RadarLab

Thus far we have considered radar mainly as a surveillance senser that detects target over a region of space.A radar not only recognizes the presence of a target, but it determines the target's location in range and in one or two angle coordinates.As it continues to observe a target over time,the radar can provide the target's tragectory, or track and predict where it will be in the future.Various surface and airborne radars constitute a major sensor input to computerized command and control systems such as the Marine MACCS and Navy NTDS. These facilities perform multiple functions, including surface, subsurface, and airspace surveillance; identification; threat evaluation; and weapons control. To operate in today's environment, command and control systems require near real-time (as it occurs) target position and velocity data in order to produce fire control quality data for a weapon system. Until recent times, the interface between the raw data supplied by the sensor and the command and control system was the human operator; however, this limited the data rate and target handling capacity. Additional personnel were added as a partial solution to these problems, but even under ideal conditions of training, motivation, and rest, the human operator can only update a maximum of six tracks about once per two seconds and that only for a short time. To supply fire control quality data, the classical solution has been to provide separate precision tracking sensors. Although the systems have widespread use, they are limited to tracking at a single elevation and azimuth. Data from these systems were provided to an analog computer to solve the fire control problem and compute launcher and weapon orders.

Track-while-scan (TWS){Limited Sector Scan}systems are tracking systems for surveillance radars whose nominal scan time (revisit time) is from 4 to 12 s for aircraft targets. If the probability of detection (PD) per scan is high, if accurate target location measurements are made, if the target density is low, and if there are only a few false alarms, the design of the correlation logic (i.e., associating detections with tracks) and tracking filter (i.e., filter for smoothing and predicting track positions) is straightforward. However, in a realistic radar environment these assumptions are seldom valid, and the design of the automatic tracking system is complicated.With the advent of the high-speed electronic digital computer, computational capacity exceeded the data-gathering capacity of the fire control sensors; however, target handling continued to be limited by the non-scanning tracking methods employed. The solution to both problems was to interface a high-scan-rate (15-20 rpm) search radar with the electronic digital computer through a device capable of converting the pattern of echo returns detected by the receiver into usable binary information. In addition, using the digital computer to direct and compose radar beam patterns, as described in the next chapter, created a flexible system capable of multimode simultaneous operation in search, designation, track, and weapon control. This ability to perform such functions as multitarget tracking, prediction of future target positions, and terrain avoidance for aircraft navigation is transforming previously fragmented sensor and control capability into a dynamic and powerful sensing, command and control, and weapon employment system. The combination of automatic detection and tracking (ADT) functions within a search radar is sometimes referred to as Track-While-Scan (TWS). ADT is being retrofitted into older equipment and is included in most new equipment. The central concept of TWS is to maintain target tracks in a computer with periodic information updates from a scanning radar.This radar rapidly scans a limited angular sector to maintain tracks ,with a moderate data rate, one more than one target within the coverage of antenna.it has been used in the past for air defense radar ,aircraft landing radars,and in some airboune intercept radar to hold multiple targets in track.

TWS radar used to rapidily scan a relatively narrow angular sector, usually in both azimuth and elevation.it combines the search function and the track function.Scanning may be performed with a single narrow beamwidth pencil beam(or a monopulse cluster of beams) that might cover a rectangular sector in a raster fashion.Scanning can also be performed with two orthogonal far beams,one that scan in azimuth and the other in elevation. TWS radar have ben used in airport landing radars,airbone interceptors,and air-defense systems.A difference between a continuous tracker and the TWS radar is that the angle error signal in a continuous tracker is used in a closed-loop servo system to control the pointing of antenna beam.In the TWS radar,however there is no closed-loop positioning of the antenna.its angle output is sent directly to data processor.Another significant difference is that the TWS radar can provide simultaneous tracks on a number of target within its sector of coverage,while the continuous tracker observes only a single target which is why it is sometimes called a single-target tracker(STT).With comparable transmitter and antennas, the energy available to perform tracking is less in a TWS radar than a STT since the TWS store its radiated energy over an angular sector rather than concentrate it in the direction of single target.In airborne-interceptor applications,the TWS radar might be preferred when multiple target have to be maintained in track and the tracking accuracy only has to good enough to launch missiles which contain their own guidance system to home on the target.On the other hand,of highly accurate tracking is needed ,a single target tracker might be preferred.

Limited Sector Scan TWS radar have been used in Precision Approach Radar(PAR) or Ground - Controlled Approach(GCA) system that guide aircraft to a landing.These radars allow a ground controller to direct an aircraft to a safe landing in bad weather by tracking it as its lands.The ground controller communicates to the pilot directions to change his or her heading up,down,right,or left.In the control of aircraft landing,fan beams have been used which are electromagnetically scanned over a narrow sector at a rate of twice per second.The azimuth sector that is scanned might be 20 degree and the elevation sector 6 degree or 7 degree.Landing radar using limited scan phased array antennas,such as the AN/TPS-19, operate differently then scanning fan beams since they electronically scan a pencil beam over a region 20 degree in azimuth 15 degree in elevation at a rate of twice per second.The AN/TPN-19 used multiple recieve beams to obtain a monopulse angle measurement.Being a phased array, the AN/TPN-19 could simutanously track up to six aircraft at a 20HZ data rate.In the past radars for the control of landing aircraft have beeen mainly used by the military.Civilian pilots prefer to use landing systems in which the control of landing is in the aircraft's cockpit rather than from a voice orginating from the ground.

Track while scan radar have also been used successfully for the control of weapons in surface to air miscile systems for both land and ship based air defense especially by the former Soviet Union.Generally ,TWS System using for fan beams have some limtation compared to system that operate that one or more pencil beams.The fan beam system can see more rain and surface clutter it is more vulnerable to electronic countermeasures and there might be problems with associating multiple targets that appear in the two beams.

