Patent Publication Number: US-11047971-B2

Title: Radar system and control method for use in a moving vehicle

Description:
INTRODUCTION 
     The subject disclosure relates to a radar system and a control method for a radar system. 
     One aspect of a radar system is detecting an angle of arrival of a signal reflected from a detection target. A wide antenna aperture allows for detection of the arrival angle with high resolution. However, populating a wide antenna aperture with a large number of antenna elements results in high cost, high complexity, large space requirements, and high power consumption of the radar system. At the same time, populating a wide antenna aperture with a small number of antenna elements results in a high level of ambiguity in the detected angle of arrival. 
     Accordingly, it may be desirable to provide a radar system and control method that provides a wide antenna aperture for high resolution and that reduces ambiguity while reducing a total number of antenna elements to minimize complexity, cost, and power consumption. 
     SUMMARY 
     In one exemplary embodiment, a radar system for use in a vehicle structured to move in a first direction may include a plurality of antenna elements and a controller operably connected to the plurality of antenna elements. The plurality of antenna elements may pe spaced apart in a second direction different from the first direction. The controller may be configured to record signals received by each antenna element at each time instant of a plurality of time instants. The controller may be further configured to calculate a position in the first direction of each antenna element for each time instant based on a velocity hypothesis. The controller may be further configured to calculate a virtual two-dimensional antenna array response based on the signal received by each antenna element at each time instant and the position in the first direction of each antenna element at each time instant. The controller may be further configured to calculate a beamforming spectrum based on the virtual two-dimensional antenna array response The controller may be further configured to identify a peak in the beamforming spectrum to identify an elevation angle from the vehicle to a target relative to the first direction. 
     In another exemplary embodiment of the radar system, the controller may be further configured to calculating a velocity score of the beamforming spectrum. The controller may be further configured to iteratively adjust the velocity hypothesis until an optimal velocity hypothesis is determined. The optimal velocity hypothesis may be a velocity hypothesis for which the velocity score of the beamforming spectrum reaches an optimal velocity score, and an optimal beamforming spectrum may be a beamforming spectrum generated using the optimal velocity hypothesis. The controller may be further configured to identify a peak in the optimal beamforming spectrum to identify the elevation angle. 
     In another exemplary embodiment of the radar system, the velocity score may be given by the equation S E =E−αΣ i |s i | 2 , where s i  is the beamforming spectrum at index i. α may be a normalization factor. N is the number of beamforming angles. E is given by the equation E=−Σ i γ i  log(γ i ), where γ i  is given by the equation 
     
       
         
           
             
