Abstract:
A process for correcting data from a three dimensional bistatic synthetic aperture radar system to eliminate distortions and resolution limitations due to the relative positions and motions of the radar transmitter and receiver with respect to a target.

Description:
BACKGROUND 
     1. Field 
     This invention relates to the processing of synthetic aperture radar data and, in particular, to the processing of three dimensional bistatic synthetic aperture radar data. 
     2. Prior Art 
     In early radar systems, fine range resolution was obtained by transmitting a narrow pulse, while fine azimuth resolution was obtained by radiating a narrow beam. In more modern radars, pulse compression techniques or active linear FM techniques, commonly referred to as Stretch techniques, are used to produce a long pulse which can, through appropriate processing, provide the range resolution of a narrow pulse. 
     For example, in the Stretch technique, a linear FM signal is transmitted. The return signal is mixed with a sample of the transmitted signal to produce an output signal at a frequency which is dependent on the range of the target. The range resolution in the Stretch technique is a function of the bandwidth of the transmitted linear FM signal, or alternatively, the resolution may be considered dependent on the range to frequency scale factor and the degree to which the frequency of the output signal for a particular target can be ascertained. 
     A narrow beamwidth was previously obtained through the use of a large antenna; however, in a more modern technique known as synthetic aperture radar, or SAR, a small antenna can provide the effective aperture of a large antenna. This is accomplished by moving the antenna along a path, the length of which determines the synthetic aperture within the limitation imposed by the antenna beamwidth. The data received along the path is stored and later processed to produce a high resolution image. Conventional SAR radar and its processing are discussed in a number of publications, including  The Radar Handbook , by M. Skolnik, McGraw-Hill, New York 1970. 
     A simple Stretch system is shown in FIG. 1. A transmitter  11  generates a linear FM signal which is passed through a coupler  12 , to a first antenna  13 , where it is radiated. The radiated signal is returned by a target  15  to a second antenna  14 , where it is passed to a mixer  16 . The mixer also receives a portion of the transmitted signal by way of the coupler  12  to serve as a local oscillator signal. The mixer output signal, at port  17 , often is converted to range information at port  19  by a Fourier transformer  18 , which is usually a pulse compression network, also referred to as a matched filter. A matched filter is a more general term, however, including any system for converting received radar signals to Stretch type frequency response signals. 
     The operation of the system of FIG. 1 can be understood with the aid of FIG.  2 . FIG. 2 is a graph containing a plot of a transmitted signal  24 , a received signal  25 , and an output signal  26 . The ordinate  22  represents frequency while the abscissa  23  represents time. The frequency axis is divided in two, with the upper portion being the normalized transmitted frequency, while the lower portion is the output frequency, which is calibrated in megahertz (MHz). 
     FIG. 2 is an example of the performance characteristics of a typical Stretch system. The transmitted signal varies linearly from 0.8 to 1.2 times the center frequency over a normalized period, while the received signal varies over the same frequency range, but is shown delayed by a time corresponding to the time for the signal to traverse the round trip radar path length to the target and back to the receiver. When these two signals mix in the mixer  16 , shown in FIG. 1, they produce a constant low output frequency such as a 4 MHz signal designated by drawing numeral  26 , in FIG.  2 . 
     The frequency difference between the transmitted and return signals is proportional to the delay which, in turn, is proportional to the target range. Therefore, an output signal representing a target at a fixed range will be at a frequency that is proportional to the range of the target. In a practical Stretch system, the mixer reference signal obtained via coupler  12  may be delayed a known amount to establish a reference range. The output signal for a target at that range will then be zero frequency. 
     A rudimentary SAR system is shown in FIG.  3 . In this Figure, antennas  31 ,  32 , and  33  produce antenna patterns  34 ,  35 , and  36 , respectively, illuminating a target  37 . 
     The antenna and antenna patterns shown in FIG. 3 may be considered a simple phased array antenna system. The signals from each antenna may be phase shifted, weighted and then combined with the signals from the other antennas to produce an effective pattern  39 , which is narrower and higher in gain than any of the individual antennas. The narrow beamwidth of the effective pattern may then be used for improved angle resolution. 
     A SAR system is, in effect, a special case of a phased array system. Although a SAR system usually contains only a single antenna, this antenna is moved to simulate a number of antennas. For example, a single antenna of a SAR system could first be positioned at the location of antenna  31 . In this location, it would produce a pattern similar to pattern  34 . The signal received at this location containing target information is stored. The single SAR antenna is then moved to the location of antenna  32  and later to that of antenna  33  with the received data at each location being stored. The stored data is then weighted, phase shifted and combined to provide the same results which would have been obtained with three separate antennas. 
     In practical applications, the single SAR antenna is often located aboard an aircraft. The antenna is continually in motion rather than being shifted in incremental fashion from position to position; however, the transmission period is relatively short, making the distance moved per transmission period equally short. In this case, each transmission may be considered as occurring at a single location. 
     FIG. 8 illustrates the operation of a typical SAR in an aircraft  81 , flying along a flight path  86 . The SAR antenna is directed constantly to the side of the aircraft, as shown by directional arrows, such as arrow  82 . The radar beam  84  illuminates a swath on the ground  83  and at a particular instant illuminates an area, such as area  85 . 
     A commonly used method for recording SAR data is to print the data on photographic film, such as on the film shown in FIG. 4, using intensity modulation. In this Figure, a film  41  contains a series of lines, such as lines  42  and  43 . Each line contains data received as a result of a single transmitted pulse. If Stretch is combined with SAR the peaks of the sinousoidal return signals are recorded as short, dark portions along the line. 
     FIG. 9A shows the coverage by a SAR radar of two targets. The radar beamwidth  91  illuminates two targets,  93  and  94 , in an area of illumination on the ground indicated by the swath  95  of width  92 . When the signals from the targets are recorded on film  99  via a series of range sweeps, such as range sweep  96 , they appear as shown in FIG. 9B, where signal history  98  corresponds to target  94  and signal history  97  corresponds to target  93 . The target histories are curved and cannot be both correct by simple geometric manipulation. 
     For a Stretch SAR, a single target at a constant range would produce a single frequency, such as the 4 MHz signal illustrated in FIG.  2 . This type of return signal would simply produce a series of constant size dots separated by a constant spacing. The dots correspond to the peaks of a 4 MHz signal. A return signal produced by a number of targets generates a more complex darkening of the line. 
     The data may be recorded on the film simply by displaying the received signal on an oscilloscope and focusing the display on the film. In such a system, the sweep speed corresponds to the rate at which the return signal is printed while the separation between the raster lines corresponds to the distance the antenna travels between pulses. 
