Patent Publication Number: US-11378674-B2

Title: Object position and movement estimation using radar

Description:
The subject disclosure relates to estimation of object position and movement using radar. 
     Vehicles (e.g., automobiles, trucks, aircraft, construction equipment, farm equipment, automated factory equipment) are increasingly equipped with detection systems for monitoring surrounding environments. Radar systems may be used for detection and tracking of objects, for example, to avoid obstacles. Radar devices may be used in vehicles to alert a driver or user and/or to take evasive action. Detection and tracking systems are also useful in autonomously operated vehicles. The position of a moving object may not be detected accurately through conventional radar processing within typical integration times. Accordingly, it is desirable to provide a system for accurate position estimation of a moving object using radar. 
     SUMMARY 
     In one exemplary embodiment, a system for estimating a range and a velocity of an object includes a receiver configured to detect a return signal including reflections of a radar signal transmitted by a transmitter. The radar signal includes a series of transmitted pulses emitted over a time frame, the return signal includes a series of successive return pulses, and each return pulse corresponds to a transmitted pulse and is associated with a respective time interval in the time frame. The system also includes a processing device configured to perform, for each return pulse, applying a first Fourier transform to the return pulse to transform the return pulse into a range spectrum and to calculate a range intensity value for each of a plurality of range hypotheses associated with the respective time interval, calculating a range variation for each of a plurality of hypothesized Doppler frequency values, and for each hypothesized Doppler frequency value, applying a second Fourier transform to the series of return pulses based on the range intensity values and the range variation. The processing device is further configured to perform outputting range and Doppler frequency data including a range-Doppler intensity value for each range hypothesis and hypothesized Doppler frequency, and estimating a range and a velocity of the object based on the range-Doppler intensity values. Estimating the range and the velocity includes selecting one or more range-Doppler intensity values associated with a reflection from the object, each of the selected one or more range-Doppler intensity values corresponding to a range of the object and a Doppler frequency value associated with a velocity of the object. 
     In addition to one or more of the features described herein, the first Fourier transform and the second Fourier transform are applied by fast Fourier transform (FFT) algorithms. 
     In addition to one or more of the features described herein, applying the first Fourier transform includes generating a two-dimensional matrix including a plurality of matrix elements, the matrix having a first dimension defined by a plurality of time elements representing each time interval, and a second dimension defined by the plurality of range hypotheses, each matrix element having a range intensity value. 
     In addition to one or more of the features described herein, the second Fourier transform is performed on a series of range intensity values, the series of range intensity values selected from range hypotheses based on the range variation. 
     In addition to one or more of the features described herein, the range variation (R(t)) is calculated based on the following equation: 
                 R   ⁡     (   t   )       =         cf   d     ⁢   t       2   ⁢     f   c           ,         
where f d  is a hypothesized Doppler frequency, f c  is a carrier frequency of the transmitted pulses, t is time and c is the speed of light.
 
     In addition to one or more of the features described herein, each return pulse corresponds to a transmitted pulse time interval index (n), and applying the first Fourier transform includes calculating a vector of range intensity values at each time element, each intensity value in the vector corresponding to a respective range hypothesis. 
     In addition to one or more of the features described herein, wherein applying the second Fourier transform includes calculating the range-Doppler intensity value y(r, f d ) as a function of range (r) and Doppler frequency (f d ) for each hypothesized Doppler frequency, wherein the calculating is performed based on the following equation:
 
 y ( r,f   d )=Σ n=0   N-1   x   n ( r+R ( nT   c )) e   −j2πnT     c     f     d   ,
 
where T c  is a time interval between transmitted pulses, n is the transmitted pulse time interval index, x n  is a vector of range intensity values for an n-th transmitted pulse, N is a number of the time interval indexes, r is an initial range hypothesis, j is an imaginary unit, and x n (r+R(nT c )) is an element having an intensity value at a range defined by (r+R(nT c )), wherein R(nT c ) is the range variation calculated for the hypothesized Doppler frequency at the time interval index n.
 
