Patent Document

GOVERNMENT INTEREST 
       [0001]    The embodiments herein may be manufactured, used, and/or licensed by or for the United States Government without the payment of royalties thereon. 
     
    
     BACKGROUND 
       [0002]    1. Technical Field 
         [0003]    The embodiments herein generally relate to law enforcement technologies and security, and more particularly to techniques for determining the origin and direction of travel of supersonic projectiles. 
         [0004]    2. Description of the Related Art 
         [0005]    Systems and methods are known that can determine the general location of the sniper and trajectory of supersonic projectiles, such as bullets and artillery shells, by measuring parameters associated with the shockwave generated by a projectile. Current systems, however, generally require a distributed array of acoustic sensors to detect the arrival times, amplitudes and frequency characteristics of a projectile&#39;s shockwave and the muzzle blast from a firearm. In addition, such systems typically require real-time operating systems (e.g., UNIX, LINUX, etc.) and generally require an elaborate time synchronization mechanism so that all the deployed sensors have a common time reference to determine the exact time of arrival for both shockwave and the muzzle blast for supersonic gun fire with common reference time frame. 
         [0006]    Conventional systems typically employ an antenna with a plurality of acoustic sensors, which can be relatively closely spaced (e.g., 1 meter apart) or widely dispersed (e.g., mounted on a vehicle or carried by soldiers on a battlefield), with each sensor measuring shockwave pressure omni-directionally at its respective location. One exemplary antenna may include, for example, a total of seven omni-directional microphones, with six microphones distributed over the surface of a sphere (approx. diameter 1 m) and the seventh microphone located in the center of the sphere. Other exemplary antennae employ four microphones arranged in a tetrahedron formation that are amenable to carry on by a person. In general, the microphones in existing systems are arranged in an array configuration and their output is collected by a common data collection unit with a common time reference. Additionally, existing array based sniper localization systems require the orientation of the array with respect to true north to be determined in order to estimate the location of the sniper and the trajectory of the bullet. 
         [0007]    Such array based sensors systems, however, are less suitable for man-wearable systems. In addition, when the sensors are distributed, time synchronization among the sensors is critical. If an operating system that is not a real-time operating system is employed (e.g., Windows® CE operating system), the synchronization among the sensors is not guaranteed. As a result, the time of arrival estimates of different events (e.g., shockwave and the muzzle blast from a firearm) could be off by 1-2 seconds making the localization of the firearm impossible. 
       SUMMARY 
       [0008]    In view of the foregoing, an embodiment herein provides a system comprising several acoustic sensors distributed over an area, for determining the trajectory of a supersonic projectile and firing position of the trajectory. Each acoustic sensor comprising: an internal clock; an acoustic sensor positioned in a known location and encountering pressure waves generated by the supersonic projectile passing proximate to the acoustic sensor, the acoustic sensor comprising a transducer collecting pressure wave information and generating pressure wave information signals in response to the pressure waves, the pressure wave information signals having a time series information signal calibrated to the internal clock; a first processor processing the time series information signal to provide parameter information for determining the trajectory of the supersonic projectile; the first processor comprising: a shock threshold detector receiving the time series information and recording arrival times of shock wave components of the pressure waves; a blast threshold detector receiving the time series information and recording arrival times and amplitude information of potential blast wave components of the pressure waves; a discrimination processor discriminating the potential blast wave components to classify each of the potential blast wave components as a blast wave, a shock wave, or neither, and storing arrival times of each of the potential blast wave components classified as the blast wave; and a localization estimation processor calculating an estimated trajectory of the projectile based on the blast wave and the shock wave arrival times and calculating an estimated firing position of the projectile. 
         [0009]    In such a system, each acoustic sensor would determine the time difference of arrival (TDOA) between the muzzle blast and shockwave time of arrivals. These measured TDOAs from several acoustic sensors are used to estimate the location of the sniper and the trajectory of the bullet. In such a system, the localization estimation processor may calculate an initial miss-angle (gamma γ), the angle between the trajectory and the line joining the sensor and the firing position, and may estimate and calculate the estimated trajectory of the projectile and the estimated firing position of the projectile from the initial gamma estimate. Additionally, the initial gamma estimate may equal 0.01. In addition, the localization estimation processor may estimate a minimum and a maximum distance of the estimated firing position of the projectile. Furthermore, the maximum distance of the estimated firing position of the projectile may be set to a maximum detection range of the acoustic sensor. Moreover, the maximum distance of the estimated firing position of the projectile may be set to 500 meters. 
         [0010]    Furthermore, the localization estimation processor may calculate the trajectory vector Ũ, further it may calculate {hacek over (γ)} k  using Ũ, further it may calculate the firing position {tilde over (Z)} using {hacek over (γ)} k  and TDOAs, which is further used to estimate the miss-angle {circumflex over (γ)} which is further utilized to compute Ũ in a continuous loop until the 
         [0000]    
       
         
           
             
               g 
                
               
                 ( 
                 num 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 N 
               
                
               
                  
                 
                   
                     
                       γ 
                       ⋒ 
                     
                     k 
                   
                   - 
                   
                     
                       γ 
                       ⋓ 
                     
                     k 
                   
                 
                  
               
             
           
         
       
     
