visualized in  FIG.  3   .  FIG.  3    is an illustration of the shape geometry at the intersection of two element clusters for Tangent Vector Stitching in accordance with an embodiment of the invention. 
     To “stitch” both the position and the direction of the k+1 cluster to the k cluster, two positions may be solved for. The equations above may be utilized to solve for these two positions. Rearranged, it can be seen that: 
         r   1   k+1   =r   N     k     −1   k   +t   N     k     k   (15)
 
         r   2   k+1   =r   N     k     k   +t   1   k+1   (16)
 
     r N     k-1     k  and r N     k     k  it may be solved for, but the edge tangent vectors t N     k     k  and t 1   k+1  may still not be solved for. To solve for these, continuity of curvature is assumed within a cluster and the angle between successive tangent vectors is solved for to obtain a rough estimate of the curvature: 
     
       
         
           
             
               
                 
                   
                     θ 
                     
                       
                         
                           N 
                           k 
                         
                         - 
                         2 
                       
                       , 
                       
                         
                           N 
                           k 
                         
                         - 
                         1 
                       
                     
                     k 
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           t 
                           
                             
                               N 
                               k 
                             
                             - 
                             2 
                           
                           k 
                         
                         · 
                         
                           t 
                           
                             
                               N 
                               k 
                             
                             - 
                             1 
                           
                           k 
                         
                       
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             t 
                             
                               
                                 N 
                                 k 
                               
                               - 
                               2 
                             
                             k 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             t 
                             
                               
                                 N 
                                 k 
                               
                               - 
                               1 
                             
                             k 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     θ 
                     
                       2 
                       , 
                       3 
                     
                     
                       k 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           t 
                           2 
                           
                             k 
                             + 
                             1 
                           
                         
                         · 
                         
                           t 
                           3 
                           
                             k 
                             + 
                             1 
                           
                         
                       
                       
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             t 
                             2 
                             
                               k 
                               + 
                               1 
                             
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             t 
                             3 
                             
                               k 
                               + 
                               1 
                             
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Thus, the edge tangent vectors is solved for by rotating their neighboring tangent vectors: 
     
       
         
           
             
               
                 
                   
                     t 
                     
                       N 
                       k 
                     
                     k 
                   
                   = 
                   
                     
                       R 
                       
                         θ 
                         
                           
                             
                               N 
                               k 
                             
                             - 
                             2 
                           
                           , 
                           
                             
                               N 
                               k 
                             
                             - 
                             1 
                           
                         
                         k 
                       
                     
                     ( 
                     
                       t 
                       
                         
                           N 
                           k 
                         
                         - 
                         1 
                       
                       k 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     t 
                     1 
                     
                       k 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       R 
                       
                         - 
                         
                           θ 
                           
                             2 
                             , 
                             3 
                           
                           
                             k 
                             + 
                             1 
                           
                         
                       
                     
                     ( 
                     
                       t 
                       2 
                       
                         k 
                         + 
                         1 
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     where R θ (t) is an operator on tangent vector k which induces a rotation of θ about the origin. Thus, two adjacent antenna clusters have been stitched together. This process can continue for k+2, k+3, . . . until all clusters have been stitched together. 
     There is the possibility of error propagation through this process, given that each successive stitching iteration depends on the last and all position vectors are based on measurements that are error-prone. Without additional physical constraints (e.g. the ends are fixed, the shape cannot bend beyond a limit, etc.) it is difficult to prevent or anticipate this. However, if there is a system constraint that can be optimized for, error can be accounted for and minimized. 
     In order to calculate the error in Angle Stitching, the final expression may be refined for the two unknown directions, including the possibility of measurement noise or error: 
       θ k,k+1 =θ N-1,N   k +Δθ N   k +δθ 1   k   (21)
 
       θ 1,2   k+1 =θ k,k+1 +Δθ 1   k+1 +δη 2   k   (22)
 
     where δθ 1   k  and δθ 2   k  are (small) measurement error parameters for the k th  stitching iteration. 
     Consider a system where the first element r 1   1  may be adjacent to the last element r N     K     K . This has two implications. Firstly, an additional stitching iteration may provide two more equations for θ K,1  and θ 1,2   1 . Secondly, measurement noise or error may be detected if this constraint is not met. To ensure this constraint is met, the optimization process below may be followed: 
     
       
         
           
             
               Minimize 
               : 
               L 
             
             = 
             
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     2 
                   
                   
                     
                       ( 
                       
                         δ 
                         ⁢ 
                         
                           θ 
                           i 
                           k 
                         
                       
                       ) 
                     
                     γ 
                   
                 
               
               γ 
             
           
         
       
       
         
           
             
               Subject 
               ⁢ 
                   
               
                 to 
                 
                   : 
                 
                 
                   | 
                   
                     
                       r 
                       
                         N 
                         K 
                       
                       K 
                     
                     - 
                     
                       r 
                       1 
                       1 
                     
                   
                   | 
                 
               
             
             = 
             
               p 
               
                 K 
                 , 
                 1 
               
             
           
         
       
     
     in addition to the constraints posed by the stitching process for the previous K−1 clusters. L, the loss function, a generalization of an I-norm, may be minimized where a high norm order, γ, penalizes kinks (or large angle error concentrated in a single stitch) in the global shape. 
     Similar to angle stitching, error in the position vectors propagates through the process and prevents a physical constraint from being met, if there is one. Invoking the same example as above, if the array is a loop, there is a possibility that the final (K) cluster may not correctly stitch to the original cluster. To correct for this, the above equations may be generalized to account for small errors: 
     
       
         
           
             
               
                 
                   
                     t 
                     
                       N 
                       k 
                     
                     k 
                   
                   = 
                   
                     
                       R 
                       
                         
                           θ 
                           
                             6 
                             , 
                             7 
                           
                           k 
                         
                         + 
                         
                           δ 
                           ⁢ 
                           
                             θ 
                             1 
                             k 
                           
                         
                       
                     
                     ( 
                     
                       
                         t 
                         
                           N 
                           k 
                         
                         k 
                       
                       - 
                       1 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     t 
                     1 
                     
                       k 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       R 
                       
                         - 
                         
                           ( 
                           
                             
                               θ 
                               
                                 2 
                                 , 
                                 3 
                               
                               
                                 k 
                                 + 
                                 1 
                               
                             
                             + 
                             
                               δ 
                               ⁢ 
                               
                                 θ 
                                 2 
                                 k 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         t 
                         2 
                         