     In addition to one or more of the features described herein, the range and Doppler frequency data includes a two-dimensional range-Doppler frequency spectrum having an output value calculated via the second Fourier transform for each of a plurality of Doppler frequencies and ranges. 
     In addition to one or more of the features described herein, selecting the one or more range-Doppler intensity values includes comparing each output value to a selected threshold, and identifying the output value as a reflection from the object based on the output value being greater than or equal to the threshold. 
     In addition to one or more of the features described herein, the processing device is further configured to estimate a direction of the object by applying beamforming to the range-Doppler intensity values from multiple antennas to estimate an azimuth and elevation angle of the object. 
     In one exemplary embodiment, a method of estimating a range and a velocity of an object includes detecting a return signal including reflections of a radar signal transmitted by a transmitter. The radar signal includes a series of transmitted pulses emitted over a time frame, the return signal includes a series of successive return pulses, and each return pulse corresponds to a transmitted pulse and is associated with a respective time interval in the time frame. The method also includes, for each return pulse, applying a first Fourier transform to the return pulse to transform the return pulse into a range spectrum and to calculate a range intensity value for each of a plurality of range hypotheses associated with the respective time interval, calculating a range variation for each of a plurality of hypothesized Doppler frequency values, and for each hypothesized Doppler frequency value, applying a second Fourier transform to the series of return pulses based on the range intensity values and the range variation. The method further includes outputting range and Doppler frequency data including a range-Doppler intensity value for each range hypothesis and hypothesized Doppler frequency value, and estimating a range and a velocity of the object based on the range-Doppler intensity values. Estimating the range and the velocity includes selecting one or more range-Doppler intensity values associated with a reflection from the object. Each of the selected one or more range-Doppler intensity values corresponds to a range of the object and a Doppler frequency value associated with a velocity of the object. 
     In addition to one or more of the features described herein, the first Fourier transform and the second Fourier transform are applied by fast Fourier transform (FFT) algorithms. 
     In addition to one or more of the features described herein, applying the first Fourier transform includes generating a two-dimensional matrix including a plurality of matrix elements, the matrix having a first dimension defined by a plurality of time elements representing each time interval, and a second dimension defined by the plurality of range hypotheses, each matrix element having a range intensity value. 
     In addition to one or more of the features described herein, the second Fourier transform is performed on a series of range intensity values, the series of range intensity values selected from range hypotheses based on the range variation. 
     In addition to one or more of the features described herein, the range variation (R(t)) is calculated based on the following equation: 
                 R   ⁡     (   t   )       =         cf   d     ⁢   t       2   ⁢     f   c           ,         
wherein f d  is a hypothesized Doppler frequency, f c  is a carrier frequency of the transmitted pulses, t is time and c is the speed of light.
 
     In addition to one or more of the features described herein, each return pulse corresponds to a transmitted pulse time interval index (n), and applying the first Fourier transform includes calculating a vector of range intensity values at each time element, each intensity value in the vector corresponding to a respective range hypothesis. 
     In addition to one or more of the features described herein, applying the second Fourier transform includes calculating the range-Doppler intensity value y(r, f d ) as a function of range (r) and Doppler frequency (f d ) for each hypothesized Doppler frequency, wherein the calculating is performed based on the following equation:
 
 y ( r,f   d )=Σ n=0   N-1   x   n ( r+R ( nT   c )) e   −j2πnT     c     f     d   ,
 
wherein T c  is a time interval between transmitted pulses, n is the transmitted pulse time interval index, x n  is a vector of range intensity values for an n-th transmitted pulse, N is a number of the time interval indexes, r is an initial range hypothesis, j is an imaginary unit, and x n (r+R(nT c )) is an element having an intensity value at a range defined by (r+R(nT c )), wherein R(nT c ) is the range variation calculated for the hypothesized Doppler frequency at the time interval index n.
 