Unfortunately, the same name track while scan was also applied in the past to what is now usually called Automatic detection and tracking (ADT).This perform tracking as part of an air surviellance radar. it is found in almost all modern civil air traffic control radar as well as military air surveillance radar. The rate at which observtions are made depends on the time for the antenna to make one rotation(which might vary from a few seconds to as much as 12 seconds).The ADT,therefore has a lower data rate than that of the STT,but its advantage is that it can simultanously track a larger number of targets(which might be many hundred or a few thousands of aircraft).Tracking is done open loop in that the antenna position is not controlled by the processed tracking data as it is in the STT.One method of obtaining tracks with a surveillance radar is to have an operator manually mark with grgase pencil on the face of the cathode-ray tube the location of the target on each scan. The simplicity of such a procedure is offset by the poor accuracy of the track. The accuracy of track can be improved by using a computer to determine the trajectory from inputs supplied by an operator. A human operator, however, cannot update target tracks at a rate greater than about once per two seconds.Thus, a single operator cannot handle more than about six target tracks when the radar has a twelve-second scan rate (5 rpm antenna rotation rate). Furthermore an operator's effectiveness in detecting new targets decreases rapidly after about a half hour of operation. The radar operator's traffic handling limitation and the effects of fatigue can be mitigated by automating the target detection and tracking process with data processing called automatic detection and tracking (ADT). The availability of digital data processing technology has made ADT economically feasible. An ADT system performs the fluctuations of target detection, track initiation, track association, track update, track smoothing (filtering) and track termination.

The automatic detector part of the ADT quantizes the range into intervals equal to the range resolution. At each range interval the detector integrates 11 pulses, where n is the number of pulses expected to be returned from a target as tile antennas catis past. The integrated pulses are compared with a threshold to indicate the presence or absence of a target. An example is the commonly used moving window detector which examines continuously the last r-1 samples within each quantized range interval and announces the presence of a target if m out of 11 of these samples cross a preset threshold.By locating the center of the n pulses, an estimate of the target's angular direction can be obtained. This is called beam splitting. If there is but one target present within the radar's coverage, then detections on two scans are all that is needed to establish a target track and to estimate its velocity. However, there are usually other targets as well as clutter echoes present, so that three or more detections are needed to reliably establish a track without the generation of false or spurious tracks. Although a computer can be programmed to recognize and reject false tracks, too many false tracks can overload the computer and result in poor information. It is for this same reason of avoiding computer overload that the radar used with ADT should be designed to exclude unwanted signals, as from clutter and interference. A good ADT system therefore reqirires radar with a good-MTI and a good CEAE-(constant false alarm rate) receiver. A clutter map, generated by the radar, is sometimes used to reduce the load on the tracking computer by blanking clutter areas and removing detections associated with large point clutter sources not rejected by the MTI. Slowly moving echoes that are not of interest can also be removed by the clutter map. The availability of some distinctive target characteristic, such as its altitude, might also prove of help when performing track association.Thus, the quality of the ADT will depend significantly on the ability of the radar to reject unwanted signals.

A functional tracking system must provide the basic target parameters of position and rates of motion. In the systems presented both position data and velocity data were used to maintain the tracking antenna on the target at all times, thus limiting the system to the traditional single-target tracking situation. In a track-while-scan system, target position must be extracted and velocities calculated for many targets while the radar continues to scan. Obviously, in a system of this type, target data is not continuously available for each target. Since the antenna is continuing to scan, some means of storing and analyzing target data from one update to the next and beyond is necessary. The digital computer is employed to perform this function and thus replaces the tracking servo systems.

Any track-while-scan system must provide for each of the following functions:

(1) Target detection

(2) Target track correlation and association

(3) Target track initiation and track file generation

(4) Generation of tracking “gates”

(5) Track gate prediction, smoothing (filtering), and positioning

(6) Track Termination

Target detection is accomplished by special circuitry in the receiver of radars designed as TWS systems or by signal-processing equipment in a separate cabinet in retrofit installations. The functions of this ADT processing equipment is similar and will be addressed here in general terms. It should be remembered that the ADT processor is not a computer, although it does have some memory capacity. Its primary function is that of converting data to a form usable by the separate digital computer that actually performs the smooth tracking, velocity, acceleration rate generation, and prediction functions of the TWS system.

The ADT processor's memory allocates space addressed to each spatial cell that makes up the search volume of the radar. The dimensions of these cells are often defined by the horizontal and vertical angular resolution capability of the radar as well as by a distance in range equal to the pulse compression range resolution.In some systems, individual beams in the radar are overlapped such that most targets will occupy more than one beam position at any radar range. When employed, this procedure (called beam splitting) allows angular resolution less than the beamwidth of the radar. In addition, this avoids gaps in coverage and allows examination of hit patterns among several beams as a means of rejecting clutter or false targets. A radar may require N hits out of M beams at the same range interval, with all hits overlapping in elevation and azimuth before any further consideration is given to initiating a track. N and M may be operator selectable or preset during the design process. Most systems employ further means to avoid generation of tracks on clutter or false returns. These can be grouped into those associated with conventional radar receiver anti-jamming circuitry and those exclusive to ADT. It should be obvious that an effective TWS radar must retain conventional features such as Constant False-Alarm Rate (CFAR), MTI, and automatic gain-control circuitry. These will help prevent the processing of obviously false returns, allowing the ADT system to make more sophisticated decisions between valid an invalid targets. The ADT processor will generate a time history of all echo returns over several radar scans. Stationary targets remaining within a given resolution cell over a number of scans result in its identification as a clutter cell. The collected information on all clutter cells within the radar search volume is referred to as a clutter map. Returns remaining in clutter cells will be rejected as false targets. Moving targets will not remain in a given cell long enough to be eliminated. Returns that very rapidly in strength and centroid location, such as clouds and chaff in high winds, could appear non-stationary and would cause a track to be established. These returns are then tested for the known velocity characteristics of valid targets by velocity filters, thus reducing the number of false tracks that would otherwise be generated in this manner.

One approach is to first quantize the range and sometimes the azimuth angle.The quantization increment in range might be the pulse width and that is angle might be the azimuth beamwidth.At each range-azimuth quantization cell, the pulse received during the time the antenna scans past the target are integrated and a detection decision is made. CFAR generally is incorporated before the decision process inorder to prevent excessive false alarm due to clutter echoes. Pulse Integration is performed in some form of automatic detector,or integrator.

