I. INTRODUCTION
Fundamentals of Spatial and Doppler Frequencies in Radar STAP PHILIPPE RIES University of Lie` ge XAVIER NEYT FABIAN D. LAPIERRE Royal Military Academy Belgium JACQUES G. VERLY University of Lie` ge
The increasing interest for arbitrary antenna arrays in radar space-time adaptive processing (STAP) creates a need for a thorough understanding of the role of, and dependencies between, spatial and Doppler frequencies and related quantities, especially in the characterization of clutter. We successively introduce “geometrical” and statistical concepts, where we respectively emphasize the 4D direction-Doppler (DD) curve and the 4D power spectral density (PSD) that characterize the (clutter) space-time field. These descriptors, which are flight-configuration dependent, but antenna independent, are fundamental since they can be used to derive the key spectral properties of any antenna, essentially by rotations and projections. These descriptors are related in various ways, mostly because the DD curve is the support of the ridge of the clutter PSD. We also emphasize the surprising benefits of systematically considering the three spatial frequencies that are always present behind the scene, even for the customary linear antenna. A solid, simple, and elegant basis for thinking about STAP for arbitrary measurement configurations and antenna arrays is provided.
Manuscript received November 24 2006; revised May 12, 2007; released for publication July 19, 2007. IEEE Log No. T-AES/44/3/929758. Refereeing of this contribution was handled by M. Rangaswamy. This work was sponsored in part by a fellowship of the FNRS (Fonds National de Recherche Scientifique), Brussels, Belgium. Authors’ addresses: P. Ries and J. G. Verly, Dept. of Electrical Engineering and Computer Science, University of Lie` ge, Sart-Tilman, Bldg. B28, B-4000 Lie` ge, Belgium, E-mail: (
[email protected]); X. Neyt and F. D. Lapierre, Dept. of Electrical Engineering, Royal Military Academy, Avenue de la Renaissance 30, B-1000 Brussels, Belgium.
c 2008 IEEE 0018-9251/08/$25.00 ° 1118
Space-time adaptive processing (STAP) is an important radar signal-processing technique for detecting slow-moving targets in the presence of clutter [1, 2]. This paper examines some basic, general principles related to the notions of spatial and temporal (i.e., Doppler) frequencies and related quantities, with particular emphasis on clutter characterization. As a start, consider the simple case of a (receive) uniform linear array (ULA) in a monostatic measurement configuration. It is well known that, to each possible position S of a stationary scatterer at a given range from the radar, there corresponds specific values of the normalized spatial frequency ºs of the wave scattered by S, as measured along the support line of the ULA, and of the normalized Doppler frequency ºd affecting the signals received at the elements of the array. One can thus define a mapping from each point S to a corresponding point S 0 in (ºs , ºd ) space. If S moves along an isorange curve (“in the clutter” and, thus, typically on the ground), the corresponding S 0 describes a curve, called direction-Doppler (DD) curve, in the 2D (ºs , ºd ) plane. Such a curve can also be defined for bistatic configurations. As we change the range and/or the configuration, i.e., the altitude of the radar platform (respectively platforms in a bistatic (BS) scenario), its (respectively their) velocity, the crab angle [1] of the receiver array, and the relative position of the emitter with respect to the receiver in a BS scenario, the DD curve generally changes. Since the early days of STAP, researchers have plotted and studied the properties of these curves, which go by various names, such as clutter ridge [2], DD curve [3], angle-Doppler response [4], clutter power spectrum locus [5, 6], or azimuth-Doppler trajectories [7]. Deriving their general equations and their properties turned out to be quite complicated [8]. DD curves are important, in particular because they are often used to design range-dependence compensation algorithms [8, 9]. For ULAs, these algorithms attempt to register the 2D DD curves at auxiliary ranges with the curve at the range of interest. For example, this is the case in Doppler warping (DW) [10], angle-Doppler compensation (ADC) [11], adaptive angle-Doppler compensation (A2 DC) [12], high-order Doppler warping (HODW) [13], and registration-based compensation (RBC) [14]. The recent interest in conformal antenna arrays (CAA) [4], [15] and, more generally, in arbitrary antenna arrays (AAA) has led to new ways of looking at basic issues in STAP (including range-dependence compensation), even for ULAs [5, 9, 16]. While it is relatively clear that moving from linear antennas (e.g. ULAs) to planar and volume antennas requires the addition of new spatial frequencies, it is perhaps less obvious that three normalized spatial frequencies, say