                           k 
                           + 
                           1 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     where δθ 1   k  and δθ 2   k  are the same error parameters for the k th  stitching iteration as above. Error may be minimized and the physical constraint may be met using the same loss function and optimization process as above. Optimization can be performed using gradient descent, or a number of other optimization processes suited for high dimensional minimizations. 
     The discussed algorithm may be utilized in flexible phased arrays. In fact, disclosed herein is a framework for reconstructing the shape of flexible phased arrays based on mutual coupling measurements among elements. This autonomous shape reconstruction may be implemented in two steps: first, a Spiral Match algorithm and second, a semidefinite programming solution. Both steps can be replaced with other algorithms, enabling a level of modularity that may make this framework both flexible and more accurate for particular systems. Below, the accuracy of this approach is demonstrated on two 2.5 GHz passive phased arrays, one with a dipole and another with a patch antenna. This framework is also demonstrated on a flexible RFIC phased array. The accuracy achieved in the integrated circuit array reconstruction was ≈6% of one wavelength. This result enables sufficiently accurate for calibration free, real-time beamforming, beamsteering, or focusing arrays. This is included but not limited to highelement arrays, strongly asymmetric shapes, stretchable arrays, 2D arrays, and large distributed systems. Mutual coupling measurements may also be sufficient for reconstructing array shape. 
     Turning to the figures,  FIG.  4    is an image of an example of a flexible phased array. The flexible phased array may be a 2D flexible phased array antenna sheet with 256 10 GHz patch antennas. The flexible arrays may be bendable to reposition the antennas within the array antenna sheet. Flexible arrays may operate while curling with biological and mechanical joints or while fluttering in the wind if fast and accurate array shape reconstruction can be achieved. In other systems, an array that can accurately reconstruct its own shape may be used as a sensor itself in applications involving gaming, haptic human-machine interfaces, biomedical sensing, etc. Shape reconstruction may be significantly more general than shape calibration, instances of which have been shown in the past. Shape calibration refers to adjustments of element excitation phases to account for changes in relative element position and orientation within the array. Without shape calibration, the beam-forming, beam-steering, beam-focusing, arbitrary wavefront generation, and interference suppression capabilities of a flexible phased array may quickly disappear. These additional sensors increase system complexity by incorporating a new sensing domain and utilizing hardware not already present in phased array systems. Additionally, each resistive strain sensor only offers a single measurement value. For an array with complex bending geometry (having high sinuosity or lacking bilateral symmetry), a single, localized measurement will not accurately capture array shape. Designing a flexible phased array with a limited number of ancillary sensors may benefit from knowledge of the expected space of curvatures the array surface will experience during design, limiting the applications of such an array. 
     Superseding prior calibration approaches, shape reconstruction may be a more complete and more difficult task: full determination of the shape of the array and the surface to which it is conformed. Various embodiments of this disclosure include systems and methods for performing shape reconstruction only from local measurements of mutual coupling between elements in a flexible phased array. While phased arrays may be designed to direct an electronically steerable beam of microwave power towards a far-field target, some fraction of the power radiated by one element in the array may be absorbed by the other elements. This incidental power transmission may be referred to as mutual coupling. While in extreme cases it may be detrimental to array performance, mutual coupling provides information about the state of elements within the array. Mutual coupling shape reconstruction transposes the problem: using mutual coupling measurements to determine relative position of elements in the array. 
     Mutual coupling measurements may contain sufficient information for array shape reconstruction but an analytical and algorithmic framework may be utilized to turn an inter-element coupling matrix into element positions. In some embodiments, a two-step, modular framework may be utilized which is adaptable to different algorithmic approaches accounting for different sets of antenna properties and physical constraints. Some embodiments may be three different arrays instead of only one. 
     The framework may be successfully utilized on two different passive arrays with different radiators. To demonstrate a realistic system-level implementation of such algorithms, an active 10 GHz array using monolithic radio frequency integrated circuits (RFICs) may be utilized. RFICs provide timing synchronization, frequency synthesis, signal amplification, digitization, and a multitude of digital control functions with a millimeter-scale footprint. With this density of complex functions, discrete components can be removed, avoiding their bulk and rigidity which may be prohibitive for a truly flexible design. The integrated circuit-based flexible phased array is described below. 
       FIG.  5    illustrates a block diagram of a modular framework for mutual coupling-based shape reconstruction in accordance with an embodiment of the invention. The framework begins with mutual coupling data represented as a matrix  202 . A first step may include applying a physical constraint mapping (PCM)  204  which processes the mutual coupling information to produce physical constraints  206  of the array shape, such as element distances, mutual angles, or local curvatures. These physical constraints  206  can be represented in different formats (matrix, etc.), but all contain the necessary information to reconstruct the shape. A second step may include array shape construction (ASC)  208 , utilizes the physical constraints  206  to generate the actual positions  210  of all elements and, thus the array shape. The actual positions  210  of all the elements may be used to determine antenna excitation configurations of the elements. 
     This disclosure includes specific embodiments of the PCM and ASC algorithms. A mutual coupling matrix S∈CN×N may be a physical constraint matrix B∈RN×N, and a position matrix X∈Rd×N, where N denotes the number of array elements and d denotes the number of physical dimensions expected in the array (e.g. d∈{1,2,3}). In some embodiments, flexible phased arrays may include a single row of N=8 elements that only include d=2 dimensions. In some embodiments, the flexible phased array may include arbitrary N number of elements in d=3 dimensions. 
     The choice of algorithm for PCM and ASC may be dependent on the primary physical constraint used (e.g., relative distances or curvatures). The freedom to choose B gives this framework much of its modularity; different physical constraints may be better suited for different antenna systems. In some embodiments, Euclidean distance matrices (EDMs) may be utilized for the physical constraint matrix. 
     A Euclidean distance matrix, E∈R N×N  may be defined as: 
         E   mn   =|˜r   m   −˜r   n | 2   (25)
 
     where ˜r n  (n∈{1, 2, . . . , N}) is the position vector associated with point n. The matrix describes the squared distance between points in a d-dimensional point cloud. EDMs may be apt candidates for B because the EDMs can use known relationships between coupling and distance to devise the PCM algorithm and use existing algorithms for determining position from relative distance. 
     As discussed below, distance impacts both the phase and power of mutual coupling measurements. In some embodiments, a Spiral Match algorithm, may be utilized for PCM. Moreover, the problem of solving for the relative positions of an arbitrary number of points in an arbitrary number of dimensions may provide an EDM, E, which may be utilized in a number of applications. In some embodiments, a semi-definite optimization may be used to solve for the relative position. 
     Mutual Coupling Model 
     Accurate mapping of coupling measurements to physical parameters of the array (PCM in  FIG.  5   ) may be used for proper shape reconstruction. The PCM algorithm may include a Spiral Match algorithm (discussed below) and maps the measured complex-valued coupling to a distance between the phase centers of the elements. Near and far-field electromagnetic interactions among the elements and with the environment may produce a rich and non-trivial behavior for the mutual coupling. Reactive fields, occlusion (e.g. blocked line of sight), surface waves, and multipath reflections off the ground plane, other elements, and nearby objects, may affect the observed coupling between elements. Exact modeling of these effects may be challenging in simple, ideal scenarios and may be impractical if not impossible in the continuously changing context of a flexible phased array. These effects may be second order in many intra-array coupling scenarios, where a far-field-like propagation mode dominates. Instead of an electromagnetically complete coupling model, a simplified model may be created with sufficient accuracy for shape reconstruction and adaptability for use with a variety of element radiator types. 
     The Spiral Match algorithm may be used to match the measured coupling to distance using far-field properties of the element antennas as the individual antenna properties can be easily measured and are relatively insensitive to array shape. The Spiral Match algorithm may start by approximating the coupling between antennas i and j in terms of the electric field of a far-field propagating plane wave, e.g., 
     
       
         
           
             
               
                 
                   S 
                   ( 
                   
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     ≈ 
                     
                       
                         A 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       ⁢ 
                       
                         e 
                         
                           - 
                           
                             j 
                             ⁡ 
                             ( 
                             
                               
                                 ω 
                                 ⁢ 
                                 t 
                               
                               + 
                               
                                 k 
                                 ⁢ 
                                 
                                   
                                     ❘ 
                                     &#34;\[LeftBracketingBar]&#34; 
                                   
                                   
                                     
                                       l 
                                       mn 
                                     
                                     → 
                                   
                                   
                                     ❘ 
                                     &#34;\[RightBracketingBar]&#34; 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     where |{right arrow over (l mn )}| is the linear distance between antennas m and n, k is the wave number, ω is the angular frequency, and A is an arbitrary amplitude scaling factor. It may be assumed that phase grows linearly with distance between antennas (|{right arrow over (l mn )}|) and the coupled power falls off as 
     
       
         
           
             
               1 
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   
                     
                       l 
                       mn 
                     
                     → 
                   
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 2 
               
             
             . 
           
         
       
     
     To evaluate this approximation, a simple study may be performed with two 2.5 GHz ground plane backed folded dipole antennas (described below).  FIG.  6 A  is an image of an antenna pair coupling set-up. Continuous ground plane may be maintained for all antenna distances.  FIG.  6 B  is a plot of a coupling power measurement compared with two different propagation models. Measurement may be averaged over the bandwidth from 2.3 to 2.7 GHz.  FIG.  6 C  is a plot of a coupling phase measurement. Antenna pair measurements may be compared with a far-field linear phase propagation model. The measurement may be at 2.48 GHz. In both the power plot and phase plot, linear model traces may be normalized to the first measured distance. The coupling between dipole antennas may be measured vs. their distance, as shown in  FIG.  6 A . The ground plane may be modified to maintain continuity at all distances. The coupling power and phase may be plotted in  FIGS.  6 B and  6 C . The measured coupling power may be accompanied by two trendlines: the first follows the Friis 
     
       
         
           
             1 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               2 
             
           
         
       
     
     trend, the second following a 
     
       
         
           
             1 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               4 
             
           
         
       
     
     trend. This 
     
       
         
           
             1 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               4 
             
           
         
       
     
     trend may be observed for communications systems with signals travelling long distances over an infinite conductive or lossy surface (e.g. the earth). For close distances, the measured coupling follows the 
     
       
         
           
             1 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               4 
             
           
         
       
     
     trend as the ground plane approximates an infinite conductive surface. However, as the distance between antennas increases, the infinite surface approximation no longer holds as the ground plane may become relatively narrow and the measured coupling returns to the 
     
