• Detection of even slowly moving targets with low reflectivity (low RCS) against a strong clutter interference.

• Estimation of the moving targets’ true geographical positions.

• Estimation of the moving targets’ velocities and driving directions.

• Have real-time capability.

• Track the moving targets during the (increased) observation time.

• Refocus the “blurred” images of extended moving targets (an extended target is a target occupying more than one SAR resolution cell) like ships and larger land vehicles to high resolution for recognition purposes.

2.18.2.1 SAR acquisition geometry and operation

A SAR instrument consists of a pulsed transmitter, at least of one antenna which is used both for transmitting and receiving, and of a phase coherent receiver [1]. The typical side-looking imaging geometry of a SAR system is shown in Figure 18.3.

The platform carrying the radar instrument moves at constant altitude with constant velocity parallel to the x-axis. The moving direction of the radar is also denoted as along-track or azimuth direction. The antenna is mounted in a way so that the antenna beam with a certain depression angle points perpendicular to the azimuth direction towards the ground (the system in Figure 18.3 is left-looking with respect to the flight path). An area on ground with a certain swath width is illuminated by the beam. The radar is periodically emitting radar pulses of duration with the so called Pulse Repetition Frequency (PRF). The PRF typically is in the order of several 100 Hz (airborne systems) to several 1000 Hz (spaceborne systems). The pulses are backscattered from the illuminated area on ground, coherently received, down converted, digitized and stored in the mass memory of the SAR instrument. SAR processing is carried out afterwards, either onboard the platform or on ground after downloading the data.

What the SAR measures are the backscattered signal energy and the time interval between the emitted and received pulses. The pulse travel time is proportional to the two-way range, i.e., the range from the antenna phase center to the target and back. A side-looking geometry is necessary so that for each measured slant range r the corresponding ground range or across-track position y can be computed unambiguously (cf. Figure 18.3).

The SAR principle is based on a movement of the sensor with respect to the illuminated targets on ground. Due to the motion of the platform the range r between the platform and a specific stationary point target on ground changes as shown in Figure 18.4. This range change causes a Doppler frequency shift of the received signal which during SAR processing is exploited for synthesizing a large antenna along azimuth direction, resulting in a narrow synthetic azimuth beam width and, hence, in high azimuth resolution.

Due to the importance for GMTI processing discussed later, in the following the range and Doppler histories of the signal backscattered from a particular non-moving point target are derived.

The position of the antenna phase center located at the moving SAR platform with respect to the origin of the Cartesian {x, y, z} coordinate system can be written as (see also sketch in Figure 18.5)

(18.5)

where is the constant platform velocity and t the time. At time the platform is at altitude above the origin of the Cartesian coordinate system. Let now the position of a certain non-moving “stationary” target be

(18.6)

The indices “0” indicate time independent parameters. The distance between the antenna and the target can simply be computed as

(18.7)

where denotes the norm. For describing the SAR principle in common literature it is often assumed, without restriction of generality, that so that at the point target is broadside the SAR platform. The minimum range at in this case can be written as

(18.8)

so that (18.7) simplifies to

(18.9)

The quadratic approximation given after the “” sign is obtained by a second-order Taylor expansion about . The time t is proportional to the azimuth position of the platform :

(18.10)

2.18.2.2 Stationary point target signal model

One single pulse transmitted by the SAR system can be expressed as

(18.11)

where represents the pulse waveform in baseband, is the so called “fast time,” j is the imaginary unit, is the radar wavelength given by the carrier frequency and denotes the speed of light in vacuum. Conventionally in SAR a linear frequency modulated (LFM) waveform, a so called “range chirp,” with a certain bandwidth and a certain duration (in the order of microseconds) is transmitted (although SAR is not limited to such waveforms). The range chirp in baseband is given as

(18.12)

where denotes the chirp slope and the envelope. The signal received from a point-like target is then a delayed and attenuated copy of the transmitted signal which can be written as

(18.13)

The free-space attenuation, the backscattering coefficient, the elevation and the azimuth angles to the target as well as the weighting of the two-way antenna pattern are covered by the coefficient A. After coherent down-conversion to baseband using, e.g., a phase preserving quadrature demodulator, the received signal is given as

(18.14)

This signal can be separated into two parts:

The raw data signal given in (18.14) in its one-dimensional representation is in fact stored in a two-dimensional arrangement in the mass memory of the radar system according to the range and azimuth dimension. To get a better insight in this storage procedure one can make use of the start-stop-approximation. It assumes that the antenna and, hence, the SAR platform is motionless when a pulse is emitted and the scattered signal received. Afterwards the antenna moves to its next sending/receiving position along the flight track. This approximation can be made since the pulse travel time is much smaller than the time needed for the antenna to move to the next position. In range dimension the signal is sampled when the antenna is “motionless.” The range sampling frequency is determined by the analog-digital converter. For a complex signal this sampling frequency has to be at least as large as the chirp bandwidth so that the Nyquist criterion is not violated. In azimuth dimension the sampling frequency and, hence, the imaginary antenna “stops” are determined by the PRF.

The signal can therefore be written in a two-dimensional form as

(18.15)

where is the “fast time” representing the range direction and t is the “slow time” representing the azimuth direction.

Due to the importance for GMTI we will put the main focus on the azimuth signal

(18.16)

where the rectangular function , defined e.g., in [3], is introduced for pointing out that the signal duration is limited by the illumination time given by the azimuth beamwidth of the antenna pattern. A small azimuth antenna length results in a wide beam (see also (18.1) and long illumination or synthetic aperture time, respectively. For typical airborne systems this time is in the order of several seconds, for state-of-the-art spaceborne systems around one second or smaller. The longer this time, the better is the achievable azimuth resolution after SAR processing.

The phase within the exponential term can according to (18.9) also be approximated using a second-order Taylor expansion, so that

(18.17)

The azimuth phase modulation furthermore can be interpreted as azimuth frequency or Doppler frequency variation if the time derivative is taken in the following way:

(18.18)

If for the phase the quadratic approximation given in the second part of (18.17) is inserted, the linear approximation of the Doppler frequency history is obtained:

(18.19)

It is obvious that the azimuth signal in the first approximation has the shape of a LFM signal with denoting the signal’s azimuth chirp slope or Doppler slope (see also Figure 18.4 bottom).

2.18.2.3 Pulse compression and image formation

As mentioned in the previous section, the transmitted pulse typically has a time duration in the order of a few microseconds, whereas the illumination time of a particular point target is in the order of seconds. Thus, for achieving a high range and azimuth resolution pulse compression has to be employed. Pulse compression generally can be performed by convolving an uncompressed input signal u(t) with a proper reference function h(t). The pulse compressed signal I(t) in its general form is then given as

(18.20)

where denotes convolution.

The optimal filter theory says that for signals embedded in white Gaussian noise the best signal-to-noise ratio (SNR) after convolution is achieved if the reference function h(t) is the complex conjugated and time reverted version of the “expected” input signal u(t):

(18.21)

where is a constant which may be used for scaling purposes and denotes the complex conjugation. If the reference function is constructed in this way, it is denoted as “Matched Filter” [19]. The resolution improvement by applying a matched filter is sketched in Figure 18.6. The comparatively long input signal u(t) after matched filtering is compressed to a pulse I(t) of short duration. The time resolution and, hence, the spatial resolution is significantly improved compared to the input signal.

It is known that a cyclic convolution in time domain corresponds to a simple multiplication in frequency domain. For that reason the convolution in (18.20) can equivalently be written as

(18.22)

where F and denote the Fourier and inverse Fourier transforms, respectively, and U( f ) and H( f ) are the frequency domain representations of u(t) and h(t), respectively.