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ºsx , ºsy , and ºsz , may always be used in a generalized model for all three types of antennas. It is also less obvious that one gets a better understanding of STAP for linear antennas by dealing, right from the start, with all three spatial frequencies. A key observation is that, for any given configuration, there exist a “universal” 4D DD curve in 4D (ºsx , ºsy , ºsz , ºd ) space, which can be projected into appropriate subspaces to deal with the particular cases of linear and planar antennas. With such a view, the complex behaviors of 2D DD curves for linear antennas [7] can easily be understood in terms of the simpler behavior of the underlying 4D DD curve. This is an example of a situation where it is crucial to realize that one’s view and understanding are limited by the fact that one is stuck in some subspace. By moving into a space of higher dimensionality, things suddenly become much simpler. For STAP, the appropriate space is the 4D (ºsx , ºsy , ºsz , ºd ) space corresponding to wave propagation in 3D space. The limitations of the 2D view for linear antennas became clear in the authors’ work that led to the publication of [5]. The goal and contribution of the present paper are to cast, in as didactic a way as possible, the STAP problem in 4D, to emphasize the benefits of doing so, and to show that all familiar descriptions follow simply and elegantly from this more general view. In STAP, the concepts of spatial and Doppler frequencies also lead to the concepts of clutter power spectral density (PSD) and of estimates thereof. Since the four frequencies are always there, it is quite logical to consider, right from the start, a 4D clutter PSD for any type of antenna, even for a linear antenna. We will see that DD curves and PSDs have corresponding properties, in particular in terms of rotations and projections. Furthermore, the support of the clutter ridge of the 4D PSD is identical to the 4D DD curve. This is well known for ULAs in the traditional (ºs , ºd ) space; this can now be explained via a simple projection argument. The paper proceeds gradually from simpler “geometrical” concepts (Sections II—IV) to more advanced statistical concepts (Section V). It is significant that one can go quite far on the sole basis of geometrical arguments (and of the knowledge of the wavelength of the wave of interest). These arguments indeed lead to the 4D DD curve and all of its fundamental properties. In Section II, we discuss the concept of spatial frequencies, first intuitively, then formally. This is a general concept found in several disciplines, such as Fourier optics [17], statistical optics [18], radioastronomy [19], and image processing [20]. It is also found in wave propagation, but, there, the concept of spatial frequency is generally hidden behind the more common and equivalent concept of wavenumber [21]. As a result, we first present the concept in a general setting, without any reference to
antennas or STAP. We emphasize the key property of “rotation of spatial frequencies,” which underlies the key issues addressed in this paper and, more generally, the rotation properties of the N-D Fourier transform for N at least equal to two. In Section III, following a brief review of isorange surfaces and curves, we discuss the notion of 4D DD curves and the projection properties of these curves. In Section IV, we illustrate the concepts of the previous sections. We interpret the effect of the antenna crab angle in terms of changes in the projection direction of the 4D DD curve. We also examine the effect of changes in range. This effect is important because it is the source of the range-dependence problem in STAP, which applies to most configurations and arrays [8, 12, 14, 22, 23]. In Section V, we derive a model of the space-time field for clutter returns and define its 4D PSD. We discuss its projection property and the relation between DD curves and PSDs. In Section VI, we provide estimates of the unattainable, theoretical PSD and interpret them in terms of superposition or convolution integrals. In Section VII, we illustrate the concepts of Section VI. Sections VIII and IX, respectively, give a discussion and the conclusion. The reader who wishes to skip the basic concepts and simultaneously deal with the geometrical and statistical aspects of the problem may start with Section V and only refer to earlier sections when necessary. II.
SPATIAL FREQUENCIES
The spatial frequencies and the Doppler frequency play a critical role in STAP. Here, we give a brief review of the concept of spatial frequencies. We start in an intuitive way and then confirm this approach with a more formal one. A. Intuitive Definition of Spatial Frequencies The notion of spatial frequency can be introduced intuitively by considering 1) a plane wave travelling in some direction d˜ and 2) the direction defined by some vector x˜ , where d˜ and x˜ are arbitrary unit vectors in 3D space (Fig. 1). To x˜ , we associate an x-axis having its origin at some arbitrary reference point O and pointing in the direction of x˜ . The yand z-axes are then chosen according to needs, but with the constraint that the three axes are orthogonal and right-handed. In Fig. 1, the wave is represented by two successive planar wavefronts depicted via two parallel triangles perpendicular to the line going ˜ through O and parallel to d. The distance along the direction of propagation between the successive wavefronts is ¸c = c=fc , where c is the speed of light and fc the carrier frequency. The points on the x-axis for which the fields are in phase are separated by an integer multiple of
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B. Formal Definition of Spatial Frequencies A plane wave propagating in the direction of d˜ (Fig. 1) is described by T
s(r, t) = ej(!c t¡k c r)
(7)
where r = (x, y, z) is some arbitrary position in 3D space, !c = 2¼fc is the temporal pulsation, and k c ¢ is the wavevector related to d˜ by k = k d˜ [21], with c
c
jk c j = kc = 2¼=¸c being the wavenumber. Noting that ˜ (7) can be rewritten as k c =jk c j = d, s(r, t) = ej(2¼fc t+2¼fsx x+2¼fsy y+2¼fsz z) : Fig. 1. Graph supporting intuitive definition of spatial frequencies.