       
         
           
             1 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               2 
             
           
         
       
     
     trend. This change is the distance exponent has a relatively small effect on the final shape reconstruction accuracy but emerges from a known, measurable phenomenon and can be incorporated into Spiral Match for a given array without much added modelling complexity. 
     The coupling phase measurements roughly follow the linear trend predicted by the expression for a far-field propagating wave.  FIG.  7 A  is an image of a ground plane backed dipole array.  FIG.  7 B  is an unwrapped coupling phase measurement plot. In array measured phased may be compared to antenna pair measurements and far-field linear phase propagation model. The measurement may be performed at 2.48 GHz. Traces may be normalized to the first pair measurement distance.  FIG.  7 B  compares the pair measurement results to the coupling measurements between elements in a flat, 8 element 1D array of the same folded dipoles with pitch of 72 mm. Since the element pitch is fixed, in-array phase measurements only exist at multiples of 72 mm. These measurements also follow the linear model suggesting that the presence of other elements in the array does not significantly disturb the phase of the propagation. 
     In the embodiments described in connection with  FIGS.  6 A- 6 C , the elements experience relative position changes but no relative rotation. However, when a flexible phased array is bent, its elements experience changes in both position and orientation. To account for the change in coupling due to this rotation various embodiments include the far-field radiation pattern of the element antennas. Element radiation patterns can be determined through analysis, simulation, or measurement. 
     Although far-field phase propagation and radiation pattern are imperfect models of the coupling between elements within a phased array, they can be used to produce accurate shape reconstructions. 
     Embodiments Including Spiral Match Algorithm 
     A Spiral Match algorithm may be utilized in the PCM. A Spiral Match algorithm may include the spiral shape generated in a polar plot of the decaying amplitude and rotating phase vs. distance of propagating waves. 
     A bijection (e.g., a one-to-one correspondence) between mutual coupling and distance may be generated. A first step includes modeling the raw mutual coupling matrix S∈C N×N  for a 1-dimensional phased array as: 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           m 
                         
                         ⁢ 
                         
                           
                             D 
                             m 
                           
                           ( 
                           
                             θ 
                             
                               m 
                               ⁢ 
                               n 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           a 
                           j 
                         
                         ⁢ 
                         
                           
                             D 
                             n 
                           
                           ( 
                           
                             θ 
                             
                               n 
                               ⁢ 
                               m 
                             
                           
                           ) 
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         - 
                         
                           j 
                           ⁡ 
                           ( 
                           
                             
                               ϕ 
                               m 
                             
                             + 
                             
                               ϕ 
                               n 
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   
                                     l 
                                     mn 
                                   
                                   → 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     θ 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ( 
                     
                       
                         
                           
                             r 
                             m 
                           
                           ^ 
                         
                         · 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               l 
                               mn 
                             
                             → 
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     where a m  is the total fixed amplitude offset (due to line attenuation, mismatch, gain, etc.) in antenna m, D m (θ) is the directivity of antenna m for an angle θ relative to broadside, {right arrow over (l mn )} is the vector pointing from the phase center of antenna i to the phase center of antenna n, ϕ m  is the total fixed phase offset in antenna m, and   is the unit vector normal to the surface of the array, that describes the orientation of antenna m. The model for a 2-dimensional phased array may be very similar, and may involve only changing the directivity function to a function of two variables (e.g., D i (θ mn ,ϕ mn )), the azimuth and elevation angles between elements m and n. D m (θ=0)=1 may be used for all m&#39;s. No significant angle-dependence may be present for the phase response of the antenna. In some embodiments, in Equation 27, the term k|{right arrow over (l mn )}| may be |{right arrow over (l mn )}| k  where k is between 1 and 2. 
     There are amplitude and phase offset terms that can prevent a bijection between coupling and distance. To isolate the distance terms, a one-time calibration of the array may be performed in the completely flat configuration. The information gathered during this measurement is another coupling matrix, S flat : 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       m 
                       ⁢ 
                       n 
                     
                     flat 
                   
                   = 
                   
                     
                       
                         
                           a 
                           m 
                         
                         ⁢ 
                         
                           
                             D 
                             m 
                           
                           ( 
                           
                             θ 
                             
                               m 
                               ⁢ 
                               n 
                             
                             flat 
                           
                           ) 
                         
                         ⁢ 
                         
                           a 
                           j 
                         
                         ⁢ 
                         
                           
                             D 
                             n 
                           
                           ( 
                           
                             θ 
                             
                               n 
                               ⁢ 
                               m 
                             
                             flat 
                           
                           ) 
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                             flat 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         - 
                         
                           j 
                           ⁡ 
                           ( 
                           
                             
                               ϕ 
                               m 
                             
                             + 
                             
                               ϕ 
                               n 
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   
                                     l 
                                     mn 
                                     flet 
                                   
                                   → 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     Because the geometry of the flat configuration is fixed and known, basic assumptions can be made to simplify the model for flat coupling. First, in a flat array 
     
       
         
           
             
               
                 θ 
                 mn 
                 flat 
               
               = 
               
                 π 
                 2 
               
             
             , 
           
         
       
     
     when m≠n. Moreover, assuming our array is composed of identical antennas, D n (θ)=D m (θ)=D(θ) for all m&#39;s and n&#39;s. Thus, we can simplify our flat coupling model to: 
     
       
         
           
             
               
                 
                   
                     S 
                     
                       m 
                       ⁢ 
                       n 
                     
                     flat 
                   
                   ≅ 
                   
                     
                       
                         
                           a 
                           m 
                         
                         ⁢ 
                         
                           a 
                           n 
                         
                         ⁢ 
                         
                           
                             D 
                             ⁡ 
                             ( 
                             
                               π 
                               2 
                             
                             ) 
                           
                           2 
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                             flat 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         - 
                         
                           j 
                           ⁡ 
                           ( 
                           
                             
                               ϕ 
                               m 
                             
                             + 
                             
                               ϕ 
                               n 
                             
                             + 
                             
                               k 
                               ⁢ 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   
                                     l 
                                     mn 
                                     flat 
                                   
                                   → 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     Moreover, |{right arrow over (l mn   flat )}| may be known in advance. To eliminate the fixed power and phase offsets, the normalized transfer function, H may be defined: 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   ≡ 
                   
                     
                       s 
                       mn 
                     
                     
                       s 
                       mn 
                       flat 
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     H 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       
                         
                           D 
                           ⁡ 
                           ( 
                           
                             θ 
                             
                               m 
                               ⁢ 
                               n 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           D 
                           ⁡ 
                           ( 
                           
                             θ 
                             
                               n 
                               ⁢ 
                               m 
                             
                           
                           ) 
                         
                       
                       
                         
                           D 
                           ⁡ 
                           ( 
                           
                             π 
                             2 
                           
                           ) 
                         
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                             flat 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                           k 
                           ⁡ 
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 
                                   l 
                                   mn 
                                 
                                 → 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             - 
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 
                                   l 
                                   mn 
                                   flat 
                                 