         [0000]    is minimum. Preferably, if the {tilde over (Z)} corresponds to the actual firing position {hacek over (γ)} k  and {circumflex over (γ)} should be equal. Moreover, the localization estimation processor may output the estimated firing position of the projectile based on the initial gamma estimate when the localization estimation processor determines the calculated g(num) is less than the initial gamma estimate, otherwise the localization estimation processor may calculate the estimated firing position of the projectile            
         [0011]    Another embodiment herein also provides a method for estimating a firing position of a supersonic projectile, the projectile producing a pressure wave, the pressure wave comprising at least a shock wave and a blast wave, the method comprising: positioning a single acoustic sensor in a known location; generating a pressure wave information signal at the acoustic sensor in response to the pressure wave; filtering unwanted harmonics in the pressure wave information signal; converting the filtered pressure information signal into a time series signal; recording shock wave arrival times of shock wave components of the pressure wave; recording blast wave arrival times of blast wave components of the pressure wave; determining a shock wave arrival time of the shock wave components and a blast wave arrival time of the blast wave components; and calculating an estimated firing position of the projectile based on the shock wave arrival time and the blast wave arrival times. 
         [0012]    According to such a method, calculating an estimated firing position of the projectile may comprise: calculating an initial gamma estimate; and calculating the estimated firing position of the projectile from initial gamma estimate. In addition, the initial gamma estimate may equal 0.01. Furthermore, calculating an estimated firing position of the projectile may comprise estimating a minimum and a maximum distance of the estimated firing position of the projectile. Moreover, calculating an estimated firing position of the projectile may comprise calculating 
         [0000]    
       
         
           
             
               g 
                
               
                 ( 
                 num 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 N 
               
                
               
                   
               
                
               
                  
                 
                   
                     
                       γ 
                       ⋒ 
                     
                     k 
                   
                   - 
                   
                     
                       γ 
                       ⋓ 
                     
                     k 
                   
                 
                  
               
             
           
         
       
     
         [0000]    from the estimated firing position of the projectile based on the initial gamma estimate. In addition, calculating an estimated firing position of the projectile may comprise: outputting the estimated firing position of the 
         [0013]    Another embodiment herein also provides a method for estimating a sniper position firing a supersonic projectile, the projectile producing a pressure wave, the pressure wave have at least a shock wave and a blast wave and the pressure wave is detected by an acoustic sensor, the method comprising: setting an initial value for a miss angle, γ 1 ; determining a value for an intercept angle, θ; computing q, where q=1−sin(θ+γ 1 ); computing an initial estimate of the sniper position based on the γ 1  and the q; determining an initial estimate for a minimal distance of the sniper position measured from the acoustic sensor and a maximum distance of the sniper position measured from the acoustic sensor, based on the γ 1  and the q; calculating an adjusted miss angle, γ 2 , based on coordinates bounded by the minimal distance of the sniper position and the maximum distance of the sniper position; and searching an area bounded by the minimal distance of the sniper position and the maximal distance of the sniper position for a global a minimum value of γ 2 . 
         [0014]    According to such a method, setting an initial value for miss angle γ may set γ k =γ, for all k, may set γ 1  to 0.01. In addition, computing an initial estimate of the sniper position based on the γ 1  and the q may comprise computing: 
         [0000]    
       
         
           
             
               
                 
                   2 
                    
                   
                     [ 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   1 
                                 
                                 - 
                                 
                                   
                                     S 
                                     _ 
                                   
                                   2 
                                 
                               
                               ) 
                             
                             T 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   1 
                                 
                                 - 
                                 
                                   
                                     S 
                                     _ 
                                   
                                   N 
                                 
                               
                               ) 
                             
                             T 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   2 
                                 
                                 - 
                                 
                                   
                                     S 
                                     _ 
                                   
                                   3 
                                 
                               
                               ) 
                             
                             T 
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 - 
                                 
                                   
                                     S 
                                     _ 
                                   
                                   N 
                                 
                               
                               ) 
                             
                             T 
                           
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   Z 
                   ] 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 2 
                               
                               
                                 q 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 d 
                                 1 
                               
                               
                                 q 
                                 1 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               1 
                             
                              
                           
                           2 
                         
                         - 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               2 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 N 
                               
                               
                                 q 
                                 N 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 d 
                                 1 
                               
                               
                                 q 
                                 1 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               1 
                             
                              
                           
                           2 
                         
                         - 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               N 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 3 
                               
                               
                                 q 
                                 3 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 d 
                                 2 
                               
                               
                                 q 
                                 2 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               2 
                             
                              
                           
                           2 
                         
                         - 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               3 
                             
                              
                           
                           2 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 d 
                                 N 
                               
                               
                                 q 
                                 N 
                               
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 d 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               
                                 q 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                           
                           2 
                         
                         - 
                         
                           
                              
                             
                               
                                 S 
                                 _ 
                               
                               N 
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0000]    where N is the number of acoustic sensors, S k  is a position of the k th  acoustic sensor, Z is an estimate of the sniper position, d k =ν(T k −t k ), ν is the propagation velocity of sound, T k  and t k  are the time of arrival (TOA) of muzzle blast and the shockwave respectively, and q 1 = . . . =q N =q where q=1−sin(θ+γ 1 ). 
         [0015]    Moreover, the computing an initial estimate of the sniper position based on the γ 1  and the q may comprise computing: 
         [0000]    
       
         
           
             
               
                 
                   γ 
                   ⋒ 
                 
                 k 
               
               = 
               
                 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                    
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           d 
                           k 
                         
                         
                            
                           
                             
                               
                                 S 
                                 _ 
                               
                               k 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                            
                         
                       
                     
                     ) 
                   
                 
                 - 
                 θ 
               
             
             , 
             
               
                 ∀ 
                 k 
               
               = 
               
                 { 
                 
                   1 
                   , 
                   2 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   N 
                 
