Title: System for resolving velocity ambiguity in pulse-doppler radar

                     R.sub.TPR
               X.sub.R =
                     V.sub.TPR
                     a.sub.TPR

                     V.sub.TPV
               X.sub.V =
                     a.sub.TPV
                have the steps of generating estimates of predicted doppler
     velocity V.sub.TPV and generating estimates of acceleration a.sub.TPV.

Description: BACKGROUND OF THE INVENTION

This invention relates to coherent pulse-doppler tracking radar, and more particularly to a system for resolving the ambiguity in target velocity estimates made from doppler shift measurements.

The concept of pulse transmission has been basic to most tracking radar systems because range can be easily measured by the time delay of each pulse round trip. Information as to the relative radial velocity of the target is determined by measuring the doppler frequency shift between transmitted and reflected waves of each pulse, and information as to the angular coordinates of the target is determined by relating received energy properties to the sensitivity and directivity patterns of the antenna.

A relative high pulse repetition frequency (PRF) is commonly employed to provide unambiguous velocity (doppler shift) determination, but range is then usually ambiguous. To resolve the range ambiguity, it is common practice to use a staggered PRF, i.e., to alternate the period of the radar pulse cycle. Both range and doppler shift can then be tracked without ambiguity. However, it is often desirable to use a medium PRF, namely a PRF that is not sufficiently high in relation to the largest expected values of doppler-shift for velocity to be tracked unambiguously. If doppler shift cannot be unambiguously tracked, the target's velocity relative to the velocity of the aircraft carrying the radar system cannot be determined except by the range tracking loop, i.e., by effectively taking the first derivative of range measurements. Doppler-shift measurements provide more accurate determination of velocity. Moreover, the determination can be made on the basis of the most recent radar return alone. The problem is to provide a radar tracking system with the facility for resolving the ambiguity in velocity.

A typical tracking system in a tactical aircraft performs two principle functions. First, it positions the antenna and the range and velocity gates of the radar signal processor, so as to keep the target accurately centered within the "field of view" of those elements. Second, it generates estimates of the target's position and motion for use in fire-control and aircraft-steering computations. These functions must be accurately performed over an extremely broad range of operating conditions, ranging from highly dynamic dog-fight encounters at short-range to very low signal-to-noise ratios at long range.

Several factors must be dealt with in the design of the tracking system in order to obtain the required accuracies and performance capabilities. First, there is the problem of measurement errors due to factors such as noise, target scintillation, and radome refraction. Maneuvers of the target and deception techniques (countermeasures) employed by targets to prevent tracking are the second class of problems which must be dealt with.

Due to the large variations in the characteristics and magnitudes of these error sources over the wide spectrum of operating conditions, variable parameter filters are necessary to meet tracking accuracy requirements of high performance aircraft. For example, narrow bandwidth filtering is required for tracking targets at long range with low signal-to-noise ratios. But rapid response is necessary for tracking highly maneuverable targets at short range. These factors, plus the requirement for generating high accuracy estimates of target line-of-sight (LOS) rates, and the desirability of employing one unified set of general filter equations to handle any radar mode --high PRF, medium PRF, low PRF, and so forth-- have led to the use of a new approach in the design of the tracking filter using Kalman filtering.

A system employing this new approach is like a typical radar tracking system in that a target reflects pulses to an antenna which analyses the return pulses and provides the desired information. A transmitter and receiver is operated in accordance with a predetermined mode in regard to frequency and PRF, both of which are often selected automatically or by an operator for a particular target to avoid velocity and range blind zones. The receiver provides video signals to a radar signal processor which produces signals that are measurements of range, range rate (radial velocity) and angular coordinates (elevation and azimuth deflection) of the target. These measurement signals are then applied to standard antenna controllers for positioning the antenna and to range and velocity controllers for positioning range and velocity gates. A typical radar tracking is thus a multiple closed-loop feedback system. Two loops for tracking in azimuth and elevation, and two loops for tracking in range and velocity. Standard servomechanism design procedures are both applicable and practical to each of these loops, but optimal results are not always achieved by such procedures.

The difference between this new approach and the prior art is that the prior art tracking loops have employed classical single control loops, using low-pass filtering to remove noise from measurement signals, whereas the new approach employs more advanced control loops employing filters for estimating quantities from measured differences between estimates and actual measurements, such as .DELTA.R.sub.m = R - R.sub.m. The timing of a range gate, R.sub.G, represents the range estimate, R, and the time of arrival of a round-trip radar pulse represents the measured range, R.sub.m. The difference then is the discriminant, .DELTA.R.sub.m. The range servo-loop continually updates the range estimate R to drive .DELTA.R.sub.m toward zero. Analogous servomechanism procedures drive the velocity discriminant, V.sub.m, toward zero and the azimuth and elevation discriminants, .DELTA..eta..sub.m and .DELTA..epsilon..sub.m, toward zero.

In accordance with this new approach, an airborne system is provided with electromagnetic energy sensors for tracking moving targets is some or all of range, radial velocity, elevation and azimuth. Energy received from the system sensors is processed to develop signals representing actual discriminants of range, .DELTA.R.sub.m, radial velocity, .DELTA.V.sub.m, elevation, .DELTA..epsilon..sub.m, and azimuth, .DELTA..eta..sub.m. These measurements are used to generate predictions of the next measurements from current estimates of system states, where the states are vectors X.sub.R (i), X.sub.V (i), X.sub.e (i) and X.sub.d (i), using for each state an equation of the general form

X.sub.i.sub.+l = .phi..sub.i X.sub.i + L.sub.i + K.sub.i (Y.sub.i - H.sub.i X.sub.i) (1)

where X is a state vector, .phi. is a transition matrix, L is a vector of dynamical aiding terms to compensate for rotational rates and inertial acceleration of the antenna, K is a gain factor, Y is the output of a measurement structure for the dynamical system and H is a system scaling factor which accounts for the gain in the measurement structure. The four channels implemented in accordance with this general equation are cross-coupled by using outputs of one or more channels as parameters in the remaining channels. For the range channel

                       R.sub.TPR
               X.sub.R =
                       V.sub.TPR
                       a.sub.TPR

thus generating estimates of predicted target range, R.sub.TPR, radial velocity, V.sub.TPR, and radial acceleration, a.sub.TPR. For the velocity (doppler) channel

                       V.sub.TPV
               X.sub.V =
                       a.sub.TPV

thus generating estimates of predicted doppler velocity V.sub.TPV and acceleration a.sub.TPV. For the antenna angle channels

               .epsilon..sub.Pe      .epsilon..sub.Pd
               .omega..sub.LSPe      .omega..sub.LSPd
    X.sub.e =  a.sub.Td     X.sub.d =
                                     a.sub.Te
               S.sub.e               S.sub.d

thus generating estimates of pointing errors, .epsilon..sub.e,d, target line-of-sight rates, .omega..sub.LSPe,d, target accelerations normal to the line of sight, a.sub.Td,e, and target angular scintillation, S.sub.e,d, all referenced directly to antenna coordinates. The pointing error estimates, .epsilon..sub.Pe and .epsilon..sub.Pd, and the target line-of-sight rates .omega..sub.LSPe and .omega..sub.LSPd are used to produce antenna elevation and azimuth commands .omega..sub.Ce and .omega..sub.Cd. The range estimate R.sub.TPR and the velocity estimate V.sub.TPV are used to produce a range gate command, R.sub.GC, and a velocity gate command, V.sub.GC, both to be used by a radar controller.