                                 → 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     The dependence of the transfer function, H, on the angle matrix, θ presents a complication. Angles θ mn  and θd are two additional degrees of freedom, theoretically independent of |{right arrow over (l mn )}|, that present a challenge to finding a bijection between coupling and distance. In the case of two mechanically detached radiators, each of the three variables |{right arrow over (l mn )}|, θ mn , and θ nm  can induce changes in the coupling independently. 
     However, because the radiators may be fixed to a nonstretchable ground plane, mechanical constraints in this system reduce the allowable space of parameters from all of    3  to a smaller subspace in    3 . Even a stretchable ground plane places restrictions on antenna mobility and introduces a mechanical constraint that reduces the allowable space of independent variables, though to a lesser degree. To illustrate this, a special two dimensional case of only |{right arrow over (l mn )}| and θ mn  may be utilized; the mechanical constraint has the effect of partitioning    2  into “forbidden” and “allowed” subspaces, where pairs of {|{right arrow over (l mn )}|, θ mn } in the “forbidden” subspace are incompatible with the mechanical constraint, as illustrated in  FIG.  8   .  FIG.  8    is a visualization of forbidden and allowed regions of the θ mn ⊗|{right arrow over (l mn )}| vector space for physically restricted phase arrays. The “constant curvature” line is shown in dashed blue in the center of the allowed region. An example of a shape that generates an allowed {θ mn , |{right arrow over (l mn )}|} pair  502  and a forbidden {θ mn , |{right arrow over (l mn )}|} pair  504 . The forbidden {θ mn , |{right arrow over (l mn )}|} pair  504  can only be generated if the array is cut in half. The “allowed” pairs occupies a subspace in the shape of a triangular wedge: for some distances, the angular freedom may be highly restricted and for other distances, the angular freedom is large. Two caveats about the exact visualization presented: 1) This plot looks different for different antenna pairs and is characterized by |{right arrow over (l mn   flat )}|. 2) The exact appearance in  FIG.  8    may be inaccurate: The allowed region may not grow positively with increasing distance. The constant curvature assumption may not be a line. The space may not be symmetric. 
     There are multiple methods of dealing with the problem of optimizing over a high dimensional space while respecting the mechanical constraints at play. One is to computationally simulate the array and generate the {|{right arrow over (l mn )}|, θ mn , θ nm } triplets through brute force. Another is to characterize this subspace analytically. Finally, subspace may be assumed to be very thin, and treated as a line. This third option is equivalent to assuming the array ground plane only takes on shapes of constant curvature. The general framework may not depend on this assumption and implementations of the first two options are possible. In most practical cases where the surface bends smoothly without sharp kinks, the local curvature in a small neighborhood of several adjacent antennas can be considered constant. The general framework may be applied locally even for non-constant curvature cases. 
     Furthermore, the signal-to-noise ratio (SNR) of measurements between elements that are far apart, with many intervening elements, may be lower than elements within a smaller neighborhood (e.g. due to occlusion, reflections, attenuation, etc.) and may not contribute a great deal to the overall quality of the shape reconstruction. This does not preclude an ability to reconstruct the shape of phased array systems with large numbers of elements. The algorithm may only measure and model local measurements which can each have a different curvature. Although for each small neighborhood of antennas, a constant local curvature may be assumed, the curvature for an adjacent (and possibly overlapping set) can be different, leaving open the possibility for a shape with varying curvature. As discussed below, coupling measurements between far apart elements have a diminishing impact on shape reconstruction and, thus, it may not be important on how they are modeled. 
     Using the constant local curvature assumption, the space of independent variables may be reduced to R 1  and a bijection between θ mn  and |{right arrow over (l mn )}| may be achieved. The angle-dependent directivity, D (θ mn (|{right arrow over (l mn )}|)) may be replaced with a distance-dependent directivity, D (|{right arrow over (l mn )}|). This simplifies the transfer function model to: 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       
                         
                           D 
                           ⁡ 
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         
                           D 
                           ⁡ 
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                                 flat 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     ⁢ 
                     
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                             flat 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           
                             l 
                             mn 
                           
                           → 
                         
                         
                           ❘ 
                           &#34;\[RightBracketingBar]&#34; 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           j 
                         
                         ⁢ 
                         
                           k 
                           ⁡ 
                           ( 
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 
                                   l 
                                   mn 
                                 
                                 → 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             - 
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 
                                   l 
                                   mn 
                                   flat 
                                 
                                 → 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     The relationship between θ mn  and |{right arrow over (l mn )}| may be determined computationally. A geometric modeling of the array may be achieved. An example of the array model with a constant radius of curvature is illustrated in  FIG.  9   .  FIG.  9    shows a 2.5 GHz flexible phased array with 0.6λ antenna spacing and dipoles antennas that extend 0.25λ normal to the local ground plane in accordance with an embodiment of the invention. The flexible phased array has a constant curvature, R=−0.222 m. Ground plane and dipole feed in black, antennas in red. In the flexible phased array, f=2.5 GHz, antenna spacing is 0.6λ, the antennas are quarter-wave dipole antennas. 
     A large number of radii of curvature may be iterated, where R&lt;0 correspond to convex array orientation and R&gt;0 to concave array orientation. In this example, R∈±[0.025,1] m. At each iteration, a one-to-one mappings 
     
       
         
           
             R 
             → 
             
               
                 
                   ❘ 
                   &#34;\[LeftBracketingBar]&#34; 
                 
                 
                   
                     l 
                     mn 
                   
                   → 
                 
                 
                   ❘ 
                   &#34;\[RightBracketingBar]&#34; 
                 
               
               ⁢ 
                   
               and 
               ⁢ 
                   
               R 
             
             → 
             
               θ 
               mn 
             
           
         
       
     
     may be generated. To generate the bijection, K may be inverted: 
       θ mn (|{right arrow over ( l   mn )}|)= (   −1 (|{right arrow over ( l   mn )}|))  (34)
 
     Practically, this is implemented simply as a vector of θ mn  and a vector |{right arrow over (l mn )}|, which are paired, and correspond to a vector of curvature radii. 
     The result may be a bijection between distance and angle which may be utilized to incorporating directivity in the model. To quantify directivity, there may be a number of options: analytical antenna model, simulated directivity pattern, direct measured results. Any of these options may yield a function D(θ), which completes the model for the transfer function, H. 
     Embodiments Including Iterative Spiral Match 
     The model that draws a bijection between mutual coupling and distance may be utilized in a Spiral Match algorithm. The Euclidean distance between the elements m and n, as defined by (25), can be estimated by minimizing the magnitude of the difference between the measured transfer function, H mn   meas =S mn   meas, bent /S mn   means, flat , and the estimate of (33): 
     
       
         
           
             
               
                 
                   
                     E 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               arg 
                               ⁢ 
                                  
                               min 
                             
                             
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   
                                     l 
                                     mn 
                                   
                                   → 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               ∈ 
                               
                                 [ 
                                 
                                   
                                     d 
                                     mn 
                                     min 
                                   
                                   , 
                                   
                                     d 
                                     mn 
                                     max 
                                   
                                 
                                 ] 
                               
                             
                           
                           ⁢ 
                           
                             H 
                             mn 
                             meas 
                           
                         
                         - 
                         
                           
                             H 
                             mn 
                           
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     where d mn   min , d mn   max  are the minimum and maximum physically possible distances for the antenna pair. This can be the projection of the measurement H mn   meas  as onto the closest location on a spiral generated by H mn , shown in  FIG.  10   , and may be the essence of Spiral Match algorithm.  FIG.  10    is a visualization of the Spiral Match algorithm in accordance with an embodiment of the invention. In black, the coupling transfer function model, H mn ({right arrow over (l mn )}), for variable distance is given. The red “X” is the measured coupling, H mn   meas . It is “projected” onto the model via the argmin function to provide the blue circle, which is the position on the model that corresponds to our predicted distance, E mn . This location corresponds to the predicted distance, |{right arrow over (l mn )}|. Unfortunately, because of measurement noise and imperfection, solving for the Euclidean distance in a single step may lead to errors, which can result in potentially non-physical solutions. The table below summarizes the steps of the iterative process: 
     
       
         
           
               
               
               
               
               
               
             
               
                   
               
               
                 Itera- 
                   
                   
                 Re- 
                   
                   
               
               
                 tion # 
                 Constraint(s) 
                 Action 
                 sult 
                 Calculation 
                 Output 
               
               
                   
               
             
            
               
                 1 
                 — 
                 — 
                 E1 
                 Calculate 
                 Polarity 
               
               
                   
                   
                   
                   
                 polarity 
               
               
                 2 
                 Polarity 
                 Add directivity 
                 E2 
                 Run RWC 
                 Phase 
               
               
                   
                   
                 if concave 
                   
                   
                 wrap 
               
               
                   
                   
                   
                   
                   
                 matrix 
               
               
                 3 
                 Polarity, 
                 Restrict distance 
                 E3 
                 — 
                 — 
               
               
                   
                 Phase wrap 
                 search space 
               
               
                   
                 matrix 
               
               
                   
               
            
           
         
       
     