     The data recorded in this form is in terms of frequency and time of receipt, rather than the more conventional target intensity and range. Obtaining the SAR Stretch data in more conventional form requires conversion of the frequency information in the vertical direction to range information. This function may be carried out in analogue fashion by means of a device such as a spectrum analyzer. It also may be carried out in digital fashion by means of a Fourier transform in the vertical direction. To complete the conversion from darkened lines to an image, the data must also be weighted, phase shifted, and properly combined in the horizontal direction. 
     An improved form of SAR referred to as Spotlight, differs from standard SAR in that the antenna is directed at a single target area, while the antenna is moved along the flight path. The data is also preprocessed by subtracting the Doppler of a reference target in the area, to simplify subsequent data combining. As long as the target area is illuminated, the path length becomes the synthetic aperture for this target. The synthetic aperture, therefore, is not limited by the beamwidth as is the case with a conventional SAR. Furthermore, data combining can now be performed approximately by horizontal Fourier transform. 
     The data from a Stretch-Spotlight SAR can be corrected so that data combining can be performed exactly with a horizontal Fourier transform, by using a fourth technique known as polar formatting. 
     FIG. 6 illustrates a Spotlight radar. In this Figure, an aircraft  60  is shown at three positions,  61 ,  62 ,  63 , along a flight path, indicated by direction arrows  68  and  69 . At each point along the flight path, the radar beam is directed at a target  67 . The radar beam for positions  61 ,  62 , and  63 , is illustrated by beams  64 ,  65 , and  66 . Since the angle of the antenna with respect to the flight path is continually changing along the path, the data should not be recorded in the simple straight line fashion, shown in FIG.  4 . To correct for the constant change in angle, the data is recorded in polar format, such as that shown in FIG.  7 . In this Figure, data lines, such as lines  72  and  73 , are recorded on a film  71  in polar coordinates. 
     It is fortunate that a simple means of providing a complete conversion of Stretch-Spotlight data to an image is available. The data in the form shown in FIG. 7 may be converted directly to an image by means of a spherical lens. This process is shown pictorially in FIG.  5 . In this Figure, a reflector  51  directs light from a light source  52  through a columnating lens  53 . The columnated light  54  passes through a film  55  containing the record data. The light emerging from the film is then passed through a spherical lens  56  to where the light rays  57  are converged produce an image on the screen or second film  58 . This simple, complete conversion only works exactly on Stretch-Spotlight polar formatted monostatic SAR data. 
     In the SAR systems described above, it is assumed that the transmitter and receiver are located on board the same platform, such as aboard a single aircraft. This type of system is referred to as a monostatic imaging radar system. It is advantageous in some applications to place the transmitter aboard one platform and the receiver aboard another. This type of system is referred to as a bistatic imaging radar system. A serious problem arises in properly recording the received data from a bistatic system so that an image may be produced by the relatively simple Fourier transform method similar to that shown in FIG.  5 . 
     Bistatic radar presents fundamental geometric processing problems which are similar in some respects, to those encountered in monostatic SAR technology. These geometric problems arise from target motion through resolution cells and are, in fact, a direct result of the geometric motion of what is referred to as the iso-Doppler and iso-range lines of the bistatic radar system during the interval required to collect the data for image formation. In particular, for most bistatic geometries, the iso-Doppler and iso-range lines are not perpendicular and, if this data is processed conventionally, it will result in a skewed image, such as an square object being imaged as a parallelogram. Furthermore, since the image skewing varies with time, changes in the skew of the parallelogram will limit the resolution that can be achieved. 
     Iso-Doppler and iso-range lines for a monostatic SAR system are shown in FIG.  10 . In this Figure, an aircraft  101  carrying a SAR is moving at a velocity V a    103 . The vector V a  is the relative velocity with respect to the aircraft of a target directly ahead. The SAR may be considered the center of a series of concentric spheres, such as sphere  102 , which represents a constant range from the SAR. Where these spheres intersect a plane, such as the earth surface, they produce circles which are referred to as iso-range lines. 
     FIG. 17 shows the transmitter circling the field of view. The transmitter starts at angle  1  designated by drawing numeral  1701  and fly along flightpath  1706  to angle  2   1702 , angle  4   1704 , angle  3   1730 , and finally, back to angle  1 . The FOV is  1707  containing illustrative targets A  1708 , C  1709  and B  1710 . FIG. 18 is a superposition FIG. 17 of the iso-Doppler and iso-range lines with the transmitter at angle  2  which is designated  1802  in FIG.  18 . 
     All targets along a line emanating from the SAR have at one instant in time, a constant relative velocity toward the SAR. For example, targets  105  and  108  on line  106  have a relative velocity of V R , represented by vector  104 , towards the SAR. These targets will produce the same Doppler frequency because of their same relative velocity towards the SAR and therefore the line  106  is referred to as an iso-Doppler line. 
     All targets on lines making the same angle with the aircraft velocity vector, V a  will have the same property because they will all have the same relative velocity V R  towards the SAR. These lines form a cone. Lines  106  and  107  represent the intersection of the cone with a plane, such as the earth&#39;s surface. 
     An infinite number of cones are possible. For example, lines  110  and  111  represent the intersection of another iso-Doppler cone with the earth surface. For clarity we will begin with such a description in two dimensions. The concept will be extended later into three dimensions. 
     Conventionally, a sharp gradient in iso-Doppler lines is represented by a high density of lines, while the direction of the gradient is taken as being perpendicular to the iso-Doppler lines. A similar representation is used for iso-range lines. 
     Note that in the monostatic system depicted in FIG. 10, the iso-doppler lines are radial and therefore intersect the iso-range circles at right angles. As will be shown, the configuration of iso-range and iso-Doppler lines in most instances is quite different for bistatic radar. 
     FIG. 12 shows the change in iso-Doppler and iso-range lines as the transmitter position is changed. The transmitting aircraft  1201  is shown prior to crossing the flightpath of the receiving aircraft&#39;s  1202  in FIG. 12A, while the transmitting aircraft is shown after having crossed the receiving aircraft&#39;s flightpath in FIG.  12 B. Idealized iso-Doppler and iso-range lines  1204  are shown within the field of view  1203 . Actual iso-range lines are perpendicular to the bistatic angle bisector, while actual iso-Doppler lines are parallel to the lines  1205  or  1206 . 
     The central problem in bistatic radar is target location. It is necessary to determine how range delay and Doppler shift relates to target position. 
     To simplify the discussion, it will be assumed that a Stretch system will be used throughout and in this Stretch system the definition of range delay is the relative time between the received signal from the target and the signal from a reference point of known position. This approach requires the least knowledge of transmitter and receiver positions. 
     It also will be assumed that a Spotlight system will be used throughout and in such a system a Doppler shift in bistatic radar is due to the movement of both the transmitter and receiver. Doppler will be taken with respect to a reference point, which is usually located within the field of view. The Doppler at this point will be set to zero; that is, the Doppler at the reference point will be subtracted from the Doppler at all other points in the field of view. 
