Abstract:
A method and apparatus is provided for correcting the phase difference estimate derived from a two-tone CW radar to correct velocity-induced range estimate phase errors by offsetting the phase difference estimate with a phase correction equal to either of the Doppler frequencies associated with returns from an object multiplied by the time interval between the samplings of the returned waveforms. The correction effectively eliminates the velocity-induced slippage between the phases of the retuned waveforms so that a comparison between the phases of the waveforms can be made to reduce or substantially eliminate range estimate bias

Description:
FIELD OF THE INVENTION  
       [0001]     This invention relates to two-tone CW radars and more particularly to a method for correcting velocity-induced range estimate phase errors.  
       BACKGROUND OF THE INVENTION  
       [0002]     As discussed in a Patent Application Serial entitled Method and Apparatus for Improved Determination of Range and Angle of Arrival Utilizing a Two-tone CW Radar by Paul D. Fiore, filed on even date herewith, assigned to the assignee hereof and incorporated herein by reference, a system is provided for providing range and angle of arrival estimates from the output of a two-tone CW radar. In this system, the range of an object from the radar is computed from the phase angle between returns from the object in which the phase of the Doppler return of one tone is compared with the phase of the Doppler return of the second tone.  
         [0003]     This system uses a two-tone CW radar in which the two tones are sequentially projected or propagated towards a target. In one embodiment the switching rate between the two tones is on the order of 100 KHz, which corresponds to 5 milliseconds of the f 1  tone followed by 5 milliseconds of the f 2  tone.  
         [0004]     When used for a fire control system to detect the range of a moving target, the system works relatively well for slow targets. However, when the target&#39;s speed approaches 300 meters per second, as in the case with rocket-propelled grenades, range estimates degrade significantly.  
         [0005]     While initially a plurality of causes was investigated to ascertain the cause of the range error, it was noticed that the Doppler frequency associated with the 300 m/sec. target was about 49 KHz. This was found to be quite close to the 50 KHz Nyquist rate associated with the 100 KHz switching. The result with uncompensated systems was wide swings in the range estimate for incoming targets, whether the target was a rocket-propelled grenade, a projectile or a missile.  
         [0006]     By way of background, the theory of two-tone continuous wave range estimation radar shows that target range is proportional to the difference in the complex phase angle between the signal returns corresponding to the two tones. In the above-mentioned sequential transmission of tones, known as a diplexing method, the two tones are transmitted sequentially, and it was assumed that the target Doppler frequency was small compared to the switching rate. With this assumption, an acceptably small bias in the range estimate results. However, it was found that the bias rate increases as the target speed increases, thus limiting the ability to accurately obtain the range of high-speed targets.  
         [0007]     For a radar to measure range, it is typically thought that some sort of amplitude or phase modulation of the carrier is required. However, as mentioned above there is a method using more than one CW signal that can in fact provide range, which involves a tellurometer and is available for geodetic survey work. The geodetic system makes use of the fact that the survey equipment is not moving and therefore has a zero Doppler shift.  
         [0008]     Radar designs for the case where there is target velocity and it is low can produce desired range estimates when using two-tone CW-transmitted signals. Additionally, approaching or receding targets can be distinguished through proper choice of CW frequencies.  
         [0009]     Thus, those two-tone CW radars provide accurate range measurements if the motion during one Doppler period is small. This means that the phases of the wave forms will not appreciably “slip” relative to each other and a comparison between the phases of the wave forms can be made.  
       SUMMARY OF INVENTION  
       [0010]     It has now been found that one can correct the measured phase difference in the Doppler-shifted returns from high-speed targets to eliminate the range estimate bias by providing an offset or correction that is applied to the measured phase difference. This correction has been found to be the Doppler frequency of either tone times the time difference between samples. This offset has been found to be linearly related to the target velocity and the time delay between the samplings for the two tones. Since it is a relatively simple matter to ascertain the time at which samples are collected for each of the two wave forms, one can derive a phase correction that is simply the frequency of one of the tones times the time difference between the samples.  
         [0011]     By correcting the phase difference originally calculated from the two-tone CW radar returns with this phase correction, the so-called slippage between the two waveforms due to the speed of the target is canceled. The result is a range estimate that is correct, independent of the speed of the approaching target.  
         [0012]     In summary, a method and apparatus is provided for correcting the phase difference estimate derived from a two-tone CW radar to correct velocity-induced range estimate phase errors by offsetting the phase difference estimate with a phase correction equal to either of the Doppler frequencies associated with returns from an object multiplied by the time interval between the samplings of the returned waveforms. The correction effectively eliminates the velocity-induced slippage between the phases of the returned waveforms so that a comparison between the phases of the waveforms can be made to reduce or substantially eliminate range estimate bias. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]     These and other features of the subject invention will be better understood in connection with a Detailed Description, in conjunction with the Drawings, of which:  
         [0014]      FIG. 1  is a block diagram of a multiple-barrel shotgun-type countermeasure system that has its firing command based on the range of an incoming rocket-propelled grenade or RPG;  
         [0015]      FIG. 2  is a block diagram of a two-tone CW monopulse radar in which sequential two tones are projected towards an object and in which the sum and difference Doppler returns from the object are analyzed to provide a range estimate;  
         [0016]      FIG. 3  is a block diagram of a simplified version of the two-tone CW radar of  FIG. 2 , illustrating that the phase difference between the Sum channels of the two tones can be used in obtaining the phase difference, from which range can be determined;  
         [0017]      FIG. 4  is a waveform diagram illustrating the time intervals at which the two tones of the radar of  FIG. 1  are transmitted, the time at which samples are taken, and the phase delay between the samples of the wave forms from which range can be estimated;  
         [0018]      FIG. 5  is a waveform diagram illustrating that for the two frequencies or tones of  FIG. 4 , the time difference between samples results in a phase angle correction that is equal to either one of the two Doppler frequencies times the time difference between the samples;  
         [0019]      FIG. 6  is a block diagram showing the phase difference measurement from the output of a singular value decomposition of a Rank One two-by-two matrix used to provide a range estimate that is corrected by an offset that offsets the phase difference measurement with the offset of  FIG. 5 , with the result being a corrected phase difference measurement applied to a range calculator; and,  
         [0020]      FIG. 7  is a block diagram of the operation of the error compensator of  FIG. 6 , illustrating the calculation of the phase error between two complex numbers followed by calculation and application of a base correction factor, such that the phase difference between the two complex numbers is offset by the appropriate correction to cancel out the effect of slippage.  
     