     Single step Spiral Match projection may be prone to issues in the presence of non-line-of-sight (NLOS) element coupling for convex configuration, where NLOS and surface-mode coupling can dominate. In this case, power changes do not follow the simple directivity model. Additionally, amplitude ambiguity at a given phase for an imperfect measurement can lead to erroneous projection of the measurement onto an incorrect segment of the spiral (introducing distance error on the order of λ). 
     Small distance error may be acceptable and can potentially be fixed in the second step of the framework, SDR (semidefinite relation). Large distance error, however, often makes SDR unworkable and results in a completely erroneous shape prediction. Mitigating large distance error may be provided by the iteration process discussed below. 
     To mitigate these issues, the Spiral Match algorithm may be performed in a number of iterations, each time adding more constraints to the system to help ensure the accuracy of the predicted Euclidean distance. These iterations and the associated constraints are shown below. In the first iteration, it may be assumed no curvature polarity (concave/convex) to minimize constraints on the problem. Without a polarity, there may be no assumption of the geometry, and, thus, there may not include directivity in the model. For the first pass, there may be assumed D(|{right arrow over (l mn )}|=1, ∀i, |{right arrow over (l mn )}| and the optimization function in (35) may be used to compute the first prediction of the EDM, E 1 . The result of this first iteration may be a very crude prediction of Euclidean distance that may be sufficiently accurate to determine the polarity of the shape. The polarity p∈{−1,+1} (−1 being convex and +1 being concave) may be decided using a voting scheme: 
         p =sign(Σ i≠j (sign|{right arrow over ( l   mn   flat )}|− E   mn   1 ))  (36)
 
     N 2 —N pairs is an even number and it&#39;s possible that p=0. If this happens, |{right arrow over (l mn   flat )}|−E mn   1 ≈0 and a small degree of error in E mn   1  may be pushing the difference above and below zero for different (i,j). In this case, the array may be in a flat or an approximately flat shape and the choice of polarity may be almost irrelevant. In the algorithm, p=0 cases may be overwritten to have p=+1 for these incredibly rare circumstances. 
     In the second iteration, p may be used to inform whether or not to include directivity in the coupling model. In the second prediction of the EDM, E 2  may be computed using the optimization function in (35). Having eliminated ambiguity about polarity and including the directivity for concave shapes, the next step may be to calculate phase wrapping-induced distance error. These errors are on the order of λ, which is often large enough to violate physical constraints, and so it becomes germane to develop such a solution that will intelligently identify violations. This may be referred to as “Recursive Wrap Correction” (RWC). 
     RWC, the details for which are provided below, is an algorithm that converts a matrix of integers, R∈   2 , representing the number of complete phase wraps associated with the distances in E 2 , to a “corrected” phase wrap matrix, R′. It can also correct for outlier measurements not caused by phase-wrapping that may carry large distance error. 
     In the third iteration of the Spiral Match algorithm, the search space for distance is truncated to a single wavelength range as follows: 
       |{right arrow over ( l   mn )}|∈[ R′   mn λ,( R′   mn +1)λ)  (37)
 
     to ensure that all predicted distances have the same number of wraps as RWC predicted. If R′ is correct, this has the effect of ensuring there are no phase wrapping-induced distance errors, which improves Spiral Match&#39;s accuracy. Spiral Match culminates with a final minimization, using the reduced search space: 
     
       
         
           
             
               
                 
                   
                     E 
                     mn 
                     3 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             arg 
                             ⁢ 
                                
                             min 
                           
                           
                             
                               
                                 ❘ 
                                 &#34;\[LeftBracketingBar]&#34; 
                               
                               
                                 
                                   l 
                                   mn 
                                 
                                 → 
                               
                               
                                 ❘ 
                                 &#34;\[RightBracketingBar]&#34; 
                               
                             
                             ∈ 
                             
                               [ 
                               
                                 
                                   
                                     R 
                                     mn 
                                     ′ 
                                   
                                   ⁢ 
                                   λ 
                                 
                                 , 
                                 
                                   
                                     ( 
                                     
                                       
                                         R 
                                         mn 
                                         ′ 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   λ 
                                 
                               
                               ] 
                             
                           
                         
                         ⁢ 
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               H 
                               mn 
                               meas 
                             
                             - 
                             
                               
                                 H 
                                 mn 
                               
                               ( 
                               
                                 
                                   ❘ 
                                   &#34;\[LeftBracketingBar]&#34; 
                                 
                                 
                                   
                                     l 
                                     mn 
                                   
                                   → 
                                 
                                 
                                   ❘ 
                                   &#34;\[RightBracketingBar]&#34; 
                                 
                               
                               ) 
                             
                           
                           
                             ❘ 
                             &#34;\[RightBracketingBar]&#34; 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     At this point, it&#39;s unlikely E 3  is perfectly symmetric. This is acceptable for the purposes of the next step—ASC—but it can be advantageous because two candidates for distance for each element pair may be used. In some embodiments, the two candidates may be compare and the better one may be chosen. An obvious candidate for this metric is the “projection error,” ΔH, defined as: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       H 
                       mn 
                     
                   
                   = 
                   
                     
                       min 
                       
                         
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ∈ 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   mn 
                                   ′ 
                                 
                                 ⁢ 
                                 λ 
                               
                               , 
                               
                                 ( 
                                 
                                   
                                     R 
                                     mn 
                                     ′ 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           H 
                           
                             m 
                             ⁢ 
                             n 
                           
                           meas 
                         
                         - 
                         
                           
                             H 
                             
                               m 
                               ⁢ 
                               n 
                             
                           
                           ( 
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               
                                 l 
                                 mn 
                               
                               → 
                             
                             
                               ❘ 
                               &#34;\[RightBracketingBar]&#34; 
                             
                           
                           ) 
                         
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     which corresponds to the distance in the complex plane between the measurement and the model at the predicted point, E mn   3 . A large projection error is an indication that the model does not possess a complex value close to the measurement—implying large measurement error. However, this implication only holds some of the time. The correlation between prediction error and projection error occurs for the patch antennas discussed below, but not for the dipole antennas. Because this is another “switch,” it may be turn flipped or not. To “flip it,” the EDM may be forced to be symmetric using the projection error as the deciding metric: 
     
       
         
           
             
               
                 
                   
                     E 
                     
                       m 
                       ⁢ 
                       n 
                     
                     
                       3 
                       ⁢ 
                       ′ 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             E 
                             
                               m 
                               ⁢ 
                               n 
                             
                             3 
                           
                         
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                 H 
                                 
                                   m 
                                   ⁢ 
                                   n 
                                 
                               
                             
                             ≤ 
                             
                               H 
                               
                                 n 
                                 ⁢ 
                                 m 
                               
                             
                           
                         
                       
                       
                         
                           
                             E 
                             
                               n 
                               ⁢ 
                               m 
                             
                             3 
                           
                         
                         
                           
                             
                               Δ 
                               ⁢ 
                               
                                 H 
                                 
                                   m 
                                   ⁢ 
                                   n 
                                 
                               
                             
                             &gt; 
                             
                               Δ 
                               ⁢ 
                               
                                 H 
                                 
                                   n 
                                   ⁢ 
                                   m 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     where E 3′  is the symmetrized version of E 3 . E 3  (or E 3′ ) is the input to the second phase of the shape reconstruction framework algorithm, discussed below. 
     Embodiments Including Semi-Definite Relaxation 
     Using a prediction of the EDM corresponding to antenna phase centers on a flexible phased array, E 3 , relative position, or shape may be reconstructed. Like many other high-dimensional problems, this can be solved with a single eigenvalue decomposition. Solving for the position with an EDM in this way is called “Classical Multi-Dimensional Scaling” (MDS), which involves few matrix operations. MDS, simple as it is, typically fails for imperfect EDMs. If, for example, E mn =|r i −r j | 2 +α mn , where α mn  is some noise-adding term, the eigenvalue decomposition may yield complex (e.g. non-physical) eigenvalues. Additionally, sparse or mislabeled EDMs cannot be used with MDS. A number of algorithms have been written and studied for different purposes and qualify as candidates for the second transformation, ASC. 
     In this system, sparse or mislabeled EDMs may not occur and choose to use semi-definite relaxation (SDR) as the candidate for ASC, for the following reasons:
         1. SDR offers the option to use a mask matrix W to weight elements in the EDM individually. This allows control of how much each EDM entry “matters.” Because local coupling measurements have higher quality than distant measurements, local measurements may be weighted more.   2. SDR searches a much smaller, convex vector space and, thus, guarantees a physical result while also decreasing runtime.   3. SDR showed the best performance along multiple metrics.
 
Some embodiments implement SDR written in MATLAB using “cvx,” a framework for performing convex optimization in MATLAB.
       