     Referring to FIG. 11, to simplify the description, it sill first be assumed that in FIG. 11A the receiver aircraft  1103  is always directed toward the field of view  1104  so it appears nonrotating by an observer  1110  located in the field of view, while the transmitter  1101  is always broadside to the field of view, so it appears to be rotating. 
     In FIG. 11B, the receiver aircraft  1107  is shown flying on a path parallel to the transmitter aircraft  1105 . The receiver aircraft is still directed towards the field of view  1109  so it appears nonrotating by an observer  1111  located in the field of view, while the transmitter aircraft  1105  remains broadside to the field of view so that it appears to rotate. 
     In order to understand the purpose of the present invention it is necessary to compare the iso-Doppler and iso-range line configurations for monostatic and bistatic radar. First note that the position of the nonrotating, receiving aircraft does not influence the position of the iso-Doppler lines. 
     The reason the receiving aircraft does not produce a Doppler gradient is explained with the aid of FIGS. 13A and 13B. In FIG. 13A, a receiving aircraft  1301  is shown approaching a field of view  1303  which contains two targets  1304  and  1305 . The angle that target  1304  makes with the line of flight  1302  is designated φ. 
     Since the receiving aircraft is almost directly approaching the target  1304 , its Doppler is high; however, there is little change in the Doppler throughout the approach because the target  1304  essentially remains directly ahead and the aircraft remains at a constant velocity. 
     On the other hand, if the field of view were shifted to another target off to the side, such as target  1307 , the Doppler due this target would change at the same rate as the velocity component of the aircraft toward the target. 
     FIG. 13B is a graph of the Doppler with respect to the angle φ. In this Figure, the ordinate  1308  is the magnitude of the Doppler shift while the abscissa is the angle φ. The plot  1309  is a cosine function, which corresponds to the component of velocity towards a target at an angle φ from the aircraft. It can be seen from this graph that, although the highest absolute Doppler value is produced where φ is equal to zero, the greatest rate of change is produced where φ is 90 degrees. Therefore, for this simplified example the receiving aircraft with φ nearly equal to zero introduces a large Doppler shift to all targets in the field of view, but produces little change in the Doppler shift. Consequently, the receiving aircraft does not influence the position of the iso-Doppler lines, whereas the transmitting aircraft, with a value of φ of nearly 90 degrees, produces virtually all the change in Doppler across the field of view and essentially establishes the iso-Doppler lines. 
     Since the transmitter aircraft is the only aircraft producing the iso-Doppler lines, they will be radial lines drawn from the transmitter to a target and as the aircraft moves, these lines will rotate about the target. 
     The direction of the iso-range lines in the target area will also rotate, but not at the same rate as the iso-Doppler lines. This can be seen with the aid of FIG.  14 . 
     In this Figure, a transmitter  1401  and a receiver  1402  form the foci for a series of eliptical iso-range lines  1408 . A target  1403  is shown located on one of the iso-range lines. An angle θ, referred to as the bistatic angle, is located at the target and is defined by lines  1404  and  1405  drawn from the target to the transmitter and a receiver, respectively. Line  1407  is the angle bisector of the bistatic angle. 
     By definition, all targets which produce signals that arrive at the receiver at the same time are located on an iso-range line. For a signal to arrive at the same time as another signal, its total path length from the transmitter to a target and then to the receiver, must be the same, or the sum of the length of line  1404  and  1405  must be a constant. A constant sum for these two line lengths is one definition of an ellipse and accordingly explains the elliptical contours of the iso-range lines for the bistatic case. 
     The line  1404  is an iso-Doppler line as it connects the transmitter with the target. The angle of the iso-Doppler line may be taken with respect to any reference line, such as the line  1405 . If the line  1405  is chosen, then the angle of the iso-Doppler line  1404  is equal to the bistatic angle. 
     If in FIG. 14 it is assumed that the transmitter is moved on any flight path, except the one directed toward target  1403 , the bistatic angle will change, as will the angle of the iso-Doppler line. The direction of the iso-range line, which is the tangent to the iso-range ellipse at the target, is perpendicular to the bistatic angle bisector  1407 . The direction of the iso-range lines in the target area therefore changes with the angle bisector  1407 . 
     As long as the receiver is either stationary or moves toward or away from the target  1403 , the change in the angle of the iso-Doppler lines can be taken as being equal to the change in bistatic angle, while the change in the direction of the iso-range lines is equal to the change in one-half the bistatic angle. 
     If in the general case, both receiver and transmitter rotate about the field of view, the iso-range lines behave in the same way, however, the iso-Doppler lines are affected by the receiver rotation. In the extreme case in which the transmitter and receiver rotate at the same rate and direction about the field of view, the iso-range lines remain perpendicular to the bistatic angle bisector, but the iso-Doppler lines become parallel to the bistatic angle bisector, and therefore, remain perpendicular to the iso-range lines at all time. Thus in the extreme, no change in monostatic polar processing is required, as the bistatic system behaves monostatically. The intermediary cases can be considered as a combination of the present invention with conventional polar format processing. 
     Returning now to the simple example of the non-rotating receiving station, the initial orientation of the iso-Doppler and iso-range lines in the field of view for the bistatic case is generally not orthogonal except for the special case when both the aircraft and target are on the same straight line; however, it might be possible to correct for the skewed image caused by the lack of orthogonality by conventional topographical image reconstitution techniques if the bistatic angle changes only a small amount over the synthetic aperture. 
     Since the direction of the iso-Doppler and iso-range lines change at different rates, it would also be possible to make some additional correction in processing the recorded data for larger bistatic angle changes by, in effect, rotating the data at a speed halfway between the rate of change in the direction of the iso-Doppler lines and the iso-range lines. The improvement provided by this correction is obviously limited because neither the rotation of the iso-Doppler nor that of the iso-range lines is completely corrected. 
     Also note from FIG. 14 that increase of the bistatic angle  1406  is accompanied by a spreading apart of the iso-range ellipses as the iso-range lines are closer together for target  1409  than for target  1403 . This phenomonon is equivalent to a loss or dilution of range resolution caused by a change in the scale factor between distance and time delay. The simple corrections described above would not correct for the loss of range resolution. 
     SUMMARY 
     It is an object of the present invention to provide a system for correcting and storing the data received from a three dimensional bistatic imaging radar system to permit simple Fourier transform processing to produce images directly from the recorded data. 
     In the present invention, the bistatic angle φ, the receiver angle α, the direction of the bistatic angle bisector, and a pulse which is coincident with the transmitted pulse, are the only real time input required to produce the correction of the bistatic data for proper recording. Other possible inputs, such as the start frequency of the transmitted pulse, are usually constant values which may be set into the equipment as one-time-only initial adjustments. 
    