In operation of the tracking system, measurements of SNR are made coincidentally with the discriminant measurements. The discriminant noise variance and the discriminant slope are then determined from the measured SNR, and these quantities are utilized in the calculations of the filter gains. Thus, the filter is provided with data on discriminant used by the filter, thereby adapting the filter to the characteristics of each measurement sample.

SUMMARY OF THE INVENTION

In accordance with the present invention, a coherent radar tracking system is provided with separate range and velocity (doppler) tracking filters implemented in accordance with the iterative equation

X.sub.i.sub.+1 = .phi..sub.i X.sub.i + L.sub.i + K.sub.i [Y.sub.i -H.sub.i X.sub.i]

where X is the estimate of range or velocity, .phi. is a transition matrix, L is a vector of dynamical aiding terms to compensate for rotational rates and inertial acceleration of the radar antenna, K is a gain factor, Y is the measurement structure for the dynamical system, and H is a system scaling factor which accounts for gain in the measurement structure. A bar over a symbol indicates a matrix quantity or value used in the filter for the parameter represented by the symbol, and the subscript i indicates a value at a given time i. The subscript i+1 then indicates an estimate of X at the next iteration time. The states of the range filter are

              R.sub.TPR
    X.sub.R = V.sub.TPR
              a.sub.TPR

where R.sub.TPR is the estimated (predicted) range of the target and the remaining states V.sub.TPR and a .sub.TPR are effectively the respective first and second derivatives of R.sub.TPR. The states of the velocity filter are

              V.sub.TPV
    X.sub.V =
              a.sub.TPV

where V.sub.TPV is the estimated (predicted) velocity of the target and the remaining state is effectively the first derivative of V.sub.TPV. The velocity state V.sub.TPV is derived ambiguously from doppler shift of the signal return from radar pulses transmitted at a medium PRF, but the state V.sub.TPR is unambiguous once ambiguity in range, R.sub.TPR, is resolved at the time the target is acquired for tracking using conventional techniques. Ambiguity is then resolved in the state V.sub.TPV when uncertainty in range channel velocity estimate, V.sub.TPR, is less than .lambda.PRF/4, i.e., when the difference between the actual doppler shift, f.sub.DP, derived from the unambiguous velocity, V.sub.TPR, and the ambiguous doppler shift, f.sub.DAP, derived from the ambiguous velocity, V.sub.TPV, is less than PRF/2. At that time, a corrected velocity V.sub.TCORR is computed in accordance with the following equation

V.sub.TCORR = .lambda./2 PRF {[(f.sub.MLC +f.sub.DAP -f.sub.DP) PRF.sup..sup.-1 +1/2]*-(f.sub.DAP +f.sub.MLC)}

where the asterisk by the bracketed quantity indicates that the quotient (f.sub.MLC +f.sub.DAP -f.sub.DP) .div. PRF is to be truncated by dropping the fractional part after the quotient has been rounded out by adding 1/2 to the quotient. The truncated number N is the number of spectra from a reference frequency f.sub.o in which the doppler shift of the target return should be. The quantity f.sub.MLC is an offset of the reference introduced to avoid main lobe clutter in the center of the doppler shift spectrum where the target is most likely to be, and is predetermined as a function of ground speed of the intercepting aircraft and elevation angle of the radar antenna. The actual doppler shift velocity, V.sub.TCORR, is thus computed by multiplying PRF by N, subtracting the sum of f.sub.DAP and f.sub.MLC and converting the difference to feet per second by multiplying by half the wavelength, .lambda., of the radar frequency in feet per second (half because of the round trip path of the returned signal). V.sub.TCORR is then used to reinitialize the velocity filter during the next filter cycle by substituting V.sub.TCORR for V.sub.TPV that one time. Thereafter, the velocity filter tracks V.sub.TPV accurately and unambiguously.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating an airborne radar monopulse tracking system embodying the present invention.

FIG. 2 is a general block diagram of the general form of the filters for each of the four filter channels in an estimator for the elevation, azimuth, range and velocity tracking system of FIG. 1.

FIGS. 3a through 3d are block diagrams of the four filter channels for tracking in elevation, azimuth, range and velocity, wherein each filter represented by a block is of the form illustrated in FIG. 2.

FIGS. 4, 5 and 6 illustrate typical range discriminant characteristics.

FIG. 7 is a timing diagram of tracking filter functions.

FIGS. 8a-8d illustrate a functional flow diagram for a programmed tracking filter routine.

FIGS. 9a -9f illustrate a detailed flow diagram for the routine of FIGS. 8a-8d.

FIGS. 10-18 illustrate flow diagrams of programs for subroutines called out in detailed flow diagrams of FIGS. 9a-9f.

FIGS. 19 and 20a-20d illustrate flow diagrams for azimuth and elevation command computations.

FIGS. 21a and 21b illustrate the techinque employed for computing signal-to-noise ratios for use in the flow diagrams of FIGS. 9a-9f.

FIG. 22 is a block of a digital computer employed as a radar data processor (RDP) in the system of FIG. 1.

FIG. 23 illustrates the storage allocation assignments for the RDP of FIG. 22.

FIG. 24 illustrates schematically the working registers of the RDP of FIG. 22.

FIG. 25 illustrates instruction formats for the RDP of FIG. 22.

FIG. 26 is a diagram useful in understanding how velocity ambiguity is resolved.

FIG. 27 is a flow chart of a routine for resolving velocity ambiguity.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Before proceeding with a detailed description of a preferred embodiment, a typical environment for the present invention will be described with reference to FIGS. 1 through 3.

In the most general terms for describing the environment, means are provided for iteratively computing a prediction Xt.sub.(i.sub.+1) of the next measurement sample Yt.sub.(i.sub.+1) from the difference (residual) between a current estimate Xt.sub.(i) and measurement Yt.sub.(i) of a tracking system state. The residual is multiplied by a gain factor Kt.sub.(i) and the product is improved by available external aiding signals Lt.sub.(i), such as for compensation of antenna motion and/or aircraft motion as they affect tracking. Means for extrapolating the next estimate Xt.sub.(i.sub.+1) from the current estimate Xt.sub.(i) and the measurement residual is provided as a minor loop in the residual loop. The two loops constitute a tracking filter.

In the foregoing, t.sub.(i.sub.+1) denotes the time of the updated measurement and t.sub.(i) denotes the time of the previous measurement. For convenience, the letter t will be omitted hereinafter and only the subscripts i and i+1 will be used to denote the times.

The extrapolation loop is a dynamic model, .phi..sub.i, of the system state aided by the available external aiding signals, L.sub.i. The general filter structure is then defined by equation (1) set forth hereinbefore in the summary of the invention. In operation, the estimates are generated by adding measurement corrections K.sub.i [Y.sub.i - H.sub.i X.sub.i ] to extropolated values .phi..sub.i X.sub.i of past estimates improved by an aiding term L.sub.i. Thus, in the absence or rejection of a measurement, extrapolation is automatically performed until the next measurement is obtained, thereby tracking through blind regions. Correlation between the measurement residuals and the errors in estimated states is important to the present invention. Such correlation can arise in one of two possible ways -- direct measurement correlation or indirect dynamical correlation.