Another approach to automatic detection is the moving window detector which examines continuosly the last n pulses and announces the presence of a target if it at least m out of n of the pulses exceed a present threshold.A by product of the automatic detection decision with a moving window detector or something similar is an angle measurement made by beam splitting. if n pulses expected to be recieved from a target ,beam splitting involves recognizing the begining and end of the n pulses and locating their centre.Angle accuracy depends on how well the begining and end of the tram of n pulse can be determined,as well as the number of pulses available and their signal to noise ratio.The beam splitting decision logic usually has no prior knowledge of targets begining.The logic must be sufficiently sensitive to quickly recognize the increased density region that signifies the start of an echo-signal pulse train , yet it must not be so sensitive it generates false starts due to noise alone.Once a target's beginning is recognized,the device must sense the end of the increased density region .If the decision logic is too sensitive to change,it could cause a single target to split into two. A rough rule of thumb often quoted is that the accuracy of beam splitting is about one-tenth of a beamwidth when the signal to noise ratio is high enough to provide a good probability of detection.

Target observation on each radar scan that survives hit pattern recognition and clutter rejection functions is initially treated as new information prior to comparison with previously held data. Logic rules established in the digital computer program, which perform correlation and association of video observations with currently held tracks, are the key to preventing system saturation in high-density environments. Generally speaking, target observations that fall within the boundary of the tracking gate are said to correlate with that track. Each observation is compared with all tracks in the computer's memory and may correlate with one track, several tracks, or no tracks. Conversely, a track gate may correlate with one observation, several observations, or no observations. Whenever there is other than a one-to-one correlation of observations with tracking gates, tracking ambiguity results.

When a new detection is recieved that is not at the location of a clutter echo stored in the clutter map, an attempt is made to associate it with an existing track.Association with an existing track is aided by establishing for each track a small search window or gate ,within which the detection of target on the next scan of the radar antenna is predicted to appear. The gate should be as small as possible in order to avoid having more than one echo fall within it when the traffic density is high or when two tracks are close to one another. On the other hand, a large gate region is needed if the tracker is to follow target turns or maneuvers. More than one gate size is used to over come this dilemma.Figure 1 shows a small nonmaneuvering gate situated around the predicted position of the target in track. The size of the gate is determined by the estimated errors in the predicted position and the estimated errors in speed and direction of the track. The detection threshold might be lowered in the gate region to increase the probability of detection. When an echo is not found within the maneuvering gate ,the larger region encompassing the maneuvering gate is then searched. The size of the maneuvering gate is determined by the estimate of the maneuvering capability of the target under track.

One reason the target might , not appear in the nonmaneuvering gate is that its radar cross section might decrease, or fade,so that it is not detected.When this is the case, it is possible for a false track to occur when a noise spike or an echo from another target is found in the maneuvering gate.To avoid the problem caused by a target fade and a false indication appearing in the larger maneuvering gate, the tracks can be divided into two tracks.(This is known as bifurcation of the track). One in the original track with no new detection in the nonmaneuvering gate.The other is a new track based on the signal found in the maneuvering gate. After recieving the target position on the next scan of the radar (or sometimes after two scans), a decision is made as to which of two tracks should be dropped.Tracking is usually done in cartesian coordinates, but the coorelation gates are defined in polar (r,θ) coordinates.

Resolution of track ambiguity:Track ambiguity arises when either multiple targets appear within a single track window or two or more gates overlap on a single target. This occurrence can cause the system to generate erroneous tracking data and ultimately lose the ability to maintain a meaningful track. If the system is designed so that an operator initiates the track and monitors its progress, the solution is simply for the operator to cancel the erroneous track and initiate a new one. For automatic systems, software decision rules are established that will enable the program to maintain accurate track files, resolve ambiguity, and prevent saturation by false targets. Depending upon the specific use, many different sets of rules are employed to resolve this ambiguity, an example of which is outlined below:

1. If several video observations exist within one tracking gate, then the video closest to the center of the gate is accepted as valid.

2. If several tracking gates overlap one video observation, the video will be associated with the gate with the closet center.

3. If tow contacts cross paths, then the tracking gates will coast through one another, and re- correlation will occur when the gates no longer overlap.

4. If one video observation separates into two or more, rule I above is applied until there is clear separation in velocity and position and a new tracking gate is generated for the uncorrelated video. Rule number 2 is then applied until the gates no longer overlap.

Concurrent with the generation of the acquisition window, a track file is generated in order to store the position and gate data for each track. In addition to the basic position and window data, calculated target velocities and accelerations are also stored in each track file. For ease of calculation and interchange of information with other systems, all data is converted from polar to rectangular coordinates by the computer, as described in succeeding chapters. Track files are stored within the digitalcomputer subsystems' memory in rectangular coordinates, and the data are used to perform the various calculations necessary to maintain the track. Note that within the computer, position data has been converted from polar coordinates to rectangular coordinates. Each track file occupies a discrete position in the digital computer's high-speed memory. As data are needed for computation or new data are to be stored, the portion of memory that is allocated for the required data will be accessed by the system software (programs). In this manner, tactical data, in addition to the tracking data, may be stored in the “track” file–for example, ESM data, engagement status, and IFF information. (It is equally important to track friendly forces as well as hostile forces.) The generation of the track file begins with the initial storage of position data along with a code to indicate that an acquisition window has been established. If target position data is obtained on subsequent scans of the radar, the file is updated with the coordinates, the velocities and accelerations are computed and stored, and the acquisition window code is canceled. The acquisition window is then decreased in size relative to that of the tracking window, and the track code is stored, which indicates an active track file. As the radar continues to scan, each input of data is compared with the gate positions of active track files until the proper file is found and updated. The techniques of computer data file sorting and searching are beyond the scope of this text; however, it should be noted that the search for the proper track file is generally not a sequential one-to-one comparison. This method is much too slow to be used in a system where speed of operation is one of the primary goals.

In principle, a track can be initiated from the target location information obtained on two successive scans of the radar antenna. In practice, however target information from three or more scans is usually needed to intiate a track.Two scans would be adequate when there is only one or a few aircraft within view, but when the radar has in view a larger number of echoes, one or more additional scan may be needed to prevent false tracks from being initiated. Thus it is more usual to require three or more scan before establishing a track.A clutter map is used to store, the locations of fixed clutter echoes and prevent tracks from being initiated based on a clutter echo combined with a real target detection.Such tracks can eventually be recognized as false and can be dropped , but it takes time and computer capacity to do so when there are a large number of them.Clutter echoes for inclusion in the clutter map are those echoes that do not change their location with time or that change loction too slowly to be targets of interest.