    
     BRIEF DESCRIPTION OF THE FIGURES 
     FIG. 1 is a block diagram of a Prior Art Stretch radar. 
     FIG. 2 is a graph showing the relation of the transmitted and received signals of a Stretch radar. 
     FIG. 3 is a pictorial diagram illustrating the antenna pattern produced by a SAR antenna. 
     FIG. 4 is a film strip illustrating the method of recording conventional SAR data. 
     FIG. 5 is a pictorial diagram of a method of producing an image from a SAR film. 
     FIG. 6 is a diagram illustrating the operation of a Spotlight radar. 
     FIG. 7 is a pictorial diagram of SAR data recorded in polar format. 
     FIG. 8 is a pictorial illustration of the ground area covered by a conventional side-looking SAR radar. 
     FIGS. 9A and 9B diagrams showing the coverage by a SAR radar of two targets and the appearance of these targets when recorded on film. 
     FIG. 10 is a pictorial diagram illustrating iso-range and iso-Doppler lines for a monostatic SAR. 
     FIGS. 11A and 11B are diagrams illustrating in-line and parallel path geometry. 
     FIG. 12 is a diagram illustrating the orientation of iso-Doppler and iso-range line as the position of the transmitter is changed. 
     FIG. 13 is a diagram illustrating the change in Doppler at the receiver as a function of the angle φ. 
     FIG. 14 is a diagram illustrating the elliptical iso-range lines for the bistatic radar case. 
     FIG. 15 is a diagram illustrating a polar format. 
     FIG. 16 is a diagram illustrating a format for the bistatic case. 
     FIG. 17 is a diagram illustrating for the bistatic case the location of three targets, a receiver, and a circular flight path about the target site for the transmitter. 
     FIG. 18 is a diagram illustrating the relationship between the transmitter-receiver angle bisector and the iso-range lines. 
     FIG. 19 is a diagram illustrating the bistatic recording format and the relation of the direction of the recording trace to the bistatic angle. 
     FIG. 20 is a diagram illustrating the geometric relations for the bistatic case for a moving transmitter. 
     FIG. 21 is a block diagram illustrating a first embodiment of the present invention. 
     FIG. 22 is a block diagram illustrating a second embodiment of the present invention. 
     FIG. 23 is a modification of FIG.  20 . 
     FIG. 24 is a plot of two flight path examples. 
     FIG. 25 is a first vector diagram for the three dimensional case. 
     FIG. 26 is a second vector diagram for the three dimensional case. 
     FIG. 27 is a third vector diagram for the three dimensional case. 
     FIG. 28 is an example of three dimensional data storage. 
     FIG. 29 is a block diagram illustrating an embodiment of the three dimensional version of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     It is possible to correct completely for both iso-Doppler and iso-range line rotations by means of the present invention. This correction can be explained by considering the monostatic polar recording format previously shown in FIG.  7  and expanded in FIG.  15 . In FIG. 15, a series of radial recording traces  1501  are drawn from the center of the recording medium  1503 . The trace angle  1502  is referred to as the monostatic polar angle. 
     Briefly, the procedure used for bistatic data correction is to rotate the Doppler frequency direction of the storage surface (or volume in three dimensions) in synchronism with the apparent rotation of the iso-Doppler lines (or surfaces) across the field of view. For the present we will limit the description to two dimension. Then this is accomplished by rotating not only the storage surface, as in conventional polar format storage of FIG. 15, but also by rotating the read-in scan line. Either digital storage for real-time processing, or film storage for optical processing can be used. In an optical recorder, read-in scan line rotation can be performed by rotating the cathode-ray tube trace. 
     As a first example, the data storage format for the present invention using Stretch-Spotlight data obtained from a bistatic radar with one terminal rotating is shown in FIG.  16 . In this Figure, a series of traces  1609  are drawn about the center of a storage surface  1601 . These traces are at an angle θ/2  1604  with the vertical axis of this drawing. The trace position θ,  1602  is equal to the angle between the line drawn from the transmitter to the target and the line drawn from the receiver to the target, which is the bistatic angle, θ. The direction and length of the trace varies for each different position of the transmitter and receiver. 
     The corresponding simplified radar configuration for FIGS. 16 and 17 is shown in FIG.  18 . In this Figure, a transmitter traverses a circular flight path  1806  about field of view (FOV) center  1807 . Four transmitter positions,  1801 ,  1802 ,  1803 , and  1804 , are indicated about the flight path. The receiver  1805  is assumed to be either fixed or moving directly towards the FOV. Iso-Doppler lines  1808  are shown as produced by the aircraft at position  1802 . These lines are shown as being essentially parallel across the field of view because of the long range of the transmitter, as indicated by drawing numeral  1813 . The iso-range lines  1810  are shown perpendicular to transmitter-to-FOV-to-receiver angle bisector  1812 . At short range the iso-range lines are ellipses  1811 . However, in the small area of the FOV they appear as the parallel lines  1810 . 
     The data is stored upon a surface which counterrotates in synchronism with the rotation of the iso-Doppler lines, by a recording trace or scan line which counterrotates in synchronism with the rotation of the iso-range lines, as in FIG.  16 . The recording trace positions  1605 ,  1606 ,  1607 ,  1608  correspond to transmitter positions  1802 ,  1804 ,  1803 , and  1801 , respectively. 
     The reason that the storage geometry of FIG. 16 matches the radar geometry of FIG. 18 is most easily understood with the aid of FIG.  19 . In FIG. 19, a vertical axis is drawn from an origin  1903  to a point A  1924 . Three circles  1907 , centered on a vertical axis  1924  are drawn through the origin  1903 . These circles represent the boundaries of the start, middle and end of the scan lines of FIG.  16 . The inner circle represents the start-of-scan boundary while the outer circle represents the end-of-scan boundary. A typical scan line  1909  is drawn from the origin  1903  to the outer circle. Only the portion between the inner and outer circle is darkened as this is the only portion along which information will be printed. The direction of the scan line  1910  is parallel to the range direction. The angle Θ/2  1905  is the angle the scan line makes with the vertical axis. Radii of the three circles  1920  drawn from the center of each circle on the vertical axis intersect the circles along the typical scan line  1909 . A Doppler direction arrow  1908  is drawn at the end of the typical scan line tangent to the outer circle. 
     Note that the locus of points having the same relative time on the scan line, such as, the beginning, middle or end of the trace, is a circle, centered on the vertical or Y-axis, and the trace direction is always radial from the origin  1903 . The Doppler direction is tangent to the locus of the point of constant time, or perpendicular to the radius of each circular locus. 
     Consider the outer circle, centered at  1904  with a diameter equal to the line length from  1903  to  1911 . Using radii  1904  to  1911  and  1904  to  1912 , draw the triangle  1911 - 1904 - 1912  with altitude  1904  to  1913  perpendicular to line  1911 - 1912 . Since radius  1904 - 1911  equals radius  1904 - 1912  triangle  1911 - 1904 - 1912  is isosceles. Then angle  1911 - 1904 - 1913  is equal to angle  1912 - 1904 - 1913 . Angle  1911 - 1904 - 1913  equals one-half angle  1911 - 1904 - 1912  and therefore also equals one-half angle  1906 . Now triangle  1903 - 1912 - 1911  is inscribed in a semicircle, and is therefore a right triangle with line  1903 - 1912  perpendicular to line  1912 - 1911 . Therefore, line  1903 - 1912  is parallel to line  1904 - 1913  and angle  1906  is proven equal to twice angle  1905 . From plane geometry, therefore, the Doppler direction is perpendicular to the Θ direction for every point on the typical scan line, while the range direction is parallel to Θ/2. 
     A recognized property of a two dimensional Fourier transform is it exhibits polar symmetry; that is, rotating the data causes only a corresponding rotation of the image and there is no distortion or resolution loss. It follows that rotation of the range-frequency direction of the data causes only a corresponding rotation of the iso-range lines in the image; likewise, rotation of the Doppler frequency direction causes a corresponding rotation of the iso-Doppler lines in the image. 
     If the angle Θ is synchronized to the instantaneous bistatic angle, the iso-Doppler lines will be maintained in a constant direction on the image. The iso-range lines, which rotate at half the rate of the transmitter-to-FOV-to receiver bistatic angle, will automatically be synchronized to the angle Θ/2 which is the trace or range-frequency direction; hence, the iso-range lines will be maintained in a constant direction on the image. If Θ is chosen to be zero when the transmitter-to-FOV-to receiver angle is zero, the iso-range and Doppler lines will be perpendicular when the transmitter is at position  1801  and will therefore be perpendicular everywhere, that is independent of 9. 
     Also not that FIG. 19 shows that the trace length is proportional to cos Θ/2, and shrinks as the bistatic geometry deviates from nearly monostatic to compensate for the bistatic dilution of range resolution. 
     To clarify the geometry in the more general case when the receiver is allowed to move, it is useful to consider an equivalent configuration in which the receiver-to-field of view direction is used as a reference, and the field of view is imagined to rotate, rather than the receiver. Then it is clear that the required processing is that of the simple nonrotating receiver case previously described plus an additional rotation of the recording medium equal to the apparent rotation of the field of view. These solutions are verified by the following mathematical analysis; first for a nonrotating receiver, then for a rotating one. 
     The data received from the target is to be processed using the present invention followed by two-dimensional Fourier transform to obtain an image. In FIG. 20, a target  2004  is located near the center of a field of view (FOV)  2003 . A transmitter  2001  moves about the FOV while a receiver  2002  holds a fixed direction towards the center of the FOV. The symbols φ o , Θ o , R o , and Y o  denote the angles and points of the geometric construction shown in the figure. In particular, X o  and Y o  are the coordinates of the target. The angle the transmitter makes with the FOV center is Θ o , previously denoted Θ, the bistatic angle, while the angle the target makes with the FOV center is φ o . The distance of the target from the FOV center is designated R o . To simplify the analysis the following assumptions will be made. 
     The receiver and transmitter are coplanar with and at long range from the FOV; that is, the angle subtended by the FOV as seen from both the receiver and the transmitter is small. Thus, the iso-range lines are taken as straight and parallel, as are the iso-Doppler lines. 
     The receiver is fixed in angle relative to the FOV. The Y o -axis is defined as the receiver-FOV direction. However, the receiver-to-FOV distance can be changing. 
     The transmitter is moving in angle relative to the FOV, and potentially can circumnavigate the FOV. The transmitter to FOV distance can also be changing. 
     A Stretch-Spotlight receiver with a long pulse and wide bandwidth is assumed. 
     In the Stretch-Spotlight receiver a reference signal is generated which models the expected return from a target location at the FOV center. The phase of this reference can be taken as 
     