SAR processing or SAR image formation, within the GMTI community often denoted as “Stationary World Matched Filtering,” can be described briefly by the following three steps:

After performing these three steps a focused SAR image is obtained. These steps are visualized for a single point target in Figure 18.7. Details on state-of-the-art SAR processing algorithms and on the RCMC can e.g., be found in [3,20].

The focused image of a single point target is also denoted as impulse response function (IRF). The simulated IRF of a perfectly focused stationary point target is shown in Figure 18.8.

The IRF has the shape of a two-dimensional sinc function. The geometric resolution is determined by the 3 dB width of the IRF. The best achievable azimuth resolution is given in (18.4). If as transmitted waveform the LFM pulse given in (18.12) with a rectangular envelope is used, the best achievable range resolution is [3]

(18.25)

The larger the chirp bandwidth , the better is the range resolution.

Due to the nature of SAR processing, the IRF of any target (independent whether it is moving or not) always is imaged at the position where the Doppler frequency of its uncompressed azimuth signal is zero. For a stationary target this position corresponds to the minimum range and to the actual azimuth position . In contrast a moving point target, which is discussed in the next section, is displaced in azimuth and to a little extend in range, depending on the motion parameters. For investigating the displacements and additional effects the azimuth and range axis of the IRF plots shown in Figure 18.8, and in some of the Figures provided later in Section 2.18.4, are labeled as azimuth shift and range shift, respectively. The origins of the axes are centered around the positions and . This has the advantage that the displacement quantities easily can be read off.

2.18.3.1 Single-channel signal model

In many of the publications related to GMTI a target in linear motion with constant acceleration during the observation time is assumed. This is especially for short observation times a valid assumption. Furthermore, it is assumed that the target does not change its altitude, i.e., that it does not move in z-direction. This is reasonable since the slopes of common roads only may cause a z-velocity component negligibly small compared to the large x- and y-components. The general acquisition geometry to consider is similar to that sketched in Figure 18.5, apart from the “stationary point target” which has to be replaced by a “moving point target.”

Under the afore mentioned assumption of linear motion the position of the moving point target (i.e., the motion equation) can in contrast to the stationary point target in (18.6) be written as

(18.26)

where , and are the positions at and are the along-track and across-track velocity components at , and and are the constant along-track and across-track acceleration components. If the target moves during the illumination time along a straight line its position can also be expressed using the moving direction or road angle and the velocity and acceleration magnitudes and a (see right part of (18.26)). In this case the velocity and acceleration magnitudes are given as and . The moving direction is measured counter-clockwise from the x-axis towards the y-axis as depicted in Figure 18.9.

Even an acceleration change might be considered in the motion equations [21–23]. However, changing accelerations are neglected in this tutorial. They only play a significant role at long illumination times in the order of several seconds and for ISAR imaging purposes.

The position vector of the platform is the same as in (18.5). The range history of the moving target can then be written as

(18.27)

An analytical treatment of this expression is difficult because of the square root. For investigating effects on SAR imagery caused by moving targets it is appropriate to use the third-order Taylor expansion of the range history about which is given as

(18.28)

where the terms in the order of have been dropped. The range between the antenna and the target at is represented by . In contrast to in (18.8) the range corresponds not to the range of closest approach since now the target is in motion and in the most general case not located at broadside position at (i.e., ):

(18.29)

With either a target track not centered in the azimuth beam or a squinted geometry as depicted in Figure 18.9 on the right can be considered. The target position at time in this case can be expressed in terms of the squint angle of the antenna beam: . If the squint angle is zero the antenna points perpendicular to the flight path so that . A squint angle is either caused by a platform yaw due to crosswind or due to antenna beam steering. Typical squint angles caused by platform yaw are in the order of a few degrees. Thus, the terms in (18.28) as well as the term have no significant contribution and therefore can be neglected. Also the term can be dropped so that the range equation for the moving target simplifies to

(18.30)

The azimuth phase of the moving target signal in the monostatic case (one common transmit (TX) and receiving (RX) antenna) is given by . The third-order Taylor expansion of the moving target’s Doppler frequency computed with (18.18) is then

(18.31)

where denotes the Doppler shift, the Doppler slope and q the quadratic Doppler coefficient.

By comparing (18.31) with (18.30) the approximated range history also can be expressed in terms of Doppler parameters:

(18.32)

Using this range approximation the single-channel moving target azimuth signal can be written as

(18.33)

where the constant phase determined by due to its unimportance has been dropped.

2.18.3.2 Multi-channel signal model

So far the range and Doppler histories of a moving target signal have been derived for the single-channel (monostatic) case where a common antenna is used for both TX and RX of the radar pulses. Now a system with M antennas is considered, where each antenna is separated from its neighbor in the along-track direction by a certain along-track baseline as depicted in Figure 18.10.

The first antenna is used for TX and RX, antenna 2 and all others for receive only. Following the derivation in [23] the ranges , for can be written as

(18.34)

The range difference is given by [24]

(18.35)

where is the directional cosine measured from the x-axis. The expression after the “” sign is the result of a first-order Taylor expansion. The multi-channel azimuth signals corresponding to the ranges in (18.34) are then

(18.36)

For the classical GMTI techniques ATI and DPCA treated in Section 2.18.5, the multi-channel azimuth signals in (18.36) need to be aligned or co-registered with the fore channel (RX1) so that the antenna phase centers are at the same spatial location at different times. Thus, a shift of the signals is necessary. For co-registering with the signal the signal needs to be shifted by , and by .

For bistatic operation (only the fore antenna transmits, all others receive) the effective phase center separation is so that the time difference can be written as

(18.37)

The co-registered signals are then (to keep the equations shorter the functions have been omitted in the following)

(18.38)

It can be shown that the range difference can be approximated as

(18.39)

and the range also can be written as [24]

(18.40)

with

(18.41)

In practice all azimuth signals are sampled with a frequency given by the PRF. The sampling interval corresponds to so that e.g., the time discrete representation of the signals in (18.36) can be written as

(18.42)

where N is the total number of azimuth samples. The received discrete azimuth signals (either un-registered or co-registered) can also be collected in a data matrix given as

(18.43)

where M is the total number of receiving channels. Vectorizing (18.42) by stacking each succeeding column beneath the other yields

(18.44)

where ^T means vector transposition.

2.18.4.1 Residual range cell migration

Depending on the motion parameters the range histories of a moving target (18.30) and a stationary target (18.7) located at the same position at are quite different. Examples are shown in Figure 18.11, where the range history of a moving target signal¹ (red) is compared with that of a stationary target (blue).

If the target travels in across-track direction with a certain across-track velocity the range history is shifted in azimuth by and in range by (Figure 18.11, left). The curvature itself is not changed significantly.

When the target travels in along-track direction or accelerates in across-track direction (Figure 18.11, right) the curvature is changed but the range history is not shifted. The range curvature change is equivalent to a quadratic phase error which after conventional SAR processing results in a blurred IRF.

A SAR processor performs a RCMC adapted only for stationary targets as depicted in the third image from the left in Figure 18.7. This RCMC is conventionally performed in the frequency domain [20]. For computing the residual range cell migration of the moving target signals it is necessary to express the range history as a function of Doppler frequency. The relation between time and Doppler frequency may be easily obtained from (18.31) if the quadratic Doppler coefficient q is neglected (for small illumination times the introduced error is negligibly small):

(18.45)

By inserting this relationship in (18.32) the quadratic approximation of the moving target range history as a function of Doppler frequency is obtained:

(18.46)

For a stationary target located at the same position as the moving target at time the quadratic approximation is

(18.47)

where and are the Doppler parameters of the stationary target. These are obtained from (18.31) by setting the motion parameters , and to zero so that

(18.48)

and

(18.49)

For a non-squinted acquisition geometry and, hence, are zero.

