Abstract:
Modern tactical radars frequently use phase shifters to electronically specify or steer the spatial position of the antenna beam without requiring mechanical motion of the antenna. These phase shifters can only be set correctly for a specific frequency. If a waveform is transmitted through the antenna which consists of multiple segments which differ in frequency or modulation from that frequency used to steer the position of the beam, errors are introduced into the monopulse measurement. These monopulse errors are reduced or eliminated by correction factors. The monopulse errors are corrected by pre-computed factors or terms which result from the differences in frequency and modulation used in the waveform from the frequency used to steer or position the beam. Correction is also provided for radar altitude. These correction factors are easily pre-computed and applied only when needed to minimize the computational requirements.

Description:
This invention was made with Government Support under Contract No. N00024-03-C-6110 awarded by the Department of the Navy. The Government has certain rights in this invention. 
    
    
     BACKGROUND 
     The present disclosure relates to radar monopulse signal processing. A monopulse signal processing system or arrangement determines the angle between the radar receive beam axis of a radio-frequency (RF) antenna and a line extending to the apparent source of a received RF signal. The received RF signal may originate as a “skin” or surface reflection of electromagnetic energy impinging on a radar “target”, or it may originate from a transmitter of a signal from the target itself, as might be the case with an identification transponder. Thus, both active and passive sources are included in the generic term “source.” 
     In the past, the term “radio frequencies” was interpreted to mean a limited range of frequencies, such as, for example, the range extending from about 20 KHz to 2 MHz. Those skilled in the art know that “radio” frequencies as now understood extends over the entire frequency spectrum, including those frequencies in the “microwave” and “millimeter-wave” regions, and up to, and including, light-wave frequencies. Many of these frequencies are very important for commercial purposes, as they include the frequencies at which radar systems, global positioning systems, satellite cellular communications and ordinary terrestrial cellphone systems operate. 
     Many modern tactical radars use phase shifters to electronically specify the spatial position of the antenna beam without requiring mechanical motion of the antenna. These phase shifters can only be set correctly for a specified frequency. If a waveform is transmitted through the antenna which differs in frequency from that used to steer the position of the beam, an error will be introduced into the monopulse measurements. A monopulse waveform may consist of many subpulses or segments, each of which can provide information about the location of the target. Each segment may have different modulation and different frequencies, which introduce monopulse errors. By proper application of correction factors these errors can be mitigated. 
       FIG. 1A  is a highly simplified representation of a monopulse radar system  8 . System  8  of  FIG. 1A  includes a monopulse transmitting and receiving antenna  10  which has a center  11 , an antenna plane  10   p , an antenna “broadside” axis which lies along a line  12  perpendicular to antenna plane  10   p , and a beam axis which can be steered away from broadside to lie along an another direction line  13 . In many reflector antenna systems, the lines  12  and  13  always coincide. In many phased array antennas the lines  12  and  13  coincide only for one beam pointing angle and diverge for all others. A signal source  14 , which may result from reflection, lies on a line  16  extending from the antenna center  11 . In order to assign directions from the location from which the signal source  14  propagates, it is necessary to assign coordinate axes for the antenna  10 . In  FIG. 1A , coordinate axis Y  12 , is the line extending from the center  11  perpendicular to the plane  10   p  of the array  10 . Two axes X and Z that are perpendicular to axis Y and to each other are defined in the antenna plane  10   p . These axes are labeled X and Z. By convention, the positive X axis is defined to the right looking outward from the antenna  10 . The positive Z direction is defined in the upward direction. 
     Spatial directions can be measured by direction angles. The direction angle α 0  in  FIG. 1A  is measured from the positive X axis to the commanded beam direction line  13 . The direction angle β 0  measured from the positive Z axis to the beam direction line  13 . The direction angles to the target  14  may be measured as departures from the direction angles α 0  and β 0  by the angular differences δα and δβ, respectively. That is, if the target direction lies along the line  16 , the departures from the beam direction line or axis  13  are labeled or designated δα and δβ. More commonly, angular departures are measured as differences in direction cosines. In the alpha a coordinate, the departure in the α angle cosine is denoted by
 
Δ u =cos α−cos α 0  
 
The departure in the beta angle cosine is denoted by
 
Δ v =cos β−cos β 0  
 
As viewed from the center  11  of antenna  10 , the target  14  is displaced from the beam axis  13  by δα from the α 0  direction and by δβ from the β 0  direction. The angular cosine departures Δu and Δv are determined in a monopulse system by monopulse signal processing performed upon the total antenna received signal, in complex envelope form comprised of three complex signals: Δα, Δβ, Σ. Signal Δα is obtained as the difference of the two vertical halves of the antenna received output. Signal Δβ is obtained as the difference of the two horizontal halves of the antenna received output. The Σ signal is the entire received output of the antenna. These three signals are conceptually separated from each other by circuits associated with the antenna, which separating circuits are illustrated as a block  6 . The three signals separated by block  6  are coupled to a monopulse signal processing system included in a receiver processing (PROC) portion  20   a  of radar  8 . The processing is performed on the complex envelopes. Given the antenna signals
 
     Σ, the total antenna output; 
     Δα, the difference of the half-antenna outputs corresponding to the α direction; and 
     Δβ, the difference of the half-antenna outputs corresponding to the β direction, the α monopulse ratio ρ α , and the β monopulse ratio ρ β  are formed as follows 
               ρ   α     =     Re   (     Δα   Σ     )                   ρ   β     =     Re   (     Δβ   Σ     )           
From these quantities, the increments in the direction cosines are obtained. These are, as indicated earlier
 
Δ u =cos α−cos α 0  
 
Δ v =cos β−cos β 0  
 
where, as indicated above, α 0  and β 0  are the commanded “steering” angles, and cos α 0  and cos β 0  are the corresponding angle cosines. Then Δu and Δv are obtained by insertion of ρ α  and ρ β  into odd degree polynomials
 
 P   α (ρ α )
 
and
 
 P   β (ρ β )
 
These polynomials are obtained in known fashion by antenna calibration.
 