    
     DETAILED DESCRIPTION  
       [0021]     Referring now to  FIG. 1 , in one application of the subject invention, a fire control system  10  for a shotgun  12  mounted on a gimbal  13  is provided by using an Ka-band CW two-tone monopulse radar  14  coupled to a planar antenna  16 , which carries a transmit element  18  and two receive elements  20  and  22  from which are derived sum and difference signals related to returns from, for instance, a high-velocity rocket-propelled grenade  26  fired by an individual  28 .  
         [0022]     In order to provide an initial gross aiming of gun  12 , the plume  34  from rocket-propelled grenade  26  is detected by the plume imaged by a lens system  32  onto an IR plume detector  30  that provides a gross bearing at unit  36  used at unit  38  to initially aim the radar and gun.  
         [0023]     Radar returns from the target result in the generation of Sum f 1  and Sum f 2  signals  42  and Diff. f 1  and Diff. f 2  signals  44  that are coupled to a unit  50  that calculates range and bearing used by a re-aim and shoot module  52 . Unit  50  provides bearing  53 , range  54  and velocity  60  to unit  52 , from which gimbal  13  is actuated to re-aim gun  12  in accordance with the more refined bearing range and velocity estimates from unit  50 .  
         [0024]     At an appropriate range, a pellet cloud  57  is projected towards RPG  26  so that as the RPG arrives at position  26 ′, it meets with an optimal pellet pattern. The firing signal for the gun is critical so that the pellet cloud meets the RPG at the correct range for establishing an optimal pellet cloud density to effect a kill.  
         [0025]     In one embodiment, this range is seven meters so that given the cone of the pellet cloud, its density will be optimal as it impacts the rocket-propelled grenade.  
         [0026]     It should be noted that the entire time that is allocated for the aiming and firing sequence is less than 150 milliseconds as illustrated by line  62 , which is from the time that the RPG is launched to the time that it arrives at its intended target.  
         [0027]     Note that, in one embodiment, if the target velocity is below that which is associated with a moving projectile, then a velocity threshold unit  58  coupled to an inhibit unit  59  cancels the re-aiming and shooting process if, for instance, the detected velocity is detected below 100 meters per second.  
         [0028]     Referring to  FIG. 2 , in one embodiment of the range and bearing calculation unit  50  a sequence of two tones, here illustrated at  72 , is propagated or projected by a transmit antenna  74  towards an object  76 , with returns from the object being detected by receive antennas  78  and  80  to generate respective sum and difference signals coupled to unit  50 . The choice of the difference of frequencies depends on the range ambiguity that is acceptable. Typically, one uses a difference of between 500 kHz and 1.5 MHz. In one embodiment, a typical set of frequencies is 24.7290 GHz and 24.7300 GHz, a difference of 1.0 MHz. Here in one embodiment the sum and difference signals are processed by a down-convert, low-pass filter and sampling unit  82 , the output of which is coupled to a demultiplexing unit  84  and a Fast Fourier transform module  86 , in turn coupled to a magnitude-square module  88 , which makes available the magnitude-squared amplitudes of the FFT bins. Thus the digital time domain data stream y 11  . . . y 22  is converted by FFT module  86  to frequency domain data stream Y 11  . . . Y 22  in terms of frequency bins, with the magnitude-squared output of module  88  accumulated at  90  so as to permit the finding of a peak by peak detection unit  92 , from which the particular bin having the target is selected as shown at  94 .  
         [0029]     Having selected the frequency bin most likely to contain the target, a two-by-to matrix A is formed from the sum and difference signals associated with this frequency bin, with the sum and difference signal matrix A being coupled to a singular value decomposition processor  96  that outputs a two-by-two matrix U, here illustrated by reference character  98  from which range can be estimated and a two-by-two matrix V, here illustrated by reference character  100  from which angle of arrival can be derived.  
         [0030]     The two-by-two matrix U is applied to a range estimation unit  102  from which a range estimate is made.  
         [0031]     The range estimate comes from analyzing the first column of the two-by-two matrix U, which when processed provides the aforementioned phase angle between the two tones.  
         [0032]     Note that matrix V is applied to an angle of arrival estimator  104 .  
         [0033]     Referring now to  FIG. 3 , to summarize what is happening in the system of  FIG. 2 , two tones are alternately generated as illustrated at  120 , with a five-microsecond duration for each of the tones. The switching time between going from frequency f 1  to frequency f 2  or vice versa is negligible. The switching rate is thus determined by the 5μ/sec. pulse durations. The two-tone radar  121  projects a beam with two tones towards object  122 , which may have a velocity, for instance of 300 meters per second. Sum f 1  and Sum f 2  signals are developed by radar  121 , which are used at  124  to determine range in terms of their phase difference.  
         [0034]     As mentioned hereinbefore, the range error rate is proportional to velocity.  