       
         Reference Phase=ω o t=γt 2    (1)  
       
     
     where ω o  and γ are the radian frequency and one-half the radian chirp slope. The return from a target if delayed by τ from the reference. Its phase is 
     
       
         Signal Phase=ω o (t−τ)+γ(t−τ) 2   (2)  
       
     
     In the receiver, the reference and signal are mixed to obtain the video phase, which is then stored for processing. Subtracting equation 2 from 1: 
     
       
         Video Phase=ω o τ+2 γτt−γτ 2    
       
     
     For high time-bandwidth the last term vanishes: 
     
       
         Video Phase=(ω o +2γτ)τ  (3)  
       
     
     Using the geometry of FIG. 28 we can relate the differential range delay, τ, to target position (X o , Y o ). Total differential range is the sum of the transmitter and receiver path lengths. 
     
       
         τ={fraction (1+L /C)}[R o  cos (Θ o −φ o )+X o ]  (4)  
       
     
     Using the trigonometric identity 
     
       
         cos (Θ o −φ o )=cos Θ o  cos φ o +sin Θ o  sin φ o    (5)  
       
     
     and the polar coordinates of the target 
     
       
         X o =R o  cos φ o , Y o =R o  sin φ o    (6)  
       
     
     we obtain from equations 5 and 6 
     
       
         R o  cos (Θ o −φ o )=X o  cos φ o +Y o  sin Θ o    (7)  
       
     
     and from equations 4 and 7 
     
       
         τ={fraction (1+L /C)}[Y o  sin Θ o +X o  (1+cos Θ o )]  (8)  
       
     
     The intuition leading to the present invention format has been described previously. The key provisions are that the polar storage angle (Θ p ) should rotate at half the rate of the transmitter angle (Θ o ), and that the scan length should diminish as the geometry deviates from monostatic. Hence, we intuitively let: 
     
       
         Θ o =2 Θ p    (9)  
       
     
     
       
         
           
             
               
                 
                   
                     R 
                     p 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           v 
                           s 
                         
                          
                         cos 
                          
                         
                             
                         
                          
                         
                           
                             θ 
                             o 
                           
                           2 
                         
                       
                       ) 
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         K 
                         + 
                         t 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     where the polar and rectangular coordinates of the storage point (R p , Θ p ) or (X p , Y p ) are related by 
     
       
         X p R p  cos Θ p , Y p =R p  sin Θ p    (11)  
       
     
     and v s  is the maximum scan velocity, K is the storage trigger delay which will be evaluated later. 
     Combining equations 8 and 9 gives 
     
       
         τ={fraction (1+L /C)}[X o (1+cos 2 Θ p )+Y o  sin 2 Θ p ]  (12)  
       
     
     Using the double angle trigonometric identities 
     
       
         τ= {fraction (2+L /C)}[X   o  cos 2 e p +Y o  sin Θ p  cos Θ p ]  (13)  
       
     
     Using equation 11,              τ   =       2   C          [         X   o            X   p       R   p          cos                   θ   p       +       Y   o            Y   p       R   p          cos                   θ   p         ]               (   14   )                                
     Using equation 10              τ   =       2       Cv   s          (     K   +   t     )              [         Y   o          Y   p       +       X   o          X   p         ]               (   15   )                                
     Using equation 3                Video                 Phase     =         2        (       ω   o     +       2   γ        t       )           Cv   s          (     K   +   t     )              [         Y   o          Y   p       +       X   o          X   p         ]               (   16   )                                
     To eliminate the dependence on time, we must let the trigger delay 
     
       
         K=ω o /2 γ  (17)  
       
     
     To determine the locus of the points of zero video phase, we set video phase equal to 2 τn is any integer. Combining equations 16 and 17 gives the lines of zero phase:            2   π        n     =         4      γ       Cv   s            [         Y   o          Y   p       +       X   o          X   p         ]                              
     or in standard form,                Y   p     =           X   o       Y   o            X   p       +         n   π          Cv   s         2        Y   o        γ                 (   18   )                                
     Equation 18 represents a family of straight lines of slope        -     [       X   o       Y   o       ]                            
     with intercepts separated by            π                   Cv   s         2        Y   o        γ       ,                          
     which is the correct grating to Fourier transform into an image of the object point. Equation 17 is equivalent to requiring f 1 /f 2 =R p1 /R p2  where f 1,2  and R p,1,2  are the start and finish transmitter frequencies and trace positions. 
     Now we will consider the more general case, when both transmitter and receiver are allowed to rotate. First, FIG. 20 is modified, as shown in FIG. 23, to allow for arbitrary receiver angle α o . The last two digits of the drawing numerals in FIG. 23 are the same as those in FIG. 20 for the corresponding elements. 
     Equations 1 through 3 are unchanged. 
     
       
         Reference Phase=ω o t+γt 2    (1)  
       
     
     
       
         Signal Phase=ω o (t−τ)+γ(t−τ) 2   (2)  
       
     
     
       
         Video Phase=(ω o +2 γt)τ  (3)  
       
     
     In the figure, Θ o  remains defined as the bistatic angle. The angle α o  is introduced as the receiver direction in the object space coordinate system. Then the path length difference between the point and the center of the field of view, and the corresponding delay difference becomes: 
     
       
         τ={fraction (1+L /C)}[R o  cos (Θ o +α o −φ o )+R o  cos (α o −φ o )]  (4)  
       
     
     The equation is denoted 4′ as it is the modified version of equation 4 presented previously. Since the target position in rectangular coordinates is still: 
     
       
         X o =R o  cos φ o  Y o =R o  sin φ o    (6)  
       
     
     we can use the trigonometric identity 5 to obtain:                      cos        (       θ   o     +     α   o     -     φ   o       )       =                    cos        (       θ   o     +     α   o       )          cos                   φ   o       +       sin        (       θ   o     +     α   o       )          sin                   φ   o                     =                      X   o       R   o          cos                   (       θ   o     +     α   o       )       +         Y   o       R   o          sin                   (       θ   o     +     α   o       )                       cos        (       α   o     -     φ   o       )       =                      X   o       R   o          cos                   α   o       +         Y   o       R   o          sin                   α   o                       (     7   ′     )                                
     and from equations 4′ and 7′ 
     
       
         τ=1/C[Y o  { sin (θ 0 +α o )+ sin α o }+X o  { cos (θ o +α o )+ cos α o }] 
       
     
     or using the trigonometric identities 
     
       
         τ=1/C[Y o (sin θ o  cos α o + cos θ o  sin α o + sin α o )+X o (cos θ o  cos α o − sin θ o  sin α o  +cos α o )]  (8′)  
       
     
     The general storage trace equations again were discovered intuitively. They are:                θ   p     =         θ   o     2     +     α   o               (     9   ′     )                 R   p     =       (       v   s        cos                     θ   o     2       )          (     K   +   t     )               (     10   ′     )                                
     where, as before, the storage point coordinates are related by 
     
       
         X p =R p  cos θ p , Y p =R p  sin θ p    (11)  
       