the absolute value jTx j of the signed distance Tx = ¸c = cos ®x , where ®x is the angle between the direction ˜ and the x-axis. We of arrival (DOA), i.e., d˜ a = ¡d, then treat jTx j as a spatial period and define the corresponding spatial frequency as the signed quantity 1 fsx = cos ®x : ¸c ¢
(1)
Similarly, the expressions for the spatial frequencies corresponding to the y- and z-axes are ¢
1 cos ®y ¸c
(2)
¢
1 cos ®z ¸c
(3)
fsy = fsz =
where ®y and ®z are the angles between the DOA and the y- and z-axis, respectively. We find it useful to define the spatial frequency vector fs = (fsx , fsy , fsz ) =
1 ˜ d ¸c a
a
In STAP, it is customary to normalize the spatial frequencies so they each take their values in [¡0:5, 0:5]. The normalized counterpart of fsx is ºsx = 0:5¸c fsx . The normalized spatial frequency vector is defined as º = (º , º , º ) = 0:5¸ f = 0:5d˜ : (4) sx
sy
sz
c s
a
As a result we have 2 2 2 ºsx + ºsy + ºsz = (0:5)2
(5)
jº s j = 0:5:
(6)
and
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This relation confirms that the quantities fsx , fsy , and fsz defined in (1)—(3) and used in (7) should indeed be interpreted as spatial frequencies just as fc is a temporal frequency. C.
Relation with Antenna Arrays
The relation between spatial frequencies and antenna arrays is as follows. A linear antenna array can “sense” the spatial frequency along the support line of the antenna array. Similarly, a planar antenna array can “sense” two spatial frequencies along two (preferably orthogonal) axes in the support plane of the antenna array. Finally, a volumetric antenna array can “sense” three spatial frequencies along three (preferably orthogonal) axes in 3D space. Of course, a wave induces specific spatial frequencies on any antenna whether it is an array or not. However, the array is generally the practical mean used to measure these spatial frequencies. D. Effect of Rotation of (x, y, z) Axes on Spatial Frequencies
where d˜ a = (cos ®x , cos ®y , cos ®z ) and fs and d˜ a are expressed in the (x, y, z) coordinate system. The components of d˜ a are the customary direction cosines of this vector. These cosines obey the well-known relation cos2 ®x + cos2 ®y + cos2 ®z = 1 [24], which ensures that d˜ is indeed a unit vector.
s
(8)
Since the values of the three spatial frequencies fsx , fsy , and fsz depend upon the choice (i.e., orientation) of the (x, y, z) axes attached to O, it is legitimate to ask how the spatial frequencies change as we rotate the (x, y, z) axes about O, while keeping the DOA fixed. Let us consider an arbitrary rotation matrix T such that the new coordinates r0 = (x0 , y 0 , z 0 ) are related to the initial coordinates r = (x, y, z) by r0 = T r. It is easily shown from (7) that f0s = T fs
(9)
where f0s is the spatial frequency vector observed in the new axes. This last relation shows that the spatial frequencies change under rotation of the coordinate system exactly in the same way as the coordinates themselves, i.e., by multiplication by T. This observation is worth being formulated as “the rotation property of spatial frequencies.”
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An illustration of this property occurs in the context of the 3D Fourier transform (FT). Consider a function f(x, y, z) and its 3D FT. First, rotate f(x, y, z) to get f(x0 , y 0 , z 0 ) and then compute the 3D FT of f(x0 , y 0 , z 0 ). Second, compute the 3D FT of f(x, y, z) and then rotate it. In both cases, the results are the same. Later, we apply the exact same reasoning to the situation where the operator is not the FT, but the process of computing the 4D DD curve from an isorange. E. Important Note Whether one deals with 1D, 2D, or 3D antennas, there are always three spatial frequencies. However, a given antenna array may only sense a subset of them. Fig. 2. Measurement configuration with transmitter T, receiver R, and scatterer S.
III. DIRECTION-DOPPLER CURVES A. Measurement Configuration The measurement configuration is depicted in Fig. 2. The ground, which is considered here to be the source of clutter, is assumed to be a (horizontal) plane. T and R are fixed reference points on the transmit (Tx) and receive (Rx) platforms, respectively. By abuse of language, we may use T to refer to both the reference point and the whole Tx platform. Ditto for R. T and R have respective velocity vectors vT and vR that are both parallel to the ground plane. The x-axis is aligned with vT and its origin is at T. The y-axis is also chosen parallel to the ground plane. T is at a height H above the ground. The receiver R is located at (xR , yR , zR ) in the (x, y, z) axes. The receiver velocity vector vR is assumed to make an angle ®R with respect to a line going through R and parallel to the x-axis (and thus to vT ). We also introduce the (xr , yr , zr ) axes with origin at R and fixed with respect to the Rx platform. The (xr , yr ) plane is chosen parallel to the ground, and the angle between the xr -axis and vR is denoted by ±. In the case of an aircraft, if the xr -axis is parallel to the longitudinal axis of the aircraft, ± coincides with the customary crab angle of the aircraft. ± can be non-zero for a variety of reasons and, in particular, if the aircraft is trying to compensate for side wind. When dealing with a ULA that is rigidly mounted on the Rx platform, one may want to align the xr -axis with the ULA. In this case and if the platform is an aircraft, the crab angle ± of the ULA (still with respect to vR ) may not coincide with the crab angle of the aircraft. The generic scatterer point S (on the ground and projecting into Sxr yr in the (xr , yr ) plane) is characterized by an azimuth angle ÁR (S), referred to the half-line going through R and pointing in the same direction as the x-axis, and by a depression
angle μR (S) (positive in the downward direction). The wavevector and the spatial frequency vector corresponding to the plane wave scattered by S are, respectively, denoted by k c (S) and º s (S). B. Isorange Surface and Curve Consider the locus S(Rb ) of the points P for which the sum of the distances from T to P and P to R is constant and equal to Rb . S(Rb ) is called an isorange surface. Its intersection with the ground defines an isorange curve C(Rb ) such that the distance from T to any point S of C(Rb ) to R is equal to Rb . C.