     In addition to one or more of the features described herein, the range and Doppler frequency data includes a two-dimensional range-Doppler frequency spectrum having an output value calculated via the second Fourier transform for each of a plurality of Doppler frequencies and ranges. 
     In addition to one or more of the features described herein, selecting the one or more range-Doppler intensity values includes comparing each output value to a selected threshold, and identifying the output value as a reflection from the object based on the output value being greater than or equal to the threshold. 
     In addition to one or more of the features described herein, the processing device is further configured to estimate a direction of the object by applying beamforming to the range-Doppler intensity values from multiple antennas to estimate an azimuth and elevation angle of the object. 
     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 top view of a motor vehicle including a radar system; 
         FIG. 2  depicts a radar system, in accordance with an exemplary embodiment; 
         FIG. 3  is a flowchart depicting a method of estimating an object position, direction and/or velocity using a radar system, in accordance with an exemplary embodiment; 
         FIG. 4  depicts examples of a transmitted radar signal, frequency domain transforms of a return signal and estimation of an object position and Doppler frequency information associated with an object; 
         FIG. 5  depicts an example of a radar detection scenario; 
         FIG. 6  depicts an example of a range map generated according to a conventional radar detection method; 
         FIG. 7  depicts an example of a range map generated according to the method of  FIG. 3 ; and 
         FIG. 8  depicts an example of azimuth signals associated with the range map of  FIGS. 6 and 7 . 
     
    
    