Since the SAR processor only performs the RCMC correctly for stationary targets a residual range cell migration remains for moving target signals. Its quadratic approximation is given as

(18.50)

This expression is only valid for signals which are not aliased in Doppler. The residual range cell migration is the reason why the IRF of the moving target also may be blurred in range direction after azimuth compression. A detailed explanation on the range blur is given in Section 2.18.4.2.

An example for the residual range migration for an airborne system is shown in Figure 18.12.

The results show that for targets accelerating in along-track direction (Figure 18.12, third image from the left) and for targets moving with constant velocity in across-track direction (second image from the right) almost no residual range cell migration exists. Thus, the major part of the signal energy is distributed along a single azimuth line. Such signals can easily be extracted from the range-compressed and RCMC data array for parameter estimation purposes discussed later.

A couple of years ago the so called “Keystone Transform” has been introduced with the aim to remove the linear range cell migration of the moving target signals, independent of their motion parameters [25]. However, the final result is the same as obtained by a conventional SAR processor based on chirp scaling [20] with omitted azimuth compression: the linear range cell migration of moving target signals is removed. Thus, if anyhow SAR processing is carried out the application of the Keystone transform is not necessary.

If the Doppler parameters and of a particular moving target signal are known (e.g., after estimation using proper techniques) a RCMC adapted to this target can be performed. An example is shown in Figure 18.13, where the same signals as for Figure 18.12 are used. In this case almost no residual range cell migration for the moving target signal remains. However, each moving target signal has different Doppler parameters and therefore requires the application of a different adapted RCMC which requires high computational power.

With the proper moving target parameters additionally an adapted azimuth matched filter can be constructed so that a perfectly focused moving target image is obtained [26].

2.18.4.2 Along-track velocity

Equation (18.31) can be used for investigating the effects on the Doppler history. The major effect caused by the along-track velocity is a change of the Doppler slope with respect to the stationary target as sketched in Figure 18.14. Here is the clutter bandwidth and is the illumination time.

The Doppler slope change is equivalent to a change of the quadratic part of the range history. After azimuth compression using the SWMF with Doppler slope the mismatch corresponds to a quadratic phase error in time domain. The IRF of the moving target is therefore defocused in the azimuth direction as shown in Figure 18.15. Unfortunately no analytical description of the defocused IRF exists.

The spread of the blur in azimuth direction can be approximated as

(18.51)

If the target only has an along-track velocity component (all other motion parameters are zero) for a system with (i.e., a spaceborne system) the simpler equation

(18.52)

can be used [27]. This equation shows that the moving target is smeared by twice the distance it has moved in the along-track direction during the illumination time . The backscattered signal energy is distributed over a larger area. With increasing the signal amplitude decreases as shown in Figure 18.15 left. A decreased signal amplitude leads to a decreased SCNR and a lower probability of detection. The target’s IRF additionally may be blurred in range due to the residual range cell migration.

For targets with (i.e., zero across-track velocity component) the spread of the blur in range direction can be computed using (18.50) and the relation (non-squinted acquisition geometry assumed so that and ):

(18.53)

After inserting the Doppler parameters given in (18.31) and (18.49) as result

(18.54)

is obtained. Especially for airborne systems the range blur may become significant. Let us assume for instance a system with and an extremely long illumination time . In this case the range blur of the IRF of a target moving in along-track direction with 100 km/h (all other motion parameters are assumed to be zero) is 21 m. However, for spaceborne systems with and the range blur given in (18.53) and (18.54) can be neglected, especially under the aspect that the typical illumination time is in the order of one second.

The relation between the range blur, the residual range cell migration and the motion parameters clearly can be recognized by looking again at Figure 18.12. Especially the across-track acceleration and the along-track velocity are the dominant motion parameters responsible for the quadratic phase errors (i.e., the mismatch with the Doppler slope of the SWMF) and, hence, for the residual range cell migration and the azimuth and range blur.

2.18.4.3 Across-track velocity

The major effect caused by the across-track velocity is a change of the Doppler shift given in (18.31). A secondary effect is a slight change of the Doppler slope . In Figure 18.16 the Doppler history of a target moving in across-track direction is sketched. After SAR processing using the SWMF all targets, independent if stationary or moving, are imaged at the positions corresponding to their zero Doppler frequencies. For the moving target signal in Figure 18.16 this may either be the position marked with or corresponds to the position of an ambiguity that is caused by an aliasing of the signal sampled by the PRF).

The azimuth time where the target is imaged can be computed by setting (18.31) to zero and substituting with for taking into account the Doppler slope of the SWMF. If only the quadratic approximation is used (the cubic coefficient q in (18.31) has been dropped since its contribution compared to the Doppler shift and Doppler slope is negligibly small) the following imaging time is obtained:

(18.55)

The imaging time corresponds to an along-track or azimuth displacement of

(18.56)

For a non-squinted acquisition geometry (i.e., ) the equation simplifies to

(18.57)

where , denoted as line-of-sight velocity, is the projection of the across-track velocity to the slant range direction. The relationship between the line-of-sight and the across-track velocity for a non-squinted geometry is given as

(18.58)

with being the incidence angle.

The target is displaced in the flying direction (i.e., ) if it moves towards the radar (i.e., ). It is displaced in opposite direction (i.e., ) if it moves away from the radar (i.e., ). In Figure 18.17 an IRF of a simulated point target moving with a constant velocity of 50 km/h in across-track direction is shown. It has a large azimuth displacement of about −978 m. Additionally the IRF is slightly shifted by −0.66 m in range direction (right image).

The reason for the range displacement is the residual range cell migration. It can be computed with (18.50) by taking into account the fact that the major part of moving target’s signal energy is located around :

(18.59)

For a non-squinted acquisition geometry (i.e., and ) the simplified expression

(18.60)

can be used. The range displacement is always negative. Thus, a target with a non-zero across-track velocity component is always displaced towards the radar. The range displacements are small if compared to the azimuth displacements, especially for spaceborne systems. For instance, a fast target moving with a line-of-sight velocity of 100 km/h is displaced by only −5 m (typical low-earth orbit platform parameters and assumed). For airborne systems the displacement is larger. Here the same target is displaced by −143 m ( and assumed).

As shown in Figure 18.17 the target’s IRF is severely displaced in azimuth but it is well focused. The slightly decreased peak amplitude is caused by the reduced spectral overlap of the SWMF with the moving target signal. The SWMF acts as a bandpass filter. Its Doppler bandwidth is conventionally limited to the clutter bandwidth given in (18.3). The reason for that is that the clutter for SAR imaging is the wanted signal. Everything outside the clutter bandwidth is uninteresting and therefore is filtered out. The extreme case is shown on the right side of Figure 18.16. Here no spectral overlap between the moving target signal shown in red and the bandwidth of the SWMF exists. As a consequence, fast moving targets are often not visible in conventional processed SAR images. To enable imaging and detection of fast moving targets the processing bandwidth of the SWMF has to be increased to the maximum possible bandwidth determined by the PRF. This is an important point to remember.

Due to the Doppler shift a part of the signal energy may be backfolded (i.e., aliased) so that ambiguities of the IRF appear at certain positions in the SAR image. On the left side of Figure 18.16, the primary IRF containing most of the signal energy is imaged at position whereas the ambiguity is imaged at . To compute also the ambiguous target positions (18.56) can be extended to

(18.61)

An example of an ambiguous signal is shown in Figure 18.18.