                 } 
               
             
             , 
           
         
       
     
         [0000]    where k is the acoustic sensor,  s k    is a three-dimensional position vector of the k th  sensor S, {tilde over (Z)} is the initial estimate of the sniper position. Furthermore, the computing an initial estimate of the sniper position based on the γ 1  and the q may comprise computing: 
         [0000]    
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   S 
                                   _ 
                                 
                                 1 
                               
                               - 
                               
                                 Z 
                                 ~ 
                               
                             
                             ) 
                           
                           T 
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   S 
                                   _ 
                                 
                                 2 
                               
                               - 
                               
                                 Z 
                                 ~ 
                               
                             
                             ) 
                           
                           T 
                         
                       
                     
                     
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   S 
                                   _ 
                                 
                                 N 
                               
                               - 
                               
                                 Z 
                                 ~ 
                               
                             
                             ) 
                           
                           T 
                         
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     U 
                     ~ 
                   
                   ] 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                            
                           
                             
                               
                                 S 
                                 _ 
                               
                               1 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                            
                         
                          
                         cos 
                          
                         
                           
                             γ 
                             ⋒ 
                           
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                            
                           
                             
                               
                                 S 
                                 _ 
                               
                               2 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                            
                         
                          
                         cos 
                          
                         
                           
                             γ 
                             ⋒ 
                           
                           2 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         
                            
                           
                             
                               
                                 S 
                                 _ 
                               
                               N 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                            
                         
                          
                         cos 
                          
                         
                           
                             γ 
                             ⋒ 
                           
                           N 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0000]    {tilde over (Z)} is the initial estimate of the sniper position, d is an estimated distance, Ũ is an estimated trajectory of the projectile, and {circumflex over (γ)} k  is an estimate of γ 1 . Additionally, the computing an initial estimate of the sniper position based on the γ 1  and the q may comprise computing: 
         [0000]    
       
         
           
             
               
                 
                   γ 
                   ⋓ 
                 
                 k 
               
               = 
               
                 
                   cos 
                   
                     - 
                     1 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 S 
                                 _ 
                               
                               k 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                           ) 
                         
                         T 
                       
                        
                       
                         U 
                         ~ 
                       
                     
                     
                        
                       
                         
                           
                             S 
                             _ 
                           
                           k 
                         
                         - 
                         
                           Z 
                           ~ 
                         
                       
                        
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
         [0000]    {hacek over (γ)} k  is an estimate of γ 1  that includes Ũ. 
         [0016]    These and other aspects of the embodiments herein will be better appreciated and understood when considered in conjunction with the following description and the accompanying drawings. It should be understood, however, that the following descriptions, while indicating preferred embodiments and numerous specific details thereof, are given by way of illustration and not of limitation. Many changes and modifications may be made within the scope of the embodiments herein without departing from the spirit thereof, and the embodiments herein include all such modifications. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0017]    The embodiments herein will be better understood from the following detailed description with reference to the drawings, in which: 
           [0018]      FIG. 1  illustrates a schematic diagram of a sniper localization apparatus according to an embodiment herein; 
           [0019]      FIG. 2  illustrates a schematic diagram of locating a sniper&#39;s position, with a constant projectile speed, according to an embodiment herein; 
           [0020]      FIG. 3  illustrates a schematic diagram of locating a sniper&#39;s position, with a variable projectile speed, according to an embodiment herein; 
           [0021]      FIG. 4  is a flow diagram illustrating a preferred method according to an embodiment herein; 
           [0022]      FIG. 5  is a flow diagram illustrating a method of finding a minimal distance of a sniper according to an embodiment herein; and 
           [0023]      FIG. 6  illustrates a schematic diagram of a computer architecture used in accordance with the embodiments herein. 
       