     Also in the arrangement of  FIG. 1A , a transmitter Tx illustrated as a portion  20   b  interacts with the remainder of system  8  to transmit a coded signal including a plurality of subpulses. Transmitter portion  20   b  may use the antenna  10  for transmission, or it may use some other antenna, as known in the art. In general, the number of subpulses to be transmitted in each pulse can be selected arbitrarily. The transmitted pulse is divided into a number of subpulses. Each subpulse, when properly isolated from the other subpulses and separately filtered, is selected to have the necessary bandwidth and transmitted power to satisfy the system bandwidth required for a specific or selected range resolution and sufficient power to satisfy the desired target detection range. The number of subpulses is limited, in general, by the beam-steering capability in terms of the number of beam positions that can be achieved within the constraints of the available equipment. In a particular embodiment of the disclosure described herein, the number of subpulses per pulse is selected to be four as illustrated in  FIG. 1B . More or fewer subpulses per pulse may be appropriate for systems with lesser or greater constraints. The purpose of the use of plural subpulses per pulse is to obtain several or plural values of Δu and Δv, which can be averaged in order to mitigate or ameliorate perturbations occasioned by factors including noise. The transmitter portion  20   b  of  FIG. 1A  also includes a beam steering controller BSC that provides the angle steering command to the antenna  10 . The beam steering controller and the waveform generator are illustrated as blocks  48  and  52 , respectively, in  FIG. 2 . 
     For purposes of explanation, four subpulses are assumed.  FIG. 1B  is an amplitude-time plot  21  illustrating a time-sequential set of four subpulses designated  1 ,  2 ,  3 , and  4 . The subpulses differ from each other in frequency. More particularly, subpulse  1  may be at a frequency of f BAND −A MHz, subpulse  2  may be at a frequency of f BAND −B MHz, subpulse  3  may be at a frequency of f BAND +C MHz, and subpulse  4  may be at a frequency of f BAND +D MHz, where frequencies A, B, C, and D are different offset frequencies, much smaller or less than the electromagnetic carrier frequency used for steering. The frequency f BAND  may have a multiplicity of values. A monopulse system may also operate in a passive mode in which the radar antenna acts only in a receive mode. In this passive mode, the radar acts as a passive receiver of transmissions from the target acting as source transponder, where the angle of arrival coordinates are determined by the radar in its receive mode. In such a passive or receive-only radar mode, the arriving electromagnetic wave will have a signal structure as a function of frequency f Band −A MHz, where the value of f BAND  is preselected by auxiliary communications with the radiating source. The signals structure in the passive radar mode is illustrated in  FIG. 1C . 
       FIG. 2  is a simplified block diagram illustrating details of receiving and monopulse processing system  20   a  of  FIG. 1A . A signal processor in accordance with this disclosure may operate in either an analog or digital manner, in accordance with its construction. However, a digital processor is preferred. 
     The monopulse processing system  20   a  of  FIG. 2  includes a receiver and a matched filter system, illustrated as a block  22 , for proper filtering of the received signal. The output from the receiver and matched filter system  22  is generated on a set of paths designated together as  24 , and may be viewed as including three complex envelope signals, namely Δα, Δβ, and Σ. These three complex envelope signals are the outputs from system  22  and are coupled by paths  24  to a bank or set  32  of three complex analog-to-digital converters (ADCs)  32   1 ,  32   2 ,  32   3 . In response to timing signals from a controller or radar control computer  90 , the bank  32  of complex A-to-D converters simultaneously converts the complex envelopes of each of the three signals Δα, Δβ, and Σ from the matched filter system  22  into three separate complex binary (digital) values. In a typical system, the bank of A-to-D converters  32  may provide each complex envelope component in the form of the components (real and imaginary) of the complex envelope, and any number of bits may be used. Each time the controller  90  activates the bank  32  of A-to-D converters, each of these converters provides a new complex value at its output and on a signal path  28 . More particularly, the digitized output from ADC  32   1  is applied to a path  28   1 , the digitized output from ADC  32   2  is applied to a path  28   2 , and the digitized output from ADC  32   3  is applied to a path  28   3 . As a group, these A-to-D converters together provide a new set of these three complex envelope values each time a conversion is commanded. If digital beam forming is employed in a phased array antenna, then the digital beam former provides the same three complex outputs from the ADCs  32 . The complex envelope values from ADC  32   2 , representing the “sum” or Σ signal, are provided to a target detection processor  34  for use in determining whether a target is present in the portion of the return signal to which these digital values correspond. The complex digital values of the three outputs of the analog to digital converters ADC  32   1 ,  32   2 ,  32   3  are provided to a monopulse signal processing computer designated generally as  40 , which is illustrated as including a “prior art” portion  40   a  joined by a path  41  to a portion  40   b  according to an aspect of the disclosure, which together provide as outputs values of Δu TAR =Δ cos α and Δv TAR =Δ cos β, which are the corrected angle cosines between the line  16  extending to the target  14  and the antenna beam axis  13  of  FIG. 1  in the alpha and β directions, respectively. The direction angles α and β may be termed “traverse” and “co-elevation” angles (sometimes known as azimuth and elevation), respectively. The corrected values of u and v, the direction cosines, are applied to tracker  95  as Δu TAR  and Δv TAR . 
     Target detection processor  34  of  FIG. 2  determines whether the received signal values indicate the presence of a target. If they do, then the detection processor  34  provides to tracker  95  a detected-target signal which specifies the position of the target. If they do not indicate the presence of a target, then either no signal or a no-target signal is provided to tracker  95  by detection processor  34 . The monopulse processor provides angular cosine coordinates to target tracker  95 , which tracks the locations of the various targets in known manner with the aid of signals from target detection processor  34  and control signals from controller  90 . 
     The monopulse signal processing computer  40  of  FIG. 2  ultimately provides to tracker  95  a set of target angle coordinates, in the form of angle cosines Δu TAR  and Δv TAR , for each set of received input values. These target angle signals specify the angle between the beam axis and the target in the event that the processed values include target energy. If tracker  95  receives a detected-target signal in conjunction with a set of the target angle signals, then tracker  95  determines the target position from the known beam position in combination with the determined range and direction cosines of the target. Tracker  95  then determines whether this target location is a newly detected target or is the new position of an old or previously identified target. If it is a new target, the tracker establishes a new target track to begin following this target. If it is a new position of an old target, then this new position is used to update the track on that old target by providing this new position as the most recent target location. When no detected-target signal or a no-target signal is received in conjunction with a set of target angle cosine signals, tracker  95  discards those angle cosine signals without further processing. Both the target detection processor  34  and the tracker  95  are conventional, aspects of the present disclosure being concerned with the monopulse signal processing  40  which converts the received digital values to their corresponding monopulse ratios and then to target angles cosines. 
     Significant discrepancies or errors have been found when comparing the target angle as determined by skin reflections with those determined by an active source on the target, such a transponder. Improved or alternative monopulse processing is desired. 
     SUMMARY 
     Modern tactical radars frequently use phase shifters to electronically specify or steer the spatial position of the antenna beam without requiring mechanical motion of the antenna. These phase shifters can only be set correctly for a specific frequency. If a waveform is transmitted through the antenna which consists of multiple segments which differ in frequency or modulation from that frequency used to steer the position of the beam, errors are introduced into the monopulse measurement. These monopulse errors are reduced or eliminated by correction factors. The monopulse errors are corrected by pre-computed factors or terms which result from the differences in frequency and modulation used in the waveform from the frequency used to steer or position the beam. Correction is also provided for radar altitude. These correction factors are easily pre-computed and applied only when needed to minimize the computational requirements. 
     A radar system according to an aspect of the disclosure includes an antenna and an antenna beam direction controller, and also includes a transmitter for transmitting subpulses at nominal frequencies F but with modulation which may result in an actual average frequency different from frequencies F. A receiver receives monopulse signals from a target, and generates rho signals, each of which rho signals is the real component of one of eight complex monopulse ratios, four of which represent the Δα angle and the other four of which represent the Δβ angle. A polynomial processor is coupled to the receiver, for producing uncorrected cosine differences of angular offsets of the target from the commanded beam pointing direction
 