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     In another exemplary embodiment of the radar system, the controller and the plurality of antenna elements are provided in the vehicle. The vehicle may an automated driving system operably connected to the radar system, the automated driving system being structured to autonomously control the vehicle. The controller may be configured to transmit the optimal velocity hypothesis to the automated driving system. The automated driving system may be structured to control operation of the vehicle based on the optimal velocity hypothesis. 
     In another exemplary embodiment of the radar system, the controller and the plurality of antenna elements may be provided in the vehicle. The vehicle may include an automated driving system operably connected to the radar system, the automated driving system being structured to autonomously control the vehicle. The controller may be configured to transmit the elevation angle to the automated driving system. The automated driving system may be structured to control operation of the vehicle based on the elevation angle. 
     In another exemplary embodiment of the radar system, a pitch of the antenna elements in the second direction is equal to or larger than 10 times a wavelength of a radar signal transmitted by the radar system. 
     In another exemplary embodiment of the radar system, the second direction is approximately perpendicular to the first direction. 
     In another exemplary embodiment of the radar system, the first direction is approximately parallel to a ground surface and the second direction is approximately parallel to a direction of gravity. 
     In another exemplary embodiment of the radar system, the controller is configured such that the identifying a peak in the beamforming spectrum may include identifying a plurality of peaks in the beamforming spectrum to identify a plurality of elevation angles from the vehicle to a plurality of targets relative to the first direction. 
     In one exemplary embodiment, a vehicle may include an automated driving system and a radar system operably connected to the automated driving system. The automated driving system may be structured to autonomously control the vehicle to move in a first direction. The radar system may include a plurality of antenna elements spaced apart in a second direction different from the first direction and a controller operably connected to the plurality of antenna elements. The controller may be configured to record signals received by each antenna element at each time instant of a plurality of time instants. The controller may be further configured to calculate a position in the first direction of each antenna element for each time instant based on a velocity hypothesis. The controller may be further configured to calculate a virtual two-dimensional antenna array response based on the signal received by each antenna element at each time instant and the position in the first direction of each antenna element at each time instant. The controller may be further configured to calculate a beamforming spectrum based on the virtual two-dimensional antenna array response. The controller may be further configured to identify a peak in the beamforming spectrum to identify an elevation angle from the vehicle to a target relative to the first direction. The controller may be further configured to transmit the elevation angle of the target to the automated driving system. The automated driving system may be structured to control operation of the vehicle based on the elevation angle. 
     In another exemplary embodiment of the vehicle, the controller may be further configured to calculate a velocity score of the beamforming spectrum. The controller may be further configured to iteratively adjust the velocity hypothesis until an optimal velocity hypothesis is determined. The optimal velocity hypothesis may be a velocity hypothesis for which the velocity score of the beamforming spectrum reaches an optimal velocity score, and an optimal beamforming spectrum may be a beamforming spectrum generated using the optimal velocity hypothesis. The controller may be further configured to identify a peak in the optimal beamforming spectrum to identify the elevation angle. 
     In one exemplary embodiment, a control method may be used with a radar system in a vehicle moving in a first direction. The radar system may include a plurality of antenna elements spaced apart in a second direction different from the first direction. The control method may include recording signals received by each antenna element at each time instant of a plurality of time instants. The control method may further include calculating a position in the first direction of each antenna element for each time instant based on a velocity hypothesis. The control method may further include calculating a virtual two-dimensional antenna array response based on the signal received by each antenna element at each time instant and the position in the first direction of each antenna element at each time instant. The control method may further include calculating a beamforming spectrum based on the virtual two-dimensional antenna array response. The control method may further include identifying a peak in the beamforming spectrum to identify an elevation angle from the vehicle to a target relative to the first direction. 
     In another exemplary embodiment of the control method, the control method may include calculating a velocity score of the beamforming spectrum. The control method may further include iteratively adjusting the velocity hypothesis until an optimal velocity hypothesis is determined. The optimal velocity hypothesis may be a velocity hypothesis for which the velocity score of the beamforming spectrum reaches an optimal velocity score, and an optimal beamforming spectrum may be a beamforming spectrum generated using the optimal velocity hypothesis. The control method may further include identifying a peak in the optimal beamforming spectrum to identify the elevation angle of the target relative to the first direction. 
     In another exemplary embodiment of the control method, the velocity score is given by the equation: S E =E−αΣ i |s i | 2 , where s i  is the beamforming spectrum at index i. α is a normalization factor. N is the number of beamforming angles, and E is given by the equation E=−Σ i γ i  log(γ i ) where γi is given by the equation 
     
       
         
           
             