Direct measurement correlation exists between a given measurement and a given state if the measurement is itself a measure of the particular state. For example, the azimuth and elevation measurement residuals of the radar are direct measurements of the angular pointing errors, and are, therefore, directly correlated with the computational estimates of angular pointing error. Dynamical correlation arises from dynamical coupling between quantities which are directly measured and quantities which are estimated. As an example, extrapolation of the angular position of the target is performed by integrating the estimate of the target's line-of-sight (LOS) rate. Thus, any errors in the estimated LOS rate will produce errors in the extrapolated angular position, thereby resulting in a correlation between the angular position error and the LOS rate error.

Means is provided for an estimated state to be updated by any measurement which is correlated -- either directly or dynamically -- with that state. Thus, in the preceding example, since the angle measurement residuals are directly correlated with angular position error, and since angular position error is dynamically correlated with LOS rate error, the angle measurement residuals are correlated with LOS rate errors and can therefore be used to reduce LOS rate errors. In that manner, a single measurement quantity may be employed to generate estimates of several system states. Each estimated state is updated by the measurement residual weighted by a particular gain factor associated with that state. To obtain the greatest accuracy, the value of each gain factor is continually made proportional to the statistical correlation between the measurement residual and the uncertainty in the particular estimate, and inversely proportional to the variance of the measurement error. This is the form of the optimal gain for a continuous-time computational means; a slight modification is necessary in the case of discrete-time computational means. The real-time calculations of the gain factors are performed as follows: The statistical (ensemble) cross-correlations between the measurement residual and the estimation uncertainties, or errors, are computed by appropriate means in terms of the measurement model and the covariance matrix of the estimation errors, which is recomputed by appropriate means in a recursive manner once each computational cycle. The covariance matrix takes into account cross-correlation between the system states due to dynamical interactions between the states.

The tracking system is divided into the following channels:

1. Azimuth angle channel

2. Elevation angle channel

3. Range channel

4. Velocity channel

These four channels are cross-coupled to obtain the benefits of cross-channel aiding. The azimuth and elevation angle channels have Kalman filter configurations and are identical in form, each estimating the following set of four state variables: (1) pointing error, (2) line-of-sight rate, (3) target acceleration normal to the line-of-sight, and (4) angular scintillation. Each of these states is referenced directly to antenna coordinates. Both of the angle channels use the same set of gain factors, which are computed and updated once each computational cycle. Measured values of signal-to-noise ratios are used in the calculation of the filter gain factors, thereby enabling the computational means to adapt to changing signal characteristics.

The range and velocity channels also have Kalman filter configurations, but employ semi-fixed instead of time-varying gains. Both of these channels employ three sets of gains, which are selected as a function of measured signal-to-noise ratio. Range signal-to-noise ratio (SNR.sub.RNG) is used to select the appropriate set of gains for the range channels, and velocity signal-to-noise ratio (SNR.sub.VEL) is used to select the appropriate set of gains for the velocity channels. In different PRF modes, the range channel processes the measurement residuals to form estimates of target range, range rate and radial acceleration. In a high PRF mode, where no range residuals are measured and where it is unnecessary to control range rate positions, the range channel consists of a single state, the target range state, and velocity aiding is provided by the velocity channel.

In high and medium PRF modes, the velocity channel processes velocity measurement residuals to form estimates of range rate and radial acceleration. The velocity channel is not employed in low PRF mode. The angular rate commands for the antenna are obtained by passing the pointing error estimate through a compensator and adding the line-of-sight rate estimate to the compensator output. The use of the accurate line-of-sight rate estimates in the antenna commands results in a substantial reduction of both dynamical and steady-state antenna pointing errors. The fact that the estimates are generated in antenna coordinates permits the antenna commands to be generated simply and directly from the estimates without any coordinate transformation.

Although each of the four channels could function independently of one another, better performance is obtained by cross coupling. The form of the channel filters permits this cross coupling to be implemented very easily. Thus, range and range rate estimates from the range and velocity channels, respectively, are used in the angle channels, and the line-of-sight rate estimates of the angle channels can be used in the range and velocity channels. In addition, roll cross coupling is provided between the two angle channels.

Each of the four tracking loops is aided by inertial navigator outputs to compensate for aircraft accelerations. However, the loops can also function in the absence of inertial navigator aiding. A unique form of aiding, utilizing gyro outputs in addition to gyro torque commands, is employed to compensate for antenna motion.

The system also employs special timing techniques which permit sufficient time to process the radar data while avoiding dynamical problems which might otherwise result from the delays between the time of measurement and the time of generating the commands required for closed-loop tracking. These techniques include pre-averaging of the radar measurements together with one-step extrapolation of the estimates in the angle channels, and two-step extrapolation in the range and velocity channels.

At the initiation of tracking, and until the tracking channels can acquire sufficient data with which to accurately track the motion of the target, the tracking loops are iterated at a fast rate in order to avoid losing the target. Thereafter, a slow rate of iteration may be employed. However, fast iteration cannot be performed if channel gain factors are to be computed due to time required for the computations. Therefore, in an initial tracking phase of predetermined duration (.about. 1.5 sec.) after initial lock-on, constant preprogrammed gain factors are used to permit the tracking loops to iterate at fast rates. Thereafter, the gain factors used are computed to provide adaptive tracking loops.

FIG. 1 illustrates an exemplary amplitude-comparison monopulse airborne radar tracking system embodying the present invention. Four antenna horns A, B, C and D are located symmetrically around the antenna axis 10, and are fed in phase by a conventional transmitter-receiver 11 to produce a narrow beam along the axis. Transmitted pulses are timed by a conventional controller 12. Between transmitted pulses, received energy is coupled to three channels of a receiver portion of the radar receiver-transmitter. An in-phase or sum channel (A+B+C+D) permits reception of fullamplitude echo signals for use in display, ranging, radial velocity tracking and as a reference signal for angle tracking. An azimuth error channel, (A+C)-(B+D), and an elevation error channel, (A+B)-(C+D) permit angle tracking. The coupling duplexers (not shown) are conventional.

The radar controller transmits range and velocity gates R.sub.G and V.sub.G to a signal processor 13. There the received sum channel is sampled to produce range and velocity discriminants .DELTA.R.sub.m and .DELTA.V.sub.m, while at the same time the difference channels are sampled to produce azimuth and elevation discriminants, .DELTA..eta..sub.m and .DELTA..epsilon..sub.m. For ranging, two samples are taken, one on each side of the predicted peak of a return pulse, to develop the range discriminant. For velocity tracking, two samples are also taken, one on each side of the predicted doppler shift to develop the velocity discriminant. The signal processor also provides measurements of signal and noise to a signal-to-noise ratio computer 13a which computes for range, velocity, elevation and azimuth the respective ratios SNR.sub.RNG, SNR.sub.VEL, SNR.sub.e and SNR.sub.d as required for use by an estimator 14.

The outputs of the signal processor 13 and SNR computer 13a are in digital form, and are used in an estimator 14 for developing an estimate of target range R.sub.TPR, an estimate of radial velocity (range rate of target) V.sub.TPV, azimuth and elevation line-of-sight rate estimates .omega..sub.LSPd and .omega..sub.LSPe, and azimuth and elevation tracking error estimates .epsilon..sub.P and .epsilon..sub.P . These angle estimates are used in an azimuth and elevation command computer 15 to produce control signals for an antenna servo 17, thus closing a servo loop through the dynamical system. The range and velocity estimates are used in a range and velocity command computer 16 to produce control signals for the radar controller 12 which produces range and velocity gates R.sub.G and V.sub.C, to close range and velocity tracking loops.