cos(Δ 0 +δα)−cos α 0  
 
cos(β 0 +δβ)−cos β 0 ;
 
The radar system also includes a multiplicative correction processor for multiplying the uncorrected cosine differences by a factor including (a) the frequency (F) at which the polynomial is determined and (b) the actual average frequency (f act ) of the particular subpulse, to thereby generate multiplied cosine differences for each subpulse. A summing correction processor is coupled to the multiplicative correction processor for adding to the multiplied cosine differences a correction term for compensating for apparent movement of the target arising in a particular direction from beam movement at each subpulse, to thereby generate a plurality of multiplicatively and additively compensated angles or angle cosine difference signals representing the direction of the target. An averaging arrangement is coupled to the summing correction processor for averaging the angles or angle cosine difference signals over all subpulses of a pulse to thereby produce averaged corrected angular difference signals representative of the location of the target. In a particular embodiment of this aspect of the disclosure, the summing correction processor further adds to the multiplied cosine differences a correction term for compensating for the actual speed of light in the environment of the radar. The correction term for compensating for apparent movement is in the form of one of
 
               (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢   u   ⁢           ⁢   and   ⁢           ⁢     (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢     v   .           
In a version of this embodiment, a squint corrector is coupled to the averaging arrangement, for summing a squint correction with the averaged angular difference signals to generate target angular information representative of corrected direction of the target.
 
     A radar system according to another aspect of the disclosure includes a transmitter, a waveform generator coupled to the transmitter for driving the transmitter with sets of sequential pulses. Each pulse of each of the sets is jump-frequency modulated relative to other pulses of the set, to thereby define a plurality of subpulses for each of the sets of sequential pulses. The radar system comprises an antenna coupled to the transmitter, for transmitting electromagnetic signals in response to the sets of sequential subpulses, and for, in the presence of a target, generating separate return signals for each of the sequential subpulses of each set. A receiver receives the return signals and generates received signals. The received signals include a separate digital signal subpulse for each of the separate return signals. A splitting arrangement is coupled to at least one of the antenna and the receiver for splitting the received signals into co-elevation and traverse difference components and a sum component. A monopulse ratio processor is coupled to the splitting arrangement for combining the co-elevation and traverse difference components and the sum component signal to provide a monopulse ratio. A monopulse correction processor is coupled to the monopulse ratio processor for correcting the direction of the target with corrections based on at least one subpulse frequency and one of active and passive operating modes. 
     A radar system according to an aspect of the disclosure includes a transmitter and a waveform generator coupled to the transmitter for driving the transmitter with sets of sequential pulses. Each pulse of each of the sets is jump-frequency modulated relative to other pulses of the set. The radar system comprises an antenna coupled to the transmitter, for transmitting electromagnetic signals in response to the sets of sequential pulses, and a receiver for, in the presence of a target, generating separate return signals for each of the sequential pulses of each set. The receiver receives the return signals and generates received signals, which include a separate digital signal for each of the separate return signals. A splitting arrangement is coupled to the receiver for splitting the received signals into vertical and horizontal difference components and a sum component; and 
     a monopulse processor coupled to the splitting arrangement for combining the vertical and horizontal difference components and a sum component signal to provide a monopulse ratio; and 
     a monopulse ratio processor coupled to the monopulse processor for generating corrected direction of the target cos α target  by 
                     cos   ⁢           ⁢     α   target       =       cos   ⁢           ⁢     α   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   α     ⁡     (     ρ     α   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     α   des         ]         +     Δ   ⁢           ⁢     u   sq                 (   1   )               
where:
 
     cos α des =designated cos α from the beam steering controller (BSC); 
     n sp =number of subpulses; 
     F 1 =nominal frequency for which the α polynomial P α (ρ) was obtained; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f BAND +subpulse average frequency; 
     ρ αi =real part of i-th monopulse ratio for subpulse i; 
     P β (ρ)=alpha polynomial for the α angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δu sq =squint correction for cos α, as shown in  FIG. 4A , and obtained in an antenna calibration facility, 
     and for generating corrected direction of target cos β target  by 
                     cos   ⁢           ⁢     β   target       =       cos   ⁢           ⁢     β   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   β     ⁡     (     ρ     β   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     β   des         ]         +     Δ   ⁢           ⁢     v   sq                 (   2   )               
where:
 