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     In another exemplary embodiment of the control method, the vehicle may include an automated driving system operably connected to the radar system. The automated driving system may be structured to autonomously control the vehicle. The control method may further include transmitting the optimal velocity hypothesis to the automated driving system. The automated driving system may be structured to control operation of the vehicle based on the optimal velocity hypothesis. 
     In another exemplary embodiment of the control method, the vehicle may include an automated driving system operably connected to the radar system. The automated driving system may be structured to autonomously control the vehicle. The control method may further include transmitting the elevation angle to the automated driving system. The automated driving system may be structured to control operation of the vehicle based on the elevation angle. 
     In another exemplary embodiment of the control method, a pitch of the antenna elements in the second direction may be equal to or larger than 10 times a wavelength of a radar signal transmitted by the radar system. 
     In another exemplary embodiment of the control method, the second direction may be approximately perpendicular to the first direction. 
     In another exemplary embodiment of the control method, the first direction may be approximately parallel to a ground surface and the second direction is approximately parallel to a direction of gravity. 
     In another exemplary embodiment of the control method, the identifying a peak in the beamforming spectrum may include identifying a plurality of peaks in the beamforming spectrum to identify a plurality of elevation angles from the vehicle to a plurality of targets relative to the first direction. 
     The above features and advantages, and other features and advantages of the disclosure are readily apparent from the following detailed description when taken in connection with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other features, advantages and details appear, by way of example only, in the following detailed description, the detailed description referring to the drawings in which: 
         FIG. 1  is a schematic diagram of a vehicle according to an exemplary embodiment; 
         FIG. 2  is a schematic diagram of a radar system according to an exemplary embodiment; 
         FIG. 3  is a schematic diagram of a vehicle according to an exemplary embodiment; 
         FIG. 4  is a schematic diagram showing an antenna and a detection target according to an exemplary embodiment; 
         FIG. 5  is a schematic diagram showing antenna elements and a detection target according to an exemplary embodiment; 
         FIG. 6  is a schematic diagram showing antenna elements and a detection target according to an exemplary embodiment; 
         FIG. 7A  is a graph showing beamforming spectra according to an exemplary embodiment; 
         FIG. 7B  is a graph showing beamforming spectra according to an exemplary embodiment; 
         FIG. 7C  is a graph showing beamforming spectra according to an exemplary embodiment; 
         FIG. 8  is a schematic diagram of a two-dimensional array of antenna elements according to an exemplary embodiment; 
         FIG. 9  is a schematic diagram showing a virtual two-dimensional array of antenna elements according to an exemplary embodiment; 
         FIG. 10  is a graph showing a relationship between velocity score and velocity hypothesis according to an exemplary embodiment; 
         FIG. 11  is a comparison of beamforming images during velocity hypothesis correction according to an exemplary embodiment; 
         FIG. 12  is a graph showing a test scenario according to an exemplary embodiment; 
         FIG. 13  is a graph showing a relationship between velocity score and velocity hypothesis in a test scenario according to an exemplary embodiment; 
         FIG. 14  is a flowchart illustrating an exemplary embodiment of a control method for a radar system; and 
         FIG. 15  is a flowchart illustrating an exemplary embodiment of a control method of a vehicle. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is merely exemplary in nature and is not intended to limit the present disclosure, its application or uses. It should be understood that throughout the drawings, corresponding reference numerals indicate like or corresponding parts and features. As used herein, the term module refers to processing circuitry that may include an application specific integrated circuit (ASIC), an electronic circuit, a processor (shared, dedicated, or group) and memory that executes one or more software or firmware programs, a combinational logic circuit, and/or other suitable components that provide the described functionality. 