DD Curve
On the one hand, each S on C(Rb ) defines a DOA at R, and thus yields a specific spatial frequency vector º s (S) with respect to the chosen axes, here the (xr , yr , zr ) axes. On the other hand, as T and/or R are moving, a wave travelling from T to S to R will exhibit a Doppler frequency fd (S). We assume that S is stationary. In STAP, one considers mostly pulsed waveforms characterized by a pulse repetition interval, which we denote by Tp [2], [1]. It is then customary to define a normalized Doppler frequency ºd (S) = Tp fd (S), where Tp is chosen such that there are no Doppler ambiguities and thus ºd takes its values in [¡0:5, 0:5] just as the spatial frequencies do. For each position of S along C(Rb ), we can view (º s (S), ºd (S)) as the coordinates of a point S 0 in the 4D space (º s , ºd ) ´ (ºsxr , ºsyr , ºszr , ºd ), where ºsxr , ºsyr , and ºszr are the normalized spatial frequencies measured in the (xr , yr , zr ) axes. If we consider successive points S along C(Rb ), we can regard the corresponding points S 0 as tracing a curve C 0 (Rb ) in the 4D (º s , ºd ) space. C 0 (Rb ) is called the 4D DD curve. The 4D DD curve represents the dependencies between the four
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Fig. 3. Illustration of isorange curve (top) and 4D DD curve (bottom) for specific MS configuration. 4D DD curve is shown in customary pair of 3D spaces.
normalized frequencies ºsxr , ºsyr , ºszr , and ºd for all points S on C(Rb ). This is illustrated in Section IV. Since it is not possible to display directly 4D entities (in 4D spaces) and, thus, the 4D DD curves, we generally choose to represent a 4D DD curve by two companion 3D curves in the (ºsxr , ºsyr , ºd ) and (ºsxr , ºsyr , ºszr ) spaces. We can immediately tell that the 4D DD curve in the latter space will be on a sphere of radius 0:5 as a direct consequence of (6). D. Projection Interpretation of 2D DD Curves In Section IIE, we insisted on the fact that there are always three spatial frequencies. As a consequence, there always exists a 4D DD curve. So, one might wonder why 2D DD curves (or simply DD curves) have almost exclusively been shown in STAP to date [6, 7, 12, 25—32]. The 2D DD curves of STAP almost always appear in the context of ULAs. The reason for this is that a linear antenna array can only sense a single spatial frequency, i.e., the one corresponding to some axis parallel to the antenna array. Here, we assume that the xr -axis is parallel to the antenna array. In this case, we can get knowledge of ºsxr , but not of ºsyr and ºszr . As a result, the only curve we can hope to create from the signals received at R is a curve in the (ºs , ºd ) ´ (ºsxr , ºd ) space. This is the (2D) DD curve. It is legitimate to ask whether the 2D and 4D DD curves are related and what this relation might be, if any. From the discussion above it is clear that going from the 4D curve to the 2D curve consists in dropping the ºsyr and ºszr coordinates of the curve. Hence, the 2D DD curve is simply the restriction of the 4D DD curve to the (ºsxr , ºd ) space, which can also 1122
be viewed as a projection along both the ºsyr and ºszr axes. A similar reasoning holds for planar antenna arrays, which can sense only two spatial frequencies. Although planar antennas can only sense two spatial frequencies, the value of the third one can be deduced from (5) up to a sign ambiguity. If the planar array is horizontal, the sign ambiguity is equivalent to a high-low ambiguity, which does not affect the ground moving target indicator (GMTI) operating mode. It is critical to understand that, in all cases, there exists an underlying 4D DD curve and that all other DD curves of lower dimensionality are related to this fundamental 4D DD curve via a projection. 4D DD curves tend to have much simpler shapes than 2D DD curves. In fact, the complex behavior of 2D DD curves [8, 14, 6] can generally be understood from the simpler behavior of the 4D DD curves. IV. ILLUSTRATION AND ANALYSIS OF DIRECTION-DOPPLER CURVES A. Complete Picture, with both Isorange and DD Curves, for an MS Configuration Fig. 3 provides a complete illustration of the construction and appearance of DD curves for a monostatic (MS) measurement configuration. The top subfigure shows the MS flight measurement configuration with T ´ R. The crab angle ± is assumed to be zero. The bottom subfigure shows the 4D DD curve in terms of its two companion 3D curves. The primary observation is that the first 3D curve is an ellipse located in an oblique plane. A secondary observation is that the second 3D curve is the circle
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Fig. 4. Illustration of isorange curve (top) and 4D DD curve (bottom) for specific BS configuration. The 4D DD curve is shown in customary pair of 3D spaces.