         [0035]     Referring to  FIG. 4 , the transmitted tones are as shown by waveform  130 , which has a period T. The round trip travel time from which no samples are allowed is indicated by double-ended arrows  132 . This leaves a time interval  134  in which it is appropriate to take samples. Samples are taken of the returns as illustrated at  136  and  138 , with waveforms  140  and  142  respectively defining the wave forms of the two tone returns at Doppler frequencies f 1  and f 2 . It will be appreciated that what is measured is the phase delay  144  between waveforms  140  and  142  to be able to determine range, with the phase delay being from the samples y 1  and y 2 .  
         [0036]     As illustrated in  FIG. 5 , the actual phase difference between waveforms  140  and  142  corresponding to f 1  and f 2  respectively is illustrated by double-ended arrows  150 , whereas the time difference between the samples, ΔT, is illustrated by double-ended arrow  152 .  
         [0037]     It will be shown that the appropriate phase correction or offset that may be applied to the phase difference calculation at  124  is equal to either one of the two Doppler frequencies multiplexed by ΔT, the time difference between the samples.  
         [0038]     Why this simple phase correction works will be discussed hereinafter. However, it is a finding of the subject invention that by simply knowing the time difference between the samples and knowing the Doppler frequency of the target, one can offset the phase difference provided by the two-tone CW radar and by this offset to be able to eliminate the effects of slippage between the f 1  and f 2  waveforms due to the velocity of the incoming target.  
         [0039]     Referring now to  FIG. 6 , in one embodiment of the subject invention, the aforementioned Rank One two-by-two matrix  160  is formed by the sum and difference signals associated with this frequency bin having the target. Here, Sum f 1 * and Sum f 2 * refer to the Sum f 1  and Sum f 2  signals that have been established as being from the target. Matrix  160  is applied to a singular value decomposition unit  162  from which the phase difference φ is available on line  164 . This is the phase difference that is calculated from the first row of Matrix U above, which is one of the results of the singular value decomposition.  
         [0040]     Having derived Δφ from the output of singular value decomposition unit  162 , an offset is applied at unit  166 , which provides an error compensation that offsets Δφ. This error compensation unit is provided with the time difference between samples as illustrated on line  168  and the measured Doppler frequency containing the target on line  170 , such that knowing the Doppler frequency and the time difference between samples, one can calculate how many degrees the sine wave should be changed to eliminate slippage.  
         [0041]     The corrected phase difference is available on line  172 , coupled to a range calculation unit  174  that multiplies the corrected phase difference by a constant to obtain range.  
         [0042]     Referring now to  FIG. 7 , error compensation unit  166  is provided with a unit  176  to which is applied the two-by-two matrix U. This unit calculates the phase difference between two complex numbers in the first column of this matrix, with the output being the phase difference on line  178 . This phase difference is applied to a calculator  180 , which applies a base correction knowing the Doppler bin number on line  182 , thereby establishing the Doppler frequency of the target. The time difference between samples is applied on line  184 , with the phase-corrected signal being outputted on line  186 . This signal is the original calculated phase Δφ offset by the phase change θ pc =f 1 ×T such that θ pc  defines how many degrees the sine wave should change in order to correct the measurement.  
       Theory of Operation  
       [0043]     The theory of two-tone continuous-wave range estimation shows that target range is proportional to the difference in complex phase angle between the signal returns corresponding to the two tones. In the prior art, the two tones are transmitted sequentially, and it was assumed that the target Doppler frequency was small compared to the switching rate. With this assumption, an acceptably small bias in the range estimate resulted. However, as noted above, the bias increases as target speed increases, thus limiting the prior art to low speed targets.  
         [0044]     The subject invention provides a method to correct the estimated phase difference for high speed targets, thereby eliminating the range estimate bias. The correction is related linearly to the target velocity and the time delay between the sampling instants for the two tones.  
         [0045]     As noted above, a tellurometer is available for geodetic survey work, and makes use of the fact that the survey equipment is not moving (i.e., zero Doppler shift). As has been discussed, the utilization of multiple CW transmitted signals can produce the range estimates for moving targets, assuming the target velocity is low.  
         [0046]     How this is accomplished is as follows: to simplify the description, first assume that the two different frequencies f k , for k=1, 2 are simultaneously transmitted. Without loss of generality, it can be assumed that the transmitted signals are of the form 
 