       FIG. 1  illustrates an exemplary embodiment of a vehicle  10  including an automated driving system  12  and radar system  14 . Automated driving system  12  may be operably connected to radar system  14 , and may also include or be operably connected to additional sensors  16  configured to detect a driving environment. Sensors  16  may include a camera, an additional radar system, a LIDAR system, or any combination of these systems. In response to driving environment information detected by radar system  14  and/or sensors  16 , automated driving system  12  may calculate a vehicle path plan for vehicle  10 . 
     Automated driving system  12  may further include or be operably connected to vehicle controllers  18  configured to control speed, acceleration, braking, steering, or other operations necessary for operating vehicle  10 . Automated driving system  12  may control vehicle controllers  18  to operate vehicle  10  according to the calculated vehicle path plan. It will be understood that vehicle  10  may be a fully autonomous vehicle in which automated driving system  12  controls all aspects of the vehicle operation, or vehicle  10  may be a vehicle in which the driver retains some control and automated driving system  12 , as part of a driver assist system, is configured to assist with a subset of vehicle operations. 
       FIG. 2  illustrates an exemplary embodiment of radar system  14  of  FIG. 1 . Radar system  14  may include an antenna  26  structured to transmit a radar signal and receive reflected radar signals from targets. Antenna  26  may include a plurality of antenna elements  27  and may further include a controller  28  operably connected to antenna  26  and configured to operate on signals received by antenna  26 . 
       FIG. 3  illustrates an exemplary embodiment of how antenna elements  27  may be arranged in vehicle  10 . For example, if normal vehicle motion is in a first direction  30 , antenna elements  27  may be spaced apart in vehicle  10  in a second direction  32 . In the embodiment shown in  FIG. 3 , first direction  30  is approximately parallel to a ground surface  40 , and second direction  32  is approximately perpendicular to first direction  30 . However, it will be understood that this convention is merely illustrative and that the system and method described herein may be implemented in other orientations as long as antenna elements  27  are spaced apart in a direction different from a direction of motion of vehicle  10 . 
       FIG. 4  illustrates an exemplary embodiment of a relationship between antenna  26  and detection target  50 . The relationship between antenna  26  and detection target  50  may be characterized by a vector V defined by a distance D between antenna  26  and detection target  50  and elevation angle θ from antenna  26  to detection target  50  relative to first direction  30 . 
       FIG. 5  is similar to  FIG. 4  and illustrates an exemplary embodiment of antenna  26  represented by discrete antenna elements  27 . As seen in  FIG. 5 , antenna elements  27  may be spaced apart in second direction  32  at a pitch d 1 . While  FIG. 5  illustrates an equal pitch d 1  between all antenna elements  27 , it will be understood an exemplary embodiment may include antenna elements  27  that are unevenly spaced in the second direction. It will be understood that for large distances to target D, the elevation angle from each antenna element  27  to detection target  50  will be substantially the same. 
     It will be understood than an angular resolution of radar system  14  will increase as an aperture of antenna  26  increases (i.e., as a width of antenna  26  facing detection target  50  increases). This can be achieved by spacing antenna elements  27  along second direction  32 . However, while increased antenna aperture may increase resolution, there may be ambiguity (i.e., false detections, at angles close to the true elevation angle θ). In the embodiment of  FIG. 5 , in which antenna elements  27  are spaced apart in second direction  32 , the phase offset between antenna elements may be given by the following equation: 
                       ϕ   offset     =         2   ⁢     π   ⁡     (     d   ⁢           ⁢   1     )         λ     ⁢   sin   ⁢   θ       ;           (   1   )               
where γ is the wavelength of the radar signal output by radar system  14  (see  FIG. 2 ). Due to this phase offset, there may be ambiguities in the detected elevation angle of detection target  50 , as indicated by the following equation:
 