     cos β des =designated cos β from the beam steering controller (BSC) indicated in  FIG. 2 ; 
     n sp =number of subpulses; 
     F 1 =nominal frequency for which the β polynomial P β (ρ) was obtained; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f Band +subpulse average frequency; 
     ρ βi =real part of i-th monopulse ratio for subpulse i; 
     P β (ρ)=beta polynomial for the β angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δv sq =squint correction for cos β, and obtained in an antenna calibration facility. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a simplified block diagram of a monopulse antenna and radar system including a transmitter and signal processor,  FIG. 1B  is a simplified amplitude/time plot of a transmitted waveform, and  FIG. 1C  is a simplified amplitude/time plot of the received waveform in the radar passive mode; 
         FIG. 2  is a simplified block diagram of a monopulse signal processing system of  FIG. 1  which includes both prior-art monopulse signal processing and processing according to aspects of the disclosure; 
         FIG. 3  is a simplified block diagram illustrating the prior-art monopulse signal processing of  FIG. 2 , and in particular the processing of the complex envelopes of the traverse and co-elevation signals; and 
         FIGS. 4A and 4B  together represent a simplified block diagram illustrating processing of signals of  FIG. 3  according to aspects of the disclosure, for determining the angle cosines of the target from the commanded or directed angle cosine of the antenna of  FIG. 1 . 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 3  represents a simplified block diagram illustrating some prior art processing of monopulse processor  40   a  of  FIG. 2 , for generating a plurality of 2N pairs of rho (ρ) signals, where N=4. Each ρ is the real component of one ratio out of four pairs of complex monopulse ratios (alternatively, each p may be the imaginary component). Four of these represent the Δα angle and the other four represent the Δβ angle. Thus, each rho is the real component of one of the eight complex monopulse ratios. As mentioned, the number of subpulses may be selected at will. In  FIG. 3 , the complex digitized receiver outputs are applied to monopulse processor  40   a  over a set  28  of complex buses or paths. More particularly, the complex Δα and Δβ signals are applied over paths  28   1  and  28   3 , respectively, to digital subpulse filters  310   a  and  310   c , respectively. The complex Σ signal is applied by way of path  28   2  to digital subpulse filter  310   b . Each output of subpulse filters  310   a  and  310 , is coupled with the corresponding output of digital subpulse filter  310   b  in a rho block of a set  314  of rho blocks, to produce the real part of the corresponding ratio ρ α . This results, at the outputs of the rho blocks of set  314 , in the quantities 
               ρ     α   1       =     Re   (       Δα   1       Σ   1       )                   ρ     α   2       =     Re   (       Δα   2       Σ   2       )                   ρ     α   3       =     Re   (       Δα   3       Σ   3       )                   ρ     α   4       =     Re   (       Δα   4       Σ   4       )                   ρ     β   1       =     Re   (       Δβ   1       Σ   1       )                   ρ     β   2       =     Re   (       Δβ   2       Σ   2       )                   ρ     β   3       =     Re   (       Δβ   3       Σ   3       )                   ρ     β   4       =     Re   (       Δβ   4       Σ   4       )           
More particularly, the Δα 1  output on path  312   a   1  of subpulse filter  310   a  is coupled with the Σ 1  output on path  312   b   1  of subpulse filter  310   b  in a rho block  314   a   1  to produce
 
               ρ     α   1       =     Re   (       Δα   1       Σ   1       )           
The other ρ&#39;s are formed in a similar way. Thus, the Δα 2  output on path  312   a   2  of subpulse filter  310   a  is coupled with the Σ 2  output on path  312   b   2  of subpulse filter  310   b  in a rho block  314   a   2  to produce
 
               ρ     α   2       =       Re   (       Δα   2       Σ   2       )     .           
Similarly, the Δα 3  output on path  312   a   3  of subpulse filter  310   a  is coupled with the Σ 3  output on path  312   b   3  of subpulse filter  310   b  in a rho block  314   a   3  to produce
 
                 ρ     α   3       =     Re   (       Δα   3       Σ   3       )       ,         
and the Δα 4  output on path  312   a   4  of subpulse filter  310   a  is coupled with the Σ 4  output on path  312   b   4  of subpulse filter  310   b  in a rho block  314   a   4  to produce
 
               ρ     α   4       =       Re   (       Δα   4       Σ   4       )     .           
Also, the Δβ 1  output on path  312   c   1  of subpulse filter  310   c  is coupled with the Σ 1  output on path  312   b   1  of subpulse filter  310   b  in a rho block  314   b   1  to produce
 
                 ρ     β   1       =     Re   (       Δβ   1       Σ   1       )       ,         
the Δβ 2  output on path  312   c   2  of subpulse filter  310   c  is coupled with the Σ 2  output on path  312   b   2  of subpulse filter  310   b  in a rho block  314   b   2  to produce
 
                 ρ     β   2       =     Re   (       Δβ   2       Σ   2       )       ,         
the Δβ 3  output on path  312   c   3  of subpulse filter  310   c  is coupled with the Σ 3  output on path  312   b   3  of subpulse filter  310   b  in a rho block  314   b   3  to produce
 
                 ρ     β   3       =     Re   (       Δβ   3       Σ   3       )       ,         
and the Δβ 4  output on path  312   c   4  of subpulse filter  310 , is coupled with the Σ 4  output on path  312   b   4  of subpulse filter  310   b  in a rho block  314   b   4  to produce
 
               ρ     β   4       =       Re   (       Δβ   4       Σ   4       )     .           
Put another way, Δα 1  of  312   a   1  is coupled with Σ 1  of  312   b   1  to produce
 
                 ρ     α   1       =     Re   (       Δα   1       Σ   1       )       ,         
Δα 2  of  312   a   2  is coupled with Σ 2  of  312   b   2  to produce
 