To keep the ambiguities at a low level commonly a high PRF should be chosen. To ensure that at least half of the signal energy of a point target lies within the PRF band its Doppler shift has to be smaller than . Using this requirement with (18.31) and again assuming for simplicity a non-squinted geometry (i.e., ) the upper bound for the across-track velocity is

(18.62)

This bound ensures that at least half of the signal energy is unambiguously available. The equation can be considered as a criterion for selecting the minimum required PRF of a SAR-GMTI system depending on the highest “expected” target across-track velocity:

(18.63)

For targets with (i.e., for targets moving with a certain across-track velocity) the spread of the range blur can be approximated as ( and assumed)

(18.64)

With the inserted Doppler parameters given in (18.31) this results in

(18.65)

If for instance an airborne system with , and would be used the IRF of a target moving in across-track direction with 50 km/h (all other motion parameters are assumed to be zero) has a range blur of 0.05 m. For state-of-the-art airborne SAR systems this is below the achievable range resolution and therefore can be neglected. However, if aside of the across-track velocity of 50 km/h also an along-track velocity component of 100 km/h is considered, the range blur increases to 52 m.

2.18.4.4 Along-track acceleration

The major effect of the along-track acceleration is a change of the quadratic coefficient q in (18.31) which causes a deflection of the Doppler history as sketched in Figure 18.19.

After azimuth compression the IRF of the moving target has a decreased peak amplitude and shows non-symmetric (unbalanced) sidelobes. The strength of this effect depends mainly on the synthetic aperture time . The longer this time the larger are the third-order phase errors in time domain and the more severe the effect. An example for an airborne system is shown in Figure 18.20.

For spaceborne SAR systems with typical illumination times below one second the effects caused by the along-track acceleration are negligibly small.

2.18.4.5 Across-track acceleration

The across-track acceleration causes a similar major effect as the along-track velocity : a change of the Doppler slope given in (18.31) (see also Figure 18.14). This results again in a blurred IRF with decreased peak amplitude. The IRF looks similar as the response shown in Figure 18.15 and is therefore not separately depicted. The azimuth blur caused by the across-track acceleration can be derived with (18.51). If all other target motion parameters are zero it is given as

(18.66)

Even an allegedly small acceleration my cause a azimuth blur in the order of several meters. For an airborne system with , and an acceleration of results in an azimuth blur of approximately 15 m. For a spaceborne system with , and the azimuth blur is 4.5 m. Neither for airborne nor for high resolution spaceborne SAR systems this effect should be neglected. By measuring the spread of the azimuth blur of the moving target’s IRF the effects caused by across-track accelerations and along-track velocities cannot be separated.

Additionally to the azimuth blur the IRF is also blurred in range. The range blur caused by the across-track acceleration can be computed with (18.54). Using the same parameters for an airborne and spaceborne system as before, the range blurs caused by an acceleration of are 0.6 m and 7 mm. Thus, at least for the spaceborne system, the range blur can be neglected.

2.18.4.6 Summary of effects

The effects treated so far caused by moving targets on SAR imagery are summarized in Table 18.1. The knowledge of these effects and their origins are fundamental for developing suitable GMTI systems and algorithms [9].

2.18.5.1 Along-track interferometry

For along-track interferometry (ATI) two receiving antennas displaced in azimuth direction by a certain baseline are necessary. The receiving antennas can either be mounted on the same platform or on separate platforms flying in formation along the same track (separate platforms allow for larger baselines). Anyhow, each antenna observes the scene from the same point in space at slightly different times and . During the time lag the signals from stationary targets remain the same whereas the signals from targets moving in range direction experience a phase shift . This phase shift is called ATI phase. It is proportional to the line-of-sight velocity of the moving target. The time lag has to be sufficiently short to avoid the effect of temporal decorrelation caused by slight changes in the scene (e.g., due to wind) between the acquisitions of an interferometric image pair. The first application of ATI was oceanography where it was used to measure tidal currents with a velocity estimation accuracy in the order of several cm/s [28].

The ATI principle is sketched in Figure 18.21. At time the target is observed by the fore antenna at range . At time it is observed by the aft antenna at a different range . The range difference (t) is proportional to the ATI phase. The ATI signal is computed by multiplying the signal of the fore antenna with the complex conjugate and co-registered signal of the aft antenna (remember the co-registered multi-channel signals given in (18.38):

(18.67)

If both channels are well calibrated and the RCS of the target does not change between both observations the complex coefficients and are identical (i.e., same phases and amplitudes), so that the ATI phase may be computed as:

(18.68)

The phases of both signals are given as

(18.69)

with where is the antenna phase center separation in along-track direction (= effective along-track baseline). If both antennas receive and transmit independently (= monostatic operation) the phase center separation corresponds to the physical antenna separation, i.e., . In case of bistatic operation where only the fore antenna is used for transmission but both antennas for signal reception the effective along-track baseline reduces to half the physical distance, i.e., .

By inserting (18.69) in (18.68) the ATI phase may be written as

(18.70)

If the target moves with constant line-of-sight velocity the ATI phase of the target at broadside position for monostatic operation in the simplest case can be approximated as (the squint angle and all other motion parameters of the target are assumed to be zero) [28]

(18.71)

If the line-of-sight velocity is computed from the measured ATI phase blind velocities and ambiguities may occur. The reason is that the ATI phase can only be measured in fractions of 2. A blind velocity occurs if the measured ATI phase becomes zero. This is the case for

(18.72)

The blind velocities can be computed by inserting (18.71) in the previous equation:

(18.73)

If the target moves with a blind velocity its ATI phase cannot be discriminated from the ATI phase of a stationary target and, hence, it cannot be detected in the interferogram.

An ambiguity occurs if the ATI phase is equal or larger than . In this case neither the motion direction (away or towards the radar) nor the velocity can be determined unambiguously. The maximum line-of-sight velocity causing no ambiguities is given as (again derived with (18.71))

(18.74)

The maximum line-of-sight velocity can be increased by using larger radar wavelengths or smaller time lags (i.e., smaller baselines).

In Figure 18.22 a polar plot of an azimuth line of interest containing a moving target signal is shown. The azimuth line was focused with the SWMF before the polar plot was generated. Several dots in the polar plot belong to the target since it spans multiple resolution cells. It also can be seen that the target dots have slightly different phases. The reason is that the target is not perfectly focused (remember the potential mismatch of the Doppler slope of the target and the SWMF). Furthermore, since ATI performs no clutter suppression also the clutter contribution disturbing the ATI phase clearly can be recognized. As a consequence the ATI phases of moving target signals are biased towards zero so that velocities are generally underestimated. For low SCNR values this bias is more significant.

Even if there is no temporal decorrelation and if the clutter would be suppressed somehow prior to ATI phase computation, the minimum detectable velocity (MDV) and the accuracy of the line-of-sight velocity estimate is limited by the thermal receiver noise. The receiver noise in both RX channels causes a phase decorrelation which results in a noisy estimate of the ATI phase. The amplitude of the complex correlation coefficient due to thermal receiver noise between both channels is given as [29]

(18.75)

where SNR is the signal-to-noise ratio after SAR focusing (i.e., after pulse compression). The standard deviation of the ATI phase for point-like scatterers for values of close to one can be approximated as [30,31]

(18.76)

This equation can be considered as a lower bound of the ATI phase standard deviation. For extended objects (i.e., targets larger than a single SAR resolution cell) the standard deviation is larger but it can be reduced to a certain degree by ATI phase averaging (=multi-looking).

The lower bound of the standard deviation of the line-of-sight velocity computed with (18.71) and (18.75) can be expressed as

(18.77)

The standard deviation of the azimuth position computed using (18.56) is then

(18.78)

By having a closer look at (18.77) and (18.78) it is obvious that the standard deviation can be decreased by increasing the time lag and, hence, the antenna phase center separation . However, it has to be kept in mind that by increasing the maximum unambiguously detectable line-of-sight velocity in (18.74) and the blind velocities in (18.73) decrease. Techniques for phase unwrapping are necessary if the motion and position parameters of faster moving targets shall be estimated with high accuracy. Promising candidates are for instance multi-baseline ATI techniques [32,33]. Ambiguities might also be resolved if the range cell migration of the target is exploited for a rough line-of-sight velocity estimation which afterwards is refined with the information obtained from the ATI phase [21]. For large time lags, temporal decorrelation may also have a negative influence, but this aspect is not treated here.