     
     Rearranging equation 11, substituting equation 9′, and using the trigonometric identities gives                  X   p       R   p       =       cos                     θ   o     2        cos                   α   o       -     sin                     θ   o     2        sin                   α   o                 (     12   ′     )                   Y   p       R   p       =       cos                     θ   o     2        sin                   α   o       +     cos                   α   o                   sin                     θ   o     2                                                  
     Using the half-angle trigonometric identities in equation 8′ gives                    τ   =                  1   C          {       Y   o     [       2                 sin                     θ   o     2        cos                     θ   o     2        cos                   α   o       +       (       2                   cos   2                       θ   o     2       -   1     )        sin                   α   o       +                                      sin                   α   o       ]     +       X   o     [         (       2                   cos   2                       θ   o     2       -   1     )        cos                   α   o       -                                    2                 sin                     θ   o     2        cos                     θ   o     2        sin                   α   o       +       cos      α     o       ]     }                 (     13   ′     )                     τ   =                  1   C          {         Y   o          [       2                 sin                     θ   o     2        cos                     θ   o     2        cos                   α   o       +     2                   cos   2                       θ   o     2        sin                   α   o         ]       +                                      x   o          [       2                   cos   2            θ   o     2        cos                   α   o       -     2                 sin                     θ   o     2        cos                     θ   o     2        sin                   α   o         ]       }                                                    
     Combining equation 12′ with equation 13′ gives              τ   =       1   C          {           Y   o          (     2                 cos                     θ   o     2       )            (       Y   p       R   p       )       +         x   o          (     2                 cos                     θ   o     2       )            (       X   p       R   p       )         }               (     14   ′     )                                
     Substituting equation 10′ in equation 14′ gives              τ   =       2       Cv   s          (     K   +   t     )              [         Y   o          Y   p       +       X   o          X   p         ]               (     15   ′     )                                
     which is identical to equation 15. Hence, the remainder of the analysis applies, and the data is a family of straight lines which is the correct grating to Fourier transform into an image of the target point. The lines are again independent of bistatic angle, θ o , or receiver direction α o , verifying the correctness of the general equations 9′ and 10′. 
     The general analysis shows how to store the data for any receiver or transmitter motion. Two examples are presented in FIGS. 24A and 24B. In each case the receiver  2416  executes a 90-degree turn ,  2401  and  2402 . In FIG. 24B, the transmitter  2403  circles at long range along a flightpath  2405 . In FIG. 24A the transmitter  2404  flys a straight line  2406 . Both slow and fast receiver speeds are illustrated. Data  2407 ,  2408  and flightpaths  2409 ,  2410  prior to the turns  2411  and  2414  are shown dashed. Drawing numerals  2415  and  2412  indicate the start of the data collection and  2413  indicates the end of the data collection. 
     The bistatic angle θ and the receiver direction angle α are obtained from calculations based on the location of the transmitter and receiver with respect to the field of view. The location of the transmitter and the receiver can be obtained from inertial navigational systems aboard the air-craft or this information may be obtained from a ground based radar which is capable of tracking both the transmitter and receiver aircrafts. 
     The preceding analysis is for two dimensional geometry. Usually, even if the radar geometry is not planar, the simplified planar processing still eliminates the most severe errors, and a distortion-free image will be obtained for those target scatterers that are located in a fixed plane, commonly called the “image projection plane”. However, the invention can readily be extended to three dimensions to eliminate errors resulting from non co-planar scatterers. 
     The two dimensional mathematical analysis was based upon the proof of a hypothesis; a solution was proposed and then demonstrated to result in an error-free result. For the most general three dimensional analysis that follows, we will use a development; that is, starting with the laws of physics, the required processing will be derived. Using this approach, the invented processing concept may be summarized as follows: 
     Store the radar data such that the data range frequency direction is aligned to the radar range gradient vector direction, and the range frequency scale is proportional to the gradient vector magnitude, and the data Doppler frequency direction and scale is aligned and proportional to the radar Doppler gradient. Then an undistorted image can be obtained from the stored data by orthonormal Fourier transform. 
     Since the Stretch radar can be viewed as a continuous frequency scanning of the target space during each pulse, data entered into storage during each frequency scan should likewise be positioned by continuous scanning. Imagine, then, that data received during one pulse is stored along a line, where each point on the line has a scalar property related to the instantaneous radar frequency or wavelength. Note that a Stretch radar is not necessary; only that the data is in a frequency versus time (frequency response or Fourier transform) form. Data received in the format will be referred to herein as being in Stretch format or as being Stretch data. 
     The present invention may be used with any radar transmitted waveform by merely translating the received data to Stretch format. A matched filter is a device which can be used to convert any radar transmitted waveform into a long frequency coded pulse or a Stretch pulse. Such devices are described in Chapter 20 of the “Radar Handbook”, M. Skolnik, McGraw Hill, New York, 1970. 
     On the next radar pulse, the FOV is scanned again in frequency but from a different radar direction, hence another line of data is stored in a different position. Each point on the new line corresponds to a point on the old line in that the radar wavelength was the same. Hence each point can be considered to have moved between pulses, and therefore also has a vector property, velocity. The situation is illustrated in FIG. 25, in two dimensions. 
     In this analysis vector notation is used. A vector is denoted by a bar ({overscore (a)}), its length by the letters alone (a), and its direction by a carat (â), which is the “unit vector”. Scalars have no embellishment (λ). 
     In FIG. 25, R p  and θ p  designated  2510  and  2509 , respectively, define the magnitude and angle of a vector {overscore (r)} ( 2503 ) to a point P ( 2507 ) on a graph in which drawing numeral  2501  denotes the abscissa and  2502  denotes the ordinate. The radar frequency f is shown in this Figure as f 1  ( 2508 ) and f 2  ( 2505 ). The velocity vector of the point, resulting from the radar geometry motion is shown as {overscore (v)} ( 2504 ) which is the rate of change of {overscore (r)} with time          ∂     r   _         ∂   t                            
     ( 2506 ) for a constant frequency, f. 
     Aligning the directions means:                v   _     =           k   D          (       ∇     f   D       _     )                         ∂     r   _         ∂   f         =       k   R          (       ∇   _        L     )                 (     1   ″     )                                
     where ∇f D  and ∇L are the Doppler and range gradients. The Doppler and range scale factors (k D , k R ) are determined by equating the spatial Doppler and range frequencies that would be obtained for target points displaced equally from the FOV center in either the Doppler or range gradient direction. For Doppler, temporal and spatial Doppler frequencies are scaled by velocity: 
     
       
         f DS v=f D    (2″)  
       
     
     and Doppler gradient is defined by 
     
       
         f D =({overscore (∇f D +L )})·{overscore (S)}  (3″)  
       
     
     for a target located at {overscore (S)}, (see  2604  in FIG. 26) the vector denoting R o  and Φ o . 
     If the target direction is parallel to the gradient              v   =       S     f   DS            (     ∇                f   D       )               (     4   ″     )                                
     and the Doppler scale is the ratio of target displacement to resulting stored Doppler spatial frequency.                k   D     =     S     f   DS               (     5   ″     )                                
     To determine the range scale factor, we must for the moment introduce the concept of fast (or intrapulse) time, during which radar frequency is chirped, but the geometry is stationary. 
     Then                  ∂   r       ∂   f       =       v   S     σ             (     6   ″     )                                
     where v s  is the intrapulse scan speed, or the rate of r with respect to fast time, and Σ is the radar chirp slope (equal to 2v) or the rate of f with respect to fast time. Similarly, the range spatial and (fast) temporal frequencies are scaled by the scan velocity 
     
       
         f RS v S =f R    (7″)  
       
     
     and the range temporal frequency is determined in the Stretch radar by the chirp rate and the range delay (fast time) of the target                f   R     =     σ      Δ                   L   C               (     8   ″     )                                
     where ΔL is the round trip path length change for the target {overscore (S)} from the FOV center, and C is the speed of light. The range gradient is defined by 
     
       
         ΔL=({overscore (∇L)})·{overscore (S)}  (9″)  
       
     
     Hence for a target direction parallel to the range gradient, combining (6″) (7″) (8″) and (9″) gives                  ∂   r       ∂   f       =           (       ∇   L     _     )     ·     S   _         Cf   RS       =       S     Cf   RS            (     ∇   L     )                 (     10   ″     )                                
     Thus the range scale factor is                k   R     =     S     Cf   RS               (     11   ″     )                                
     Since we wish the ratio of target displacement to recorded spatial frequency to be identical in the Doppler and range directions, let                S     f   RS       =       S     f   DS       =       k   D          =   Δ        k               (     12   ″     )                                
     then                k   R     =     k   C             (     13   ″     )                                
     where k is an arbitrary scale factor to be chosen to suit a particular storage medium resolution and FOV size. Then equations (1″) become 
     
       
         {overscore (v)}=k({overscore (∇f D +L )})   (14″)  
       
     
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         r 
                         _ 
                       
                     
                     
                       ∂ 
                       f 
                     
                   
                   = 
                   
                     
                       k 
                       C 
                     
                      
                     
                       ( 
                       
                         
                           ∇ 
                           L 
                         
                         _ 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                     ″ 
                   
                   ) 
                 
               
             
           
         
                 
         
             
         
      
     
     Now radar frequency and wavelength are related: 
     
       
         f=C/λ  (16″)  
       
     
     hence                ∂   f     =         -   C       λ   2            ∂   λ               (     17   ″     )                                
     and (15″) can be rewritten                  ∂     r   _         ∂   λ       =         -   k       λ   2            (       ∇   L     _     )               (     18   ″     )                                
     To evaluate the Doppler and range gradients, consider the transmit and receive processes separately. Then the rotation of the transmitter relative to the FOV can be treated as if the transmitter were fixed and the FOV rotating at the rate {overscore (Ω X +L )}  2601  as shown in FIG.  26 . Likewise the FOV can be considered rotating at a different rate {overscore (Ω R +L )} ( 2601 ) with respect to the receiver. In either event, the velocity of the target is {overscore (Ω)}×{overscore (S)}. The component directed toward the receiver or transmitter is: 
     