Timing for the sections 13, 13a, 14, 15 and 16, which collectively constitute a digital radar data processor, is provided by a clock pulse system that is based on a very stable high frequency clock pulse source in a conventional manner. The estimator 14 operates on a fixed cycle of a period selected for the mode of radar operation, namely high, medium or low PRF. In each mode, the cycle period is greater than the sampling period in the signal processor. In order to utilize the information contained in the intervening samples, a preaveraging system 18 implemented as part of the radar data processor, accumulates the intervening samples and divides by the number of samples taken to provide average discriminant inputs to the estimator. Assuming the discriminant values are in binary form, averaging may consist of simply adding 2.sup.N samples, where N is a whole integer, and dividing by 2.sup.N by effectively shifting down N places at the end of every 2.sup.N samples. The accumulator is reset after every 2.sup.N samples.

State vectors of the four Kalman filter channels in the estimator may be used for purposes other than tracking, such as in a display and fire control system 19. Each of the four tracking loops is aided by outputs from an aircraft inertial navigation system 20 which produces signals representing pitch angle .theta., aircraft roll angle .phi..sub.I, aircraft heading angle .psi. and north, east, and vertical components of interceptor velocity, V.sub.N, V.sub.E and V.sub.D. The antenna servo 17 also provides input signals representing antenna roll gimbal angle .phi..sub.G, antenna elevation gimbal angle .epsilon..sub.G and antenna azimuth gimbal angle .eta..sub.G. These signals are all either in digital form or immediately converted to digital form, preferably the former.

Although each of the four tracking filters could function completely and independently, better performance is obtained by cross coupling the filters in particular ways. The form of the filters permits cross coupling to be easily implemented. Range (R.sub.TPR) and range-rate (V.sub.TPR) estimates from the range and velocity filters, respectively, are used in the angle filters, and the line-of-sight rate estimates .omega..sub.LSPd and .omega..sub.LSPe are used in the range and velocity filters. In addition roll cross coupling is provided between the two angle-tracking filters.

A general block diagram applicable to each of the four tracking filters in the system will now be described with reference to FIG. 2. A block 21 represents the dynamical system of a tracking channel, such as for elevation tracking. The differential equation in the block indicates the form of the discrete tracking system. Pursuing the example of the elevation channel, the dynamical system can be more accurately defined by the following continuous state equations

.epsilon..sub.e = .omega..sub.LSe - .omega..sub.Ae + .omega..sub.Ar .epsilon..sub.d (2) ##EQU1## Where: .epsilon..sub.e is the pointing error in elevation.

.omega..sub.LSe is the angular rate in elevation of the line-of-sight to the target.

.omega..sub.Ae is the angular rate of the antenna axis in elevation.

.omega..sub.Ar is the angular (roll) rate about the antenna axis.

.epsilon..sub.d is the pointing error in azimuth.

R is the slant range to the target.

a.sub.T.sbsb.d is the acceleration of the target in azimuth.

a.sub.I.sbsb.d is the acceleration of the interceptor (radar platform) in azimuth.

.omega..sub.LSd is the angular rate in azimuth of the line-of-sight to the target.

.tau..sub.T is the correlation time of target acceleration model.

w.sub.T is white noise driving target acceleration model.

S.sub.e is scintillation of the target in elevation.

.tau..sub.s is the correlation time of the angular scintillation process.

w.sub.s is the white noise driving angular scintillation model.

These continuous state equations (2)-(5) may be more conveniently set forth in matrix form by the following substitutions in the dynamical system equation (block 21 in FIG. 2).

    ______________________________________
              .epsilon..sub.e
              .omega..sub.LSe
    X =       a.sub.T.sbsb.d
                       (6)
              S.sub.e
              0           1       0       0
                          2 R     1
              0     -                     0
                          R       R
    A =                                        (7)
                                  1
              0           0   -           0
                                  .tau..sub.T
                                          1
              0           0       0   -
                                          .tau..sub.s
              -     .omega..sub.Ae
                           +   .omega..sub.Ar .epsilon..sub.d
                                       (8)
                    a.sub.I.sbsb.d
    L =       -            +   .omega..sub.Ar .omega..sub.LSd
                    R
                               0
                               0
              0
              0
    w =       w.sub.T
                     (9)
              w.sub.s
    ______________________________________

The value of X from the differential equation in block 21 is the input to measurement structure represented by a dotted line block 22 in FIG. 2, where H is a measurement system parameter by which the input signal X is multiplied. The multiplier is represented by a block 23. Noise in the measurement system is added to the signal HX to provide a model of measurements

Y.sub.i = HX + v (10)

where v is the noise added, as represented by the circle 24 in the measurement structure. The subscript i indicates the value of the measurement during a particular cycle of the filter shown within a dotted line block 25 in FIG. 2. The filter produces an estimate of the measurement X.sub.i, from which a predicted measurement Y.sub.i is produced by multiplication of the estimate X.sub.i by H.sub.i in block 26, where H is the system measurement parameter. The subscript i indicates the value of H at time i, thus implying that H is not a constant. A bar over a symbol indicates a matrix quantity or value used in the filter as the filter's version of the parameter represented by the symbol.

The predicted measurement is subtracted from the actual measurement in a subtractor represented by a circle 27 to form a measurement residual. That residual is multiplied by a filter gain K.sub.i in block 28. Aiding signals L.sub.i are added to the product through an adder 29 to compensate for antenna motion as sensed at antenna gimbals and aircraft motion as sensed by the inertial navigation system (INS). The output of the adder 29 is fed back through a unit delay operator 30. The feedback signal X.sub.i is then used in an extrapolator 31 to form the product .PHI..sub.i X.sub.i which is added to the next measurement residual while forming the next estimate X.sub.i.sub.+1. In that manner the predicted measurement can be continually extrapolated. The extrapolator loop equation is;

X.sub.i.sub.+1 = .PHI..sub.i X.sub.i + L.sub.i (11)

where X, .PHI. and L are vectors. Observing that Y.sub.i = H.sub.i X.sub.i from the measurement structure, the filter state equation is;

X.sub.i.sub.+1 = .PHI..sub.i X.sub.i + L.sub.i + K.sub.i (Y.sub. i -H.sub.i X.sub.i) (12)

where the filter gain is given by the equations

K.sub.i = .PHI..sub.i M.sub.i H.sub.i.sup.T (H.sub.i M.sub.i H.sub.i.sup.T +R.sub.i).sup.-1 (13)

M.sub.i.sub.+1 = (.PHI..sub.i -K.sub.i H.sub.1) M.sub.i .PHI..sub.i.sup.T + Q.sub.i (14)

and Q.sub.i is the covariance matrix of the noise driving the systems dynamical model and R.sub.i is the covariance matrix of the measurement noise. The superscript T denotes the transpose of the associated matrix, and is not an exponent.

For the elevation and azimuth channels, .PHI..sub.i is given by the equation

.PHI..sub.a.sbsb.i = I + A.sub.a.sbsb.i T + A.sub.a.sbsb.i.sup.2 T.sup.2 /2 (15)

where I is the identifying matrix, T is the filter cycle period and ##EQU2## The elevation and azimuth filter parameters are given the equations

H.sub.a.sbsb.i = [k.sub.a.sbsb.i 0 0 k.sub.a.sbsb.i ] (17)

R.sub.a.sbsb.i = .sigma..sub.a.sbsb.i.sup.2 (18) ##EQU3## and .DELTA.V.sub.di is compensation for the acceleration a.sub.I.sbsb.d of the interceptor along the azimuth axis; .theta..sub.r is compensation for roll cross-coupling due to antenna roll motion and .intg. .omega..sub.A.sbsb.e dt is compensation for antenna motion about the elevation axis.