                 ρ     α   2       =     Re   (       Δα   2       Σ   2       )       ,         
Δα 3  of  312   a   3  is coupled with Σ 3  of  312   b   3  to produce
 
                 ρ     α   3       =     Re   (       Δα   3       Σ   3       )       ,         
Δα 4  of  312   a   4  is coupled with Σ 4  of  312   b   4  to produce
 
     
       
         
           
             
               
                 ρ 
                 
                   α 
                   4 
                 
               
               = 
               
                 Re 
                 ( 
                 
                   
                     Δα 
                     4 
                   
                   
                     Σ 
                     4 
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     Δβ 1  of  312   c   1  is coupled with Σ 1  of  312   b   1  to produce 
                 ρ     β   1       =     Re   (       Δβ   1       Σ   1       )       ,         
Δβ 2  of  312   c   2  is coupled with Σ 2  of  312   b   2  to produce
 
                 ρ     β   2       =     Re   (       Δβ   2       Σ   2       )       ,         
Δβ 3  of  312   c   3  is coupled with Σ 3  of  312   b   3  to produce
 
                 ρ     β   3       =     Re   (       Δβ   3       Σ   3       )       ,         
and Δβ 4  of  312   c   4  is coupled with Σ 4  of  312   b   4  to produce
 
                 ρ     β   4       =     Re   (       Δβ   4       Σ   4       )       ,         
Thus, there are a total of eight real values of the rhos. Four of them:
 
ρ α     1   ,ρ α     2   ,ρ α     3   ,ρ α     4    
 
relate to the direction angle α (the traverse angle) and the other four:
 
ρ β     1   ,ρ β     2   ,ρ β     3   ,ρ β     4    
 
relate to the direction angle β (the co-elevation angle). The rho signals are coupled by way of a path  41  to portion  40  of the monopulse processing. More particularly, the ρ α     1   , ρ α     2   , ρ α     3   , ρ α     4    outputs are coupled by way of a path  41   a  to portion  40   b   1  of processing  40 , and the ρ β     1   , ρ β     2   , ρ β     3   , ρ β     4    outputs of rho blocks  314   b  are coupled by way of a path  41   b  to portion  40   b   2  of processing  40 .  FIG. 4A  illustrates an alpha portion of monopulse processing block  40 , and  FIG. 4B  illustrates a beta portion of monopulse processing block  40 .
 
       FIGS. 4A and 4B  together illustrate details of correction processor  40   b  of  FIG. 2 .  FIG. 4A  relates to details of α processing  40   b   1 , and  FIG. 4B  relates to details of β processing  40   b   2 .  FIGS. 4A and 4B  together illustrate processing in accordance with aspects of the disclosure of the rhos produced in the arrangement of  FIG. 3 . As mentioned, each ρ is the real component of one of eight complex monopulse ratios, four of which represent the Δα angle and the other four of which represent the Δβ angle.  FIG. 4A  illustrates a set  410   a  of polynomial blocks, and  FIG. 4B  illustrates a set  410   b  of polynomial blocks. In general, each polynomial block of sets  410   a  and  410   b  of polynomial blocks produces a cosine of an angle offset from the commanded beam pointing angle cosine, and the outputs of the polynomial blocks provide the cosine differences
 
cos(α 0 +δ α )−cos α 0  
 
cos(β 0 +δβ)−cos β 0  
 
Each of these cosine differences defines an uncorrected direction of the target (along target line  16 ) relative to the beam direction (along line  13 ).
 
     The arrangements of  FIGS. 4A and 4B  implement two types of corrections.  FIG. 4A  implements the correction cosines to the alpha angle cosine and  FIG. 4B  implements the correction cosine to the beta angle cosine. The corrections in either case are applied to the corresponding polynomials, the P α &#39;s and the P β &#39;s, where each rho is the real component of one of the eight complex monopulse ratios. The correction multiplications and additions are the same for both angles, but the actual values differ based on the direction of the target relative to the beam direction. The correction operation for the cosine of the alpha angle is shown in  FIG. 4A  and the correction operation for the cosine of the β angle is shown in  FIG. 4B . The description of the operations is the same for  FIG. 4A  and  FIG. 4B . The only difference between the two corrections is the set of monopulse polynomials to which the corrections are applied. For the α direction the set of monopulse polynomials are
 
 P   α (ρ α1 ), P   α (ρ α2 ), P   α (ρ α3 ), P   α (ρ α4 ),
 
and for the β direction the set of monopulse polynomials are
 
 P   β (ρ β1 ), P   β (ρ β3 ), P   β (ρ β4 ).
 
     As illustrated in  FIG. 4A , the alpha rhos are applied by paths  41   a  to polynomial blocks of a set  410   a  of polynomial blocks. More particularly, ρ α1  is applied to a polynomial block  410   a   1  for generating the output P α (ρ α1 ), ρ α2  is applied to a polynomial block  410   a   2  for generating the output P α (ρ α2 ), ρ α3  is applied to a polynomial block  410   a   3  for generating the output P α (ρ α3 ), and ρ α4  is applied to a polynomial block  410   a   4  for generating the output P α (ρ α4 ). As illustrated in  FIG. 4B , the beta rhos are applied by way of paths  41   b  to polynomial blocks of a set  410   b  of polynomial blocks. More particularly, ρ β1  is applied to a polynomial block  410   b   1  for generating the output P β (ρ β1 ), ρ β2  is applied to a polynomial block  410   b   2  for generating the output P β (ρ β2 ), ρ β3  is applied to a polynomial block  410   b   3  for generating the output P β (ρ β3 ), and ρ β4  is applied to a polynomial block  410   a   4  for generating the output P β (ρ β4 ). 
     According to an aspect of the disclosure, additive and multiplicative corrections are made to the monopulse angle cosines (Δu, Δv) in correction blocks  40   b   1  and  40   b   2  of  FIGS. 4A and 4B , respectively, to improve the accuracy of the monopulse angles by correcting for the effects of frequency deviation and for the actual speed of light. The monopulse angle cosines for each subpulse (deviation angles δα and δβ of  FIG. 1A ) are averaged together in averaging blocks  411   a  and  411   b  after application of corrections. Ordinary prior-art squint corrections are made to the averaged corrected monopulse angle cosines in summing blocks  416   a  and  416   b.    
     The multiplicative corrections are made by sets  412   a  and  412   b  of multipliers in  FIGS. 4A and 4B , respectively. Each multiplier of sets  412   a  and  412   b  of multipliers receives a correction input at one of its input ports, because the polynomials which are used in the polynomial blocks are predicated on operation at one nominal frequency, but are used at frequencies which differ from the nominal. The multiplicative correction inputs correct for the effects of frequency in the polynomial blocks. The multiplicative correction inputs are of the form 
                 F   1       f   act       ,         
where F 1  is the frequency at which the polynomial is determined, and f act  is the actual average frequency of the particular subpulse. The value of f act  may differ from the nominal frequency of a subpulse if the modulation causes an average frequency shift.
 