The process of initiating a track in a dense environment of targets and clutter not diminated by the radar can be quite demanding in both computer software and hardware. Initiation of a new track may take more computer time and capability than any other aspect of ADT.Requiring three scans for a civil air-traffic control radar to establish a track is usually not a burden. Waiting three scans for track establishment , however may be an excessively long time for a military air-defense radar that has to direct weapon-control radars to defined against high speed trackers that “pop up” at short range over the horizon.it is possible to quickly aquire the target on the basis of a sinlge scan past the target if the radar can obtain a quick second look. This might be done with a look-back beam directed to the angle of the original detection.The quick look-back can provide confirmation of detection and an estimate of target's related velocity. A phased array radar is well suited for this purpose, but mechanical rotating radar can also be outfitted with a fixed look back beam. Look back might also be accomplished with a 3D radar whose eletronically scanning beam in elevation is returned to the elevation angle of initial detection, before the radar beam entirely scans past the target.

A “gate” can be defined as a small volume of space composed of many of the cells described previously, initially centered on the target, which will be monitored on each scan for the presence of target information. Gate dimension and position information is generated by the general-purpose digital computer and sent to the ADT processor for application. Action by the clutter map is inhibited inside these gates to avoid generation of clutter cells by valid tracks. When a target is initially detected, the algorithm receives only the position data for that initial, instantaneous target position. The acquisition gate is then generated in the following manner:

Acquisition Gate Tracking Gate

The acquisition gate is large in order to allow for target motion during one scan period of the radar. If the target is within the acquisition gate on the next scan, a smaller tracking gate is generated that is moved to the new expected target position on subsequent scans. In the event that a video observation does not appear within the tracking gate on a subsequent scan, ADT will enter some type of turn-detection routine. A common means of dealing with a turn or linear acceleration of the target is the turn-detection gate. The turn detection gate is larger than the tracking gate and is co-located with it initially, employing separate logic and algorithms different from the tracking routine. The turn-detection gate size is determined by the maximum acceleration and turn characteristics of valid tracks. Some systems will maintain one track with the original tracking gate and one with the turn-detection gate for one or more scan after initial turn detection, allowing for the possibility of inaccurate observation, clutter, or possible decoys prior to the determination of which is the valid track position. Only one of these tracks would be displayed to the operator, that one being determined by a programmed validity routine.

As was discussed in the earlier sections on servocontrolled tracking systems, (conical scan and monopulse), tracking errors were generated as a result of the target moving off the antenna axis. It was the task then of the error detectors and servo systems to reposition the antenna axis onto the target. During the process of repositioning the antenna, the system response motion was smoothed by employing rate (velocity) and position feedback. Recall that this feedback was accomplished by electrical and mechanical means within the servo system. Recall also that in general the system lagged the target, i.e., the target would move the system would respond. In a track-while-scan system, tracking errors also exist due to target motion. The tracking gate now has replaced the “tracking antenna,” and this gate must be positioned dynamically ont eh target in a manner similar to that of the “tracking antenna.” However, there is no “servo” system to reposition and smooth the tracking gate's motion. This repositioning and smoothing must be done mathematically within the TWS algorithm. To this end, smoothing and prediction equations are employed to calculate the changing position of the tracking window. Instead of the system “lagging” the target the tracking gate is made to “lead” the target, and smoothing is accomplished by comparing predicted parameters with observed parameters and making adjustments based upon the errors derived from this comparison.

The classic method of smoothing track data is by the - tracker or - filter described below. This simple filter is ill-suited to extreme target maneuvers and in most current systems is increased in complexity to what is called the Kalman filter. Among other things, the Kalman filter is capable of dealing with higher order derivatives of target motion (i.e., beyond acceleration). In some systems, complete track file information is retained in the separate command and control system computer, with the TWS computer retaining only position data, velocity, and acceleration.On the basis of a series of past target detections, the automatic tracker makes a smooth (filtered) estimate of the target's present position and velocity , as well as its predicted position and velocity.One method or accomplishing this is with the α-β(alpha-beta) tracker.

The role of the track-smoothing function is to take the current known state (i.e. position, heading, speed and possibly acceleration) of the target and predict the new state of the target at the time of the most recent radar measurement. It is a computing step to improve the estimate of the tracks position as well as to revise the errors in the former prediction. The latest track prediction is combined with the associated plot to provide a new, improved estimate of the next target state. There are a wide variety of algorithms, of differing complexity and computational load that can be used for this process. These algorithms are realized in special filters.

All of these filters work in two steps:

♦prediction, and

♦correction.

The observed position of the target is recorded for the last two or more scans. Then, the smoothed parameters (often called innovations) are computed for position, velocity and acceleration. Using these values, the predictions for position and velocity are computed.

Successive applications of this system of equations serves to minimize the tracking error in a manner similar to the repositioning of the “tracking antenna” in the servo tracking system. It should be noted at this point that although the techniques of tracking differ between the two, i.e., servo and TWS, the function and the concepts exhibit a direct correlation.If the radar does not receive target information on a particular scan , the smoothing and prediction operation can be continued by properly accounting for the missing data. (This is sometimes called coasting).When data from a target is missing for a number of consecutive scan, the track is terminated. Although the criterian to be used for determining when to terminate a track depends on the application,it has been suggested that when three target reports are used to establish a track, five consecutive misses is a suitable criterian for termination.

TWS Radar System Advantages

1)The advantage of TWS compared to a continuous tracker is that multipath target can be tracked. Becuase it shares its energy over region of space, the TWS radar needs to have a larger transmitter to obtain the same detection and tracking capabilities of a STT that dwells continuously on a single target.

2)TWS radar are also more vulnerable to angle jamming than are continuous trackers.

3)TWS radar can use monopulse angle measurements is not made with closed-loop tracking a TWS radar should not experience the wild fluctuations in angle caused by glint when there are not tiple scatterers within the resolution ceil or when there is multipath at low elevation angles.

4)Scan and track on same system.

5)Tracking information used for computation of a fire control solution.

6)Immediate fire control solution.