     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. 
     In accordance with one or more exemplary embodiments, methods and systems for radar detection and position estimation are described herein. An embodiment of a radar system is configured to estimate a position and/or velocity of an object. An object may be any feature or condition that reflects transmitted radar signals. The radar system may be included in or connected to a vehicle for detection of objects such as road features, road obstructions, other vehicles, trees, people and others. The radar system is not limited to use with vehicles, and may be used in any context (e.g., weather, aviation and others). 
     The radar system is configured to transmit radar signals from one or more transmitters, each of which includes a series of successive pulses transmitted over a selected time frame. Reflections of the transmitted pulses are detected by a receiver and multiplied or mixed with a reference signal (e.g., a waveform corresponding to the transmitted radar signal) to generate a detection signal having a series of return pulses. Each return pulse is correlated with one of a plurality of successive time intervals within the time frame (referred to as “sub-frames”), and transformed into a frequency spectrum. For example, a first Fourier transform is applied to each return pulse to calculate intensity values at a corresponding time interval. Each intensity value (also referred to as a “range intensity value”) corresponds to a pre-selected or hypothesized range (also referred to as a “range hypothesis”), and is indicative of the likelihood of a reflection at the hypothesized range. A “range” is a distance from a transmitter. A range hypothesis may be a single range value or a range interval bounded by upper and lower range values. A second Fourier transform is applied to each return pulse to estimate a frequency shift (referred to as a “Doppler frequency”) associated with a velocity of the object. 
     In one embodiment, the radar system is configured to determine a variation between ranges corresponding to successive time intervals prior to transforming the return pulses according to the second Fourier transform. The range and Doppler frequency associated with a return pulse is then calculated using the second Fourier transform in combination with the range variation. In one embodiment, the range variation is determined as a function of Doppler frequency, by calculating a range variation for each of a plurality of hypothesized Doppler frequencies. 
     For example, individual return pulses are correlated with successive sub-frames, and each return pulse is input to an element of a matrix having dimensions defined by time interval and range. For example, a matrix referred to as a “range plot” includes rows having time elements corresponding to each time frame. Each time element is part of a column of range elements (also referred to as “range bins”), and each column includes a range element for each of a plurality of range values (range hypotheses). 
     In one embodiment, for each time element, the corresponding return pulse is scanned over the range bins by performing a first fast Fourier transform (FFT) process or algorithm. This process is referred to as “range FFT.” Based on the range FFT, each return pulse is transformed to an intensity value (a “range intensity value”) for each range bin. 
     Based on the range variation, a second Fourier transform is applied to the return pulses to output the range and velocity of the object over the time frame. In one embodiment, an output corresponding to object range and velocity is generated by applying a discrete Fourier transform (DFT) on range bins corresponding to the range variation (e.g., on range bins intersected by a range line). 
     Embodiments described herein have numerous advantages. For example, the radar system according to embodiments described herein can attain accurate range and Doppler estimation while adjusting for movement of an object during a selected integration time frame. Radar processes are performed according to periodic time frames which are selected based on considerations such as desired resolution signal-to-noise ratio (SNR). Longer time frames provide higher SNRs and higher Doppler resolutions. In order to efficiently implement Doppler signal processing, conventional systems assume that the range to an object is fixed within the selected time frame. This assumption can compromise the accuracy of radar processes if an object is moving at a high enough speed, as the reflected energy is spread over different ranges and Doppler frequencies. This spread can result in low detection probability and reduced accuracy. Embodiments described herein address the above challenges by accounting for variations in range over the time period, which allows for accurate position estimation over longer integration times, while maintaining a high radar resolution (e.g., 10 cm or less). The embodiments thus provide for the ability to adjust for movement of an object at high speeds in a longer time frame while maintaining a desired resolution, without requiring excessive processing resources. In addition, the embodiments increase accuracy by generating output peaks that are sharper than peaks generated by conventional processes. 
       FIG. 1  shows an embodiment of a motor vehicle  10 , which includes a vehicle body  12  defining, at least in part, an occupant compartment  14 . The vehicle body  12  also supports various vehicle subsystems including an engine assembly  16 , and other subsystems to support functions of the engine assembly  16  and other vehicle components, such as a braking subsystem, a steering subsystem, a fuel injection subsystem, an exhaust subsystem and others. 
     The vehicle  10  includes aspects of a radar system  20  for detecting and tracking objects, which can be used to alert a user, perform avoidance maneuvers, assist the user and/or autonomously control the vehicle  10 . The radar system  20  includes one or more radar sensing assemblies  22 , each of which may include one or more transmit elements and/or one or more receive elements. The vehicle  10  may incorporate a plurality of radar sensing assemblies disposed at various locations and having various angular directions. 