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0024]    The embodiments herein and the various features and advantageous details thereof are explained more fully with reference to the non-limiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. Descriptions of well-known components and processing techniques are omitted so as to not unnecessarily obscure the embodiments herein. The examples used herein are intended merely to facilitate an understanding of ways in which the embodiments herein may be practiced and to further enable those of skill in the art to practice the embodiments herein. Accordingly, the examples should not be construed as limiting the scope of the embodiments herein. 
         [0025]    The embodiments herein provide a sniper localization system using distributed single microphone sensors or handheld computing devices (e.g., a personal digital assistant or “PDA”) using only time differences of arrival (“TDOA”) between the shockwave and muzzle blast at each sensor. In addition, a sniper localization method using only the TDOA for determination of the sniper location and does not require the time synchronization across multiple sensors is described herein. Referring now to the drawings, and more particularly to  FIGS. 1 through 6 , where similar reference characters denote corresponding features consistently throughout the figures, there are shown preferred embodiments. 
         [0026]      FIG. 1  illustrates a schematic diagram of a sniper localization apparatus  1  according to an embodiment. Sniper localization apparatus  1  is a device, preferably embodied as a handheld device; that comprises at least one processor or central processing unit (CPU)  10 . The CPUs  10  are interconnected via system bus  12  to various devices such as a random access memory (RAM)  14 , internal clock  16  and an input/output (I/O) adapter  18 . Sniper localization apparatus  1  can read inventive instructions (as described further below) on RAM  14  and follow these instructions to execute the methodology of the embodiments herein. Internal clock  16  of sniper localization apparatus  1  is sufficiently stable to determine the TDOA between the shockwave and the muzzle blast of a firearm accurately. As described in further detail below, the time of arrival information for the shockwave can be used to determine the projectile&#39;s trajectory: azimuth, elevation, and intercept with an arbitrary plane in the system coordinate frame. With additional information from the muzzle blast, an accurate location of the origin of the projectile and a line of bearing to the origin of the projectile can be determined. Even when the muzzle blast is masked, shadowed, silenced or otherwise distorted, at least the bullet trajectory can be estimated from the shockwave alone. 
         [0027]    I/O adapter  18  connects non-volatile storage devices, such as magnetic disk units  20  and solid state storage units  22 , to system bus  12 . I/O adapter  18  also connects an acoustic sensor  24  to system bus  12 . Acoustic sensor  24  is preferably configured as a single microphone sensor, which is capable of recording the acoustic signals due to firing a firearm. In addition, acoustic sensor  24  detects acoustic signals and detects (in combination with internal clock  16 ) the time of arrival of muzzle blast and the shockwave and hence the TDOA between the two. 
         [0028]    Sniper localization apparatus  1  may further include a communication adapter  26  that optionally connects bus  12  to a wireless communication network  26   a . Additionally, sniper localization apparatus  1  may also include a user interface and display adapter  28  that connects a touch screen device  28   a , or other user interface devices, to the bus  12  to display output and gather user input. 
         [0029]      FIG. 2 , with reference to  FIG. 1 , illustrates a schematic diagram of the geometry of a bullet trajectory and the shockwave cone. In  FIG. 2 , sniper Z is located a position  30  and vector  32  (or U in the equations below) is the unit vector in the direction of a bullet (not shown) shot from position  30 . As the bullet travels at supersonic speed, the shockwave created generates a shockwave cone  34  with angle  36  θ, where sin θ=1/m, m is the Mach number. The shockwave propagates perpendicular to the surface of shockwave cone  34  and reaches position  38  (or S k  in the equations below) of acoustic sensor  24 . The point where the shockwave radiates towards the sensor is denoted by position  40  (or A k  in the equations below). By the time the shockwave reaches position  38 , the bullet travels from position  40  to position  42  (or C k  in the equations below) and the miss distance is the distance between the sensor S k  and the projection of S k  on to the trajectory denoted as position  48  (or B k  in the equations below). The distance h k  is defined as h k =∥S k −B k ∥, where ∥B∥ denotes the norm of the vector B. In addition, angle  46  γ k  is the angle between the trajectory of the 
         [0030]    In the equations that follow, T k  is the time of arrival of muzzle blast and t k  is the shockwave, where T k  and t k  are described by Equation (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       k 
                     
                     = 
                     
                       
                          
                         
                           
                             S 
                             k 
                           
                           - 
                           Z 
                         
                          
                       
                       v 
                     
                   
                   ; 
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               A 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         mv 
                       
                       + 
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             
                               A 
                               k 
                             
                           
                            
                         
                         v 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ν denotes the propagation velocity of the sound. Noting that in the time the shockwave propagates from position  40  to position  38  as shown in  FIG. 2 , the bullet travels from position  40  to position  42  and that implies that the bullet travels from position  30  to position  42  during the time period t k . Therefore, t k  can be re-written as Equation (2): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                           
                              
                             
                               
                                 A 
                                 k 
                               
                               - 
                               Z 
                             
                              
                           
                           mv 
                         
                         + 
                         
                           
                              
                             
                               
                                 S 
                                 k 
                               
                               - 
                               
                                 A 
                                 k 
                               
                             
                              
                           
                           v 
                         
                       
                       = 
                       
                         
                            
                           
                             
                               C 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         mv 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               B 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         mv 
                       
                       + 
                       
                         
                            
                           
                             
                               C 
                               k 
                             
                             - 
                             
                               B 
                               k 
                             
                           
                            
                         
                         mv 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         1 
                         mv 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     S 
                                     k 
                                   
                                   - 
                                   Z 
                                 
                                 ) 
                               
                               T 
                             
                              
                             U 
                           
                           + 
                           
                             
                               h 
                               k 
                             
                              
                             cot 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where T is the transpose and noting that ∥B k −Z∥ is the projection of the vector S k −Z on to the trajectory of the bullet with vector  32 . Trigonometric relations provide that ∥B k −Z∥=∥S k −Z∥cos γ k  and h k =∥S k −Z∥ sin γ k  and sin θ=1/m. Applying these trigonometric relations to the equations above produces Equation (3): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         ( 
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             θ 
                              
                             
                                 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               γ 
                               k 
                             
                           
                           + 
                           
                             cos 
                              
                             
                                 
                             
                              
                             θ 
                              
                             
                                 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               γ 
                               k 
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             θ 
                             + 
                             
                               γ 
                               k 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
       and Equation (4): 
       [0031]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                           ) 
                         
                         T 
                       
                        
                       U 
                     
                     
                        
                       
                         
                           S 
                           k 
                         
                         - 
                         Z 
                       
                        
                     
                   
                   = 
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       
                         γ 
                         k 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Equations (1) and (3) shown above, the TDOA is shown below as Equation (5): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         k 
                       
                       - 
                       
                         t 
                         k 
                       
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             sin 
                              
                             
                               ( 
                               
                                 θ 
                                 + 
                                 
                                   γ 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                        
                       
                         
                           S 
                           k 
                         
                         - 
                         Z 
                       
                        
                     