                         ϕ   amb     &lt;       sin     -   1       ⁢     1   N         ;     ⁢     
     ⁢       where   ⁢           ⁢   d   ⁢           ⁢   1     =     N   ⁢       λ   2     .                 (   2   )               
N is a coefficient used to express the pitch d 1  between antenna elements  27  in terms of wavelength γ. ϕ amb  is the maximal unambiguous angle, which means that any θ that is larger than ϕ amb  will be ambiguous. In a Bartlett beamforming spectrum, any θ that is larger than ϕ amb  will have multiple possible hypothesis angles with equal likelihood. Based on equation (2), it will be understood that as N increases, the maximal ambiguous angle for an array spaced in the second direction will approach 0 degrees, meaning that there will be a large number of possible hypothesis angles with equal likelihood.
 
     Based on these equations (1) and (2), it will be understood that an antenna  26  with a large pitch (i.e., a high N) between antenna elements will have a high ambiguity, and an antenna  26  with a low pitch, (i.e., N=1) will have a low ambiguity. However, at typical wavelengths for a radar system (such as 4 mm), a system with a small pitch between antenna elements  27  will require so many antenna elements  27  to achieve the desired antenna aperture and resolution that the system will become cost prohibitive and resource prohibitive. In contrast, a radar system in which the spacing between antenna elements  27  is increased will increase antenna aperture and resolution, but at the cost of increased ambiguity. In an exemplary embodiment, a desired angular resolution may be achieved with antenna elements  27  having a pitch d 1  that is greater than or equal to 10 times the wavelength of the radar signal, but the resulting high ambiguity would be unsuitable for practical purposes. 
     In order to overcome the ambiguity presented by widely spaced antenna elements arranged as in  FIG. 5 , antenna elements  27  may be arranged spaced at a pitch d 2  in the moving direction (i.e., first direction  30 ) as seen in  FIG. 6 . In the arrangement of  FIG. 6 , in which antenna elements  27  are spaced apart in first direction  30 , the phase offset between antenna elements may be given by the following equation: 
                       ϕ   offset     =         2   ⁢     π   ⁡     (     d   ⁢           ⁢   2     )         λ     ⁢   cos   ⁢   θ       ;           (   3   )               
where λ is the wavelength of the radar signal output by radar system  14  (see  FIG. 2 ). Due to this phase offset, there may be ambiguities in the detected elevation angle of detection target  50 , as indicated by the following equation:
 