7)Automated tracking

8)Track prediction

9)Tracks multiple targets simultaneously

10)Tracking info used to compute FC solution

11)Tracks with only a search radar and a computer

Real World Applications

1)AN/SYS-2 IADT(Integrated Automated Detection and Tracking)

♦used on some missile ships

♦Carriers

♦CG

♦DDG

♦FFG

2)Mk 92 CAANS - FFG

3)AN/SPQ-9 – 5” Gun

4)Mk 23 TAS – NATO Sea Sparrow

5)AEGIS C&D

♦holds track files for AEGIS

♦dedicated computer

Conventional TWS procedures applicable to mechanically scanned radar systems do not fully exploit the potential of radars employing phased array beam steering in all axes. The ability of these systems to position beams virtually instantly and to conduct many types of scans simultaneously provides great flexibility in planning tracking strategies. These radars conduct what amounts to a random search, employing different pulse widths and PRTs applicable to the situation. When an echo that meets required parameters is observed, the radar immediately transmits several beams in the vicinity of the target while continuing search in the remainder of the surveillance area. This results in the immediate establishment of a track without the relatively long (.5 to 4 seconds) time period waiting for the antenna to return on the next scan, which is experienced in mechanically scanning radars. This technique, called active tracking, results in greatly decreased reaction time and more efficient use of the system.

A typical tracking radar has a pencil beam to receive echoes from a single target and track the target in angle, range, and/or doppler. Its resolution cell—defined by its antenna beamwidth, transmitter pulse length, and/or doppler bandwidth— is usually small compared with that of a search radar and is used to exclude undesired echoes or signals from other targets, clutter, and countermeasures. Electronic beam-scanning phased array tracking radars may track multiple targets by sequentially dwelling upon and measuring each target while excluding other echo or signal sources. Because of its narrow beamwidth, typically from a fraction of 1° to 1 or 2°, a tracking radar usually depends upon information from a search radar or other source of target location to acquire the target, i.e., to place its beam on or in the vicinity of the target before initiating a track. Scanning of the beam within a limited angle sector may be needed to fully acquire the target within its beam and center the range-tracking gates on the echo pulse prior to locking on the target or closing the tracking loops. The primary output of a tracking radar is the target location determined from the pointing angles of the beam and position of its range-tracking gates. The angle location is the data obtained from synchros or encoders on the antenna tracking axes shafts (or data from a beam-positioning computer of an electronic-scan phased array radar). In some cases, tracking lag is measured by converting tracking-lag-error voltages from the tracking loops to units of angle. This data is used to add to or subtract from the angle shaft position data for real-time correction of tracking lag.

There are a large variety of tracking-radar systems, including some that achieve simultaneously both surveillance and tracking functions. A widely used type of tracking radar and the one to be discussed in detail is a ground-based system consisting of a pencil-beam antenna mounted on a rotatable platform which is caused by motor drive of its azimuth and elevation position to follow a target in Figure 6. Errors in pointing direction are determined by sensing the angle of arrival of the echo wavefront and corrected by positioning the antenna to keep the target centered in the beam.

The principal applications of tracking radar are weapon control and missile-range instrumentation. In both applications a high degree of precision and an accurate prediction of the future position of the target are generally required. The earliest use of tracking radar was in gunfire control. The azimuth angle, the elevation angle, and the range to the target were measured, and from the rate of change of these parameters the velocity vector of the target was computed and its future position predicted. This information was used to point the gun in the proper direction and to set the fuzing time. The tracking radar performs a similar role in providing guidance information and steering commands for missiles. In missile-range instrumentation, the tracking-radar output is used to measure the trajectory of the missile and to predict future position. Tracking radar which computes the impact point of a missile continuously during flight is also important for range safety. Missile-range instrumentation radars are normally used with a beacon to provide a point-source target with high signal-to-noise ratio. Some of these systems achieve a precision of the order of 0.1 mil in angle and a range accuracy of 5 yd.

By extending the concepts described in this chapter and adding additional computer capability, it is possible to combine the outputs of several radars that are co-located, as on ships or at shore-based command and control systems such as MACCS. Systems such as the AN/SYS-2 IADT, now entering limited use aboard missile ships, develop a single track file based on the outputs of several radars. When radars are employed with different scan rates, a separate external timing reference must be employed, which becomes the scan rate for the IADT and the synthetic video display. Track updating and smoothing occur as previously described, except updates are considered relative to narrow sectors of the search volume–typically a few degrees in width. System software updates tracks a few sectors behind the azimuth position of the synthetic scan. Observations are accepted from the radar having the oldest unprocessed position data sector by sector as the synthetic scan passes. This provides the first available position report, no matter which radar produced it. Thus, position reports are used as they occurred in real time, and no position report is accepted out of order. IADT reduces loss of data due to individual radar propagation and lobing characteristics while allowing quality weighting of data relative to it source. IADT systems can accept the output of TWS or non- TWS radars as well as that derived from IFF transponders.When multiple radar view a common volume, there can be improved tracking , the data rate can be greater than any of the radar acting alone ,there is less vulnerability to electronic countermeasures, and less likelihood of having missed detection due to reduced echo-signal strength caused by null in one of the antenna pattern or changes in the target aspect.

Integrated Tracking From Collocated Radar at a Single Site:When more than one radar covering approximately the same volume in space, are located in the same vicinity , their individual output can be combined to form a single track.The radar might operate in different frequency bands, have different antenna characteristics , and diferent data rates. There is more than one way to combine the output of multiple radars.A good approach is to combine all the detections from each radar to form a single track and to update the track rather than develop seperate tracks at each radar and either select the best track or combine them is some other manner.The data from the various radars do not arrive at the tracker at a uniform rate.The development of a single track file by the use of the total data available from all radars produce a better track than combining the tracks developed individually at each radar.it reduces the likelihood of a loss of data as might be caused by antenna lobing, target feeding, interference, and clutter since integrated processing permits the favourable weighting of the better data and lesser weighting of poorer data. This method of combining data from multiple radar has been known as either automatic detection and integrated Tracking (ADIT) or integrated automatic detection and Tracking (IADT).