                     = 
                     
                       
                         d 
                         k 
                       
                       
                         q 
                         k 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d k =ν(T k −t k ) and q k =[1−sin(θ+γ k )]. As discussed in further detail below, solving Equations (4) and (5) above describes the sniper location at position  30  and the trajectory of the bullet. In addition, ν is known, the TDOA (T K −t k ) is measured by each sensor S k  and the cone angle θ is also known. If γ k  is known for all k, then Equations (4) and (5) can be solved exactly. Since the γ k  is dependent on the location of the sniper Z, an iterative solution is derived that involves estimation of γ k , Z and U. 
         [0032]      FIG. 3 , with reference to  FIGS. 1 and 2 , illustrates a schematic diagram of the geometry of a bullet trajectory and the shockwave cone, where the bullet (not shown) is subjected to resistance. In  FIG. 3 , angle  46  varies with respect to distance. Angle  50  is the angle that shockwave cone  54  propagates to position  40 , but angle  50  is different than angle  52  when shockwave cone  54  reaches acoustic sensor  24 . The difference in angles is attributable to a decrease in speed of the bullet (e.g., due to wind resistance). In spite of the decrease in speed, the bullet may be assumed to travel at an average of speed of Mach m 1  from position  30  to position  48  (or B k  in the equations below) and sin θ 1 =1/m 1 . Therefore, a bullet travels from position  48  to position  42  at an average speed of Mach m 2    
         [0033]    θ=and a corresponding shockwave cone  56 . 
         [0034]    From  FIG. 3 , the time of arrival of shockwave cone  54  is given by Equation (6): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               B 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         
                           
                             m 
                             1 
                           
                            
                           v 
                         
                       
                       + 
                       
                         
                            
                           
                             
                               C 
                               k 
                             
                             - 
                             
                               B 
                               k 
                             
                           
                            
                         
                         
                           
                             m 
                             2 
                           
                            
                           v 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         ( 
                         
                           
                             
                               cos 
                                
                               
                                   
                               
                                
                               
                                 γ 
                                 k 
                               
                             
                             
                               m 
                               1 
                             
                           
                           + 
                           
                             
                               
                                 sin 
                                  
                                 
                                     
                                 
                                  
                                 
                                   γ 
                                   k 
                                 
                               
                               
                                 m 
                                 2 
                               
                             
                              
                             cot 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         [ 
                         
                           
                             sin 
                              
                             
                                 
                             
                              
                             
                               θ 
                               1 
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               γ 
                               k 
                             
                           
                           + 
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               θ 
                               2 
                             
                              
                             sin 
                              
                             
                                 
                             
                              
                             
                               γ 
                               k 
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       t 
                       k 
                     
                     = 
                     
                       
                         
                           β 
                            
                           
                              
                             
                               
                                 S 
                                 k 
                               
                               - 
                               Z 
                             
                              
                           
                         
                         v 
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             α 
                             + 
                             
                               γ 
                               k 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             β 
             = 
             
               
                 
                   
                     
                       
                         sin 
                         2 
                       
                        
                       
                         θ 
                         1 
                       
                     
                     + 
                     
                       
                         cos 
                         2 
                       
                        
                       
                         θ 
                         2 
                       
                     
                   
                 
                  
                 
                     
                 
                  
                 and 
                  
                 
                     
                 
                  
                 α 
               
               = 
               
                 
                   
                     sin 
                     
                       - 
                       1 
                     
                   
                    
                   
                     ( 
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         
                           θ 
                           1 
                         
                         / 
                         β 
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
         [0000]    Then, the TDOA is described by Equation (7): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         k 
                       
                       - 
                       
                         t 
                         k 
                       
                     
                     = 
                     
                       
                         
                            
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                            
                         
                         v 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             β 
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   α 
                                   + 
                                   
                                     γ 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                        
                       
                         
                           S 
                           k 
                         
                         - 
                         Z 
                       
                        
                     
                     = 
                     
                       
                         d 
                         k 
                       
                       
                         w 
                         k 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where w k =1−β sin (α+γ k ) and d k =ν(T k −t k ). Similar to what was described above for  FIG. 2 , the unit vector in the direction of bullet trajectory is related to γ k  by Equation (8): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               S 
                               k 
                             
                             - 
                             Z 
                           
                           ) 
                         
                         T 
                       
                        
                       U 
                     
                     
                        
                       
                         
                           S 
                           k 
                         
                         - 
                         Z 
                       
                        
                     
                   
                   = 
                   
                     cos 
                      
                     
                         
                     
                      
                     
                       
                         γ 
                         k 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0035]    As previously mentioned, solving Equations (4) and (5), shown above in relation to  FIG. 2 , identifies sniper position  30  (shown in  FIGS. 2 and 3 ) and the trajectory of a bullet (not shown) shot from position  30 . Similarly, solving Equations (7) and (8) identifies the sniper position  30  shown in  FIG. 3  for two velocity models. While in the case of Equations (4) and (5) one would require to know one Mach number for the bullet, in the case of Equations (7) and (8) one would require to know two Mach numbers m 1  and m 2 . 
         [0036]    In the solution to Equations (4) and (5) discussed below,  Z =[x z , y z , z z ] T  and is a three-dimensional vector depicting position  30 . In addition, the bullet is traveling along the trajectory that is defined by the unit vector 
         [0000]    Ū=[sin φ cos φ sin φ sin φ cos φ] T  where φand φ represent the elevation and azimuth angles of the trajectory and T denotes the transpose. In addition, ∥Ū∥ 2 =Ū T Ū=1. Equation (8), shown above in relation to  FIG. 3 , requires a similar solution to what is described below for  FIG. 2 . Therefore, Equation (8) will not be discussed further because those skilled in the art would capable of applying the discussion below to the Equation (8) without undue experimentation. 
         [0037]    As discussed in further detail below, vector Z is describe by using Equation (5) and substituting in the estimated value of Z in Equation (5) to estimate the angles γ k  for all k. Squaring both sides of Equation (1) produces Equation (9): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 S 
                                 _ 
                               