                         ϕ   amb     &lt;       cos     -   1       ⁢     1   N         ;     ⁢     
     ⁢       where   ⁢           ⁢   d   ⁢           ⁢   2     =     N   ⁢       λ   2     .                 (   4   )               
As described in detail herein, ϕ amb  is the maximal unambiguous angle. Based on equation (4), it will be understood that as N increases, the maximal ambiguous angle for an array spaced in the first direction will approach 90 degrees; meaning that ambiguity will be low.
 
     As seen from equations (3) and (4), it will be understood that antenna elements  27  with a large pitch in the first direction  30  (i.e., a high N), will have a low angular resolution and a low ambiguity, which is the opposite of the response of the arrangement shown in  FIG. 5 . 
       FIGS. 7A and 7B  illustrate a graphical representation of the different Bartlett beamforming response of antenna arrays based on the vertical array arrangement of  FIG. 5  and the horizontal array arrangement of  FIG. 6 . For example,  FIG. 7A  includes curve  60 , which represents a Bartlett beamforming spectrum based on the response of antenna elements  27  spaced apart in second direction  32  as shown in  FIG. 5 . As seen in  FIG. 7A , curve  60  has a high angular resolution (indicated by the narrow width of the peaks  61  of curve  60 ) but also a high ambiguity (indicated by the high number of closely spaced peaks with similar amplitude). In contrast,  FIG. 7B  includes curve  62 , which represents a Bartlett beamforming spectrum based on the response of antenna elements  27  spaced apart in first direction  32  as shown in  FIG. 6 . As seen in  FIG. 7B , curve  62  has a low resolution (indicated by the wide width of the peak  63  of curve  62  centered at 0 degrees) and a low ambiguity (indicated by the lack of similar peaks to the main peak of curve  62 ). 
       FIG. 8  illustrates an exemplary embodiment of an antenna  26  in which antenna elements  27  are spaced apart in both first direction  30  and second direction  32 . By arranging antenna elements  27  in both the first direction  30  and the second direction  32 , the resulting beamforming spectrum can achieve both high resolution and low ambiguity. For example,  FIG. 7C  includes curve  64 , which represents a Bartlett beamforming spectrum based on the response of antenna elements  27  arranged as shown in  FIG. 8 . As seen in  FIG. 7C , curve  64  has a high angular resolution (indicated by the narrow width of the primary peak  65  of curve  64 ) and a low ambiguity (indicated by the existence of only one peak  65  of curve  64  at the highest amplitude). 
     However, it will be understood that providing an antenna  26  with antenna elements  27  spaced in both first direction  30  and second direction  32 , as shown in  FIG. 8 , increases the total number of required antenna elements  27 , which would increase the cost and resource consumption of the radar system. Additionally, it may not be physically feasible to provide a large two-dimensional array of antenna elements  27  due to space limitations in a vehicle. Accordingly, the motion of vehicle  10  can be used to generate a virtual two-dimensional array of antenna elements  27 . 
     For example, as seen in  FIG. 3 , vehicle  10  may move in first direction  30 . As vehicle  10  is moving, controller  28  (see  FIG. 2 ) may record the responses of antenna elements  27  at a plurality of time instants t 1 , t 2 , t 3 , . . . t T  to generate a virtual two-dimensional antenna array. 
       FIG. 9  illustrates an exemplary embodiment of a virtual two-dimensional antenna array  60 . Antenna elements  27   1 ,  27   2 ,  27   3 , . . .  27   T  correspond to positions of antenna elements  27  at time instants t 1 , t 2 , t 3 , . . . t T . Pitch d 1  between antenna elements  27  in the second direction  32  is known from when antenna elements  27  are installed in vehicle  10 . Relative positions of antenna elements  27   1 ,  27   2 ,  27   3 , . . .  27   T  in first direction  30  can be calculated based on a velocity of vehicle  10  in the first direction  30  and a predetermined time difference Δt between time instants t 1 , t 2 , t 3 , . . . t T . The velocity of vehicle  10  may be initially determined by velocity hypothesis calculated from instrumentation of vehicle  10  such as a speed sensor, GPS, or other suitable instruments, or may be based on a previously calculated velocity hypothesis of vehicle  10 . 
     The responses of antenna elements  27  at time instants t 1 , t 2 , t 3 , . . . t T  can be used in conjunction with the calculated positions of antenna elements  27  to calculate a virtual two-dimensional antenna array response of virtual two-dimensional antenna array  60 . Bartlett beamforming can then be performed on the virtual two-dimensional antenna array response to generate a beamforming spectrum such as curve  64  in  FIG. 7C . The elevation angle of detection target  50  can be determined from the beamforming spectrum. 
     The accuracy of the velocity hypothesis can affect calculation of the positions of antenna elements  27   1 ,  27   2 ,  27   3 , . . .  27   T  within the virtual two-dimensional antenna  60  of  FIG. 9 , and consequently, affect subsequent beamforming based on virtual two-dimensional antenna  60 . To determine the optimal velocity hypothesis, the initial velocity hypothesis may be iteratively adjusted, as described in detail herein. 
     For example, controller  28  (see  FIG. 2 ) may be configured to calculate a velocity score for the beamforming spectrum calculated from the virtual two-dimensional antenna array response. The velocity score may be given by the following equation:
 