     The option to use a mask matrix W to window entries that have not been collected (e.g. in mic/speaker localization in a large theatre) is an option that allows this method to be utilized for large array systems, or ones where some elements have only either a receiver or a transmitter. In some embodiments, a completely dense EDM may be collected, and a mask matrix maybe used to weight entries based on SNR. 
     Recognizing that SNR 
     
       
         
           
             
               
                 ( 
                 
                   S 
                   
                     m 
                     ⁢ 
                     n 
                   
                 
                 ) 
               
               ∝ 
               
                 1 
                 
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     
                       
                         l 
                         mn 
                       
                       → 
                     
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                   2 
                 
               
             
             , 
           
         
       
     
     it is likely that EDM error, δE, defined as: 
       δ E   mn =∥{right arrow over ( l   mn )}|− E   mn   3 |  (41)
 
     follows the general trend: 
     
       
         
           
             
               
                 
                   
                     δ 
                     ⁢ 
                     
                       E 
                       
                         m 
                         ⁢ 
                         n 
                       
                     
                   
                   ∝ 
                   
                     
                       
                         ❘ 
                         &#34;\[LeftBracketingBar]&#34; 
                       
                       
                         
                           l 
                           mn 
                           flat 
                         
                         → 
                       
                       
                         ❘ 
                         &#34;\[RightBracketingBar]&#34; 
                       
                     
                     k 
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     where k is some unknown exponent. This known relationship may be utilized by weighting higher quality EDM entries more than lower quality ones. The intuitive approach to implement this is to have entries in matrix W decrease as they move away from the diagonal. 
     The freedom to design W may be chosen, based on the system, to minimize shape reconstruction error. A number of structures for W may be explored, some of which are presented in Table I. 
                                     Type   Equation   Variables                  Uniform   W mn  = l   —               Binary             W   mn     =     {             1   ⁢            ❘   &#34;\[LeftBracketingBar]&#34;       m   -   n       ❘   &#34;\[RightBracketingBar]&#34;         &lt;   a                 0   ⁢            ❘   &#34;\[LeftBracketingBar]&#34;       m   -   n       ❘   &#34;\[RightBracketingBar]&#34;         ≥   a                     a               Linear   W mn  = a|m − n| + b   a, b       Quadratic   W mn  = a|m − n| 2  + b|m − n| + c   a, b, c       Exponential   W mn  = a −b|m−n|  + c   a, b, c                    
In most of these structures, there is a design variable that can be turned to adjust for different systems and optimized to minimize shape reconstruction error. Superior performance may be produced with a mask matrix where entries are uniform for close pairs and exponential for far pairs.
 