     For example, each radar sensing assembly  22  includes a transmit portion and a receive portion. The transmit and receive portions may include separate transmit and receive antennas or share an antenna in a transceiver configuration. Each radar sensing assembly  22  may include additional components, such as a low pass filter (LPF) and/or a controller or other processing device. 
     The radar sensing assemblies  22  communicate with one or more processing devices, such as processing devices in each assembly and/or a remote processing device such as an on-board processor  24  and/or a remote processor  26 . The remote processor  26  may be part of, for example, a mapping system or vehicle diagnostic system. The vehicle  10  may also include a user interaction system  28  and other components such as a GPS device. 
     The radar system  20  is configured generally to acquire radar signals and analyze the radar signals to estimate a position and/or a velocity of an object. The position and/or velocity are estimated by integrating acquired signal pulses over a selected time frame. The length of the time frame is selected to provide for a desired resolution. As discussed further below, the radar system  20  analyzes the radar signals over the time frame while adjusting the signals to account for motion of the object over the time frame. 
       FIG. 2  illustrates aspects of an embodiment of a computer system  30  that is in communication with or is part of the radar system  20 , and that can perform various aspects of embodiments described herein. The computer system  30  includes at least one processing device  32 , which generally includes one or more processors for performing aspects of radar detection and analysis methods described herein. The processing device  32  can be integrated into the vehicle  10 , for example, as the on-board processor  24 , or can be a processing device separate from the vehicle  10 , such as a server, a personal computer or a mobile device (e.g., a smartphone or tablet). For example, the processing device  32  can be part of, or in communication with, one or more engine control units (ECU), one or more vehicle control modules, a cloud computing device, a vehicle satellite communication system and/or others. The processing device  32  may be configured to perform radar detection and analysis methods described herein, and may also perform functions related to control of various vehicle subsystems. 
     Components of the computer system  30  include the processing device  32  (such as one or more processors or processing units) and a system memory  34 . The system memory  34  may include a variety of computer system readable media. Such media can be any available media that is accessible by the processing device  32 , and includes both volatile and non-volatile media, removable and non-removable media. 
     For example, the system memory  34  includes a non-volatile memory  36  such as a hard drive, and may also include a volatile memory  38 , such as random access memory (RAM) and/or cache memory. The computer system  30  can further include other removable/non-removable, volatile/non-volatile computer system storage media. 
     The system memory  34  can include at least one program product having a set (e.g., at least one) of program modules that are configured to carry out functions of the embodiments described herein. For example, the system memory  34  stores various program modules  40  that generally carry out the functions and/or methodologies of embodiments described herein. For example, a receiver module  42  may be included to perform functions related to acquiring and processing received signals, and an analysis module  44  may be included to perform functions related to position estimation and range finding. The system memory  34  may also store various data structures  46 , such as data files or other structures that store data related to radar detection and analysis. Examples of such data include sampled return signals, frequency data, range-Doppler plots, range maps, and object position, velocity and/or azimuth data. 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. 
     The processing device  32  can also communicate with one or more external devices  48  such as a keyboard, a pointing device, and/or any devices (e.g., network card, modem, etc.) that enable the processing device  32  to communicate with one or more other computing devices. In addition, the processing device  32  can communicate with one or more devices that may be used in conjunction with the radar system  20 , such as a Global Positioning System (GPS) device  50  and a camera  52 . The GPS device  50  and the camera  52  can be used, for example, in combination with the radar system  20  for autonomous control of the vehicle  10 . Communication with various devices can occur via Input/Output (I/O) interfaces  54 . 
     The processing device  32  may also communicate with one or more networks  56  such as a local area network (LAN), a general wide area network (WAN), and/or a public network (e.g., the Internet) via a network adapter  58 . It should be understood that although not shown, other hardware and/or software components could be used in conjunction with the computer system  30 . Examples include, but are not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, and data archival storage systems, etc. 
       FIG. 3  illustrates aspects of an embodiment of a computer-implemented method  70  of radar detection and analysis, which includes estimating an object location or position, direction and/or velocity. The method  70  may be performed by a processor or processors disposed in a vehicle (e.g., processing device  32 , as an ECU or on-board computer) and/or disposed in a device such as a smartphone, tablet or smartwatch. The method  70  is discussed in conjunction with the radar system  20  of  FIG. 1  and components shown in  FIG. 2  for illustration purposes. It is noted that aspects of the method  70  may be performed by any suitable processing device or system. 
     The method  70  includes a plurality of stages or steps represented by blocks  71 - 75 , all of which can be performed sequentially. However, in some embodiments, one or more of the stages can be performed in a different order than that shown or fewer than the stages shown may be performed. 