In equations (17) and (18), k.sub.a.sbsb.i is the angle discriminant slope and .sigma..sub.a.sup.2 is the angle discriminant noise variance determined from signal-to-noise ratio (SNR). An SNR is determined for both the range (SNR.sub.RNG) and radial velocity (SNR.sub.VEL) measurements of discriminants, as well as elevation (SNR.sub.e) and azimuth (SNR.sub.d) measurements of discriminants, and ideally each SNR would be used in the respective filters. However, as will be noted more fully hereinafter, the SNR is very nearly the same in both elevation and azimuth so that an average SNR can be used with sufficient accuracy in both angle filters. Another possibility is to compute the angle discriminant slope separately for the two angle filters from SNR.sub.e and SNR.sub.d and to use an average of those values in computing the gain factor K.sub.a.sbsb.i to be used in both angle channels. The average SNR of the angle filters is used to determine the discriminant noise variance to be used in both channels. This alternative is the one adopted for the specific exemplary embodiment to be described because of the desire to minimize computing time without significantly degrading performance. Thus, separate measurements of SNR.sub.e and SNR.sub.d are made by the signal processor 13 (FIG. 1) during each filter channel period while the discriminant measurements are being made. The angle discriminant noise variance, .sigma..sub.a.sbsb.i.sup.2, and the angle discriminant slope, k.sub.a.sbsb.i, are then computed as the average values for the elevation and azimuth channels. In the specific exemplary embodiment to be described, they are computed according to the following equations:

k.sub.a.sbsb.i = 1/2 (k.sub.e.sbsb.i + k.sub.d.sbsb.i) (23)

where k.sub.e and k.sub.d are predetermined functions of SNR.sub.e and SNR.sub.d, respectively.

.sigma..sub.a.sup.2 = C.sub.o for SNR.sub.a < 1 (24a) ##EQU4## where C.sub.o is a predetermined constant and SNR.sub.a = 1/2(SNR.sub.e +SNR.sub.d).

For the range and velocity filter channels, sets of predetermined constant gain factors, K.sub.r and K.sub.v, are selected based on SNR.sub.RNG and SNR.sub.VEL to reduce computation time in the specific exemplary embodiment to be described. However, for the present, it is assumed that each of the range and filter channels are fully implemented in a manner analogous to the implementation of each of the angle filter channels.

The value .DELTA.V.sub.di in Equations (21) and (22) is computed from INS velocities with transformation of those velocities to the antenna coordinate system. Assuming the transformation, .DELTA.V.sub.di is computed from the equation

.DELTA.V.sub.di = V.sub.di - V.sub.d(i.sub.-1) (25)

and the value .theta..sub.r is computed from the equation ##EQU5## where all symbols are as defined hereinbefore except .eta..sub.G = time derivative of azimuth gyro angle .eta..sub.G.

.omega..sub.a.sbsb.d = antenna inertial rate about azimuth, d, axis.

.epsilon..sub.G = elevation gyro angle.

.omega..sub.RG = angular rate of the aircraft roll gimbal.

The quantity in the curly brackets is a measurable expression for .omega..sub.A.sbsb.r, and .theta..sub.r is mechanized by measuring each of these quantities separately and combining as shown. Note that if a rate gyro were mounted on the r axis of the antenna, its output would directly measure .omega..sub.A.sbsb.r.

From the foregoing, it is evident that the full elevation channel equation is

X.sub.e(i.sub.+1) = .PHI..sub.ai X.sub.ei + L.sub.ei + K.sub.ai .DELTA.e (27a)

where the residual .DELTA.e = Y.sub.ei - H.sub.ai X.sub.ei - k.sub.a.sbsb.i r.sub.e.sbsb.i, and

             .epsilon..sub.ei
             .omega..sub.LSei
    X.sub.ei =       ,
             a.sub.Tdi
             S.sub.ei

The computed radome compensations r.sub.ei and r.sub.di are determined as a function of gimbal angle. The full azimuth (deflection) channel equation is

X.sub.d(i.sub.+1) = .PHI..sub.ai X.sub.d (i) + L.sub.di + K.sub.ai .DELTA.d (27b)

where the residual .DELTA.d = Y.sub.di -H.sub.ai X.sub.di - k.sub.ai r.sub.di, and

             .epsilon..sub.di
             .omega..sub.LSdi
    X.sub.di =
             a.sub.Tei
             S.sub.di

In both angle channels, the filter input, Y.sub.i, is the average of discriminants over the ith measurement interval. The measurement residual is then the difference between the input Y.sub.i and the predicted measurement Y.sub.i = X.sub.i H.sub.i. Roll cross-coupling between channels is provided between the two angle filters.

In the range filter, the continuous system Equation (1) evolves into the following discrete equivalent

X.sub.r(i.sub.+1) = .PHI..sub.ri X.sub.ri + L.sub.ri + K.sub.ri [Y.sub.ri -H.sub.ri (X.sub.ri -X.sub.Gi)] (28)

where:

Y.sub.ri = .DELTA.R.sub.mi

X.sub.Gi = R.sub.Gi

             R.sub.i
    X.sub.ri =
             V.sub.ri
             a.sub.Tri

.PHI..sub.ri = I + A.sub.ri T + A.sub.ri.sup.2 T.sup.2 /2 ##EQU6## where .DELTA.V.sub.ri is the change in interceptor (own ship) velocity along the radar boresight axis, and ##EQU7## where in practice .omega..sub.LSi.sup.2 may be substituted by a constant set to approximate the square of the line-of-sight rate.

By replacing X.sub.ri with V.sub.vi, Y.sub.ri with .DELTA.V.sub.mi, and X.sub.Gi with V.sub.Gi, the continuous system equation for the velocity filter evolves, where ##EQU8## The discriminant .DELTA.V.sub.mi is the average for the ith cycle of radial velocity determined by measuring doppler shift. The parameters used in .PHI..sub.ri, L.sub.ri and N.sub.ri are analogous to the parameters of the range filter. The parameter H.sub.ri = [k.sub.ri 0 0 k.sub.ri ] contains the discriminant slope for range. The corresponding parameter H.sub.vi = [k.sub.vi 0 0 k.sub.vi ] for the velocity filter contains the discriminant slope for velocity. The parameters k.sub.ri and k.sub.vi have the same form as the parameter k.sub.ai for the angle filters, as noted hereinbefore, and are stored in read only memories 48 and 49 for retreival as functions of SNR.sub.RNG and SNR.sub.VEL, respectively.

Referring to the general Equation (1), the last term is the measurement residual Y.sub.i - H.sub.i X.sub.i. The measurement residual is seen to be the actual measurement Y.sub.i minus the computed estimate or predicted measurement H X.sub.i. In the case of the angle filters, the measured discriminant Y.sub.i is the deviation of the antenna axis from the line-of-sight, i.e., the pointing error .epsilon. in elevation or azimuth. Consequently, the estimate computed as a filter output for control of the antenna is, after scaling, in a form to be subtracted directly from the measured value Y.sub.i, i.e., H.sub.i X.sub.i is an estimate of the measured value Y.sub.i. Therefore, the direct difference is the desired measurement residual. This residual is analogous to an error signal in a conventional servo control loop.