 s   k ( t )=cos(ω k   t +ψ t ),  (1) 
 
 where ψ k  is an unknown phase angle, and ω k =2πf k . Assume that a target has a range that varies with time as r(t)=r−vt, where r is the initial range (in meters) and v the radial velocity magnitude (in meters/sec). A positive v corresponds to a closing target. The received signals are given by  
                       x   k     ⁡     (   t   )       =     2   ⁢   α   ⁢           ⁢       s   k     ⁡     (     t   -       2   ⁢           ⁢     r   ⁡     (   t   )         c       )                     =     2   ⁢   α   ⁢           ⁢       s   k     ⁡     (         (     1   +       2   ⁢   υ     c       )     ⁢   t     -       2   ⁢   r     c       )                       =     2   ⁢     αcos   ⁡     (           ω   k     ⁡     (     1   +       2   ⁢   υ     c       )       ⁢   t     -       2   ⁢     ω   k     ⁢   r     c     +     ψ   k       )           ,                 (   2   )             
 
 where 2a is some unknown attenuation factor, and c is the speed of light. After multiplication by the transmitted waveform and low pass filtering, the signal becomes  
                   y   k     ⁡     (   t   )       =     α   ⁢           ⁢     cos   ⁡     (           2   ⁢     ω   k     ⁢   υ     c     ⁢   t     -       2   ⁢     ω   k     ⁢   r     c       )           ,           (   3   )             
 
 where use has been made of the formula cos(a) cos(b)=½cos(a−b)+½cos(a+b). If the frequencies f k  are close to each other, the periods of the waveforms in Equation 3 will be very close. Additionally, if the motion during one Doppler period is small, the phases of waveforms will not appreciably “slip” relative to each other. Thus, a comparison between the phases of the waveforms may be made. Note that in the development above, if a had in fact been associated with a complex attenuation, i.e. a phase shift, then this error is common to both phases and is therefore cancelled out when the phase difference is calculated. 
 