                               k 
                             
                             - 
                             
                               Z 
                               _ 
                             
                           
                           ) 
                         
                         T 
                       
                        
                       
                         ( 
                         
                           
                             
                               S 
                               _ 
                             
                             k 
                           
                           - 
                           
                             Z 
                             _ 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             d 
                             k 
                           
                           
                             q 
                             k 
                           
                         
                         ) 
                       
                       2 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                            
                           
                             
                               S 
                               _ 
                             
                             k 
                           
                            
                         
                         2 
                       
                       - 
                       
                         2 
                          
                         
                             
                         
                          
                         
                           
                             S 
                             _ 
                           
                           k 
                           T 
                         
                          
                         
                           Z 
                           _ 
                         
                       
                       + 
                       
                         
                            
                           
                             Z 
                             _ 
                           
                            
                         
                         2 
                       
                     
                     = 
                     
                       
                         
                           ( 
                           
                             
                               d 
                               k 
                             
                             
                               q 
                               k 
                             
                           
                           ) 
                         
                         2 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0038]    Setting the value of k={1, 2, . . . , N} in Equation (9) and taking the difference between pairs, produces the linear equations shown below as Equation (10): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       2 
                        
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       1 
                                     
                                     - 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       2 
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       1 
                                     
                                     - 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       N 
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       2 
                                     
                                     - 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       3 
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         S 
                                         _ 
                                       
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                     - 
                                     
                                       
                                         S 
                                         _ 
                                       
                                       N 
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       [ 
                       Z 
                       ] 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       q 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     
                                       q 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     1 
                                   
                                    
                                 
                                 2 
                               
                               - 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     2 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       N 
                                     
                                     
                                       q 
                                       N 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     
                                       q 
                                       1 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     1 
                                   
                                    
                                 
                                 2 
                               
                               - 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     N 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       3 
                                     
                                     
                                       q 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       2 
                                     
                                     
                                       q 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     2 
                                   
                                    
                                 
                                 2 
                               
                               - 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     3 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       N 
                                     
                                     
                                       q 
                                       N 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               - 
                               
                                 
                                   ( 
                                   
                                     
                                       d 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                     
                                       q 
                                       
                                         N 
                                         - 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                 
                                 2 
                               
                               - 
                               
                                 
                                    
                                   
                                     
                                       S 
                                       _ 
                                     
                                     N 
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0039]    In Equation (10),  S   k  is position  38  of acoustic sensor  24  and d k  is the product of propagation velocity ν and the difference in the time of muzzle blast and shockwave (T k −t k ) at acoustic sensor  24 . Solving the linear equations shown above in Equation (10) requires values of q k  for all k. However, from Equation (6) above, q k =1−sin(θ+γ k ), where θ=sin −1 (1/m), where the Mach number (m) is assumed to be known. In general, γ k  is small when the distance between the sniper location  30  and the sensor  24  is greater than 200 m and the distance between the bullet&#39;s path and the sensor is less than 10 m. For example, to calculate an initial estimate of the sniper location at position  30 , based on the equations above, γ k =0.01 is arbitrarily set for all k. In Equation (11) below, {tilde over (Z)} is the initial estimate of the sniper location at position  30  and {tilde over (Z)} can be used in Equation (5) to estimate the approximate values of γ k , which is {circumflex over (γ)} k  and shown as Equation (11): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         γ 
                         ⋒ 
                       
                       k 
                     
                     = 
                     
                       
                         
                           sin 
                           
                             - 
                             1 
                           
                         
                         ( 
                         
                           1 
                           - 
                           
                             
                               d 
                               k 
                             
                             
                                
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   k 
                                 
                                 - 
                                 
                                   Z 
                                   ~ 
                                 
                               
                                
                             
                           
                         
                         ) 
                       
                       - 
                       θ 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       ∀ 
                       k 
                     
                     = 
                     
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           N 
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    In Equation (3) above, {circumflex over (γ)} k  is used to solve for approximate trajectory Ũ using the system of linear equations given by Equation (4) and re-written in a matrix form as shown below as Equation (12): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       S 
                                       _ 
                                     
                                     1 
                                   
                                   - 
                                   
                                     Z 
                                     ~ 
                                   
                                 
                                 ) 
                               
                               T 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       S 
                                       _ 
                                     
                                     2 
                                   
                                   - 
                                   
                                     Z 
                                     ~ 
                                   
                                 
                                 ) 
                               
                               T 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       S 
                                       _ 
                                     
                                     N 
                                   
                                   - 
                                   
                                     Z 
                                     ~ 
                                   
                                 
                                 ) 
                               
                               T 
                             
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         U 
                         ~ 
                       
                       ] 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                                
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   1 
                                 
                                 - 
                                 
                                   Z 
                                   ~ 
                                 
                               
                                
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               
                                 γ 
                                 ⋒ 
                               
                               1 
                             
                           
                         
                       
                       
                         
                           
                             
                                
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   2 
                                 
                                 - 
                                 
                                   Z 
                                   ~ 
                                 
                               
                                
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               
                                 γ 
                                 ⋒ 
                               
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                                
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   N 
                                 