In the range and velocity filters, the measured discriminator outputs .DELTA.R.sub.m and .DELTA.V.sub.m are the difference between the respective slant range and radial velocity of the target and the anticipated range R.sub.G and velocity V.sub.G set by the radar controller from the estimates in range and velocity from the filters. It is therefore necessary to add R.sub.G -R and V.sub.G -V to the range and velocity discriminator outputs to form the measurement residuals. Consequently, in the range and velocity filters, the measurement residuals are given by the differences .DELTA.R.sub.mi - H.sub.ri (X.sub.ri - X.sub.rGi) and .DELTA.V.sub.mi - H.sub.vi (X.sub.vi - X.sub.vGi).

FIGS. 3a through 3d illustrate separately four filter channels 41, 42, 43 and 44 for the estimator 14 of FIG. 1 for elevation, azimuth, range and velocity, each with its separate controller, where the discriminants are

.DELTA..eta..sub.mi = Y.sub.di

.DELTA..epsilon..sub.mi = Y.sub.ei

.DELTA.R.sub.mi = Y.sub.ri

.DELTA.V.sub.mi = Y.sub.vi

The equations for the elevation and azimuth angle state vectors in an exemplary embodiment are ##EQU9## where all terms except .delta..sub.Re and .delta..sub.Rd are as defined hereinbefore. The constant C.sub.10 is a constant which appropriately scales the radar discriminant measurement. The terms .delta..sub.Re and .delta..sub.Rd are radome error corrections which can, for example, be computed or determined empirically for antenna gimbal angles and stored in a read-only memory 45 (FIG. 3a) for use in the elevation and azimuth angle filters to correct for refraction of electromagnetic energy by the radome for any position of the antenna during the ith iteration. These equations are used at all times for initial and final tracking at high, medium or low PRFs. However, for initial tracking some simplification may be achieved by omitting radome compensation.

The Kalman gain factor K.sub.ai is computed by iterating the following two equations, first the equation for M.sub.ai and then the equation for K.sub.ai.

M.sub.ai = [.PHI..sub.ai.sub.-1 - K.sub.ai.sub.-1 H.sub.ai.sub.-1 ] M.sub.ai.sub.-1 .PHI..sub.ai.sub.-1.sup.T + Q.sub.ai (31)

K.sub.ai = .PHI..sub.ai M.sub.ai H.sub.ai.sup. T . [H.sub.ai m.sub.ai H.sub.ai.sup. T + R.sub.ai ].sup..sup.-1 (32)

where all terms are as defined hereinbefore. The value H.sub.a is a matrix containing the angle discriminant slope k.sub.a, and R.sub.a is equal to the angle discriminant noise variance .sigma..sub.a.sup.2. The variables k.sub.a and .sigma..sub.a.sup.2 could be predetermined empirically as a function of SNR.sub..sub..SIGMA. = 1/2(SNR.sub.e + SNR.sub.d), and stored in a read-only memory for use in computing M.sub.ai and K.sub.ai. Computation of these equations is carried out by a block 46 (FIG. 3b) for both the elevation and the azimuth channels. This assumes the discriminant slopes k.sub.e for elevation and k.sub.d for azimuth are nearly the same to permit using an average k.sub.a = 1/2(k.sub.e + k.sub.d). In some systems, that may not be sufficiently true, in which case separate computations may be required for unique Kalman gain factors in each of the two angle filters.

As noted hereinbefore, there is cross-coupling between the angle filters in the aiding vectors L.sub.e and L.sub.d as reflected in Equation (8) for L.sub.e. The aiding vector L.sub.d is the exact symmetric version of the aiding vector L.sub.e. One need only substitute subscripts in Equation (8) to provide the equation for L.sub.d.

The system driving noise covariance matrix Q.sub.a set forth in Equation (19) can be simplified to the following form ##EQU10## where Q.sub.s and .alpha..sub.4 are single valued constants determined analytically or empirically, T is the filter iteration period, and the angular scintillation parameters .tau..sub.s and .tau..sub.s.sup.2 are updated when the Q.sub.a matrix is to be updated according to the following equations

.tau..sub.s = min {.alpha..sub.1, .alpha..sub.3 R.sub.TPR /.vertline.V.sub.TPV .vertline..sup.1/2 } (34a)

.tau..sub.s.sup.2 = .alpha..sub.2 / R.sub.TPR.sup.2 (35 b)

where .alpha..sub.1, .alpha..sub.2 and .alpha..sub.3 are single valued constants determined analytically or empirically. This matrix appears only in the angle filters and is the same for both azimuth and elevation. Its function is to reduce tracking error caused by angular scintillation, i.e., to estimate angular scintillation of the target.

The angle filter outputs to the azimuth and elevation command computers 15a,b are .omega..sub.LSP.sbsb.e .sbsb.d and .epsilon..sub.p.sbsb.e .sbsb.d as shown in FIGS. 3a,b. Note that for convenience, the subscripts i+1 have been omitted, it being understood that the outputs are for the time T.sub.i.sub.+1. The computers 15a,b calculate the antenna commands .omega..sub.c.sbsb.e and .omega..sub.c.sbsb.d as follows:

.omega..sub.c.sbsb.e (i.sub.+1) = .omega..sub.LSP.sbsb.e (i.sub.+1) + K.sub.2 . [l.sub.e (i.sub.+1) + a.sub.2 .epsilon..sub.P.sbsb.e (i.sub.+1) ] (36a)

.omega..sub.c.sbsb.d(i.sub.+1) = .omega..sub.LSP.sbsb.d (i.sub.+1) + K.sub.2 . [l.sub.d (i.sub.+1) + a.sub.2 .epsilon..sub.P.sbsb.d (i.sub.+1) ] (36b)

where K.sub.2 and a.sub.2 are single valued constants. Parameters l.sub.e and l.sub.d are intermediate antenna command variables computed in accordance with the following equations:

l.sub.e (i.sub.+1) = b.sub.2 l.sub.e (i) + c.sub.2 .epsilon..sub.P.sbsb.e .sbsb.i (37 a)

l.sub.d (i.sub.+1) = b.sub.2 l.sub.d(i) + c.sub.2 .epsilon..sub.P.sbsb.d .sbsb.i (37 b)

where b.sub.2 and c.sub.2 are single-valued constants. The l-parameters provide filtering (sloping) of the tracking errors .epsilon..sub.P.sbsb.e .sbsb.d. The computations provide for different selection of b.sub.2 and c.sub.2 during initial track and final track.

The computations of these Equations 36a and 36b consist of adding compensated elevation and azimuth LOS rate estimates to respective elevation and azimuth tracking errors to maintain angle tracking. The addition is represented in FIGS. 3a and 3b by an encircled summation symbol, .SIGMA.. The second term added to the tracking errors, the compensated LOS rate estimates, is formed by the indicated computation represented in FIGS. 3a and 3b by blocks becoming the legends COMP.sub.e and COMP.sub.d. As will be noted in an exemplary implementation (to be described hereinafter) of the present invention in the form of a digital radar data processor programmed to carry out the functions of at least sections 14, 15 and 16 of FIG. 1, the computation of Equations 37a and 37b is included in this angle tracking filter routine, rather than as part of an antenna control routine. That is done merely as a matter of convenience, to avoid having to store .epsilon..sub.P.sbsb.ei and .epsilon..sub.P.sbsb.di in a buffer during cycle i+1.