The Integrated Automated Detection and Tracking (IADT) System AN/SYS-2 is a computer-based radar data processor with automated radar target detection, tracking, and correlation capabilities. The AN/SYS-2 correlates contact data from the 2-D and 3-D air-search radars to provide a single, unduplicated, highly-accurate, surveillance picture output in various operational environments, including clutter and electronic countermeasures. The AN/SYS-2 accomplishes this by taking advantage of the mutually supporting aspects of the 2-D and 3-D radars surveillance volume of coverage and by exploiting their special modes of radar operation. The AN/SYS-2 is designed to enhance significantly the effectiveness of the combat system by reducing reaction time, improving threat assessment, providing earlier warning, and providing a prompt and reliable detection capability in the presence of high target density and electronic countermeasures. Systems such as the AN/SYS-2, now entering limited use aboard missile ships, develop a single track file based on the outputs of several radars. When radars are employed with different scan rates, a separate external timing reference must be employed, which becomes the scan rate for the IADT and the synthetic video display. Track updating and smoothing occur as previously described, except updates are considered relative to narrow sectors of the search volume – typically a few degrees in width. System software updates tracks a few sectors behind the azimuth position of the synthetic scan. Observations are accepted from the radar having the oldest unprocessed position data sector by sector as the synthetic scan passes. This provides the first available position report, no matter which radar produced it. Thus, position reports are used as they occurred in real time, and no position report is accepted out of order. IADT reduces loss of data due to individual radar propagation and lobing characteristics while allowing quality weighting of data relative to it source. IADT systems can accept the output of TWS or non-TWS radars as well as that derived from IFF transponders.

The Model 4.0 Combat Direction System software programs for the FFG-7 class Guided Missile Frigate are hosted in either AN/UYK-7 or AN/UYK-43 computers. The software interfaces digitally with other components of the Combat System, including the Mk92 Weapons System, Harpoon, AN/SYS-2 Integrated Automatic Detection and Tracking System, AN/SLQ32 Electronic Warfare System and AN/SQQ-89 Anti-Submarine Warfare System.

An early tracking approach is the so called Alpha-beta tracker. It is a simplified form of observer for estimation, data smoothing and control applications. It is closely related to Kalman filters and to linear state observers used in control theory. Its principal advantage is that it does not require a detailed system model. This tracker assumed fixed Gaussian errors and a constant-speed, non-maneuvering target model to update tracks. On the basis of a series of part target detections, the automatic tracker makes a smoothed (filtered) estimate of the target's position and velocity , as well as its predicted position and velocity.One method for accomplishing this is with the α-β(alpha -beta) tracker that computes the present smoothed target position $ \bar{x_{n}}$ and smoothed velocity $ \bar{\dot{x_{n}}}$ with the following operations.

\begin{equation} Smoothed position , \bar{x_{n}} = x_{pn} + α( x_{n} - x_{pn}) \end{equation} \begin{equation} Smoothed velocity , \bar{\dot{x_{n}}} = \bar{\dot{x_{n-1}}} + \frac{β}{T_{s}}( x_{n} - x_{pn}) \end{equation}

The predicted position on the next scan( the n+1st ) is then,

\begin{equation} x_{p(n+1)} = \bar{x_{n}} + \bar{\dot{x_{n}}} T_{s} \end{equation}

where $x_{pn}$ = predicted position of the target on the nth scan, $x_{n}$ = measured position on the nth scan, α = position smoothing parameter, β = velcity smoothing parameter , and $T_{s}$ = time between observations.

if α = β = 0, the tracker uses no current target information, only the smoothed data from prior observations. When α = β = 1, no smoothing of the data is included at all.Thus the closer α and β are to zero, the more important is the track in determining the predicted track. The closer they are to 1, the more important is the currently measured data.if target acceleration is significant , a third equation can be added to describe an α,β,γ tracker , where γ = acceleration smoothing parameter.

Benedict and Border show that if the transient response to a maneuvering target can be modeled by a ramp function , the output noise variance at steady state is minimized in an α - β tracker when $ β = \frac {α^{2}}{2-α}$. it was stated that the analysis does not , and cannot , specify the optimum value of α. The value of α is determined by the bandwidth and will depend on the system application. In selecting α, a compromise usually must be made between good smoothing of the random measurement errors (requiring a narrow bandwidth) and a rapid response to maneuvering target (wide bandwith). Trunk states that an α and a β satisfying the above relation can be chosen so that the tracking filter will follow a specified g turn.

Another criterian for selecting the α - β values is based on the best linear track fitted to the radar data in a least square sense:

\begin{equation} α = \frac {2(2n-1)}{n(n+1)} \end{equation}

\begin{equation} β = \frac {6}{n(n+1)} \end{equation}

where n is the number of the scan or target obsrevation(n>2). The above equations for α and β are also called the kalman gain components.

The classical α - β tracker is designed to minmise the mean - square error in the smoothed position and velocity. This type of tracker is said to be relatively easy to implement , but it does not handle the maneuvering target. Some means has to be included to detect maneuvers and change the values of α and β accordingly.The two tracking gate described in connection with Fig 1 is one example of how to deal with a large maneuvers. Another example is an adaptive α - β tracker which varies the smoothing parameter to achieve a variable bandwidth that allows the radar to follow target maneuvers. When the target is not maneuvering the adaptive tracking algorithm provides heavy smoothing. if the target maneuvers or makes a turn , the filter bandwidth is widened so as to allow the track filter to follow. As the selection of the values of α and β become more sophesticated and requires knowledge of the statistics fo the measurement errors and the prediction errors, the α - β tracker approaches the Kalman Filter.The selection of (α , β) is a design tradeoff. Small gains make a small correction in the direction of each detection.As a result ,the tracking filter is less sensitive to noise but is more sluggist to respond to maneuvers - deviation from the assumed target model. Conversily, large gains produce more tracking noise but quicker response to maneuvers. These errors are readly calculated as a function of α and β using formula's shown in Table 1.

To tune the α - β filter for radar tracking , one uses the radar parameters to calculate the tracking errors listed in Table 1 as a function of the tracking gain α and β. Then one selects the gains that best meet the needs of the application. For example, consider a radar that has 50 - meter range measurement accuracy and a two second constant update interval.The application of this radar system is to track a target that moves linearly but with occasional unpredictable maneuvers of up to 1g ($9.8 m/s^2$).