     At block  71 , a radar signal is transmitted by one or more transmit elements in a radar system such as the radar system  20 . Each transmit element transmits radar signals having a series of pulses. As described herein, “pulses” refer to a series of repeating waveforms, which are not limited to those described herein. In one embodiment, the transmit element transmits a linear frequency-modulated continuous wave (LFM-CW) signal. This signal may be referred to as a “chirp signal,” and each pulse may be referred to as a “chirp.” 
     An example of a chirp signal  80  is shown in  FIG. 4 , which includes a series of chirps  82 . As shown, each chirp  82  has a frequency f that increases linearly within a sub-frame. Each sub-frame has a duration T c , which is equal to the time between adjacent chirps  82 . 
     Although only one chirp signal  80  is discussed herein, it is to be understood that the method  70  is applicable to multiple radar signals and/or chirp signals, from a single transmit element or multiple transmit elements. For example, the method  70  may be performed for multiple chirp signals  80  emitted by the same transmit element, for multiple transmit elements having the same location and orientation, and/or from multiple chirp signals  80  from transmit elements having different locations and/or orientations. 
     Characteristics of the chirp signal  80  may be selected, for example, to distinguish signals from different transmit elements. Examples of characteristics include bandwidth (BW), chirp slope (rate of increase or decrease of frequency) and chirp time (temporal length of each chirp). Other characteristics may be selected, such as field of view and detection threshold (minimum detectable amplitude). 
     Returning to  FIG. 3 , at block  72 , a return signal is detected or measured by one or more receive elements as a measurement signal. For example, analog signals detected by the receive elements are sampled and converted to digital signals, referred to herein as detection signals. In one embodiment, the detection signals are multiplied by or mixed with a reference signal to generate a return signal. For example, a detection signal including reflections of the transmitted chirps  82  is mixed with a reference signal such as the waveform of the chirp signal  80 . The resulting return signal includes a series of return pulses, which may be correlated with the chirps  82 . 
     At block  73 , a processing device, such as the processor  32 , transforms each return pulse into the frequency domain by using a Fourier transform. In one embodiment, the processing device  32  uses a fast Fourier transform (FFT) algorithm (also referred to as “range FFT”) to generate range spectra associated with each return pulse. The range FFT is a one-dimensional FFT configured to transform the return pulses into range intensity values that can be used to estimate the range (referred to as the “range domain”) of a reflection. 
     Referring again to the example of  FIG. 4 , the processing device  32  performs a range FFT process, which includes performing Discrete Fourier transforms on the samples making up each return pulse to generate range spectra  86 . Examples of range spectra  86  are shown correlated in time with their respective chirps  82 . 
     Each range spectrum  86  is scanned over a series of range bins, each of which is a filter configured to detect an intensity of a range spectrum  86  at frequencies f r  associated with a given hypothesized range value. The range value may be a single value or multiple values. For example, a range spread of about 30 meters is selected and divided into successive range bins representing 10 cm intervals. A range spectrum  86  may be assigned to a range bin based on the range spectrum  86  having a spike (an amplitude exceeding some threshold) that corresponds to a frequency (f r ) in the range bin. For example, a range intensity value is calculated for a range of a given range bin, and the range intensity value assigned to the range bin is zero if the intensity of the range spectrum  86  at the range bin is zero or below a threshold intensity. 
     In one embodiment, each range spectrum  86  is stored in a matrix  88  that forms rows  90  having a number N of time elements n. Each time element has a width equal to the temporal length of a sub-frame of the time frame. For each time element n in the row  90 , the matrix  88  includes a column  92  of successive range bins. 
     Each range bin in the columns  92  is associated with a range value r, which indicates a distance from the transmit element to an object. The range r can be expressed, for example, as range bin numbers, actual range values, frequency values or any other suitable index. 
     The range FFT process outputs, for each time element n (which can be represented by a transmitted pulse time interval index n or a chirp index n), a vector x n  with complex elements. Each complex element in the vector x n  corresponds to a frequency f r  associated with a given range bin. 
     As an illustration, the matrix  88  shows eight range spectra  86 , which are input to corresponding time elements n in the row  90  (N=8). Each range spectrum  86  is given an integer, so that the first spectrum is n=0, the second adjacent spectrum is n=1, and so on. 
     Each return pulse is transformed and scanned along a corresponding column  92 . One or more peaks are identified for each range spectrum  86 , and each range spectrum  86  is assigned to a range bin. In the example of  FIG. 4 , each range spectrum  86  includes one peak representing reflections from one object. However, the range spectra  86  may have multiple peaks representing multiple objects. If a range spectrum  86  has multiple peaks, it may be each assigned to multiple range bins. 
     At block  74  of  FIG. 3 , the processing device  32  calculates a range variation R(t) between at least two adjacent range spectra  86  as a function of time t. In one embodiment, the range variation is calculated for one or more hypothesized Doppler frequencies f d . The Doppler frequency f d  can be expressed as: 
                       f   d     =         2   ⁢     f   c       c     ⁢     v   r         ,           (   1   )               
where f c  is the carrier frequency (source frequency) of a transmitted pulse, c is the speed of light, and v r  is the radial velocity of a moving object (i.e. the projection of the velocity vector to the direction pointing from the object to the radar sensing assembly  22 ). The velocity v r  can thus be expressed as:
 