The equations for the range and velocity state vectors for blocks 43 and 44 of FIGS. 3c and 3d are ##EQU11## where R.sub.BIN is a scaling factor for converting .DELTA.R.sub.m into feet. ##EQU12## where f.sub.w is the filter band width for a radar frequency f=1/2 used to convert .DELTA.V.sub.m into feet per second. The superscripts and subscripts a in the range filter, and the superscripts and subscripts d in the velocity filter, indicate an initial track phase of a fixed time T.sub.j following acquisition of a target. In final track, which follows a fixed period of initial track, the superscripts and subscripts a and d are substituted by b and e, respectively. C.sub.g is a constant equal to 0 for initial track and to 1 for final track. The range and velocity gate command computations of the computer 16 are represented by blocks 16a and 16b in FIGS. 3c and 3d. In an exemplary embodiment, these computations (which will be described hereinafter) are carried out by a radar data processor, programmed as noted hereinbefore.

Referring first to just the range filter, computation of the transition matrix .PHI..sub.R.sbsb.a is as follows:

.PHI..sub.R.sbsb.a = I.sub.3 + IA.sub.Ra + 1/2T.sup.2 A.sub.Ra.sup.2 (40)

where I.sub.3 is a 3 .times. 3 identity matrix and A.sub.R.sbsb.a is the system dynamical matrix given by ##EQU13## C.sub.1 is a constant with one of two fixed values, depending upon target range at the time of entry into the initial track phase, and C.sub.2 is a single valued constant. These constants may be determined analytically or empirically.

For the initial phase, the operating filter period T is an integer multiple of the phase period T.sub.j, and the constant C.sub.9 is set equal to zero. That omits the dynamical aiding terms to simplify the range filter computations.

K.sub.1.sup.a, K.sub.2.sup.a and k.sub.3.sup.a are constants, each with one of three fixed values, depending upon one of three ranges (high, medium and low) of SNR.sub.RNG. At the time of entry into initial track, the middle Kalman gain K.sub.2.sup.a is used until the first valid SNR.sub.RNG measurement is made after entry.

A.sub.I.sbsb.r is the interceptor (own ship) acceleration about its r (roll) axis given by ##EQU14## where T.sub.1 denotes the time interval between the time of the updated motion compensation and the time of the last motion compensation, and .DELTA.V.sub.r is a velocity vector about the r axis computed from INS data in a manner to be described hereinafter.

In the final track phase, the constant C.sub.1 in the system dynamical matrix is replaced by a constant C.sub.3, a constant with one of two values depending upon target range. The resulting transition matrix .PHI..sub.R is then denoted .PHI..sub.R.sbsb.b. The time T is then the filter iteration period and the set of Kalman gain constants are replaced by K.sub.1.sup.b, K.sub.2.sup.b and K.sub.3.sup.b. The constant C.sub.9 is set equal to one to reinstate the dynamical aiding term.

Referring now to the velocity filter equation, the transition matrix .PHI..sub.R.sbsb.d is as fillows:

.PHI..sub.R.sbsb.d = I.sub.2 + TA.sub.R.sbsb.d + 1/2 T.sup.2 A.sub.R.sbsb.d (43)

where I.sub.2 is a 2 .times. 2 identity matrix and the system dynamical matrix A.sub.R.sbsb.d is given by ##EQU15## K.sub.2.sup.d and K.sub.3.sup.d are constants, each with one of three fixed values depending on SNR.sub.VEL as in the range filter. All other terms are as defined hereinbefore.

For final track in the velocity filter the constants K.sub.2.sup.d and K.sub.3.sup.d are replaced by constants K.sub.2.sup.e and K.sub.3.sup.e, each of which is again a set of three fixed values selected as a function of SNR.sub.VEL for every filter iteration. This provides a Kalman gain factor which is a function of SNR.sub.VEL, just as a Kalman gain factor is provided in the range filter as a function of SNR.sub.RNG.

The manner in which own-ship motion compensation calculations are made will now be described. Interceptor motion data including velocity components in the north (N), east (E) and down (D) coordinate system, and interceptor pitch (.theta.), roll (.phi..sub.I) and true heading (.psi.) are provided by the inertial navigation system (INS). A vector of first differences is then formed using the corresponding measurements taken approximately 0.2 second earlier: ##EQU16## where t.sub.i denotes the time of the updated measurement and t.sub.i.sub.-1 denotes the time of the previous velocity measurement used for motion compensation. Let T.sub.1 represent the time interval between these measurements, as noted hereinbefore. The INS and antenna angular motion data, (.theta., .phi..sub.1, .psi., .phi..sub.G, .epsilon., .eta.) measurements occur midway (.+-.5 m-sec) between the two times at which the above velocity first difference data are taken. The pitch, roll and true heading data from this intermediate measurement are used for motion compensation as shown below. The remaining data needed for this computation are: .epsilon. = antenna elevation gimbal angle; .eta. = antenna azimuth gimbal angle, and .phi..sub.G = antenna roll gimbal angle. Thus, all data for the tracking filter interceptor motion compensation will be available when the second velocity measurement is made. Coordinate transformation is used to obtain stabilized antenna coordinates .epsilon..sub.s, .eta..sub.s and .rho..sub.s from the above INS and antenna angular data. ##EQU17## Solving the above matrices will result in the following general equations.

.DELTA.V.sub.r = .DELTA.V.sub.N cos (.eta..sub.s + .psi.) cos .epsilon..sub.s + .DELTA. V.sub.E sin (.eta..sub.s + .psi.) cos .epsilon..sub.s - .DELTA.V.sub.D sin .epsilon..sub.s (50)

.DELTA.V.sub.e = .DELTA.V.sub.N [cos (.eta..sub.s + .psi.) sin .epsilon..sub.s sin .zeta..sub.s - sin .eta..sub.s cos .zeta..sub.s + .DELTA. V.sub.E cos (.eta..sub.s + .psi.) cos .zeta..sub.s + .DELTA.V.sub.D cos .eta..sub.s sin .zeta..sub.s (51)

.DELTA.V.sub.d = .DELTA.V.sub.N [cos(.eta..sub.s +.psi.) sin .epsilon..sub.s cos .zeta..sub.s + sin (.eta..sub.s +.psi.) sin .zeta. .sub.s ] - .DELTA.V.sub.E cos (.eta..sub.s +.psi.) sin .zeta..sub.s + .DELTA.V.sub.D cos .epsilon..sub.s cos .zeta..sub.s (52) The terms .DELTA.V.sub.e and .DELTA.V.sub.d are also used by the angle tracking filter channels.

To simplify other computations define: ##EQU18##

The manner in which SNR values are computed will be described, but first the following description will cover the manner in which they are used to determine the actual discriminant noise variances and actual discriminant slopes corresponding to each discriminant measurement utilized by the filters.

SNR data from the computer 13a are used to set the gains of the tracking filters. In both of the velocity and range filters, the gains available are quantized in initial track and final track, as just noted above. In the angle filters, the gains are continuous functions of SNR. Reasonably high SNRs are required for acceptable tracking. Initially the filters will be tracking (mostly extrapolating) at a relatively low SNR because the target will not be centered in the tracking gates. At low SNR levels, the current discriminant data is weighted by the filter gains lighter than under normal tracking because the return signal is weak relative to the noise in the signal.