Figure 18.23 shows the standard deviations of the line-of-sight velocity and the azimuth position for exemplary airborne and spaceborne acquisition geometries. It is obvious that with airborne systems even with small baselines in the fraction of a meter good estimation results can be achieved. For spaceborne systems comparatively large baselines are required to push down the standard deviations and, hence, to improve the parameter estimation accuracy. However, the size and weight of SAR satellites are limited by launcher restrictions. For instance, the German TerraSAR-X satellite has an antenna length of 4.8 m which can be split into two RX sub-apertures of 2.4 m each. In bistatic operation (i.e., TX with full antenna and RX with both halves simultaneously) the phase center separation or the effective along-track baseline, respectively, is only 1.2 m. For high SNR values of 20 dB the azimuth position estimation accuracy is in the order of 150 m. The accuracy decreases to 490 m if the SNR decreases to 10 dB. Compared to airborne systems, state-of-the-art spaceborne systems generally achieve worse position and motion parameter estimation accuracies. The two main reasons are the too short baselines and the worse SNR values.

For an in-depth ATI performance analysis also the clutter contribution has to be considered [34, 35]. However, for understanding the basic ATI principles and performance limitations the previous explanations are sufficient.

In reality a moving target seldom has only a line-of-sight velocity component with all other motion components being zero as assumed in the derivation of (18.71). Although ATI is primarily sensitive to the line-of-sight velocity also the other motion components contribute to the ATI phase. Unfortunately there exists no analytical description for the ATI signal focused with a SWMF. However, if the Doppler slope of the matched filter used for azimuth compression is adapted to the Doppler slope of the target an analytical solution exists (matched filters adapted to target motion’s parameters are discussed in Section 2.18.7.1). In case of a non-squinted geometry (i.e., and ) the ATI phase of the peak of the refocused target signal is given as [23]

(18.79)

The ATI phase depends not only on the across-track velocity but also on the along-track velocity . This has to be taken into account for accurate parameter estimation.

2.18.5.2 Displaced phase center antenna technique

The displaced phase center antenna (DPCA) technique is one of the simplest clutter suppression techniques in radar [36]. The antenna configuration is exactly the same as in ATI: at least two receiving antennas are needed (see also Figure 18.21). The only difference is the signal processing. Instead of a complex conjugate multiplication the co-registered signal received by the aft antenna is subtracted from the signal received by the fore antenna for obtaining the DPCA signal

(18.80)

Signals from stationary targets are canceled since they are identical in both successive observations. Targets moving with sufficient line-of-sight velocity cause a certain phase shift and, thus, will not be canceled. As a consequence even slowly moving targets otherwise masked by the clutter can be detected in the “clutter-suppressed” DPCA image. If the clutter is homogeneous DPCA can be considered as the optimal linear filter for target detection [37]. Although DPCA originally was developed for dual-channel systems, it also can be used with more than two channels [38–40].

In older literature it is often stated that the so called “DPCA Condition” has to be fulfilled. That means that the PRF has to be selected in a way that the spatial location of the fore antenna phase center when an echo is received is the same as the location of the aft antenna phase center when the next echo is received:

(18.81)

Nowadays this restriction is relaxed. By modern signal processing techniques co-registration can be performed with sub-sample accuracy using interpolation or resampling, respectively. Often the co-registration is done by applying a phase ramp on the signal after transforming it via an FFT to the Doppler domain (remember the displacement law of the Fourier transform). The phase ramp to be applied in Doppler domain for co-registration along azimuth generally is given as

(18.82)

This leads to accurate co-registration results as long as the Nyquist sample theorem is fulfilled for both clutter and moving target signals. In other words, a sufficiently high PRF is required for avoiding aliasing of signals in Doppler. However, if the PRF is too low the aliased clutter signals after co-registration have a constant phase error [18]. A coarse co-registration without interpolation can be done by shifting the data by an integer amount of samples. The time in (18.82) for fine co-registration with sub-sample accuracy can then be decreased to ). The constant phase error derived in [18] in this case can be written as

(18.83)

Due to this constant phase error aliased clutter might be mistaken as false moving targets. Especially for spaceborne systems the Doppler bandwidth is large compared to the PRF so that Doppler aliasing cannot be excluded. It generally is more severe than in airborne systems. Therefore, for spaceborne systems it is advisable to meet the DPCA condition in (18.81) as accurate as possible so that no phase ramp in Doppler for fine-co-registration needs to be applied. In this case is zero and the phase error in (18.83) vanishes.

For achieving a good detection performance it is essential that the receiving channels are well calibrated. For calibration the “Adaptive 2D Channel Balancing” method originally proposed by Ender [7] has established itself in the GMTI community. The method operates in the 2D frequency domain. The channel transfer functions are adapted to a reference channel, conventionally the fore channel. The method is not limited to two channels. Also co-registration is performed between the channels since any phase ramps in frequency domain are removed [41]. In Figure 18.24 an example with three moving targets is shown. It can be seen that after channel balancing (right column) the SCNR as well as the correlation coefficient is increased. Even targets with lower RCS are clearly visible in the balanced DPCA image. Theoretically the clutter can be suppressed down to noise level. The application of a channel balancing method is absolutely crucial, especially for airborne GMTI algorithms based on classical ATI and DPCA.

If the channels are well calibrated so that , the DPCA signal can also be expressed as

(18.84)

where is the phase of the complex coefficient . For detecting a moving target the magnitude of the DPCA signal is of interest:

(18.85)

If the target moves with constant line-of-sight velocity (all other motion parameters are zero) the DPCA magnitude can be approximated as

(18.86)

It is obvious that the DPCA magnitude drops to zero if the sine becomes zero. This is the case if the target moves with a blind velocity given in (18.73). Thus, neither with ATI nor with DPCA targets moving with blind velocities can be detected.

2.18.6.1 Preprocessing

As input for a SAR-GMTI processor single- or multi-channel raw data acquired with air- or spaceborne sensors are used. These data need to be preprocessed (step 2 in Figure 18.25) before the “GMTI Kernel” can be applied. Depending on the particular GMTI algorithm to be used in the “GMTI Kernel” several preprocessing options are conceivable (no claim to completeness):

2.18.6.2 Clutter suppression

Most of the multi-channel GMTI algorithms found in the literature use either DPCA or STAP techniques for clutter suppression.

2.18.6.3 Detection

Conventionally moving targets are detected on a pixel by pixel basis. A threshold is set to discriminate between one of the two hypotheses (moving target signal not present) and (moving target signal present):

(18.93)

where denotes the vector of the received data, is the clutter vector and the noise vector.

Generally a so called “Constant False Alarm Rate (CFAR)” detector is envisaged. The threshold is computed in a way that the percentage of the image pixels which lie above the threshold is constant. For computation the clutter statistics have to be known precisely.

If the probability density function (PDF) of the clutter is known, the false alarm rate or the probability of a false alarm is obtained by integrating the PDF:

(18.94)

where is the threshold and the PDF of the clutter metric to be tested (e.g., the clutter DPCA amplitude, the clutter ATI phase, the clutter ATI phase combined with the clutter ATI amplitude, etc.). If a certain false alarm rate is desired this equation needs to be solved for the threshold . Depending on an analytical solution is not always possible so that numerical methods may be necessary.

The probability of detection can be expressed as

(18.95)

where is the target plus interference PDF.