     The sum or additive correction is applied to each summing circuit of a set  414  of summing circuits of  FIGS. 4A and 4B  for each subpulse. The additive corrections are of the form 
               (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢   u   ⁢           ⁢   or   ⁢           ⁢     (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢     v   .           
The additive corrections are off-broadside, off-frequency corrections. The sequential subpulses are transmitted in directions established by the beam steering control (BSC) signals. Each subpulse of a sequence will be directed in a slightly different direction than the previous and subsequent subpulse. Consequently, the apparent location of the target moves with time (or correspondingly frequency) if not corrected. The additive correction compensates for this deviation. The additive correction includes compensation for the change in frequency imposed upon the subpulses and also for the velocity of light. The speed of light which is used is that for transmission through the atmosphere rather than assuming vacuum speed of light. This allows a radar to correct for the actual light speed regardless of its elevation.
 
     As mentioned, additive and multiplicative corrections are made to the monopulse angle cosines according to an aspect of the disclosure to improve the accuracy of the monopulse angles. The monopulse angles cosines for each subpulse (deviation angles) are averaged together after application of corrections. More particularly, alpha corrections  40   b   1  of  FIG. 4A  are applied to the polynomial outputs
 
 P   α (ρ α     1   ), P   α (ρ α     2   ), P   α (ρ α     3   ), P   α (ρ α     4   )
 
as illustrated in  FIG. 4A , and beta corrections  40   b   2  are applied to the polynomial outputs
 
 P   β (ρ β     1   ), P   β (ρ β     2   ), P   β (ρ β     3   ), P   β (ρ β     4   )
 
as illustrated in  FIG. 4B . The polynomials P α  and P β  are odd functions of their arguments:
 
 P   α (ρ)=− P   α (−ρ)
 
and
 
 P   β (ρ)=− P   β (−ρ)
 
     As shown in  FIGS. 4A and 4B , the alpha corrections  40   b   1  and the beta corrections  40   b   2  include both multiplicative (X) and additive (Σ) corrections. More particularly, the P α (ρ α1 ) output of polynomial block  410   a   1  of  FIG. 4A  is applied to a multiplier  412   a   1  of a set  412   a  of multipliers, for multiplication by 
                 F   1       f     ACT   A         ,         
the P α (ρ α2 ) output of polynomial block  410   a   2  of  FIG. 4A  is applied to a multiplier  412   a   2  of set  412   a  of multipliers, for multiplication by
 
                 F   1       f     ACT   B         ,         
the P α (ρ α3 ) output of polynomial block  410   a   3  of  FIG. 4A  is applied to a multiplier  412   a   3  of set  412   a  of multipliers, for multiplication by
 
                 F   1       f     ACT   C         ,         
and the P α (ρ α4 ) output of polynomial block  410   a   4  of  FIG. 4A  is applied to a multiplier  412   a   4  of set  412   a  of multipliers, for multiplication by
 
                 F   1       f     ACT   D         .         
Similarly, the P β (ρ β1 ) output of polynomial block  410   b   1  of  FIG. 4B  is applied to a multiplier  412   b   1  of a set  412   b  of multipliers, for multiplication by
 
                 F   1       f     ACT   A         ,         
the P β (ρ β2 ) output of polynomial block  410   b   2  of  FIG. 4B  is applied to a multiplier  412   b   2  of set  412   b  of multipliers, for multiplication by,
 
                 F   1       f     ACT   B         ,         
the P β (ρ β3 ) output of polynomial block  410   b   3  of  FIG. 4B  is applied to a multiplier  412   b   3  of set  412   b  of multipliers, for multiplication by,
 
               F   1       f     ACT   C             
and the P β (ρ β4 ) output of polynomial block  410   b   4  of  FIG. 4B  is applied to a multiplier  412   b   4  of set  412   b  of multipliers, for multiplication by
 
                 F   1       f     ACT   D         .         
Following the multiplicative corrections in sets  412   a  and  412   b  of multipliers, the multiplied outputs of the sets  412   a  and  412   b  of multipliers are applied to corresponding sets  414   a  and  414   b  of adding or summing (Σ) circuits.
 
     The multiplied output of multiplier  412   a   1  of  FIG. 4A  is applied to a first input port of a summing circuit  414   a   1 , which receives at its second input port the quantity 
               (           f   Band     ⁢   c         f   actA     ⁢     c   0         -   1     )     ⁢     u   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a first subpulse and for the frequency of the first subpulse, the multiplied output of multiplier  412   a   2  is applied to a first input port of a summing circuit  414   a   2 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   B       ⁢     c   0         -   1     )     ⁢     u   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a second subpulse and for the frequency of the second subpulse, the multiplied output of multiplier  412   a   3  is applied to a first input port of a summing circuit  414   a   3 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   C       ⁢     c   0         -   1     )     ⁢     u   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a third subpulse and for the frequency of the third subpulse, and the multiplied output of multiplier  412   a   4  is applied to a first input port of a summing circuit  414   a   4 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   D       ⁢     c   0         -   1     )     ⁢     u   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a fourth subpulse and for the frequency of the fourth subpulse, which represents corrections to the value of the polynomial for a fourth subpulse. Also, the multiplied output of multiplier  412   b   1  of  FIG. 4B  is applied to a first input port of a summing circuit  414   b   1 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   A       ⁢     c   0         -   1     )     ⁢     v   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a first subpulse and for the frequency of the first subpulse, the multiplied output of multiplier  412   b   2  is applied to a first input port of a summing circuit  414   b   2 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   B       ⁢     c   0         -   1     )     ⁢     v   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a second subpulse and for the frequency of the second subpulse, the multiplied output of multiplier  412   b   3  is applied to a first input port of a summing circuit  414   b   3 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   C       ⁢     c   0         -   1     )     ⁢     v   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a third subpulse and for the frequency of the third subpulse, and the multiplied output of multiplier  412   b   4  is applied to a first input port of a summing circuit  414   b   4 , which receives at its second input port the quantity
 