that is the intersection of a horizontal plane with the constraining sphere jº s j = 0:5. The 2D DD curve in the (ºsxr , ºd ) space is a diagonal “line.” This line is actually a closed curve and the point S 00 moving along it in synchrony with S makes roundtrips on the diagonal. This “diagonal” 2D DD curve is the one STAP researchers are familiar with in the context of sidelooking ULAs [7, 8]. Indeed, ºsxr corresponds to the spatial frequency that would be sensed by a sidelooking ULA for ± = 0. A key point is that the “diagonal” pattern is now clearly seen to be the projection into the 2D (ºsxr , ºd ) subspace of the 3D DD curve present in the (ºsxr , ºsyr , ºd ) subspace . The linear nature of the 2D DD curve is a result of the fact that the 3D DD curve is in a plane orthogonal to the (ºsxr , ºd ) plane. Of course, the diagonal pattern is also the projection of the 4D DD curve directly into the (ºsxr , ºd ) space. The 2D DD curve in the (ºsyr , ºd ) space is an ellipse. This elliptical 2D DD curve is the one STAP researchers are familiar with in the context of forwardlooking ULAs [7], [8]. Indeed, ºsyr corresponds to the spatial frequency that would be sensed by a forwardlooking ULA for ± = ¼=2. The point made above regarding the projection is applicable here too. B. Complete Picture, with both Isorange and DD Curves, for a BS Configuration Fig. 4 is the counterpart of Fig. 3 for a BS wing-to-wing configuration. The description and discussion of the present case can easily be obtained by adapting those of the previous case. Of course, the main points made previously apply here too.
The main difference between the MS and BS cases lies in the shape of the 4D DD curves and their various projections in 3D and 2D subspaces. In particular, we see that the curve in the (ºsxr , ºsyr , ºd ) subspace is no longer confined to a plane. Its twisted shape gives rise to the “eight” in the projection in the (ºsxr , ºd ) plane. STAP researchers will immediately recognize the projections into the (ºsxr , ºd ) and (ºsyr , ºd ) spaces as curves that are typically found in connection with sidelooking and forwardlooking ULAs, respectively [7, 8, 12, 22]. C.
Effect of Antenna Crab Angle ±
We now have the tools to explain easily the well-known changes in the appearance of the 2D DD curve resulting from changes in the antenna crab angle ± in the case of a ULA. Let us consider a ULA placed along the xr -axis. As ± is varied or, equivalently, as the (xr , yr , zr ) axes are rotated, the same rotation also takes place in the axes in which the 4D DD curve is described (see the rotation property of spatial frequencies in Section IID). Let us focus on the (ºsxr , ºsyr , ºd ) subspace. To obtain the 2D DD curve corresponding to a ULA whose elements are placed along the xr -axis, we have to project the 4D DD curve into the (ºsxr , ºd ) plane, i.e., along the ºsyr and ºszr directions. As ± changes, the ºsxr and ºsyr axes rotate. Thus, the projection of the 4D DD curve into the (ºsxr , ºd ) plane changes. But this projection is precisely the 2D DD curve. Consequently, the different 2D DD curves obtained as ± varies are the projections of the unique 4D DD curve along different directions. In Fig. 5, we clearly see that the projections of the 4D
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Fig. 5. Effect of crab angle ± on otherwise identical BS configurations. Successive columns correspond to increasing values of ±: 0, 15, 30, and 45 deg. The top row shows the 4D DD curves, but only through their projection in the (ºsxr , ºsyr , ºd ) axes. The bottom row shows the corresponding 2D DD curves, shown in the usual way. Observe that the curves in the (ºsxr , ºd ) axes in the top plots and the curves in the bottom plots are indeed identical.
DD curves match the corresponding 2D DD curves for ULAs. D. Effect of Range Rb Fig. 6(a) illustrates the variation of 4D DD curves with range Rb for an MS configuration. Fig. 6(b) does the same for a BS configuration. The projection in the various subspaces are not shown to avoid cluttering the diagrams. As for ULAs, the geometry-induced range-dependence problem for CAAs (as well as for AAAs) leads to erroneous covariance matrix estimates and thus to losses in detection performance [5]. The variations with range of the various DD curves is one of the clearest manifestations of the range-dependence problem in STAP. Correctly handling this problem is critical for improving the detection performance of STAP-based systems. Discussing range-dependence compensation is beyond the scope of the present paper. The configurations for which there is no range dependence can be found by physical reasoning about the spatial and Doppler frequencies. This reasoning closely follows the one given in [5]. The first condition to be fulfilled for the 4D DD curve to be independent of range is that the curves at different ranges must overlap in the 3D space of the spatial frequencies. Since the height ºszr of points along the curves in the 3D space of the spatial frequencies only depends on the elevation angle at which the scatterers along the isoranges are seen from the 1124
Fig. 6. Evolution of 4D DD curves for increasing range Rb in (a) MS configuration and (b) BS wing-to-wing configuration.