       
         {circumflex over (R)}·({overscore (Ω)} R ×{overscore (S)}) or {circumflex over (X)}·({overscore (Ω)} X ×{overscore (S)})  
       
     
     where {circumflex over (R)}  2603  is the receiver direction vector (denoting α o ) and {circumflex over (X)} ( 2603 ) the transmitter direction vector (denoting α o +θ o ) 
     The Doppler shift is the sum of the receiver directed component with the transmitter directed component:            λω   D       2      π       =         R   ^     ·     (         Ω   _     R     ×     S   _       )       +       X   ^     ·     (         Ω   _     X     ×     S   _       )                                
     The bistatic Doppler gradient is the vector sum of the monostatic gradients:                ∇     (       λω   D       2      π       )       =     ∇     [         R   ^     ·     (         Ω   _     R     ×     S   _       )       +       X   ^     ·     (         Ω   _     X     ×     S   _       )         ]                   =       ∇     [       R   ^     ·     (         Ω   _     R     ×     S   _       )       ]       +     ∇     [       X   ^     ·     (         Ω   _     X     ×     S   _       )       ]                                      
     Since              R   ^     ·       Ω   _     R       ×     S   _       =             Ω   _     R     ·     S   _       ×     R   ^       =         S   _     ·     R   ^       ×         Ω   _     R          (   identity   )                     ∇     (       λω   D       2      π       )       =       ∇     [         S   _     ·     R   ^       ×       Ω   _     R       ]       +     ∇     [         S   _     ·     X   ^       ×       Ω   _     X       ]                                
     which is the vector sum of the two monostatic gradients. Expanding the gradients gives the Doppler gradient:          -     ∇     (       λω   D       2      π       )         =           Ω   _     R     ×     R   ^       +         Ω   _     X     ×     X   ^                                
     To determine the range gradient, let L be the distance from the transmitter to the target to the receiver, as shown in FIG.  27 . 
     In this Figure, drawing numeral  2701  denotes the abscissa,  2702  denotes the ordinate,  2703  denotes the vector {overscore (R)},  2704  denotes the vector {overscore (X)},  2705  denotes the vector {overscore (S)},  2706  denotes the Target, and  2708  denotes the Receiver (R). 
     Now L=|{overscore (R)}−{overscore (S)}|+|{overscore (X)}−{overscore (S)}| 
     The range gradient ∇L is: 
     
       
         ∇L=∇|{overscore (R)}−{overscore (S)}|+∇|{overscore (X)}−{overscore (S)}| 
       
     
     Since, 
     
       
         |{overscore (R)}−{overscore (S)}|≅R−{overscore (S)}·{circumflex over (R)} 
       
     
     
       
         |{overscore (R)}−{overscore (S)}|≅X−{overscore (S)}·{circumflex over (X)} 
       
     
     
       
         ∇L≅∇R−∇{overscore (S)}·{circumflex over (R)}+∇X−∇{overscore (S)}·{circumflex over (X)} 
       
     
     Also, since distances R and X are constant, independent of target position: 
     
       
         ∇R−0 ∇X=0  
       
     
     So 
     
       
         −∇L≅∇{overscore (S)}·{circumflex over (R)}+∇{overscore (S)}·{circumflex over (X)}={circumflex over (R)}+{circumflex over (X)} 
       
     
     If the bistatic angle bisector is {overscore (B)} the range gradient is:            -     ∇   L       ≅     B   _       =         R   ^     +     X   ^       =       (     2                 cos                     θ   o     2       )          B   ^                                
     Where θ o  is the bistatic angle between {circumflex over (X)} and {circumflex over (R)}. 
     Thus the range and Doppler gradients for a generalized bistatic radar are: 
     
       
         ∇(λf D )={overscore (Ω)} R ×{circumflex over (R)}+{overscore (Ω)} X ×{circumflex over (X)}  (19″)  
       
     
     
       
         −∇L={circumflex over (R)}+{circumflex over (X)}  (20″)  
       