Various techniques may be employed to compute the discriminants and the SNRs. By way of example, the range gate command, R.sub.GC, used to place the range gate at the predicted range bin of the target may be used to produce two adjacent range gates. The sign of a target signal from one gate is arbitrarily set negative, and the sign of a target signal from the other is set positive. The range discriminant is then essentially the algebraic sum of two range gated signals. The range tracking filter attempts to center the target between the adjacent gates. This type of tracker is referred to as the "split gate" type. To compute the SNRs, noise measurements are effectively made by averaging signal return in an equal number of range bins, or doppler filters, on each side of the gated range bins, or doppler filters, and subtracting that average from an average value of the two gated signal returns used to form the discriminants, taking into account noise present in the signal. The SNR's are then effectively computed as the ratio of the sum of the split gate signals used to compute the discriminants (less noise) and the noise thus measured. SNR measurements are averaged in the computer 13a over the same number of sampling cycles (typically four) as the preaveraging system 18 for the discriminants.

Split-gate tracking is used for all discriminants. For the velocity discriminants, the technique is directly analogous to that described for range. Two doppler filters are gated and the tracker attempts to maintain the target centered between the two filters. For the azimuth and elevation tracking, range-gated signals from the monopulse radar system are processed as the discriminants. The angle trackers attempt to maintain the target at the antenna axis by driving the antenna to a position such that (A+C)-(B+D) = 0 and (A+B)-(C+D) = 0. In that manner coherent subtraction of the left and right half of the antenna forms the azimuth discriminant, and the coherent subtraction of the upper and lower half of the antenna forms the elevation discriminant. Note that microwave plumbing on the back of the antenna does the coherent subtraction. Range and velocity discriminants are formed with only the sum channel signal (A+B+C+D).

In each case, the discriminant is a bipolar signal. In the noise free case, its magnitude is dependent on the tracking error as well as target signal magnitude which varies with range and size of target. To reduce the dependence on signal magnitude, the discriminants can be normalized by dividing by some function, f (M.sub.1, M.sub.2), of the target signal, where M.sub.1 and M.sub.2 are the signals used to form the discriminant. The present system uses the sum as follows: ##EQU19## Normalizing by dividing by the largest of M.sub.1 and M.sub.2 would give 50 percent better gain stability over SNR varying from 6 to 30 db, but that normalizing technique would only be possible in range and velocity tracking because in angle tracking M.sub.1 and M.sub.2 are not known separately, but their sum (M.sub.1 + M.sub.2) is known.

In the noise free case the normalized discriminant magnitude is dependent on the error, and its sign indicates the direction of the error. Receiver noise not only produces measurements with dispersion around where they would be in the noise free case, but actually causes the mean output to be lower. The mean suppression becomes greater with lower signal-to-noise ratio.

Typical range discriminant characteristics are shown in FIGS. 4, 5 and 6. FIG. 4 shows range discriminant means, i.e., the mean value of discriminants as a function of range error in bin widths for different SNR's of -5, 0, +7, +12 and +20 db. The standard deviations, .sigma., are shown in FIG. 5 for the same range errors and SNR's. FIG. 6 shows small signal discriminant slopes. These data can be determined analytically, but are preferably determined experimentally, for use in the selection of the gains in the filters. As noted hereinbefore, each filter can provide optimal tracking performance under instantaneous signal characteristics if provided data on the actual discriminant noise variances, .sigma..sub.i.sup.2, and actual discriminant slopes, k.sub.i, corresponding to each discriminant measurement, i.e., on a cycle by cycle basis. Once the data of FIGS. 4, 5 and 6 have been established, the noise variances and discriminant slopes can be determined during each filter cycle as a function of discriminants and SNR. The discriminant slope, k.sub.i, and the noise variance, .sigma..sub.i.sup.2, are used in the calculations of the Kalman gain, K.sub.i. This is, in theory, to be done in each filter during each cycle, but in practice it is fully done only in the angle tracking filters, and even in the angle filters an average discriminant slope k.sub.a.sbsb.i = 1/2 (K.sub.e.sbsb.i + K.sub.d.sbsb.i) and an average noise variance .sigma..sub.a.sbsb.i.sup.2 = 1/2 (.sigma..sub.e.sbsb.i.sup.2 + .sigma..sub.d.sbsb.i.sup.2) of the elevation (e) and azimuth (d) channels are used to calculate a Kalman gain, K.sub.a.sbsb.i, for use in both angle filters. That is done because the Kalman gain factor is an input to the calculations of the one step covariance matrix, M.sub.i, which is quite complex and would require too much time to compute separately for each channel. Instead an average covariance matrix, M.sub.a.sbsb.i, is computed for use in both filters using an average Kalman gain.

In the range and velocity filters, the computations are further simplified in the exemplary embodiment to be described hereinafter. That is done by simply providing three precalculated Kalman gain factors for selection based on SNR. The SNR used in each filter is ideally SNR.sub.RNG for range and SNR.sub.VEL for velocity, but in practice both SNR's may be assumed to be very nearly the same at all times, at least sufficiently so for the accuracy required in using just one SNR for precomputed Kalman gain factors in the range and velocity channels. Consequently, in the exemplary embodiment to be described, SNR.sub.RNG could be used to select the Kalman gains K.sub.r.sbsb.i and K.sub.v.sbsb.i for the range and velocity channels. The theory otherwise remains the same for all filters as described hereinbefore in general terms.

In an exemplary embodiment a digital radar data processor is programmed to implement all four filters. Consequently, the estimator 14 will now be described with reference to functional flow charts of a functional filter (FT) Routine in FIGS. 8a, 8b and 8c, and with reference to detailed flow charts of that routine in FIGS. 9a through 9f.

The FT Routine provides quantities necessary to maintain track. Specifically, the variables generated by FT and passed on to other routines to maintain track are listed below.

.epsilon..sub.p.sbsb.e -- Elevation tracking error estimate.

.omega..sub.LSP.sbsb.e -- Target elevation LOS rate estimate used to maintain elevation angle tracking.

.epsilon..sub.P.sbsb.d -- Azimuth tracking error estimate.

.omega..sub.LSP.sbsb.d -- Target azimuth LOS rate estimate used to maintain azimuth angle tracking.

R.sub.gc -- range gate command derived by radar controller from R.sub.TPR and used to generate the range gate R.sub.G.

V.sub.gc -- velocity gate command derived by radar controller from V.sub.TPV and used to generate the velocity gate V.sub.G.

To compute the quantities necessary to maintain angle tracking, FT implements a four-state Kalman filter for both the elevation and azimuth channels. The state vectors of the angle filters are:

    .epsilon..sub.P.sbsb.e
    Elevation tracking error estimate.
    .omega..sub.LSP.sbsb.e
    Target elevation LOS rate estimate.
    a.sub.Td
    Target acceleration (inertial) estimate along
             antenna d-axis.
    S.sub.e
    Elevation angular scintillation estimate.
    .epsilon..sub.P.sbsb.d
    Azimuth tracking error estimate.
    .omega..sub.LSP.sbsb.d
    Target azimuth LOS rate estimate.
    a.sub.T.sub.-e
    Target acceleration (inertial) estimate along
             antenna (-e) axis.
    S.sub.d
    Azimuth angular scintillation estimat