         [0047]     To perform the phase comparison, one measures y k (t) for a period of time and then takes the Fourier transform, often implemented as a fast Fourier transform (FFT), thereby obtaining integration gain against noise. The phase of the transform for the FFT bin corresponding to ω k  is given by  
               ϕ   k     =           -   2     ⁢     ω   k     ⁢   r     c     ⁢   mod   ⁢           ⁢   2   ⁢     π   .               (   4   )             
 
         [0048]     The difference in Fourier phase is  
                   ϕ   2     -     ϕ   1       =         2   ⁢     (       ω   2     -     ω   1       )     ⁢   r     c     ⁢   mod   ⁢           ⁢   2   ⁢   π       ⁢     
     ⁢       Δ   ϕ     =           -   4     ⁢   π   ⁢           ⁢     Δ   f     ⁢   r     c     ⁢   mod   ⁢           ⁢   2   ⁢     π   .     
     ⁢   where                 (   5   )                   Δ   ϕ     ⁢     =   Δ     ⁢       ϕ   2     -     ϕ   1         ⁢     
     ⁢       and   ⁢             ⁢             ⁢     Δ   f       ⁢     =   Δ     ⁢       f   2     -       f   1     .                 (   6   )             
 
         [0049]     To obtain an estimate for r, assuming r&lt;c/(2Δ f ) one obtains  
               0   ≤       4   ⁢     πΔ   f     ⁢   r     c     &lt;     2   ⁢   π       ,           (   7   )               0   ≤          Δ   ϕ          &lt;     2   ⁢     π   .               (   8   )             
 
         [0050]     Therefore, with the restriction r&lt;c/(2Δ f ), the phase difference is unambiguous, and one can solve for the range via  
               r   est     =         c   ⁢          Δ   ϕ              4   ⁢     πΔ   f         .             (   9   )             
 
         [0051]     The above description assumed simultaneous transmission of the two frequencies. Alternatively, the subject system uses a diplexing method, in which the frequencies are transmitted sequentially in a time multiplexed fashion. Note that the two received waveforms must be sampled synchronously to the change in transmit frequencies. Also, sufficient time must be allowed between the change in frequency and the sampling time so that the signal can propagate to the target and back.  
         [0052]     It is assumed in the prior art that the Doppler frequencies ω d −2ω k v/c (in radians) are small compared to the sampling rate. If this situation does not hold, then the resulting range error bias grows unacceptably large.  
         [0053]     With this as background, the subject invention discusses the application of the diplexing method to a system in which the Doppler frequencies can be much larger, up to the Nyquist frequency of the switching and sampling rate.  
         [0054]     Let f k  for k=1,2 denote each of the two frequencies used (in Hz). Let l=1,2 index the sum and difference channels (l=1 is the sum channel, l=2 is the difference channel).  
         [0055]     For all channels l=1,2 and for all frequencies k=1,2 let the time domain data samples be represented by y kl (n), n=1, . . . ,N. The data stream for each frequency/channel combination is sampled at a rate of f s  Hz and is transformed via a conventional windowed FFT  
                   Y   kl     ⁡     (   m   )       =       ∑     n   =   0       N   -   1       ⁢           ⁢       w   ⁡     (   n   )       ⁢       y   kl     ⁡     (   n   )       ⁢     ⅇ       -   j2π     ⁢           ⁢     mn   /   N               ,     
     ⁢     k   =   1     ,       2   ⁢           ⁢   l     =   1     ,       2   ⁢           ⁢   m     =       -   N     /   2       ,   …   ⁢           ,       N   /   2     -   1     ,           (   10   )             
 
 where w(n) is a window function. Alternatively, using well-known methods, a heavily zero-padded FFT may be used to give refined results in the processing to follow. Additionally, other well-known interpolation methods can be employed to further refine the results in the processing to follow. 
 
         [0056]     Next, the magnitude squared results of the FFTs are calculated and the results accumulated to obtain  
                 Z   ⁡     (   m   )       =       ∑     k   =   1     2     ⁢           ⁢       ∑     l   =   1     2     ⁢           ⁢              Y     k   ⁢           ⁢   l       ⁡     (   m   )            2           ,           ⁢     m   =       -   N     /   2       ,   …   ⁢           ,       N   /   2     -   1.             (   11   )             
 
         [0057]     The peak bin {tilde over (m)} of Z, such that Z({tilde over (m)})≧Z(m) is determined via a simple peak search. As is well known, the peak search is generally performed only over frequency regions where target returns can occur as determined by system design and target dynamics. Optionally, a refined peak may be calculated and used in the processing to follow. There are many well known methods to calculate refined peaks, such as a parabolic interpolation approach. For the present purposes the peak frequency is referred to as bin {tilde over (m)}. The Doppler frequency in Hz is determined as  
                     f   d     =       ω   d     /     (     2   ⁢   π     )                   =         m   ~     N     ⁢       f   s     .                     (   12   )             
 