Analytical descriptions of the PDFs of the clutter multi-look ATI phase and ATI amplitude can be found in [34]. In that paper different clutter types are modeled and verified with real data. A discrimination between homogeneous (Gaussian), heterogeneous (non-Gaussian) and extremely heterogeneous clutter is made. A novel polynomial PDF called p-distribution is introduced. This PDF matches the real data much more accurately, particularly for heterogeneous composite terrain. All clutter parameters for determining the detection thresholds are estimated from the real data. Further detection metrics are discussed in [45]. Here also a so called “Hyperbolic Detector” well suited for heterogeneous terrain such as urban areas is introduced.

For estimating the clutter statistics from the real data the moving target itself has to be excluded. This can be achieved by introducing a guard zone around the pixel under test as depicted in Figure 18.27. The clutter statistics is then estimated from the data surrounding the guard zone. The purpose of the guard zone is to exclude that moving target signal components disturb the clutter and, hence, lead to a wrong clutter PDF estimate. The size of the guard zone shall be chosen in accordance with the expected sizes of the moving vehicles to be detected. If the guard zone is too small or if a lot of targets move close together, the clutter PDF estimate is biased and the false alarm rate will not remain constant anymore.

For studying and comparing the performance of different detector types as quality measures the probability of detection versus varying SCR and versus can be used [46]. The latter quality measure is known as receiver operating characteristics (ROC) of a detector. Since the exact PDF of the target signal is generally unknown, it is conventionally assumed to be either deterministic or Gaussian distributed.

2.18.6.4 Signal extraction

Once a moving target has been detected it is of interest to estimate its motion and position parameters. For that purpose the moving target signal may be extracted from the data. An extraction also may be required if for instance the target image shall be refocused to high resolution using ISAR imaging techniques [15–17]. In the following two practicable extraction methods are discussed.

2.18.6.5 Parameter estimation

In (18.31) it is shown that the target’s motion parameters are related to the Doppler parameters shift , slope , and quadratic coefficient q. Thus, motion parameter estimation can be reduced to Doppler parameter estimation. From the known Doppler parameters the position and motion parameters of the moving target can be computed. The intention of this section is to show how this computation is principally performed by different state-of-the-art GMTI algorithms. In the following it is supposed that the Doppler slope and shift already have been estimated. Basic Doppler parameter estimation methods are discussed afterwards in Section 2.18.7.

It is also supposed that the radar parameters and the acquisition geometry are known accurately so that the parameters , and need not to be estimated. This is a valid assumption, particularly if the range is large compared to a potential range displacement (i.e., . Additionally a non-squinted acquisition geometry is assumed (i.e., and ).

2.18.6.6 Visualization

Finally, the detected target can be visualized. Often a symbol representing the moving target is overlaid on the corresponding SAR image on its estimated position. For instance, the symbol may be an arrow pointing into the estimated moving direction and the color of this symbol may correspond to the velocity. An example for such a visualization is shown in Figure 18.32.

Another elegant possibility for visualization is Google Earth. The parameters of the detected moving targets are written into a Keyhole Markup Language (KML) file. Also links to target symbols and SAR images can be incorporated easily. The KML file can then be visualized with Google Earth as shown in Figure 18.33.

Furthermore, additional information about the detected target can be provided in an interactive way. The user or operator can click on the moving target symbols to open a window. This window may contain more information about the detected target. Also data from additional data sources can be incorporated and fused with the SAR data. For instance, the estimated parameters of detected ships could be fused with “ground truth” data obtained from the automatic identification system (AIS) [54]. An example is depicted in Figure 18.34. Such ground truth data can also be used for evaluating the parameter estimation accuracy of GMTI algorithms.

2.18.7.1 Matched filter bank

The basic principle of a matched filter was discussed in Section 2.18.2.3. It was shown that a signal can be focused by performing a convolution with a proper reference function. Also a moving target signal can perfectly be focused if the proper reference function is applied [21]. The proper azimuth reference function for focusing the moving target signal is given as

(18.108)

However, the Doppler parameters , and q are unknown in advance. They need to be estimated. This can be done by convolving the range compressed moving target signal successively with different azimuth reference functions. For each of these functions different Doppler parameter are assumed. The estimated Doppler parameters , and are obtained from the reference function that after convolution results in the highest peak value of the IRF. The successive convolution with different reference functions is called “Matched Filter Bank” and can be considered as a maximum likelihood estimator for the Doppler or motion parameters [56].

Principally the moving target signal has to be extracted from the range compressed data array before the matched filter bank can be applied [47]. For this task the range history tracking method presented in Section 2.18.6.4.1 can be used for example. However, as a brute force method also two-dimensional matched filters can be constructed taking additionally into account the range history of the target signal. Each (pre-detected) pixel of the range compressed data array has then to be convolved with a set of two-dimensional matched filters. One can conceive that this method has a high computational load and is rather time consuming. However, the method may be of importance for spaceborne systems which suffer from low SNR values. The advantages are the higher compression gain and the possibility to estimate the parameters from fast moving targets aliased in Doppler [57]. The minimum PRF requirement given in (18.62) is not applicable in this case. A lower PRF resulting in less range ambiguities can be used. Additionally, the swath width (cf. Figure 18.3) of the SAR system can be increased [1] by lowering the PRF.

The moving target signal can also be extracted “by chance” using the method discussed in Section 2.18.6.4.2. With this method successive azimuth lines are extracted. The disadvantage is that no direct information about the range history is preserved. In this case the matched filter bank is not very sensitive to the Doppler shift . However, the Doppler shift or equivalently the across-track velocity can be estimated using ATI. If additionally the quadratic coefficient q is neglected (it only has a significant contribution at large observation times) a much simpler azimuth reference function can be used:

(18.109)

The equivalent in Doppler domain is given as

(18.110)

It has to be kept in mind that with this reference function only the Doppler slope of the moving target signal can be estimated. After focusing with this reference function the target still appears displaced from its actual position.

An example of the application of a matched filter bank on real dual-channel airborne data is shown in Figure 18.35. On the left side a SAR image containing four targets moving on a runway with different along-track velocities is shown. Target 2 moved with 10 km/h. The azimuth line containing “Target 2” was extracted from the clutter suppressed DPCA image and the azimuth compression was removed. Afterwards the matched filter bank was applied. The matched filter bank output, denoted as “Matched Filter Map,” is shown on the right, once in its two-dimensional representation and once in its 3D representation. The target is represented by the focused peak. The peak position corresponds to a certain estimated Doppler slope and to a certain azimuth position. In the example the target moved only in along-track direction ( and ) so that with (18.100) the along-track velocity can be computed easily (the negative sign in the image is due to the antiparallel motion with respect to the flight path).

As already explained, the application of the matched filter bank is not limited to azimuth lines containing only one moving target signal. In fact a multi-target scenario can be resolved as shown in the example depicted in Figure 18.36. Here an azimuth line of interest containing a part of the Autobahn A8 near Chiemsee, Germany, where several targets have moved was extracted. In the SAR image on the left the targets are severely blurred in azimuth and cannot be recognized and separated. However, in the matched filter map of the DPCA signal each target appears as a focused peak at a certain azimuth and Doppler slope position. Thus, a matched filter bank not only can be used for parameter estimation, but also for target detection and target separation. It can deal with multi-component LFM signals. Target detection in the matched filer map either can be performed by applying a certain amplitude threshold or by comparing the sharpness of the peaks with analytical sharpness functions [27].