Rx platform, overlap in the 3D space of the spatial frequencies occurs if and only if the Rx is on the (flat) ground. In this case, the scatterers are seen at an elevation angle of zero regardless of range. The second (and last) condition is that the Doppler frequency must be independent of range. This means that the Doppler frequency corresponding to any
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particular spatial frequency must be independent of range. In a configuration where the Rx is located on the ground, the Doppler frequency due to the Rx velocity will be constant along radial lines from the Rx. The only configurations for which the Doppler frequency induced by the Tx platform velocity is independent of range is either 1) when the Tx is static, in which case this Doppler frequency is zero and thus independent of range, or 2) when the Tx is located on the ground and at the same location as the Rx, in which case the Doppler frequency only depends on the Tx azimuth angle (and on the Tx velocity) and is thus independent of range. V. MODELS OF SIGNALS RECEIVED A. Model of Space-Time Field We have gone as far as we could on the sole basis of “geometrical” arguments. To go further, we need to introduce a model of the ground clutter signal received at each point of the Rx antenna, for each pulse, and for each range Rb of interest. We use a simple narrowband model that allows us to discuss the main properties of the power spectrum of the received signals. We thus do not attempt to model the effects of polarization, finite bandwidth, and decorrelation, as well as, in the case of array antennas, of the gain patterns of the individual array elements. For a more complete model, the reader is referred to [4]. The continuous-space, discrete-time random field due to some scatterer S on the isorange C(Rb ) for a given Rb is given by T
yS (½, m) = c(S)ej2¼º s (S)½ ej2¼ºd (S)m
(10)
where ½ = (xr , yr , zr )=(0:5¸) is the normalized position vector of the point where the value of the field is considered, m is the index of the mth pulse, and c(S) is the amplitude random variable. The discrete nature of the temporal component of the field accounts for the sampling of the pulses, as is commonly done in pulse-Doppler radar [33]. The random field yS (½, m) is assumed to be wide-sense stationary (WSS). Hence, we can write its statistical autocorrelation function (ACF) as [34]
where U is the spatial frequency variable vector (Uxr , Uyr , Uzr ), and V is the customary temporal frequency variable. Note that the 4D FT is a continuous FT over the spatial part and a discrete FT over the temporal part. The total field at some position ½ due to all points in a ground annulus A around the isorange curve C(Rb ) can conceptually be written as an integral over this annulus. The exact shape of the annulus is determined by C(Rb ), the gain patterns of the Tx and Rx antenna array elements, and the resolution of the Tx waveform. To avoid unnecessary additional mathematical difficulties, we assume that the contributions to the field at ½ come from some Nc clutter patches, where each patch i is centered around a point Si on C(Rb ) and is chosen in such a way that the Nc patches approximately cover A. Neglecting range ambiguities, our model for the clutter-induced field at ½ and for m is thus y(½, m) =
N c ¡1 X
ySi (½, m):
(12)
i=0
From the assumption that ySi (½, m) is WSS, it follows that y(½, m) is also WSS. Hence the clutter PSD, which we denote by P(U, V), is the 4D FT of the clutter ACF °(¢½, ¢m) of y(½, m). STAP commonly considers an antenna array with N elements and a pulse train with M pulses, where the elements and pulses are indexed by n 2 [0, N ¡ 1] and m 2 [0, M ¡ 1], respectively. (Note that the use of m was already anticipated in (10)). Therefore, the values of the fields received at the N antenna elements and for the M pulses are jointly denoted by the 2D array y[n, m] and given by y[n, m] = y(½n , m)
(13)
where ½n is the (normalized) position of the nth antenna element. y[n, m] is called a snapshot. It corresponds to a specific range Rb . If we discretize Rb in “range gates” Rl , l 2 [0, L ¡ 1], the 3D array y[n, m, l] is called a data cube. It contains all the data available for processing in a given coherent processing interval. The 2D array y[n, m] is often written as a vector y by lexicographically stacking the rows on top of each other [1, 2].