         [0058]     In Equations 10 and 11, the frequency bin index m is centered around zero so that both approaching targets, positive Doppler, and receding targets, negative Doppler, can be properly phase corrected. From Equation 3 one sees that since the signals input to the FFT are real, the FFT output is conjugate symmetric, and thus an approach/recede ambiguity is present. As described in Equation 3, a method for resolving this ambiguity is to properly choose the transmitter frequencies so that the target range satisfies r&lt;c/(2Δ f ). Thus, the correct sign for the velocity can be determined, thereby determining the correct sign of {tilde over (m)}.  
         [0059]     With the peak FFT bin and the corresponding Doppler frequencies identified, we now come to the central idea of the subject invention. If the samples of the two frequencies are spaced apart by an amount of time T, equal to one-half of the sample period of the data streams corresponding to the two frequencies which are each sampled at a rate of f s  Hz, during this time period T, the target has moved a distance of vT. This results in a relative phase shift between the two data streams. This previously unaccounted-for phase shift can be determined by substituting T for t in the first term inside the cosine bracket in Equation 3:  
               Δ   c     =         2   ⁢   ω   ⁢           ⁢   υ   ⁢           ⁢   T     c     .             (   13   )             
        where ω can be taken as either ω 1  or ω 2 , since the difference ω 2 -ω 1  is small relative to c.        
 
         [0061]     Since the Doppler frequency is  107   d =2ωv/c (in radians), one has  
                     Δ   c     =       2   ⁢   ωυ   ⁢           ⁢   T     c                 =       ω   d     ⁢   T                 =     2   ⁢   π   ⁢           ⁢     f   d     ⁢   T                 =     2   ⁢   π   ⁢           ⁢   T   ⁢           ⁢       m   ~     N     ⁢     f   s                   =     2   ⁢     π   ⁡     (     1   /     (     2   ⁢     f   s       )       )       ⁢       m   ~     N     ⁢     f   s                   =       π   ⁢           ⁢     m   ~       N                   (   14   )             
 
         [0062]     The result is that  
               Δ   ϕ   ′     ⁢     =   Δ     ⁢       Δ   ϕ     +       π   ⁢           ⁢     m   ~       N               (   15   )             
 
 is the corrected phase that is used to estimate the range in  
               r   est     =         c   ⁢          Δ   ϕ   ′              4   ⁢     πΔ   f         .             (   16   )             
 
         [0063]     In the subject system, the two frequencies f 1  and f 2  are time-multiplexed fashion by the transmitter. The transmitted signal reflects off the object of interest, and is received by sum and difference channels, Rsum and Rdiff. Typical radar functions such as down conversion, low pass filtering and analog-to-digital converter (ADC) sampling are performed. Next, a demultiplexing operation is performed. This produces the time-domain data streams y 11 (n), . . . ,y 22 (n). Individual FFTs are performed according to Equation 10 to produce the frequency-domain data streams Y 11 (m), . . . , Y 22 (m). The squared magnitude of the FFT bins are then accumulated according to Equation 11 to produce Z(m), which is then peak searched to produce the index m for the largest peak. This index is used to retrieve the corresponding bins Y 11 ({tilde over (m)}), . . . , Y 22 ({tilde over (m)}).  
         [0064]     There are several approaches to processing these bins to derive the Fourier phase difference φ 2 -100  1 . One simple (but suboptimal approach) is to calculate 
 
φ k =angle( Y   k1 ( {tilde over (m)} )),  k= 1, 2  (17) 
 
 and then Δ φ  via Equation 6. A better approach is to optimally combine Y 11 ({tilde over (m)}), . . . ,Y 22  ({tilde over (m)}) and directly produce Δ 100 . In either case, Δ′ φ  is next calculated using Equation 15. Finally, the range is estimated using Equation 16. 
 
         [0065]     While the present invention has been described in connection with the preferred embodiments of the various figures, it is to be understood that other similar embodiments may be used or modifications or additions may be made to the described embodiment for performing the same function of the present invention without deviating therefrom. Therefore, the present invention should not be limited to any single embodiment, but rather construed in breadth and scope in accordance with the recitation of the appended claims.