     Embodiments Including Passive Flexible Phased Arrays 
     In order to validate the proposed shape reconstruction method, several connectorized flexible phased arrays were built. While lacking the dynamic, multi-purpose capabilities of an integrated flexible phased array, connectorized arrays allow for quick measurements of mutual coupling (e.g., by a standard vector network analyzer). The connectorized array elements may be designed to have center frequencies close to 2.5 GHz with element pitch of 0.6λ. Results from eight-element, 1D arrays with folded dipole and patch antennas are presented below. 
     A block diagram of the measurement setup of a connectorized array on the rigid frame is depicted in  FIG.  11   .  FIG.  11    is a measurement setup for coupling measurements of connectorized arrays in accordance with an embodiment of the invention. In order to measure the full coupling matrix of the array without changing connectors, each element connects to an RF splitter  804  then two digitally controlled 8-to-1 RF multiplexers  802 , which are then connected to the ports of a vector network analyzer  806 . A computer  810  and script controls the multiplexers  802 , triggers the VNA  802 , and logs the coupling measurements. To flex the connectorized arrays into known shapes in a quick and repeatable fashion, rigid wooden frames of known convex and concave bend radii were designed and built. These frames are shown in  FIG.  12 A . 
     The folded dipole used in the measurements is shown in  FIGS.  12 A and  12 B .  FIG.  12 A  illustrates an example folded dipole antenna with design dimensions.  FIG.  12 B  is a plot of measured and simulated folded dipole input matching and adjacent element coupling. Ground plane backed dipoles (folded or unfolded) are well suited to mutual coupling shape reconstruction. Radiation from a dipole originates from the current distribution on the arms and is initially omni-directional (except for the direction along the arms themselves) prior to reflection off of the ground plane. Because this omni-directional radiation originates well above the ground-plane, the coupling may be less sensitive to occlusion and has propagation properties close to the ideal far field assumed by the simplified propagation model. The folded dipole presents an input impedance of 275 ohms at its arms. A high impedance edge-coupled quarter-wave transmission line extends to ground plane, transforming the antenna input impedance and connecting to a 50 ohm SMA connector. 
     The folded dipole array shape reconstruction results are shown in  FIGS.  13 A and  13 B .  FIGS.  13 A and  13 B  illustrate shape reconstruction results for passive 2.5 GHz folded dipole phased arrays. In  FIG.  13 A , the left images illustrate various concave wooden frames with dipole antennas and curvature radii. The middle images illustrate EDM error heatmaps and mean EDM error (ΔE) for concave shapes. The right images illustrate reconstructed antenna shapes (red), true antenna shapes (black), and mean shape error (Δx) for concave shapes. 
     In  FIG.  13 B , the left images illustrate convex wooden frames with dipole antennas and curvature radii. The middle images illustrate error heatmaps and mean EDM error (ΔE) for convex shapes. The right images illustrate reconstructed antenna shapes (red), true antenna shapes (black), and mean shape error (Δx) for convex shapes. 
     The far-field radiation pattern used in the algorithm was determined using a finite difference time domain (FDTD) simulator. The accuracy of the Spiral Match algorithm is measured by ΔE, the mean element pair distance error. ΔE is the mean difference between the predicted distances and the “truth” distance predicted by the known physical design dimensions of the frames. The accuracy of the shape reconstruction is measured by Δx, which is the mean difference between the reconstructed element position and the position of the elements according to the frame design dimensions. An exponential weighting matrix was used within the semidefinite relaxation algorithm. 
     The next array used for testing the shape reconstruction algorithm was an eight element linearly polarized patch antenna array. While the folded dipole antennas are near ideal candidates for shape reconstruction, patch antennas present a greater challenge. The challenges offered by patch antennas are a bellwether for if shape reconstruction could be a ubiquitous tool for a wide variety of flexible arrays or is limited to narrow subset of ideal systems. Typical patch antennas are low profile, highly resonant, and inherently directional with radiation emerging from fields located at both ends of the patch along its excitation axis (E-field line). In a flexible array, their low profile means they are more susceptible to occlusion and do not share the pseudo-omnidirectionality of the ground plane backed dipoles. The highly resonate nature (particularly when built on thin substrates) leads to low matching bandwidth. Low matching bandwidth can effect the quality of coupling measurements as a pair of patches could de-tune from each other as the array changes shape. 
     The presented patch antenna, depicted in  FIGS.  14 A and  14 B , may be made with 1.27 mm thick high dielectric substrate (∈ r  11.2) in order to reduce their size and better facilitate bending.  FIG.  14 A  is an image of a patch antenna with design dimensions. The substrate may extend 5 mm around the copper patch.  FIG.  14 B  is a plot of a measured and simulated patch antenna input matching and adjacent element coupling. The feed is inset by 6.35 mm in order to match to 50 ohms. The far-field radiation pattern used in the algorithm was determined using an FDTD simulator. The results of the patch antenna shape reconstruction is shown in  FIGS.  15 A and  15 B . The patch antennas shape reconstruction is as accurate as the dipole reconstruction with the exception of the most convex shape. The final matrix in  FIGS.  15 A and  15 B  shows higher error in the off diagonal element pair distances which would indicate that occlusion caused by the extreme bend is the culprit. 
       FIGS.  15 A and  15 B  are shape reconstruction results for passive 2.5 GHz patch antenna phased arrays. In  FIG.  15 A , the left images are concave wooden frames with patch antennas and curvature radii. The middle images are EDM error heatmaps and mean EDM error (ΔE) for concave shapes. The right images are reconstructed antenna shapes (red), true antenna shapes (black), and mean shape error (Δx) for concave shapes. 
     In  FIG.  15 B , the left images are convex wooden frames with patch antennas and curvature radii. The middle images are EDM error heatmaps and mean EDM error (ΔE) for convex shapes. The right images are reconstructed antenna shapes (red), true antenna shapes (black), and mean shape error (Δx) for convex shapes. Despite this, the results show the viability of the shape reconstruction algorithm for radiator with tightly confined near-field profile. 
     Active Integrated Flexible Phased Array 
     While the passive, connectorized arrays discussed above are useful tools for understanding mutual coupling shape reconstruction, they may not provide the utility of a fully integrated flexible phased array. A truly self-contained flexible phased array may be enabled by RFICs, which combine array element functions (e.g. frequency synthesis, phase shifting, power amplification, etc.) in a compact and low-profile package. Conventional silicon ICs are rigid, but are sufficiently small as to not significantly change flexibility of a larger flexible phased array. An integrated circuit flexible phased array prototype was created to demonstrate mutual coupling shape calibration and reconstruction. The implemented array is a 1D, eight-element transmit and receive capable array operating at 10 GHz built on a 4-layer flexible printed circuit board (PCB) shown in  FIGS.  16 A and  16 B .  FIG.  16 A  is an integrated circuit-based flexible phased array with 8 elements bent with front shown.  FIG.  16 B  is the back of the integrated circuit-based flexible phased array of  FIG.  16 A  with feed network, chips, interposers, and circuitry shown. 
     Each element circuit may include a custom single channel transmitter/receiver integrated circuit which may be implemented in a standard 65 nm CMOS process. The RF path of the RFIC starts with a 2.5 GHz phase reference signal, which is shared by all elements in the array. The phase reference may be used by the phase-locked loop (PLL) to synthesize 10 GHz IQ signals. The PLL uses programmable divider current injection to provide 360°+ phase control to its output which is fed to both the transmit and receive paths. An SPI digital interface controls all programmable sub-circuits. In the transmit path, the signal passes through a programmable IQ summation phase shifter which provides another independent 360° phase control. The phase shifter may be controlled by a programmable SRAM, allowing for rapid phase-shifting for beam steering and/or data modulation. The phase shifter can be disabled and bypassed to reduce power consumption. With the PLL phase control and the IQ summer, the transmit and receive phase can be set arbitrarily and independently. The transmit path ends in a power amplifier (PA) and balun before the signal is sent off chip to the radiator. 
     The receive path begins at the same balun which the PA is driving. A receive-enable switch between one side of the balun and ground allows the receiver to measure the power amplifier output (switch closed) and allows the receiver to see higher impedance when the PA is not operating (switch open). A bypass-able amplifier ensures the receive channel has sufficient dynamic range to measure small signals as well as the large signals from self-loop or adjacent element measurements. This amplifier is followed by a direct down-conversion IQ mixer. The resulting baseband IQ signals are digitized and processed off-chip. 
     Each element RFIC may be mounted on a thin 2-layer interposer made from a traditional RF PCB substrate. While PCB substrates are rigid when thick, a small, thin interposer does not affect the flexibility of the larger board. The supply filtering capacitors may be aligned with the axis of bending to minimize their effect on overall stiffness, ensuring that bend radius is limited by plastic deformation of the flexible PCB. The block diagram is shown in  FIG.  17    and RFIC die photo and interposer are shown in  FIGS.  18 A- 18 C .  FIG.  17    is a schematic block diagram of a an RFIC element circuit.  FIG.  18 A  is an image of an RFIC die.  FIG.  18 B  is an image of an interposer with RFIC.  FIG.  18 C  is an image of 3 interposers mounted on flexible printed circuit board. 
     The radiator, built from a single sheet of copper on polyimide, is a ground plane backed dipole. By aligning the dipole arms with the axis of bending, the array can be flexed significantly without strongly affecting radiator performance. In addition to having favorable properties for shape reconstruction, ground plane backed dipoles provide the wide, single lobe pattern that is desirable for beam-steering applications.  FIG.  19 A  illustrates a radiator mounted perpendicular to the surface of the ground plane which avoids the need for a thick, likely rigid, layer of substrate. The radiator may be mounted on a flexible printed circuit board. The feet beneath the fold line are soldered to the circuit board. Element pitch may be 0.6λ.  FIG.  19 B  is the measured and simulated far-field patterns with bending. The measured and simulated in-array element pattern may be for φ=0° cut. Pattern may be measured flat and when array is conformed to a 120 mm convex bend radius.  FIG.  19 C  is the measured and simulated far-field patterns without bending. One leg of the radiator connects to a single-ended transmission line while the other is grounded. The vertical feed acts as a balun to convert the input to a differential drive for the radiating arms. The in-array element pattern may be for φ=90° cut. All measurements are normalized to their global maximum. Overall, the simulated, flat, and bent results match closely. 
     Residual asymmetry may be observed in  FIG.  19 B  as the pattern tilts slightly towards the grounded leg. Deviation between bent and flat performance is present in the φ=90° cut where the bent result experiences slightly less lobe splitting than the flat simulated and measured results. This is consistent with the expectation that a convex bend increases element spacing, decreasing the inter-element coupling which contributes to lobe splitting. This pattern change with array shape is small and does not meaningfully affect the quality of the shape reconstruction results. 
     Shape reconstruction using the flexible integrated circuit based phased array is demonstrated in  FIG.  20   .  FIG.  20    is shape reconstruction results for integrated 10 GHz phased arrays with dipole antennas for various radii of curvature. The left images are EDM error heatmaps and error mean (ΔE). The right images are reconstructed antenna shapes (red), true antenna shapes (black), and shape error mean (Δx). Eight rigid frames (R=±100, 150, 220, 400 mm) were used to provide repeatable reference dimensions. The on-chip power amplifiers and receivers were used to collect the coupling measurements. When adjusted by a factor of four to account for wavelength difference between 2.5 GHz and 10 GHz, the final shape error, Δx, outperforms the connectorized 2.5 GHz dipole array. The minimum, maximum, and average Δx is lower. While some of this performance increase can be contributed to a smaller ratio of bend radii to array length, it shows the viability of shape reconstruction in integrated circuit arrays. The shape reconstruction framework is shown to be frequency agnostic by these results. While the framework itself is agnostic, there are practical concerns for which frequency is involved. The 10 GHz array is smaller, as such nearby objects and reflectors relatively farther away and less likely to effect local coupling measurements through reflections. As such the higher frequency may be responsible for the more accurate results. Improved quality of coupling measurements does come with a mechanical trade-off. As the operation frequency of the array rises, its flexibility could decrease as the density of rigid or semi-rigid components and the complexity and thickness of signal routing increases with finer antenna pitch. 
     The described framework is multi-staged and multi-faceted, offering many opportunities for modification and expansion. This framework can easily be implemented when other antennas, sizes, or array constraints are in use. 
     A system that has other sensors or capabilities can easily accommodate an algorithm relying on angle or curvature as the constraint. Choosing another physical constraint may include modifications to or alternatives for PCM and ASC as well. 
     The PCM algorithm (e.g. Spiral Match algorithm) is rooted in a model for the coupling. This model can easily be changed to include other constraints or known EM phenomenon to improve its versatility and our accuracy. The simplified model may be used to predict the Euclidean distance in three iterations but, again, these iterations and the constraints they leveraged may be system-dependent. Additional iterations can be added if a system is more or less constrained. Moreover, the parameters of the iterations can be easily changed for new systems. For example, the constraints used in the Recursive Wrap Correction algorithm can be easily changed. 
     New candidates for PCM could leverage frequency diversity if the system included broadband antennas. While broadband systems could use the single frequency approach disclosed herein, frequency diversity may alleviate phase wrapping ambiguity when determining distance. Mutual coupling may be measured at multiple frequencies or using a chirp similar to frequency modulated continuous wave (FMCW) radar from which a distance matrix would then be constructed. 
     EDM (e.g. the ASC algorithm) algorithms may each solve a different type of problem. In a massive 2D array (e.g. 100×100), coupling may only be investigated locally, leaving a sparse EDM. This system might include a different method to deal with highly sparse EDMs. The freedom to choose the ASC algorithm makes the framework highly modular to any number of non-idealities in the EDM. Many EDM applications include a mask matrix as a design parameter. Some embodiments may include banded mask matrices that decay exponentially. 
     Embodiments Including Recursive Wrap Correction 
     As discussed above, the iterative Spiral Match algorithm may include RWC. The input to RWC, R, may be defined as: 
     
       
         
           
             
               
                 
                   
                     R 
                     
                       m 
                       ⁢ 
                       n 
                     
                   
                   = 
                   
                     floor 
                     ( 
                     
                       
                         E 
                         
                           m 
                           ⁢ 
                           n 
                         
                         2 
                       
                       λ 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     The problem of phase wrapping implies that R mn  is occasionally 1 or 2 phase wraps greater or less than the true number. 
     RWC takes place in two steps: recursive error correction and matrix selection. 
     1) Recursive Error Correction: The root method of this step, Θ(R), is a recursive algorithm which inputs an (potentially) erroneous phase wrap matrix, R, and outputs either:
         Itself, if R is free of errors. This is the canonical “base case.”   [Θ(R 1 ),Θ(R 2 )], where R 1  and R 2  are two options for matrices that correct a single rule violation in R.
 