               (           f   BAND     ⁢   c         f     act   D       ⁢     c   0         -   1     )     ⁢     v   0           
to thereby produce a summed output, which represents corrections to the value of the polynomial for a fourth subpulse and for the frequency of the fourth subpulse.
 
     The summed outputs from the summing circuits of sets  414   a  and  414   b  of summing circuits of  FIGS. 4A and 4B  are applied to averaging circuits illustrated as  411   a  and  411   b  of  FIGS. 4A and 4B , respectively. Averaging circuits  411   a  and  411   b  each generate or form a mean or average value of the applied summed outputs. The averaged corrections are then summed with well-known antenna squint corrections in summing blocks  416   a  and  416   b . The squint-corrected target angle cosine delta is the corrected target angle cosine offset from commanded beam pointing direction. The target tracker  95  of  FIG. 2  receives the corrected target angle cosine offset information from correction processor  40   b  and sums the correction with the beam pointing direction to establish the estimated target direction. 
     According to an aspect of the disclosure, the monopulse processing solves for cos α target  which is the cosine of the alpha angle of the target  14  measured from the array X axis using the equation 
                     cos   ⁢           ⁢     α   target       =       cos   ⁢           ⁢     α   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   α     ⁡     (     ρ     α   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     α   des         ]         +     Δ   ⁢           ⁢     u   sq                 (   1   )               
where:
 
     cos α des =designated cos α from the beam steering controller (BSC) indicated in  FIG. 2 ; 
     n sp =number of subpulses; 
     F 1 =nominal frequency for which the polynomials are obtained; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f Band +subpulse average frequency; 
     f actA , f actB , f actC , f actD. =actual average frequencies of subpulses A, B, C, D; the value of f act  may differ from the nominal frequency of a subpulse if the modulation causes an average frequency shift; 
     ρ αi =real part of i-th monopulse ratio for subpulse i; 
     P α (ρ)=alpha polynomial for the α angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δu sq =squint correction for cos α, and obtained in an antenna calibration facility. 
     In a particular application, n sp =4, but in another application, n sp  may be a different number. 
     In a particular application P α (ρ) is an odd polynomial of degree 7 evaluated at argument ρ, but polynomials of different degree may be used. Thus, in the embodiment with 4 subpulses, the polynomial P α  would have four values: 
     P α (ρ α1 ), P α (ρ α2 ), P α (ρ α3 ), P α (ρ α4 ) and 
     f BAND =the nominal frequency used for transmission for a particular set of transmitted pulses; in a particular application, f BAND  may have plural or many values. 
     The monopulse processing also solves for cos β target  which is the cosine of the beta angle of the target  14  measured from the array Y axis using the equation 
                     cos   ⁢           ⁢     β   target       =       cos   ⁢           ⁢     β   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   β     ⁡     (     ρ     β   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     β   des         ]         +     Δ   ⁢           ⁢     v   sq                 (   2   )               
where:
 
     cos β des =designated cos β from the beam steering controller (BSC) indicated in  FIG. 2 ; 
     n sp =number of subpulses; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f BAND +subpulse average frequency; 
     ρ βi =real part of i-th monopulse ratio for subpulse i; 
     P β (ρ)=beta polynomial for the β angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δv sq =squint correction for cos β, and obtained in an antenna calibration facility. 
     Angular squint arises from imperfections in manufacture of antenna radiating elements and in their locations in an antenna array. 
     The resulting angle cosines are applied to a conventional target tracker, as known in the art. 
     While the description and analysis of the geometry associated with the radar and target is couched in terms of direction cosines, those skilled in the art understand that the description could instead be in terms of angles. Those skilled in the art using analysis based on angles will generate equations which may be different in form from those set forth herein, but which may be totally equivalent in principle. 
     A radar system ( 8 ) according to an aspect of the disclosure includes an antenna ( 10 ) and an antenna beam direction controller ( 90 ), and also includes a transmitter ( 20   b ) for transmitting subpulses at nominal frequencies F but with modulation which may result in an actual average frequency different from frequencies F. A receiver ( 20   a ;  40   a ) receives monopulse signals from a target ( 14 ), and generates rho (ρ) signals, each of which rho (ρ) signals is the real component of one of eight complex monopulse ratios, four of which represent the Δα angle and the other four of which represent the Δβ angle. A polynomial processor ( 410   a ,  410   b ) is coupled to the receiver ( 20   a ;  40   a ), for producing uncorrected cosine differences of angular offsets of the target ( 14 ) from the commanded beam pointing direction ( 13 )
 
cos(α 0 +δα)−cos α 0  
 
cos(β 0 +δβ)−cos β 0 ;
 
The radar system ( 8 ) also includes a multiplicative correction processor ( 412   a ,  412   b ) for multiplying the uncorrected cosine differences by a factor including (a) the frequency (F) at which the polynomial is determined and (b) the actual average frequency (f act ) of the particular subpulse, to thereby generate multiplied cosine differences for each subpulse. A summing correction processor ( 414   a ,  414   b ) is coupled to the multiplicative correction processor ( 412   a ,  412   b ) for adding to the multiplied cosine differences a correction term for compensating for apparent movement of the target arising in a particular direction from beam movement at each subpulse, to thereby generate a plurality of multiplicatively and additively compensated angles or angle cosine difference signals representing the direction of the target ( 14 ). An averaging arrangement is coupled to the summing correction processor for averaging the angles or angle cosine difference signals over all subpulses of a pulse to thereby produce averaged corrected angular difference signals representative of the location of the target. In a particular embodiment of this aspect of the disclosure, the summing correction processor further adds to the multiplied cosine differences a correction term for compensating for the actual speed of light in the environment of the radar. The correction term for compensating for apparent movement is in the form of one of
 
                 (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢   u   ⁢           ⁢   and   ⁢           ⁢     (           f   BAND     ⁢   c         f   act     ⁢     c   0         -   1     )     ⁢     v   .       ⁢                 
In a version of this embodiment, a squint corrector is coupled to the averaging arrangement, for summing a squint correction with the averaged angular difference signals to generate target angular information representative of corrected direction of the target.
 