2.18.7.2 Fractional Fourier transform

The fractional Fourier transform (FrFT) is a linear operator (no cross-terms in case of multi-component LFM signals) and can be considered as a generalization (or rotation with rotation angle of the conventional Fourier transform [59]. The FrFT has found many applications like swept-frequency filters, time-variant filtering and multiplexing, pattern recognition and the study of time-frequency distributions. It is also known, that the Radon transform [60] of the Wigner spectrum is equal to the magnitude square of the fractional Fourier transform. Filtering in the fractional Fourier domain, rather than in the ordinary Fourier domain, allows one to decrease the mean square error in the estimate of a distorted and noisy signal [61].

The principle of the FrFT is sketched in Figure 18.37. After application of the FrFT the signal energy in the fractional Fourier domain is integrated along the fractional time axis . For LFM signals the application of the FrFT with the optimum rotation angle (in this case the fractional time axis is parallel to time-frequency history of the LFM signal) results in a sharp spectrum with maximum peak amplitude. Due to this behavior, which for LFM signals is equivalent to pulse compression and matched filtering, the FrFT is also attractive for SAR-GMTI [62].

The application of the FrFT for estimating the Doppler slope is quite similar to the matched filter bank approach. Instead of applying different matched filters for focusing now the FrFT is applied successively on the extracted range compressed data whereby each time a different rotation angle is used. The rotation angle corresponds to the Doppler slope in the following way [35]:

(18.111)

where the ratio between the number of samples N and the PRF is used for normalization purposes. The estimate of the Doppler slope is then computed with this equation using the optimum rotation angle which maximizes the peak amplitude of the spectrum.

As with the matched filter bank also the FrFT can cope with multi-component LFM signals as shown in Figure 18.38. On the top the “Fractional Spectra Map” is depicted. This map is obtained by successive application of the fractional Fourier transform with different rotation angles. At the bottom for each of the three moving target signals the spectrum obtained by using the optimum rotation angle is shown.

In contrast to the matched filter bank the application of the FrFT has higher computational cost but also some advantages. For instance, bandpass filtering can be applied directly in the fractional Fourier domain as depicted in Figure 18.39. By this orthogonal interferences can be filtered out and multi-component LFM signals can be separated from each other and extracted from the data. After applying the inverse FrFT on the bandpass filtered signal using - as rotation angle, the clutter and noise suppressed moving target signal is obtained in the time domain.

2.18.7.3 Time-frequency analysis

One application of time-frequency (TF) analysis in the field of GMTI is the estimation of the instantaneous Doppler frequency of the moving target signal. From the estimated instantaneous Doppler frequency (cf. (18.18)) the phase history of the range compressed moving target signal can be computed by integration [63]:

(18.112)

where is an unimportant constant phase term. With the estimated phase a proper reference function can be constructed. With this reference function even targets in non-linear motion can be properly focused. Thus, the application of the reference function can already be considered as a simplified ISAR imaging method.

Before the phase history can be computed with (18.112) it is necessary to estimate the Doppler frequency . For this purpose a suitable TF transform has to be applied on the range compressed moving target signal. Probably the best known linear TF transform is the short-time Fourier transform. As disadvantage a trade-off between time and frequency resolution has to be made: either a small time or a small frequency resolution can be obtained. It is impossible to get both simultaneously. In contrast the Wigner-Ville distribution (WVD) is the TF distribution having the best time-frequency resolution [64,65]. However, since it is a quadratic TF transform cross-terms and interferences may occur if the WVD is applied on multi-component LFM signals. To avoid these issues or to keep at least the negative influences at a low level, clutter suppression and signal separation has to be performed prior to its use, e.g., by applying DPCA and range history tracking. In [17] it is shown how the Pseudo Wigner-Ville Distribution (PWVD) can be used for Doppler parameter estimation and ISAR imaging.

In Figure 18.40 an example is shown where the PWVD is used for estimating the Doppler slope of the moving target signal. The PWVD was applied on the same data set used in Section 2.18.7.1 for the explanation of the matched filter bank principle.

2.18.7.4 Radon transform

Also the Radon transform [60] applied on the TF representation (i.e., the Wigner-Ville spectrum) of the moving target signal can be used for target detection and Doppler slope estimation [66]. An example is shown in Figure 18.41. However, nowadays algorithms relying on the Radon transform of the Wigner-Ville spectrum are replaced increasingly by algorithms using the fractional Fourier transform.

2.18.8.1 Joint domain STAP

Originally STAP was designed to operate in the time domain (note that the data must not be co-registered). STAP processing operates on the radar data cube depicted on the right in Figure 18.43. The data cube is convenient for visualizing subsequent space-time processing although the radar processer does not store the data in the format shown in the figure [68]. Each data cube corresponds to a single coherent processing interval (CPI).

The page of the data cube corresponding to the kth range cell is

(18.113)

where M is the number of RX channels and N is the number of temporal samples per RX channel. The matrix in previous equation can be vectorized by stacking each succeeding column one beneath the other. This yields the space-time snapshot for the kth range cell, i.e.,

(18.114)

where the elements are abbreviated as . The space-time snapshot in general is composed of

(18.115)

where s denotes the moving target signal, c the clutter and n the uncorrelated component due to thermal receiver noise or sky noise. The multi-channel signal for STAP often is modeled as [8]

(18.116)

where is a complex amplitude describing the reflectivity of the scatterer, denotes the range to the antenna array center, and are the transmit and receive antenna characteristics of the mth channel, and denotes the antenna phase center positions in azimuth direction with respect to the array origin, and is the directional cosine. It is assumed that the antenna array center origin is freely chosen at the center of the array, so that . The common phase multiplier represents the conventional azimuth chirp used for classical SAR imaging via azimuth compression or matched filtering.

The space-time processor linearly combines the elements of the space-time snapshot by applying a weight vector. As result at the output of the space-time processor a scalar is obtained

(18.117)

where ^H denotes conjugate transposition and w is the complex weight vector of dimension . The optimal weight vector maximizes the output SCNR and takes, under the assumption of homogeneous Gaussian clutter, the form

(18.118)

where is the interference covariance matrix of dimension and is the steering vector of dimension .

In practice both and d are unknown and need to be estimated. Thus, instead of (18.118) an estimate of the weight vector in the form of

(18.119)

is applied, where is an estimate of and v is a surrogate for d. This approach is known as sample matrix inversion (SMI). It is common to compute the covariance matrix estimate as

(18.120)

where of dimension are known as secondary training data. Conventionally K data ranges are used for training (i.e., the averaging is performed along range). To avoid target self-whitening the cell under test as well as cells where already a target has been detected should be excluded from the data.

2.18.8.2 Post-doppler STAP

Since the classical STAP is computationally inefficient and additionally requires a large number of training cells, STAP algorithms needing lower computational power and less training cells have been developed.

With a linear transformation the space-time snapshot z can be projected into a lower dimensional subspace (= Reduced-Dimension STAP) [68]:

(18.121)

where T is the transformation matrix, has dimension with . The transformation matrix is independent of the data. In contrast to the joint-domain STAP the computational burden for matrix inversion drops from to . The transformation of the steering vector and the optimal weight vector are given as and , respectively. The estimate of the covariance matrix again can be computed in the same manner as before, but now using the transformed data instead of z. The adaptive weight vector using the estimated covariance matrix and a surrogate of the steering vector is given as .

A practically implementable and efficient Reduced-Dimension STAP algorithm is Post-Doppler STAP [4]. The space-time snapshot given in (18.114) is transformed to Doppler domain before STAP processing takes place. For the period where target and clutter remain in the same range-Doppler resolution cell, the measured space-time snapshot can be expressed by the random vector [4,8]

(18.122)

where a is a complex constant comprising the target’s RCS among others and is the clutter-plus-noise interference.