     
     Where {circumflex over (R)}, {circumflex over (X)} are the terminal directions; {overscore (Ω R +L )}, {overscore (Ω X +L )} their rotations. First, combining (18″) and (20″)                  ∂     r   _         ∂   λ       =       k        (       R   ^     +     X   ^       )         λ   2               (     21   ″     )                                
     Integrating:                ∫     ∂     r   _         =       k        (       R   ^     +     X   ^       )              ∫   λ            ∂   λ     2                 (     22   ″     )                 r   _     =         k        (       R   ^     +     X   ^       )       λ     +     K   _               (     23   ″     )                                
     where {overscore (K)} is an arbitrary constant (with respect to λ) of integration. 
     Next, combining (14″) and (19″):                v   _     =       k   λ          (           Ω   _     R     ×     R   ^       +         Ω   _     x     ×     X   ^         )               (     24   ″     )                                
     However {overscore (v)} must also be the time derivative of {overscore (r)} (time being “slow” or conventional time changing with geometry, we will have no further need for the fast time concept). Differentiating (23″) with respect to time:                v   _     =         ∂     r   _         ∂   t       =         k   λ          (         ∂     R   ^         ∂   t       +       ∂     X   ^         ∂   t         )       +            K   _            t                   (     25   ″     )                                
     But {overscore (Ω)} R  and {overscore (Ω)} X  are defined as                  ∂     R   ^         ∂   t       =           Ω   _     R     ×     R   ^                       ∂     X   ^         ∂   t         =         Ω   _     X     ×     X   ^                 (     26   ″     )                                
     Hence combining (24″) (25″) and (26″) gives                  ∂     K   _         ∂   t       =   0           (     27   ″     )                                
     and {overscore (K)} must be constant with respect to both time and wavelength. With this understanding, equation (23″) may be taken as the storage equation. The vector {overscore (K)} may be considered as defining the coordinate origin, a useful aid in understanding the result, but may also be taken as zero, hence          r   _     =     k                       R   ^     +     X   ^       λ                              
     is perfectly adequate. Replacing {circumflex over (R)}+{circumflex over (X)} by {overscore (B)} gives                r   _     =         2      k                 cos                     θ   o     2       λ          B   ^               (     28   ″     )                                
     This equation reduces to equation 9′ and 10′ when the geometry is two dimensional and the vector notation is replaced by trigonometric notation. 
     FIG. 28 shows how data from a three dimensional bistatic radar system would be stored and processed. The geometry corresponds to the ground to air imaging situation in which the target  2801  overflys the fixed receiving terminal  2802 , with the fixed transmitting terminal  2803  located at long range from the target (such as a range equal to 20 times the target altitude  2803 ). 
     The data would be stored on a three dimensional surface  2804 , then processed by three dimensional Fourier transform. The X, Y, Z coordinate of the point at which data is stored ( 2806 ) is determined by the target position along its flightpath  2805  according to equation 28″. The three dimensional transform would be performed one dimension at a time. First, for each combination of X and Y, the data would be transformed in the Z direction and the stored untransformed data would be replaced by the partially transformed data, simultaneously filling all the unoccupied storage elements. Then for each combination of X and F Z  the data would be transformed in the Y direction. 
     Again the one dimensional transformed stored data would be replaced by two dimensional transformed data. Then for each combination of F Y  and F Z  the data would be transformed in the X direction, resulting in a stored image (F X , F Y , F Z  coordinates) in three dimensions. The procedure is a simple extention of the well known two dimensional Fourier transform. 
     From the preceding discussion, it can be seen that the purpose of the invention is to record bistatic radar information so that it may be retrieved in rectangular coordinate form and be converted to an image by conventional Fourier transform means. The basic functions which must be carried out are: 
     1. The video scan must be started at a point along a radius from the origin proportional to the product of cosine θ/2 and a value equal to the start frequency of the transmitter divided by the slope of the transmitted FM signal,          ω   o       2      γ                            
     2. The scan must then be compressed by an amount proportional to cosine θ/2. 
     3. The recording medium must be set to an orientation equal to the direction of the bistatic bisector. In two dimensions this corresponds to an angle equal to        θ   2                          
     for each scan when the receiver does not rotate or          θ   2     +   α                          
     otherwise. 
     A two dimensional system for recording bistatic data in accordance with the present invention is shown in FIG.  21 . This system comprises a divider  2101 , a cosine function generator  2102 , a scan staring point product generator  2103 , a scan signal generator  2110 , a directional control generator  2109 , a scan projector  2107 , a film  2106 , a drive means  2105 , and an adder  2111 . 
     In the operation of this system, a signal representing the bistatic angle is fed to an input port of the divider  2101  to produce at its output port a signal representing one-half the bistatic angle. The signal representing one-half the bistatic angle is supplied to the Cosine Function Generator to provide at its output port a signal representing the cosine of one-half the bistatic angle. The scan starting point generator accepts this signal and a signal representing the transmitter start frequency divided by the slope of the transmitted FM signal to produce at its output port a signal representing the product of the two input signals. The Scan Signal Generator accepts the product signal to determine the point along a radius away from the origin at which the scan will start and accepts the signal representing the cosine of one-half the bistatic angle to compress the scan after the starting point by an amount which is proportional to the cosine of one-half the bistatic angle. The Scan Signal Generator also accepts the radar trigger signal to determine the start time of the scan. 
     The adder  2111  adds the receiver angle α to one-half the bistatic angle. The Directional Control Generator accepts the signal representing the sum of the receiver angle and one-half the bistatic angle and the output of the scan signal generator. From the radial scan and the angle, it generates the corresponding horizontal and vertical sweeps. 
     The Scan Projector accepts the output signals of the Directional Control Generator to produce a scan beginning at a starting point that is at a specified distance away from the origin, which is simply a reference point for the display of the received radar video. The specified distance is proportional to the product value. The scan is projected at an angle that is equal to one-half the bistatic angle plus the receiver angle from an arbitrary angular reference. The scan speed, after the initiation point, is proportional to the cosine of one-half the bistatic angle. 
     In a typical embodiment, the scan projector is a cathode-ray tube (CRT). The scan signal generator provides control signals to the CRT to produce display similar to that shown in FIGS. 24A and 24B. 
     The angle at which the trace is projected may be set by means of the CRT or alternately by means of orienting the recording medium. A typical recording medium is film, such as the film  2106  shown positioned in front of the CRT. 
     In the alternative approach, the film is oriented by the film drive unit  2105  which is driven directly by the          θ   2     +   α                          
     signal obtained from  2111 . In this approach, the initiation of each scan at a staring point away from the origin may be accomplished in the Scan Signal Generator by producing control signals which initiate the scan in synchronism with the radar trigger. The trace is rapidly moved to the starting point, after which the speed of the trace is made proportional to the cosine of one-half the bistatic angle. The video data containing the received radar information for each scan is fed to the video input port  2108  of the scan projector to modulate the beam intensity of the CRT. 
     The invention may also be implemented in digital form. It is possible to carry out digitally all the operations, including the scaling, delay, rotation, and recording functions. The former three can be considered data address change operations and may be carried out in a digital address generator, such as that shown in the system of FIG.  22 . 
     The system in this Figure comprises a radius address generator  2210 , an X-Y address generator  2216 , and an angle address generator  2215 , as well as four analog to digital converters (A/D)  2201 ,  2208 ,  2220  and  2212 , a short term memory  2204 , a long term memory  2205 , a multiplier  2209 , a divider  2213 , an internal clock generator  2218 , and a cosine function generator  2214  and adder  2221 . 
     Radar video information is received at the input port  2202  of A/D  2201 , where it is converted to digital form and stored in the short term memory  2204 . The bistatic angle θ is received at the input port  2211  of A/D  2212 , where it is converted to digital form and then supplied, through the divider  2213 , to the cosine function generator  2214  to produce the function cosine          θ   2     .                          
     The transmitter start frequency divided by the slope of the transmitted FM signal,          ω   o       2      γ                            
     is received at the input port  2207  of A/D  2208 , where it is converted to digital form and then supplied to the multiplier  2209 . The multiplier receives the cosine        θ   2                          
     signal from the cosine function generator  2214  and supplies the product          ω   o       2      γ                            
     cosine        θ   2                          
     to the radius address generator  2210 . The radar trigger and a radar clock signal, which are received at input port  2206 , are supplied to the radius address generator  2210  and clock generator  2218 . 
     The cosine function generator also supplies a signal representing cosine        θ   2                          
     to the clock generator  2218 , which produces slow clock signals at a rate equal to the input clock rate multiplied by cosine          θ   2     .                          
     This clock signal is supplied to the Radius Address Generator  2210 . 
     The radar video data is advanced through A/D  2201  by the internal clock signal supplied by generator  2218 . This clock signal also advances the radius generator starting at the advanced address determined by the shift signal obtained from multiplier  2209 , during date read-in to short term memory  2204 . During data read-out from memory  2204 , the radius address generator is advanced by the normal radar clock signal  2206  starting from the first address. 
     The receiver angle α is applied to the input port  2219  of A/D  2220  where it is converted to digital form, then added in adder  2221  to the value        θ   2                          
     supplied by divider  2213 . 
     The value          θ   2     +   α                          
     is supplied to the angle address generator  2215  by the adder  2221 . The output of the angle address generator  2215  and radius address generator  2210  as well as the output of the short term memory  2204  are supplied to the long term memory  2205 . 
     In the operation of the system of FIG. 22, video data is shifted into the short term memory  2204  from A/D  2201  in accordance with the slow internal clock signals. It is stored in this memory at addresses determined by the Radius Address Generator, starting in accordance with the product          ω   o       2      γ                            
     cos          θ   2     .                          
     The storing of video data into the memory at a starting address in accordance with the product signal accomplishes the requisite shift in starting point, which is the first step required in reformatting the data. In the second step, data compression proportional to cosine        θ   2                          
     is accomplished by providing slowly incrementing radius addresses for the short term memory determined by the clock  2218 , which is controlled by a signal representing cosine          θ   2     .                          
     The third basic step, which is setting the recording medium or the radial trace to an angle equal to          θ   2     +   α                          
     is accomplished by the angle address generator and the memory  2205 . In effecting this third function, the video data is read out from the memory  2204  into the memory  2205  using the read in angle address obtained from generator  2215  which is shifted in accordance with the value          θ   2     +     α   .                            
     Information is retrieved from the output port  2217  of memory  2205  in rectangular coordinates by entering rectangular address signals from the X-Y generator  2216 . 
     It should be noted that memories  2204  and  2205  may be random access memories (RAMS) and therefore both may be combined into a single unit, as indicated by the dashed lines about these two units. The single memory unit is designated  2203 , and is referred to as the memory and recording unit. 
     Alternately, the long term memory may be provided by other forms of data storage such as the CRT and film system of FIG.  21 . In this case, the long term memory could be a conventional polar format camera, and the short term memory can be considered a bistatic to monostatic format converter. Furthermore, if an analog long term memory is used, an accelerated internally generated clock signal can be used for data read-out from short term memory  2204  provided by generator  2218 , in which case the radar clock  2206  would be used for data read-in. 
     An implementation of a three dimensional processor is shown in FIG. 29, which is modified from FIG. 22 only in that three dimensional address are applied to the long term memory. All the drawing numerals in FIG. 29 begin with  29 . With the exception of the drawing numerals mentioned below, those drawing numerals in FIG. 29 which have the same last two digits as those in FIG. 22 have the same or an equivalent function. On read in, the radius address is controlled as before, but the two dimensional bistatic bisector single angle address          θ   2     +   α                          
     is replaced by the three dimensional (azimuth and elevation) angle addresses determined by vector additional ( 2921 ) from the receiver and transmitter three dimensional directions ( 2919 ) and ( 2922 ). On read out, the long term memory is scanned in three (rectilinear) directions, (X, Y and Z) rather than two, but still one dimension at a time.