     
       
         
           
             
               
                 
                   
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     In one embodiment, the range variation over the return pulses is calculated for each of a plurality of hypothesized Doppler frequencies f d . For example, for a given f d , R(t) is calculated between adjacent range bins in which a return pulse has been assigned. The Doppler frequencies may be individual frequency values or frequency ranges. 
       FIG. 4  shows the result of an example of a range variation calculation performed according to embodiments described herein. In the example of  FIG. 4 , for a first hypothesized Doppler frequency f 0 , the range variation R(t) between successive time elements n is calculated according to equation (3), and corresponds to a linear function range represented by a line L 1 . The range variation is similarly calculated for a second hypothesized Doppler frequency f 1 , and corresponds to a linear function range represented by a line L 2 . The range variation calculated for a third hypothesized Doppler frequency f 2  corresponds to a linear function range represented by a line L 3 . As can be seen in  FIG. 4 , the range variations corresponding to lines L 1  and L 2  intersect mostly empty cells (having an intensity value of zero or a value below a threshold) in the matrix  88 . 
     In some cases, a range value resulting from the range variation calculations falls in between range bins or are within some proximity to an adjacent range bin. In some cases, the value of R(t) at that range bin can be rounded up or down (i.e., moved to the next bin above or below), or adjusted in any other suitable manner, such as by linear interpolation. 
     At block  75  of  FIG. 3 , the range and velocity of the object is determined by applying a second Fourier transform to estimate the frequency shift (Doppler frequency) and estimate object position and velocity. The second Fourier transform is selected to calculate frequencies associated with object range and velocity and thereby transform the return pulses into the range-Doppler frequency domain. In one embodiment, the processing device  32  uses a fast Fourier transform (FFT) algorithm to generate frequency spectra associated with each return pulse, which can be used to estimate a position and velocity value associated with each frequency spectrum. This FFT is a one-dimensional FFT referred to as a “Doppler FFT.” 
     The processing device  32  calculates the range and Doppler frequency for an object at a time associated with a sub-frame by performing the Doppler FFT while accounting for range variations R(t) estimated at block  74 . 
     In one embodiment, each return pulse is analyzed and transformed using a Doppler FFT algorithm that outputs range and Doppler frequency data. For example, an output of the Doppler FFT includes an output value as function of Doppler frequency and range variation. The output may be considered a range-Doppler spectrum including intensity values as a function of range and Doppler frequency. 
     An example of a Fourier transform function that can be applied is represented by the following formula or equation:
 
 y ( r,f   d )=Σ n=0   N-1   x   n ( r+R ( nT   c )) e   −j2πnT     c     f     d   ,  (4)
 
In the above equation, N is a number of the time elements n, and n is a transmitted pulse time interval index having a temporal length equal to a time interval T c  between pulses, r is an initial range hypothesis (e.g., corresponding to the lowest range bin), and j is an imaginary unit.
 