                                 - 
                                 
                                   Z 
                                   ~ 
                                 
                               
                                
                             
                              
                             cos 
                              
                             
                                 
                             
                              
                             
                               
                                 γ 
                                 ⋒ 
                               
                               N 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which can be solved for Ũ using regression. Substituting {tilde over (Z)} and Ũ in Equation (4) above to solve for γ k  results in Equation (13): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         γ 
                         ⋓ 
                       
                       k 
                     
                     = 
                     
                       
                         cos 
                         
                           - 
                           1 
                         
                       
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     S 
                                     _ 
                                   
                                   k 
                                 
                                 - 
                                 
                                   Z 
                                   ~ 
                                 
                               
                               ) 
                             
                             T 
                           
                            
                           
                             U 
                             ~ 
                           
                         
                         
                            
                           
                             
                               
                                 S 
                                 _ 
                               
                               k 
                             
                             - 
                             
                               Z 
                               ~ 
                             
                           
                            
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       ∀ 
                       k 
                     
                     = 
                     
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           N 
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0040]    Ideally, if the values of {tilde over (Z)}={tilde over (Z)}, and Ũ=Ũ the values of {circumflex over (γ)} k  and {hacek over (γ)} k  calculated using the Equations (11) and (13) respectively, are equal resulting in determining the sniper location and the bullet&#39;s trajectory. 
         [0041]      FIGS. 4 and 5 , with reference to  FIGS. 1 through 3 , are flow diagrams illustrating a method for estimating a sniper&#39;s location based on acoustic information gathered from acoustic sensors (e.g., acoustic sensor  24 ) and accurate timing of when the acoustic information is gathered from an accurate timing device (e.g., internal clock  16 ). Step  60 , of the method shown in  FIG. 4 , initially sets γ k =0.01 for all k and computes q k =1−sin(θ+γ k ). Step  62  estimates an initial sniper location {tilde over (Z)} using Equations (10)-(13) as shown in  FIG. 5 , which is further elaborated below. Step  64  determines the initial minimal and maximum distances values from a sniper located at position  30  (shown in  FIGS. 2 and 3 ) to the acoustic sensor  24  location at position  38  (shown in  FIGS. 2 and 3 ). The minimum distance is determined by the fact that γ k ≧0, ∀k. Setting γ k =0 in Equation (5) 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     min 
                   
                   = 
                   
                     
                       max 
                       
                         
                           k 
                           = 
                           1 
                         
                         , 
                         2 
                         , 
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                         , 
                         N 
                       
                     
                      
                     
                       
                         d 
                         k 
                       
                       
                         1 
                         - 
                         
                           sin 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the D max  is determined the muzzle blast detection capability of the acoustic sensor  24 , which is approximately 500 m. 
         [0042]    Step  66  estimates the rough location of the sniper firing point and the approximate trajectory of the projectile using coarse grid search. When determining the coarse location of the sniper firing point, the following excerpt of pseudo-code may be used, as a non-limiting example: 
         [0000]                                            Let {tilde over (Z)} = [X z  Y z  Z z ] T  = [0 0 0] T  then               num = 0;               for i = X min  :X inc  :X max                  for j = Y min  :Y inc  :Y max                   for k = Z min  :Z inc  :Z max                    Z_new = [X z  + i, Y z  + j, Z z  + k] T                    
       Use                 Z_new                 in                 equation                   (   11   )                   to                 estimate                 the                     γ   ⋒     k                   for                 all                   k   .         
                  
       Use                 Z_new                 and                     γ   ⋒     k                   in                 equation                   (   12   )                   to                 estimate                 the       
                  trajectory of the bullet Ũ .                  
       Use                 Z_new                 and                   U   ~                   in                 equation                   (   13   )                   to                 estimate                 the                       γ   ⋓     k     .         
                  num = num + 1;                  
         Compute                   g        (   num   )         =       ∑     k   =   1     N                   γ   ⋒     k     -       γ   ⋓     k                  
                 end                end               end                    
where X min , X max , Y min , Y max , Z min , Z max  are determined by the D min  and D max . The X inc , Y inc , Z inc  are arbitrarily set to 10 m. The pseudo-code shown above is intended only as an example of how the initial coarse location of the sniper firing point and the trajectory may be computed and is not shown in any particular computer programming language or computer interpretable format. While not representing a specific computer programming language, shown in the pseudo-code above is an example of how it may be easily converted to any particular computer programming language by those skilled in the art. As discussed in greater detail below, while executing steps  66  and  68 , the method shown in  FIG. 4  can use the method shown in  FIG. 5  with initial estimate of {circumflex over (γ)} k  using Z new  in Equation (11). This invocation of algorithm presented in  FIG. 5  allows the search to occur around the Z new  as opposed to finding the fitness just at Z new .
 