 S   E   =E−αΣ   i   |s   i | 2 ;  (5)
 
where s i  is the beamforming spectrum at index i. α is a normalization factor that can be given by the equation:
 
                     α   =       0.25   ⁢           ⁢     log   ⁡     (   N   )             max   i     ⁢     (     s   i     )           ;           (   6   )               
where N is the number of beamforming angles (i.e., the beamforming grid points) and max i ( ) is a function returning the maximum for all i, E is given by the equation:
 
 E=−Σ   i γ i  log(γ i )  (7)
 
γ i  is given by the equation:
 
     
       
         
           
             
               
                 
                   
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     Controller  28  may be further configured to determine whether an optimal velocity score has been identified. In an exemplary embodiment, the optimal velocity score is a minimum velocity score. However, it will be understood that other types of optimal velocity scores, such as a maximum velocity score, may be used depending on the exact equations and axis polarities used to calculate the velocity score. Controller  28  may determine a minimum velocity score by using a coarse grid search over a large span of velocity hypotheses, and then a refined grid search in the vicinity of the most likely candidates. Alternatively, the adjustment may be made by an iterative gradient descent (i.e., starting from an initial guess, and each time through choose the next velocity step in a direction that recues the velocity score). 
       FIG. 10  illustrates an exemplary embodiment of an optimization process of the velocity hypothesis. Curve  220  illustrates a relationship between velocity score and velocity hypothesis in a hypothetical calculation. Each of the points on curve  220  represents a velocity hypothesis for which the velocity score was calculated. The velocity hypothesis at the minimum of curve  220  is the optimal velocity hypothesis. 
     Once the velocity hypothesis is adjusted, controller  28  may calculate new positions of antenna elements  27   1 ,  27   2 ,  27   3 , . . .  27   T  of virtual two-dimensional antenna array  60  (see  FIG. 9 ), and then calculate a new beamforming spectrum based on the revised virtual two-dimensional antenna array  60 . 
     Once controller  28  identifies a minimum velocity score, the velocity hypothesis associated with the minimum velocity score is output as the optimal velocity hypothesis, and the beamforming spectrum associated with the optimal velocity hypothesis is identified as the optimal beamforming spectrum. Controller  28  may output a peak of the optimal beamforming spectrum as an elevation angle of detection target  50 . Controller  28  may output the optimal velocity hypothesis and the elevation angle of detection target  50  to automated driving system  12  (see  FIG. 1 ), and automated driving system  12  may control vehicle  10  based on the optimal velocity hypothesis the elevation angle of detection target  50 . The optimal velocity hypothesis may also be used as an initial velocity hypothesis for subsequent detections. 
       FIG. 11  illustrates experimental data comparing a beamforming image  70  generated with a velocity error and an optimal beamforming image  72  generated with an accurate velocity.  FIG. 12  illustrates the test scenario used to generated images  70 ,  72  shown in  FIG. 11 . As seen in  FIG. 12 , the detection target  74  includes a collection of discrete elements arranged in an L-shape, and a relative velocity of 30 m/s between detection target and the radar was used.  FIG. 13  illustrates a curve of the velocity score as a function of velocity hypothesis for the test scenario, with point  76  illustrating the velocity hypothesis of 29.5 m/s used to generate beamforming image  70  (see  FIG. 11 ) and point  78  illustrating the velocity hypothesis of 30 m/s used to generate beamforming image  72 . Based on  FIGS. 11-13 , it can be seen that optimal beamforming image  72  resolved detection target  74  more clearly when an accurate velocity hypothesis was used. 
       FIG. 14  illustrates an exemplary embodiment of a control method  100  for use with a radar system  14  in a vehicle  10  structured to move in a first direction  30  (see  FIG. 1  and  FIG. 3 ). In block  106 , vehicle  10  is provided that includes radar system  14  having a controller  28  and an antenna  26  comprising a plurality of antenna elements  27  spaced apart in second direction  32 . In block  108 , vehicle  10  is moved in first direction  30 . 
     In block  110 , controller  28  records a response of antenna elements  27  at each of a plurality of time instants t 1 , t 2 , t 3 , . . . t T  (see  FIGS. 2 and 9 ). In block  114 , controller  28  calculates positions of antenna elements  27  at each of the plurality of time instants based on an initial velocity hypothesis to generate a virtual two-dimensional antenna array  60  (see  FIG. 9 ). In block  114 , controller  28  calculates a virtual two-dimensional antenna array response by combining the response of antenna elements  27  at each of the time instants with the positions of antenna elements  27  at each of the time instants. In block  116 , controller  28  performs Bartlett beamforming on the virtual two-dimensional antenna array response to generate a beamforming spectrum. 
     In block  118 , a velocity score S E  is calculated for the beamforming spectrum calculated in block  116 , as described in detail herein. In block  120 , it is determined whether a minimum velocity score has been identified. If a minimum velocity score has been identified (“Yes” in block  120 ), then the method proceeds to block  124 . If no minimum velocity score has been identified (“No” in block  120 ), then the method proceeds to block  122 . In block  122 , the initial velocity hypothesis is adjusted as described in detail herein. Once the velocity hypothesis is adjusted in block  122 , the method returns to block  112 , where new positions of antenna elements  27  are calculated based on the adjusted velocity hypothesis. Once a minimum velocity score is identified in block  120 , a peak of the beamforming spectrum associated with the minimum velocity score is identified in block  124 . Alternatively, if multiple detection objects are at a similar range but different angles of arrival, multiple peaks may be detected in block  124 . 
       FIG. 15  illustrates an exemplary embodiment of how the control method  100  shown in  FIG. 14  may be used in the context of operating vehicle  10 . For example, in block  202 , an optimal velocity hypothesis for vehicle  10  and elevation angle of detection target  50  (see  FIG. 3 ) are calculated as discussed in detail herein. In block  204 , controller  28  of radar system  14  transmits the optimal velocity hypothesis and elevation angle to automated driving system  12  (see  FIGS. 1-2 ). In block  206 , automated driving system  12  adjusts control of vehicle  10  based on the optimal vehicle hypothesis and the elevation angle of detection target  50 . 
     The exemplary embodiments described above result in significant advantages over conventional systems and methods. For example, the exemplary embodiments make it possible to achieve the combined high resolution and low ambiguity of a two-dimensional antenna array by using a single array of antenna elements combined with velocity information of the vehicle, thereby reducing cost, complexity, and power requirements of the radar system. 
     While the above disclosure has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from its scope. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the disclosure without departing from the essential scope thereof. Therefore, it is intended that the present disclosure not be limited to the particular embodiments disclosed, but will include all embodiments falling within the scope thereof.