The Kalman filter is similar to the classical α - β tracker except that it inherently provides for the dynamical or maneuvering target. In the Kalman filter a model for the measurement error has to be assumed, as well as a model of the target trajectory and the disturbance or uncertainty of the trajectory. Such disturbances in the trajectory might be due to neglect of higher-order derivatives in the model of the dynamics, random motions due to atmospheric turbulence, and deliberate target maneuvers. The Kalman filter can, in principle, utilize a wide variety of models for measurement noise and trajectory disturbance; however, it is often assumed that these are described by white noise with zero mean. A maneuvering target does not always fit such an ideal model, since it is quite likely to produce correlated observations. The proper inclusion of realistic dynamical models increases the complexity of the calculations. Also, it is difficult to describe a priori the precise nature of the trajectory disturbances. Some form of adaptation to maneuvers is required. The Kalman filter is sophisticated and accurate, but is more costly to implement than the several other methods commonly used for the smoothing and prediction of tracking data. Its chief advantage over the classical α - β tracker is its intlerent ability to take account of maneuver statistics. If, however, the Kalman filter were restricted to modeling the target trajectory as a straight line and if the measurement noise and the trajectory disturbance noise were modeled as white, gaussian noise with zero mean, the Kalman filter equations reduce to the α - β filter equations with the parameters α and β computed sequentially by the Kalman filter procedure. This classical α - β tracking filter is relatively easy to implement. To handle the maneuvering target, some means may be included to detect maneuvers and change the values of α and β accordingly. In some radar systems, the data rate might also be increased during target maneuvers. As the means for choosing α and β become more sophisticated, the optimal α - β tracker becomes equivalent to a Kalman filter even for a target trajectory model with error. In this sense, the optimal α - β tracking filter is one in which the values of α and β require knowledge of the statistics of the measurement errors and the prediction errors, and in which α and β are determined in a recursive manner in that they depend on previous estimates of the mean square error in the smoothed position and velocity.

When the Kalman filter is modeled with the target trajectory as a straight line , and the measurement noise and the trajectory disturbance are modeled as white , guassian noise with zero mean , the kalamn filter equations reduce to the α - β tracker equations with α and β computed sequentially by the kalman filter procedure.Blackman states that “ Experience with airborne radars has shown the versatility of kalman filter to be almost indespensable when dealing with problems presented by missing data and variable measurement noise statics” . The kalman filter has better performance than the α - β tracker since it utilizes more information. The α - β tracker, however might be considered when the target's maneuver statistics are not known or in a dense target environment where computational simplicity is important. The Kalman filter and the α - β tracker also can be applied to control digitally the feedback loop in the single target tracker. The Kalman filter is essentially a set of mathematical equations that implement a predictor-corrector type estimator that is optimal in the sense that it minimizes the estimated error covariance—when some presumed conditions are met. Since the time of its introduction, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. This is likely due in large part to advances in digital computing that made the use of the filter practical, but also to the relative simplicity and robust nature of the filter itself. Rarely do the conditions necessary for optimality actually exist, and yet the filter apparently works well for many applications in spite of this situation.Of particular note here, the Kalman filter has been used extensively for tracking in interactive computer graphics.

The Kalman filter addresses the general problem of trying to estimate the state x € $R^n$of a discrete-time controlled process that is governed by the linear stochastic difference equation

\begin{equation} x_{k} = Ax_{k-1}+Bu_{k}+w_{k-1} \end{equation}

with a measurement z € $R^m$ that is

\begin{equation} z_{k} = Hx_{k}+v_{k} \end{equation}

The random variables w_{k} and v_{k} represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions.

\begin{equation} p(w)∼N(0,Q) \end{equation}

\begin{equation} p(v)∼N(0,R) \end{equation}

In practice, the process noise covariance Q and measurement noise covariance R matrices might change with each time step or measurement, however here we assume they are constant.

The n×n matrix A in the difference equation equation (6) relates the state at the previous time step k-1 to the state at the current step k , in the absence of either a driving function or process noise. Note that in practice A might change with each time step, but here we assume it is constant. The n×l matrix B relates the optional control input u€R to the state x. The m×n matrix H in the measurement equation , equation (7) relates the state to the measurement z_{k}. In practice H might change with each time step or measurement, but here we assume it is constant.

We define ${\hat{x_{k}}}^{-} $ € $ R^{n}$ (note the “super minus”) to be our a priori state estimate at step k given knowledge of the process prior to step k, and $\hat{x_{k}}$ € $ R^{n}$ to be our a posteriori state estimate at step k given measurement . We can then define a priori and a posteriori estimate errors as

\begin{equation} {e_{k}}^{-}= x_{k}-\hat{x_{k}}^{-} , and \end{equation}

\begin{equation} {e_{k}}= x_{k}-\hat{x_{k}} \end{equation}

The a priori estimate error covariance is then

\begin{equation} {P_{k}}^{-} = E[{e_{k}}^{-} {e_{k}}^{-T}], \end{equation}

and the a posteriori estimate error covariance is

\begin{equation} P_{k} = E[e_{k} {e_{k}}^{T}] \end{equation}

In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that computes an a posteriori state estimate $\hat{x_{k}}$ as a linear combination of an a priori estimate ${\hat{x_{k}}}^{-}$ and a weighted difference between an actual measurement $z_{k}$ and a measurement prediction H${\hat{x_{k}}}^{-}$ as shown below in equation (14). Some justification for equation (14) is given in “The Probabilistic Origins of the Filter” found below.

\begin{equation} \hat{x_{k}} = \hat{x_{k}}^{-} + K(z_{k}-H\hat{x_{k}}^{-}) \end{equation}

The difference $(z_{k}-H\hat{x_{k}}^{-})$in equation (14) is called the measurement innovation, or the residual. The residual reflects the discrepancy between the predicted measurement $\hat{x_{k}}^{-}$ and the actual measurement $z_{k}$ . A residual of zero means that the two are in complete agreement.

The n×m matrix K in equation (14) is chosen to be the gain or blending factor that minimizes the a posteriori error covariance equation (13). This minimization can be accomplished by first substituting equation (14) into the above definition for $e_{k}$ , substituting that into equation (13), performing the indicated expectations, taking the derivative of the trace of the result with respect to K, setting that result equal to zero, and then solving for K. For more details see (Maybeck 1979; Jacobs 1993; Brown and Hwang 1996). One form of the resulting K that minimizes equation (13) is given by,

\begin{equation} K_{k} = P_{k}^{-} H^{T} (H {P_{k}}^{-} H^{T}+R)^{-1} \end{equation}

\begin{equation} = \frac {P_{k}^{-} H^{T}}{H{P_{k}}^{-} H^{T}+R)^{-1}} \end{equation}

Looking at equation (15) we see that as the measurement error covariance R approaches zero, the gain K weights the residual more heavily. Specifically,

\begin{align*} \lim_{R_{k\to 0}} K_{k} = H^{-1} \end{align*}

On the other hand, as the a priori estimate error covariance $P_{k}^{-}$ approaches zero, the gain K weights the residual less heavily. Specifically,

\begin{align*} \lim_{P_{k\to 0}^{-}} K_{k} = H^{-1} \end{align*}

Another way of thinking about the weighting by K is that as the measurement error covariance R approaches zero, the actual measurement $z_{k}$ is “trusted” more and more, while the predicted measurement $H$ $ \hat{x_{k}}^{-}$ is trusted less and less. On the other hand, as the a priori estimate error covariance $P_{k}^{-}$ approaches zero the actual measurement $z_{k}$ is trusted less and less, while the predicted measurement $H$ $ \hat{x_{k}}^{-}$ is trusted more and more.