         [0043]    In Step  68 , the method of  FIG. 4  finds the final solution, that is, the sniper firing position and the trajectory. This step is similar to the Step  66  with search space set around the solution obtained in Step  66 . If the solution in Step  66  is denoted by {tilde over (Z)}=[X z  Y z  Z z ] T , and set X min =X z −5; X max =X z +5; Y min =Y z −5; Y max =Y z +5, and Z min =Z z −5; Z max =Z z +5 in the pseudo code provided above to estimate the final position of the sniper firing location and the trajectory. 
         [0044]      FIG. 5 , with reference to  FIGS. 1 through 4 , is a schematic diagram of step  62  shown in  FIG. 4 , as related to the Equations (1) through (13) described above. In step  70  of the method shown in  FIG. 5 , an initial value for {hacek over (γ)} k  is set (e.g., {hacek over (γ)} k =0.01). Step  72  uses the initial value of γ k , set in step  70 , to compute {tilde over (Z)} as described in Equation (10) shown above. Step  74  uses the value of {tilde over (Z)} computed in step  72  to compute a value for {circumflex over (γ)} k  using Equation (11) shown above. Step  76  uses the value of {circumflex over (γ)} k  computed in step  74  to compute the value of Ũ using Equation (12) shown above. Step  78  uses the value of Ũ computed in step  76  to compute the value of {hacek over (γ)} k  using Equation (13) shown above. In step  80 , the method shown in  FIG. 5  determines whether the value of {hacek over (γ)} k  computed in step  78  is approximately equal to the value of {circumflex over (γ)} k  computed in step  74 . In Step  80 , the value 
         [0000]    
       
         
           
             
               g 
                
               
                 ( 
                 num 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 N 
               
                
               
                   
               
                
               
                  
                 
                   
                     
                       γ 
                       ⋒ 
                     
                     k 
                   
                   - 
                   
                     
                       γ 
                       ⋓ 
                     
                     k 
                   
                 
                  
               
             
           
         
       
     
         [0000]    is computed for each iteration number ‘num’ and the iterations terminate when g(num)&gt;g(num−1). Step  82  is used to find the global minima and it may involve searching several values of Z that give minimal difference of as 
         [0000]    
       
         
           
             
               ∑ 
               
                 k 
                 = 
                 1 
               
               N 
             
              
             
                 
             
              
             
                
               
                 
                   
                     γ 
                     ⋒ 
                   
                   k 
                 
                 - 
                 
                   
                     γ 
                     ⋓ 
                   
                   k 
                 
               
                
             
           
         
       
     
         [0000]    in steps  66  and  68 . 
         [0045]    The techniques provided by the embodiments herein may be implemented on an integrated circuit chip (not shown). The chip design is created in a graphical computer programming language, and stored in a computer storage medium (such as a disk, tape, physical hard drive, or virtual hard drive such as in a storage access network). If the designer does not fabricate chips or the photolithographic masks used to fabricate chips, the designer transmits the resulting design by physical means (e.g., by providing a copy of the storage medium storing the design) or electronically (e.g., through the Internet) to such entities, directly or indirectly. The stored design is then converted into the appropriate format (e.g., GDSII) for the fabrication of photolithographic masks, which typically include multiple copies of the chip design in question that are to be formed on a wafer. The photolithographic masks are utilized to define areas of the wafer (and/or the layers thereon) to be etched or otherwise processed. 
         [0046]    The resulting integrated circuit chips can be distributed by the fabricator in            
         [0047]    The embodiments herein may comprise hardware and software embodiments. The embodiments that are implemented in software include but are not limited to, firmware, resident software, microcode, etc. Furthermore, the embodiments herein can take the form of a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. For the purposes of this description, a computer-usable or computer readable medium can be any apparatus that can comprise, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. 
         [0048]    The medium can be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. Examples of a computer-readable medium include a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk. Current examples of optical disks include compact disk-read only memory (CD-ROM), compact disk-read/write (CD-R/W) and DVD. 
         [0049]    A data processing system suitable for storing and/or executing program code will include at least one processor coupled directly or indirectly to memory elements through a system bus. The memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code in order to reduce the number of times code must be retrieved from bulk storage during execution. 
         [0050]    Input/output (I/O) devices (including but not limited to keyboards, displays, pointing devices, etc.) can be coupled to the system either directly or through intervening I/O controllers. Network adapters may also be coupled to the system to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks. Modems, cable modem and Ethernet cards are just a few of the currently available types of network adapters. 
         [0051]    A more general representation of the hardware environment for practicing the embodiments herein is depicted in  FIG. 6 . This schematic drawing illustrates hardware configuration  100  of an information handling/computer system in accordance with the embodiments herein. Hardware configuration  100  comprises at least one processor or central processing unit (CPU)  110 . The CPUs  110  are interconnected via system bus  112  to various devices such as a random access memory (RAM)  114 , read-only memory (ROM)  116 , and an input/output (I/O) adapter  118 . The I/O adapter  118  can connect to peripheral devices, such as disk units  111  and tape drives  113 , or other program storage devices that are readable by the system. The system can read the inventive instructions on the program storage devices and follow these instructions to execute the methodology of the embodiments herein. The system further includes a user interface adapter  119  that connects a keyboard  115 , mouse  117 , speaker  124 , microphone  122 , and/or other user interface devices such as a touch screen device (not shown) to the bus  112  to gather user input. Additionally, a communication adapter  120  connects the bus  112  to a data processing network  125 , and a display adapter  121  connects the bus  112  to a display device  123  which may be embodied as an output device such as a monitor, printer, or transmitter, for example. 
         [0052]    The foregoing description of the specific embodiments will so fully reveal the general nature of the embodiments herein that others can, by applying current knowledge, readily modify and/or adapt for various applications such specific embodiments without departing from the generic concept, and, therefore, such adaptations and modifications should and are intended to be comprehended within the meaning and range of equivalents of the disclosed embodiments. It is to be understood that the phraseology or terminology employed herein is for the purpose of description and not of limitation. Therefore, while the embodiments herein have been described in terms of preferred embodiments, those skilled in the art will recognize that the embodiments herein can be practiced with modification within the spirit and scope of the appended claims.

Technology Category: 3