°S (¢½, ¢m) = EfyS (½, m)yS¤ (½ ¡ ¢½, m ¡ ¢m)g where ¢½ = (¢½xr , ¢½yr , ¢½zr ) and ¤ denotes complex conjugation. The corresponding PSD is the 4D FT of °S (¢½, ¢m) [34], PS (U, V) =
Z
+1 Z +1 Z +1
¡1
¡1
+1 X
¡1 ¢m=¡1 T
¢ °S (¢½, ¢m)e¡j2¼(U
¢½+V¢m)
d¢½ (11)
B. Projection Interpretation of 2D Clutter PSD In Section IIID, we showed that each 2D DD curve (typically associated with a ULA) is in fact a projection of a 4D DD curve. We now show that each 2D PSD (typically associated with a ULA) is also a projection of a 4D PSD of the type just defined. If the Rx antenna is a linear (1D) antenna located, say, along the xr -axis, we are only able to measure the (spatial) vector lags along the xr -axis. Thus, we
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only get knowledge of °(¢½xr , 0, 0, ¢m), i.e., of the 2D function ¢
°xr (¢½xr , ¢m) = °(¢½xr , 0, 0, ¢m)
(14)
which is the ACF associated with the ULA. °xr (¢½xr , ¢m) is a legitimate ACF and the corresponding PSD is Z +1 X Pxr (Uxr , V) =
¡1
statistical concept of clutter PSD. Substituting (10) in (12), we get y(½, m) =
°xr (¢½xr , ¢m)e¡j2¼(Uxr ¢½xr +V¢m) d¢½xr :
¢m
=
¡0:5
¡1
¡1
¡1
°(¢½, ¢m) = P(Uxr , Uyr , Uzr , V)
where we have expanded the vector arguments. If we set ¢½yr = 0 and ¢½zr = 0, the resulting expression can be restructured as °(¢½xr , 0, 0, ¢m) =
¡0:5
+1 ·Z ¡1
+1 Z
¡1
+1
¡1
P(Uxr , Uyr , Uzr , V)dUyr dUzr
¢ ej2¼(Uxr ¢½xr +V¢m) dUxr dV:
¸
(16)
This relation can be interpreted as meaning that the bracketted term, denoted by Iyr zr (Uxr , V), is the 2D FT of °(¢½xr , 0, 0, ¢m), which we have also denoted by °xr (¢½xr , ¢m) in (14). It follows that Pxr (Uxr , V) = Iyr zr (Uxr , V). This result shows that the 2D PSD Pxr (Uxr , V) can be obtained from the 4D PSD P(Uxr , Uyr , Uzr , V) by integrating this function along the axes Uyr and Uzr , or, equivalently, by projecting P(Uxr , Uyr , Uzr , V) into the (Uxr , V) subspace. Similar conclusions are obtained if we consider any line orientation in ¢½ space (instead of the ¢½xr -axis above) or any plane orientation in ¢½ space (this would be the case for planar antennas arrays). Showing this from scratch leads to a rather tedious analytical development. However, a simple elegant proof can be provided by invoking the 3D version of the projection-slice theorem of computed tomography [35], the two flavors of which (i.e., integration along lines and integration along planes) are useful in the present context. C. Relation Between 4D DD Curve and 4D Clutter PSD We now provide the connection between the geometrical concept of 4D DD curve and the 1126
T
´(Si )ej2¼º s (Si )¢½ ej2¼ºd (Si )¢m :
The PSD P(U, V) of y(½, m) being the 4D FT of °(¢½, ¢m), we have
(15)
+0:5 Z
N c ¡1 X i=0
¢ ej2¼(Uxr ¢½xr +Uyr ¢½yr +Uzr ¢½zr +V¢m) dUxr dUyr dUzr dV
Z
(17)
where the random process c(Si ) is assumed to be zero mean, i.e., Efc(Si )g = 0, and uncorrelated, i.e., Efc(Si )c¤ (Si0 )g = ´(Si )±ii0 , where ´(Si ) is the expected power due to the ith clutter patch and ±ij = 0 if i 6= j and ±ij = 1 if i = j. Throughout, Si is considered to be on C(Rb ). Given the above assumptions, the ACF °(¢½, ¢m) of y(½, m) is easily found to be
°(¢½xr , ¢½yr , ¢½zr , ¢m) +0:5 Z +1 Z +1 Z +1
T
c(Si )ej2¼º s (Si )½ ej2¼ºd (Si )m
i=0
Starting with the 4D FT defining P(U, V) in terms of °(¢½, ¢m) and inverting the transform, we get
Z
N c ¡1 X
P(U, V) =
+1 N c ¡1 X X
k=¡1 i=0
´(Si )±(U ¡ º s (Si ), V ¡ ºd (Si ) ¡ k),