The justification for equation (14) is rooted in the probability of the a priori estimate $ \hat{x_{k}}^{-}$ conditioned on all prior measurements $z_{k}$ (Bayes’ rule). For now let it suffice to point out that the Kalman filter maintains the first two moments of the state distribution,

\begin{equation} E[x_{k}] = \hat{x_{k}} \end{equation}

\begin{equation} E[(x_{k} - \hat{x_{k}}){(x_{k} - \hat{x_{k}})}^{T}] = P_{k} \end{equation}

The a posteriori state estimate equation (14) reflects the mean (the first moment) of the state distribution— it is normally distributed if the conditions of equation (10)and equation (11) are met. The a posteriori estimate error covariance equation (13) reflects the variance of the state distribution (the second non-central moment). In other words,

\begin{equation} p(x_{k}|z_{k}) ∼ N(E[x_{k}],E[(x_{k} - \hat{x_{k}}){(x_{k} - \hat{x_{k}})}^{T}]) = N(\hat x_{k},P_{k}) \end{equation}

The Kalman filter estimates a process by using a form of feedback control: the filter estimates the process state at some time and then obtains feedback in the form of (noisy) measurements. As such, the equations for the Kalman filter fall into two groups: time update equations and measurement update equations. The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The measurement update equations are responsible for the feedback—i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The time update equations can also be thought of as predictor equations, while the measurement update equations can be thought of as corrector equations.The specific equations for the time and measurement updates are presented below in :

\begin{equation} \hat{x_{k}} = A\hat{x_{k-1}}+Bu_{k} \end{equation}

\begin{equation} P_{k}^{-} = A P_{k-1} A^{T} + Q \end{equation}

\begin{equation} K_{k} = P_{k}^{-} H^{T} (H P_{k}^{-}H^{T}+R)^{-1} \end{equation}

\begin{equation} \hat{x_{k}} = \hat{x_{k}}^{-}+K_{k}(z_{k}-H\hat{x_{k}}^{-}) \end{equation}

\begin{equation} P_{k}=(1-K_{k}H)P_{k}^{-} \end{equation}

The first task during the measurement update is to compute the Kalman gain ,$K_{k}$ . Notice that the equation given here as equation (22) is the same as equation (19). The next step is to actually measure the process to obtain $z_{k}$, and then to generate an a posteriori state estimate by incorporating the measurement as in equation (23). Again equation (23) is simply equation (18) repeated here for completeness. The final step is to obtain an a posteriori error covariance estimate via equation (24). After each time and measurement update pair, the process is repeated with the previous a posteriori estimates used to project or predict the new a priori estimates.

With $\hat{x_{k|k}}$ we denote the estimate of state $x_{k}$ at time k.Let $P_{k|k}$ be the varaince matrix of the error $x_{k}$-$\hat{x_{k|k}}$.The goal is to minimize $P_{k|k}$ (in some defined way).

Predict Phase of the Filter.In this first phase of a standard Kalman filter, we calculate the predicted state and the predicted variance matrix as follows (using state transition matrix $F_{k}$, control matrix $B_{k}$, and process noise variance matrix $Q_{k}$, as given in the model):

\begin{equation} \hat{x_{k|k-1}} = F_{k}\hat{x_{k-1|k-1}}+B_{k}u_{k} \end{equation}

\begin{equation} {P_{k|k-1}} = F_{k}{P_{k-1|k-1}}{F_{k}^T}+Q_{k} \end{equation}

Update Phase of the Filter. In the second phase of a standard Kalman filter, we calculate the measurement residual vector $\vec{z_{k}}$ and the residual variance matrix $S_{k}$ as follows (using observation matrix $H_{k}$ and observation noise variance $R_{k}$, as given in the model):

\begin{equation} \vec{z_{k}} = y_{k} - H_{k}\hat{x_{k|k-1}} \end{equation}

\begin{equation} {S_{k}} = H_{k}{P_{k|k-1}}{H_{k}^T}+R_{k} \end{equation}

The updated state estimation vector (i.e., the solution for time t) is calculated (in the innovation step) by a filter

\begin{equation} \hat{x_{k|k}} = \hat{x_{k|k-1}}+K_{k}\vec{z_{k}} \end{equation}

The Kalman filter is a minimum mean-square error estimator.The error in the a posteriori state estimation is $x_{k}$-$\hat{x_{k|k}}$

We seek to minimize the expected value of the square of the magnitude of this vector, $E[{||x_{k}-\hat{x_{k|k}}||}^2]$ .This is equivalent to minimizing the trace of the a posteriori estimate covariance matrix $P_{k|k}$. By expanding out the terms in the equation above and collecting, we get:

\begin{equation} P_{k|k} = P_{k|k-1}-K_{k}H_{k}P_{k|k-1}-P_{k|k-1}{H_{k}}^T{K_{k}}^T+K_{k}S_{k}K_{k}^T \end{equation}

The trace is minimized when its matrix derivative with respect to the gain matrix is zero. Using the gradient matrix rules and the symmetry of the matrices involved we find that

\begin{equation} \frac{∂tr(P_{k|k})}{∂K_{k}} = -2(H_{k}P_{k|k-1})^T+2K_{k}S_{k}=0 \end{equation}

Solving this for $K_{k}$ yields the Kalman gain:

\begin{equation} K_{k}S_{k} = (H_{k}P_{k|k-1})^T = P_{k|k-1}{H_{k}}^T \end{equation}

\begin{equation} K_{k} = P_{k|k-1}{H_{k}}^T {S_{k}}^{-1} \end{equation}

This gain, which is known as the optimal Kalman gain, is the one that yields MMSE(Minimum Mean Square Error) estimates when used.