     x n  is a vector of range intensity values at a time interval index n (the n-th transmitted pulse and corresponding return pulse). The element x n (r+R(nT c )) is the intensity value at a time element n in a range bin r+R(nT c ), where R(nT c ) is the range variation calculated for the hypothesized Doppler frequency at the time element n (or transmitted pulse time interval index n). 
     The output of the above equation is a complex value y(r, f d ) (which may have arbitrary units) based on a summation of the intensity values in the range bins for each of a plurality of pre-selected Doppler frequency values or hypothesized Doppler frequencies f d . x n  is the output vector (in arbitrary units) of the range FFT for the n-th return pulse. 
     In the example of  FIG. 4 , each column  92  is the range FFT for a different return pulse or chirp (before applying the transformation to the range-Doppler spectrum), and the rows represent different successive return pulses or chirps. 
     For each hypothesized Doppler frequency f d  value, the range variation R(t) is calculated and can be represented as a diagonal line that indicates the range bins along which a pulse is scanned. For each hypothesized Doppler frequency f d , samples are taken from frequency data stored in or associated with a range bin intersected by the calculated diagonal. The summation and phase rotation performed in equation (4) is performed along the diagonal. For example, in the first time element (n=0), the range is equal to the range r associated with the first range bin. The summation proceeds to the next time element (n=1) and the range is r+R, where R is calculated for the time interval T c *n (which is equal to T c  at bin number 1). 
     Based on the Doppler FFT, the processing device  32  outputs range and Doppler frequency data indicative of the range and velocity of an object. For example, the processing device  32  outputs range and Doppler frequency data that includes a range-Doppler intensity value for each range hypothesis, such as for range interval or range bin (or at least a subset of the range hypotheses), for a hypothesized Doppler frequency. The output can include range-Doppler intensity values for one or more hypothesized Doppler frequencies. For example, the range and Doppler frequency data for each hypothesized Doppler frequency is compared to an intensity threshold to identify one or more selected hypothesized Doppler frequencies associated with reflections from an object. 
     In the example of  FIG. 4 , calculation of y(f 0 ) along the diagonal L 1  results in an output of zero or close to zero, as the range bins along this diagonal are empty (x n =0). Calculation along the diagonal L 2  results in a similar output. 
     Calculation of y(f 2 ) along the diagonal L 3 , which is populated with samples of the return pulses, results in one or more signal spikes at Doppler frequency f 2 . Based on this calculation, the velocity is estimated to correspond with the Doppler frequency f 2  and the output provides the estimated range over the time frame. 
     Additional processes may be performed as part of the method  70  or in addition to the method  70 . For example, further processing is performed to estimate the direction of the object. In one embodiment, azimuth filtering is performed, for example, applying azimuth beamforming over multiple antennas per a range and Doppler bin to show the output as a function of azimuth. 
       FIGS. 5-8  illustrate an example of range and Doppler calculations and outputs performed according to embodiments described herein, in comparison with an example of calculations performed by a conventional process. 
     In this example, radar pulses  94  are emitted toward a moving object  96 . Region  98  shows an observation window during a time frame selected for a radar signal. A sufficiently high radar range resolution is selected, for example, as about 10 cm or less. Such a range resolution is desirable in contexts such as autonomous vehicle operation, to allow sufficient time to react to a detected object. The time frame in this example is selected to be about 50 milliseconds. Also in this example, as shown in  FIG. 5 , the object  96  is moving toward a radar transmitter at a relative velocity of about 50 m/s. 
       FIG. 6  shows the result of a conventional radar detection method, in which the object range is assumed to be constant during the time frame. The result is in the form of a range map  100  that is color coded according to the legend  102 , showing an output of the detection method as a function of range and azimuth. As shown in region  104 , there is a large range spread at azimuth of zero degrees. 
       FIG. 7  illustrates a range map  110  that is color coded according to the legend  112 . As shown in region  114 , the process performed according to embodiments described herein produces a significantly smaller range spread and thus significantly better accuracy at high range resolutions and longer integration times. 
     In addition, as noted above, the embodiments described herein produce higher intensity and sharper signal peaks. For example,  FIG. 8  shows an azimuth cut  116  of the range map  100  from the conventional process, and an azimuth cut  118  of the range map  110 . As shown, the methods described herein produce significantly stronger peaks. 
     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.