     A radar system ( 8 ) according to another aspect of the disclosure includes a transmitter ( 20   b ), a waveform generator ( 91 ) coupled to the transmitter ( 20   b ) for driving the transmitter with sets ( 21   1 ,  21   2 , . . . ) of sequential pulses ( 1 ,  2 ,  3 , &amp;  4 ). Each pulse of each of the sets ( 21   1 ,  21   2 , is jump-frequency modulated relative to other pulses of the set, to thereby define a plurality of subpulses ( 1 ,  2 ,  3 , &amp;  4 ) for each of the sets ( 21   1 ,  21   2 , . . . ) of sequential pulses. The radar system ( 8 ) comprises an antenna ( 10 ) coupled to the transmitter ( 20   b ), for transmitting electromagnetic signals in response to the sets ( 21   1 ,  21   2 , . . . ) of sequential subpulses ( 1 ,  2 ,  3 , &amp;  4 ), and for, in the presence of a target ( 14 ), generating separate return signals for each of the sequential subpulses of each set. A receiver ( 22 ,  32 ) receives the return signals and generates received signals (on set  28  of paths). The received signals include a separate digital signal subpulse for each of the separate return signals. A splitting arrangement ( 6 ) is coupled to the antenna ( 10 ) for splitting the received signals into co-elevation and traverse difference (Δβ, Δα) components and a sum (Σ) component. A prior-art monopulse ratio processor ( 40   a ) is coupled to the splitting arrangement ( 6 ) for combining the co-elevation and traverse difference (Δβ, Δα) components and the sum (Σ) component signal to provide a monopulse ratio (ρ α &amp; ρ β ). A monopulse correction processor ( 40   b ) is coupled to the monopulse ratio processor ( 40   a ) for correcting the direction of the target ( 14 ) with corrections based on at least one subpulse frequency and one of active and passive operating modes. 
     A radar system ( 8 ) according to an aspect of the disclosure includes a transmitter ( 20   b ) and a waveform generator ( 52 ) coupled to the transmitter ( 20   b ) for driving the transmitter ( 20   b ) with sets ( 21 ) of sequential pulses ( 1 ,  2 ,  3 , . . . ). Each pulse of each of the sets is jump-frequency modulated relative to other pulses of the set. The radar system ( 8 ) comprises an antenna ( 10 ) coupled to the transmitter ( 20   b ), for transmitting electromagnetic signals in response to the sets of sequential pulses, and a receiver ( 20   a ) for, in the presence of a target ( 14 ), generating separate return signals for each of the sequential pulses ( 1 ,  2 ,  3 , . . . ) of each set ( 21 ). The receiver receives the return signals and generates received signals, which include a separate digital signal for each of the separate return signals. A splitting arrangement ( 6 ) is coupled to the receiver ( 20   a ) for splitting the received signals into vertical and horizontal difference components (Δβ, Δα ) ) and a sum component (Σ). A monopulse processor ( 312 ) is coupled to the splitting arrangement ( 6 ) for combining the vertical and horizontal difference components (Δβ, Δα ) ) and a sum component (Σ) signal to provide a monopulse ratio (ρ α and ρ   β ). A monopulse ratio processor ( 40   b   1 ,  40   b   2 ) is coupled to the monopulse processor ( 312 ) for generating corrected direction of the target cos α target  by 
                     cos   ⁢           ⁢     α   target       =       cos   ⁢           ⁢     α   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   α     ⁡     (     ρ     α   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     α   des         ]         +     Δ   ⁢           ⁢     u   sq                 (   1   )               
where:
 
     cos α des =designated cos α from the beam steering controller (BSC); 
     n sp =number of subpulses; 
     F 1 =nominal frequency for which the a polynomial P α (ρ) was obtained; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f BAND =subpulse average frequency; 
     ρ αi =real part of i-th monopulse ratio for subpulse i; 
     P α (ρ)=alpha polynomial for the α angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δu sq =squint correction for cos α, as shown in  FIG. 4A , and obtained in an antenna calibration facility, 
     and for generating corrected direction of target cos β target  by 
                     cos   ⁢           ⁢     β   target       =       cos   ⁢           ⁢     β   des       +       1     n   sp       ⁢       ∑     i   =   1       n   sp       ⁢           ⁢     [           F   1     ⁢       P   β     ⁡     (     ρ     β   ⁢           ⁢   i       )           f   acti       +       (         cf   BAND         c   0     ⁢     f   acti         -   1     )     ⁢   cos   ⁢           ⁢     β   des         ]         +     Δ   ⁢           ⁢     v   sq                 (   2   )               
where:
 
     cos β des =designated cos β from the beam steering controller (BSC) indicated in  FIG. 2 ; 
     n sp =number of subpulses; 
     F 1 =nominal frequency for which the β polynomial P β (ρ) was obtained; 
     f BAND =the nominal frequency of the frequency band in use; 
     f acti =actual frequency of the i-th subpulse=f BAND +subpulse average frequency; 
     ρ βi =real part of i-th monopulse ratio for subpulse i; 
     P β (ρ)=beta polynomial for the β angle derived for frequency F 1 , and evaluated at real part ρ of the monopulse ratio; 
     c=light speed at the antenna; 
     c 0 =vacuum light speed; 
     Δv sq =squint correction for cos β, and obtained in an antenna calibration facility.