A matrix, R, has potentially many rule violations; each violation is associated with two potential corrections. Thus, recursive error correction involves generating and traversing a binary tree of matrices, where R is the highest parent node and the terminal nodes form a set, {R 0 }, of error-free alternatives to R.
       

     Θ(R) begins by searching R, starting on the two inner diagonals to the right and left of the main diagonal ({R mn ,∀|m−n|=1}) and ending at the two corner elements ({R mn ,∀|m−n|=N−1}). Each element is checked for the following two possible rule violations: 1) Discontinuity (ie. |R m,n −R m,n+1 |&gt;a); 2) Wrong direction (ie. R m,n &gt;R m,n+1 ) The exact conditions of these violations (how discontinuity is defined, in what circumstances the direction is “wrong”, etc.) may depend on the system this algorithm is being implemented in. The system includes antennas separated by 0.6λ, thus defining “discontinuity” as a difference in wrap count between mutual matrix elements greater than 1 (a=1). The wrap counts may strictly increase for increases in |m−n| (“Wrong direction”). 
     When an error is discovered, the algorithm breaks out of the search and begins a new iteration of Θ(R) with two alternatives as inputs. These two alternatives are new matrices, R 1  and R 2  each with a single element changed that simultaneously corrects the rule violation and does not introduce a new violation. 
     Recursion continues until all alternatives are error free or a stop condition is reached. The rule violations and corrections are defined in a way that the stop condition is rarely reached. The result of this process is a set, {R′}, of correct phase wrap matrices that are alternatives to the original input, R. After, the next phase maybe matrix selection, to pick the final phase wrap matrix, R′. 
     2) Matrix Selection: The first phase of matrix selection is to cut matrices that violate any of the following “global” criteria (ordered from most important to least): 
     1) Duplication (ie. R′ k =R′ l , k≠l); 
     2) Absolute distance limits; and 
     3) Symmentry (i.e. R′ k ≠R′ k   T ). 
     Cuttings take place in these three stages. If any stage results in the cutting of all but one option, that option becomes R 0 . If any stage results in the cutting of all options, the cuts do not take place and the algorithm moves to the next stage. 
     If after these 3 rounds of cuts there are still multiple options voting takes place based on a set metric. Examples of metrics include:
         Curvature Discontinuity: favor matrices that reflect constant curvature conditions.   Symmetry: favor the “most symmetric” matrix. This obviously only applies if the third round of cuts does not take place.   Change: favor the matrix that is closest to the original.
 
If the system possesses constraints that can be leveraged, a metric can be designed to pick the phase wrap map that most aligns with that constraint. In this case, symmetry may be utilized if the last cut does not take place and curvature discontinuity otherwise.
       

     The conditions for arriving to this point may be rare. In the vast majority of cases, the original input, R, is free of rule violations and, thus, is also the output R′—these metrics and conditions having absolutely no impact on the outcome. RWC may be seldom used because a dataset large enough to accurately test for the best metric may be rare. On the other hand, in the case where an erroneous phase wrap matrix is produced by the second iteration of Spiral Match, RWC may provide a realistic alternative that can correct for phase wrapping errors. 
     The above discussed processes can be generalized for higher dimensional arrays or constellations of elements where there are known constraints (such as element pitch) between clusters. In the case of a 2D distributed array, there may be more constraints than degrees of freedom and the process may be overconstrained. This may rule out the possibility of using a simple geometric approach in an error-prone system. In some embodiments, the system may be formulated performed in the 1D case, and use an error minimization to optimization for the shape. 
     Some systems may be so highly constrained that it may not be desirable to use all the information: A large number of degrees of freedom implies a computationally expensive optimization that may have a higher potential to fall into local minima and fail; decreasing the number of degrees of freedom may help make the system less overconstrained. If the field of constraints are culled intelligently, the above formulation may be used to stitch clusters in 2D while mitigating concerns about falling into local minima or expanding the runtime and power consumption of the stitching algorithm. 
     The above discussed algorithm constitutes a simple geometric method to “stitch” clusters of elements which are locally, but not globally, shape reconstructed. This method is flexible, allowing for the integration of physical constraints unique to different systems to minimize error propagation. This may have great advantages for a number of applications. For example, the systems may be utilized in wireless power transfer (WPT) and/or communications, where phased arrays have seen most of their utility and where phase setting for beamforming is critical. Here are some contexts in which this method can be used toward these ends: deployable WPT arrays for space; deployable communications platforms for space; WPT arrays integrated into fabrics; communication arrays integrated into fabrics; deployable WPT and/or communication arrays for boats/cars/planes/large vehicles; and/or conformal arrays for planes/boats/vehicles. Different applications are associated with different system time constants—a reflection of how long the system has before the shape changes. This time will set an upper limit on how long the system may operate, placing demands on the runtime and hardware. The system may be implemented in large-scale wireless power transfer arrays integrated into normal objects (window shapes, drapes, bed sheets, etc.) in a residential or commercial environment. 
     As opposed to using shape reconstruction as a means to beamform, the shape can be the end itself. This method can make use of electromagnetic resonators for the purpose of shape reconstruction—as a shape sensor—in the following contexts: biosensors to detect body configuration/conformation; for athletes; for people/animals with injuries—as a therapeutic tool; for VR applications, in gaming, in simulations, or in VR entertainment; sensing the configuration of cars on the road; detecting the configuration of drones/planes/satellites in a constellation; sensing topology of nodes on a local wireless network; sensing topology of nodes on a large wireless network; prediction of channel state information for efficient wireless networks; and/or use as an electromagnetic tool (e.g. with conformal arrays, to detect conformation shape after installation, to detect antenna&#39;s directivity pattern, to compute phase offset at the end-point of propagation). The biosensing applications would require the array to be biocompatible (e.g. safe, flexible) with changes to the electromagnetic coupling model to account for mode transmission across the body. The car configuration and topology sensing would require the coordination of independent nodes, or communication with some centralized node. In some embodiments, the shape reconstruction systems may include large-scale shape sensors for humans, animals, or other objects for athletics, gaming, VR, physical therapy; large-scale communications platforms that are flexible, collapsible, or otherwise reconfigurable; and/or large-scale, flexible power transfer systems in for space solar power applications. 
     Very large array systems, may exceed the dimensions that can be serviced by singular clock lines or single RFICs, can benefit greatly from the disclosed system and methods, which may enable large-scale shape reconstruction on distributed platforms. Bridging the gap to the next generation of large-scale and distributed systems may include high-power, high accuracy array systems that demand accurate and reliable distributed shape reconstruction methods. 
     Though the embodiments disclosed above include reconstruct shape without any “additional” hardware, other sensors can help add information to improve performance or offer new possibilities. For example, the system may include use of parasitic Tx/Rx radiators in addition to those on the array. Also, the system may include environmental sensors (e.g. temperature, strain, light sensors) to detect changes to array. The environmental sensors may be utilized to estimate changes to devices/transmission line that can be de-embedded. 
     Example Operating Procedure 
     As discussed above, various embodiments of the invention involve recalibrating the shape of an antenna array based on a desired functionality.  FIG.  21    illustrates an example operating procedure for performing the calibration of the shape of the antenna array. The procedure includes generating ( 2102 ) calibration information of the array in advance with a known shape. In some embodiments, the known shape may be a flat configuration of the antenna array. This calibration produces phase offsets inherent to the array. The procedure further includes using ( 2104 ) the calibration information to generate a transfer function, H meas . The procedure to generate this calibration information is previously discussed in detail. The procedure further includes calculating ( 2106 ) an ideal shape of the array based on a desired functionality and the transfer function, H meas . The shape of the array may produce different functionality for the array which may be dictated by the transfer function. The procedure further includes implementing ( 2108 ) the ideal shape of the array to provide the desired functionality. 
     DOCTRINE OF EQUIVALENTS 
     While the above description contains many specific embodiments of the invention, these should not be construed as limitations on the scope of the invention, but rather as an example of one embodiment thereof. It is therefore to be understood that the present invention may be practiced in ways other than specifically described, without departing from the scope and spirit of the present invention. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their equivalents. Littrow angle, mechanical baffle and beam dump can be placed to eliminate the spectral overlap.