The optimum detection for one range-Doppler resolution cell, under the assumption of homogeneous Gaussian clutter, is achieved by comparing [4]

(18.123)

to a threshold. Clutter suppression is performed by multiplying each Doppler frequency bin of the signal vector z (i.e., the space-time snapshot transformed to Doppler domain) with the inverse of the clutter-plus-noise covariance matrix . A target match (i.e., matched filtering) is performed afterwards by multiplying the intermediate result with the complex conjugated and transposed steering vector (i.e., the expected moving target signal). With this matched filtering operation the Doppler shift (which is proportional to the line-of-sight velocity , cf. (18.31)) and the direction-of-arrival angle of the moving target signal (from which the true azimuth position can be computed).

The clutter covariance matrix for each Doppler frequency bin can be estimated from the data by performing averaging in range:

(18.124)

The resulting detection performance in the optimum case is directly proportional to the remaining output SCNR of the optimum filter given as

(18.125)

It can be used for analyzing the detection performance of a given system design [4,8]. Maximizing the SCNR results in a maximized probability of detection for a fixed false alarm rate .

A visualization example of the SCNR is given in Figure 18.44. Here and are substituted by the direction-of-arrival angle and the Doppler shift .

Under assumption of homogeneous Gaussian clutter and non-fluctuating target RCS (Swerling-0 case) the probability of detection can be computed analytically as [68]

(18.126)

where is the modified zero-order Bessel function of the first kind and is given as

(18.127)

The normalized detection threshold calculates to

(18.128)

where is the desired false alarm rate.

The probability of detection for a given false alarm rate can either be plotted as a function of SCNR, or, more relevant in the field of GMTI, as a function of line-of-sight velocity. An example is shown in Figure 18.45. The minimum detectable velocity (MDV) can directly be read off. For achieving a probability of detection of 0.9 for given SNR and CNR values of 20 dB, the MDVs are approx. 1 m/s (3.6 km/h) and 1.5 m/s (5.4 km/h) for false alarm rates of and , respectively.

Most STAP techniques where developed to be used with multi-channel airborne systems. Airborne STAP techniques have in contrast to spaceborne techniques the following advantages:

The successful application of STAP techniques on multi-channel spaceborne systems is more challenging. Since spaceborne systems suffer from low SNR significantly more temporal samples of the moving target signal need to be integrated coherently for ending up with a sufficient target detection performance. Thus, especially for systems with high range resolution, the range cell migration of the target signal has to be taken into account (cf. Section 2.18.4.1) and a matched filter bank has to be applied (cf. Section 2.18.7.1). Promising novel techniques are the ISTAP (imaging STAP) and EDPCA (extended DPCA) techniques introduced recently in [39,40].

2.18.8.3 EDPCA

The Extended Displaced Phase Center Antenna Technique (EDPCA) is an extension of the DPCA and ATI methods to three or more channels [39,40]. The flow chart of the algorithm is shown in Figure 18.46. The SAR compression filter is matched to the moving target parameters for maximizing target’s SNR. For this task, the application of a bank of SAR processing filters is necessary. That means that for each of the M complex SAR images several times an adapted range cell migration correction and azimuth compression has to be performed. The number of necessary iterations depends on the range resolution and the accepted loss in SCNR compared to the optimum case.

The clutter cancellation filter is either derived from the estimated clutter-plus-noise covariance matrix or pre-computed using the known system, instrument and geometry parameters. EDPCA is partially adaptive and can be used with an arbitrary number of RX channels.

The empirical clutter-plus-noise covariance matrix is estimated by averaging the measured data vector over range cells and azimuth cells (note that EDPCA operates in contrast to Post-Doppler STAP on fully compressed data) [69]

(18.129)

where in the parameter vector all moving target parameters (i.e., along-track, across-track or line-of-sight velocity) used for adapted SAR are condensed. The clutter-plus-noise covariance matrix is only of dimension ( number of RX channels) and is thus invertible with low computational power ().

For achieving a good performance the clutter-plus-noise covariance matrix should be estimated for each image pixel under test. A large number of range and azimuth cells should be used for averaging. The pixel under test and a guard zone should be excluded from training. Already detected targets should also be excluded.

The normalized test statistics is given as [69]

(18.130)

where is the steering vector and is a CFAR threshold.

What is not shown in the flow chart in Figure 18.46 is a clustering stage, where several pixel-based detections of the same target are clustered to only one physical target.

2.18.8.4 ISTAP

Imaging space-time adaptive processing (ISTAP) is a combination of Post-Doppler STAP and SAR [39,40]. As with Post-Doppler STAP clutter cancellation is performed in the Doppler domain. However, there is no segmentation into short coherent processing intervals so that all the data are coherently processed. Thus, high SCNR values can be achieved so that the ISTAP technique is well suited for spaceborne systems.

The flow chart of ISTAP is shown in Figure 18.47. As with EDPCA the SAR compression filter is matched to the moving target parameters for maximizing the target SNR. Again a bank of SAR processing filters is necessary. However, ISTAP requires less computational power since the clutter-plus-noise covariance needs only to be estimated once and not for every iteration. Furthermore, also clutter cancellation needs to be performed only once.

The empirical Doppler-dependent clutter-plus-noise covariance matrix is estimated by averaging the measured data vector over range cells, identical to conventional Post-Doppler STAP [70]:

(18.131)

Again, the clutter-plus-noise covariance matrix is only of dimension and is thus invertible with low computational power ().

The clutter-plus-noise covariance matrix needs to be estimated for each Doppler bin.

The normalized test statistics for ISTAP is given as [70]

(18.132)

where is the SAR transfer function (only the Doppler slope is considered but not the shift) which maximizes the SNR of the target with parameter is the steering vector and is the CFAR threshold.

• Improvement of the target detection performance and the reduction of false alarms.

• Accurate and robust SAR-GMTI algorithms for spaceborne SAR-MTI systems.

• Improvement of the position and motion parameter estimation performance. This is of special importance for spaceborne SAR-GMTI where due to low SNR and SCNR values large estimation errors may arise. First promising techniques are ISTAP and EDPCA introduced in [39,40]. A different method requiring a large along-track baseline was presented in [71].

• Reduction of the system complexity and processing time. This is of special importance for affordable real-time systems to be used for civilian traffic monitoring. A suitable technique which also can be combined with classical STAP was introduced in [72].

• Trend to shorter wavelengths (e.g., Ka-band with wavelengths in the order of 8 mm): this allows for smaller instruments enabling both high resolution and good GMTI performance.

• Low PRF GMTI algorithms which can be used together with high-resolution wide swath (HRWS) SAR imaging.

2.18.9.1 Summary of algorithms

In Table 18.2 a summary of the algorithms treated in this tutorial is given. Also the parameter estimation accuracy and the required computational load are assessed in a qualitative way.

Applications:

Open Issues and Problems:

Glossary

ATI along-track interferometry; the phase differences of the signals received by two antennas separated in along-track or flight direction are measured. The phase difference is related to moving target parameters.

Clutter unwanted radar echos; for GMTI echos from the stationary background scene are considered as clutter

DPCA displaced phase center antenna; similar antenna arrangement as with ATI but instead of the phases the amplitudes are evaluated. Commonly DPCA is used for suppressing the clutter, i.e., the signals of stationary targets.

IRF impulse response function; focused signal of a point target

(G)MTI (ground) moving target indication; the detection of targets moving on ground and the estimation of their geographical positions, their velocities and moving directions.

SAR synthetic aperture radar; a side-looking imaging radar system exploiting the Doppler effect due to platform motion for imaging

SCR signal-to-clutter ratio

SCNR signal-to-clutter-plus-noise ratio

SNR signal-to-noise ratio

STAP space-time adaptive processing

SWMF stationary world matched filter; a two-dimensional filter used for SAR processing; every conventional SAR processor is considered as a SWMF in this tutorial

Relevant Theory: Signal Processing Theory and Array Signal Processing