Si 2 C(Rb ) where ±(U, V) is the 4D Dirac delta “function,” and the summation over k accounts for the periodicity inherent to the discrete-time FT [36]. From here on, we assume that there is no aliasing and we focus on the case where V is limited to the interval [¡0:5, 0:5]. We clearly see that, as Si proceeds along C(Rb ), the location of the corresponding “impulse” ±(U ¡ º s (Si ), V ¡ ºd (Si )) describes a curve in the (U, V) space. From the definition of the 4D DD curve given in Section III, it is clear that this curve is precisely the 4D DD curve. Hence, the 4D DD curve is the support of the clutter PSD. VI. ESTIMATION OF THE POWER SPECTRAL DENSITY So far, we have dealt, on the one hand, with the geometrical concept of DD curves and, on the other hand, with the continuous-space, discrete-time random process y(½, m) and the corresponding ACF and PSD. However, none of these theoretical quantities can be observed directly. In practice, one can only obtain estimates of the PSD. This is due to 1) the finite extent of the antenna array, which covers only a set of discrete spatial locations and the finite length of the train of pulses, and 2) the random nature of the clutter snapshots. For simplicity, we consider here only the clairvoyant case, i.e., the case where estimation errors are introduced only because of the sampling of the signal y(½, m). The problems due to the random nature of the clutter snapshots must be treated by a covariance matrix estimation algorithm, as e.g. in
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[5]. Considering the clairvoyant case means that we have access to the true covariance matrix of the vector snapshot y, R = Efyy† g: As an aside, the Toeplitz-Block-Toeplitz structure of the covariance matrix observed in the ULA case comes from the uniform sampling along the temporal axis and from the uniform spacing of the antenna elements along the corresponding spatial axis. For antenna arrays with non uniformly spaced elements, such as is generally the case for CAAs (and AAAs), this structure disappears. One might then ask whether the DD curves remain useful in studying properties such as the range dependence of the clutter statistics. We will see that this is indeed the case by showing that the clutter PSD and the expected value of its estimates are linked by a superposition integral [37] such that, in the clutter PSD estimate, the energy due to clutter is located around the DD curve. We will show that this is true both for Fourier-based estimates [21, 38] and for the minimum-variance estimate (MVE) [21, 38, 34].
Z
P(U, V) =
+1 Z +1 Z +1
¡1
¡1
+1 X
¡1 ¢m=¡1 T
¢ °(¢½, ¢m)e¡j2¼(U Z
°(¢½, ¢m) =
+0:5 Z
¡0:5
+1 Z
¡1
+1 Z
¡1
¢ P(U, V)ej2¼(U
T
¢½+V¢m)
(20)
+1
¡1
¢½+V¢m)
dU dV:
Using (20), evaluated at ¢½ = ½n ¡ ½n0 and ¢m = m ¡ m0 , in (19), and interchanging the integrals and the sums leads to ˆ V)g EfP(U, ÃZ +0:5 Z +1 Z +1 Z +1 1 = (NM)2 ¡0:5 ¡1 ¡1 ¡1 ¢ P(U0 , V0 ) 0T
¢ ej2¼[U
N¡1 X M¡1 X N¡1 X M¡1 X
n=0 m=0 n0 =0 m0 =0
(½ ¡½ 0 )+V0 (m¡m0 )] n
n
¡j2¼(UT ½ +Vm) j2¼(UT ½ 0 +Vm0 )
¢e
A. Fourier-Based Estimation As a Fourier-based estimate we can e.g. use [38] ¯N¡1 M¡1 ¯2 ¯X X ¯ T 1 ¯ ¡j2¼(U ½ +Vm) ¯ ˆ n P(U, V) = y[n, m]e ¯ ¯ ¯ (NM)2 ¯ n=0 m=0
(18)
also known as the periodogram in the case of uniform sampling [34]. Considering the clairvoyant case is equivalent to considering the expected value ˆ ˆ EfP(U, V)g of P(U, V). By some straightforward ˆ V)g algebraic manipulations, one can express EfP(U, as 1=(NM)2 v† (U, V)Rv(U, V), where v(U, V) = [ej2¼(U ej2¼(U
T
½ +V0) 0
T
½ +V(M¡1)) 0
j2¼(UT ½
¢¢¢e
N¡1
+V0)
T
j2¼(U ½
¢¢¢e
N¡1
¢¢¢
(NM)2
n=0 m=0 n0 =0 m0 =0
¢e
n
e
]
°(½n ¡ ½n0 , m ¡ m0 )
n
!
:
(19) ˆ We now relate EfP(U, V)g to the PSD defined by the FT relations
e
n
dU dV
Let us introduce the space-time beampattern B(U, V, U0 , V0 ) defined by [21] B(U, V, U0 , V0 ) =
1 † v (U, V)v(U0 , V0 ): NM
0
!
:
(21)
(22)
B(U, V, U0 , V0 ) represents the gain for a plane wave giving rise to the spatial frequency vector U0 and the Doppler frequency V0 if the array is steered in the space-time direction giving rise to (U, V). Noting that (22) can also be written as N¡1 M¡1 1 X X j2¼(U0 ¡U)T ½ j2¼(V0 ¡V)m ne e NM n=0 m=0 (23) we can rewrite (21) as
B(U, V, U0 , V0 ) =
+V(M¡1)) T
¡j2¼(UT ½ +Vm) j2¼(UT ½ 0 +Vm0 )
n
0
ˆ EfP(U, V)g
is the space-time steering vector for the array under consideration [4]. Taking the expected value of (18), one finds à N¡1 M¡1 N¡1 M¡1 XXXX 1 ˆ EfP(U, V)g =
d¢½
=
Z
+0:5
¡0:5
Z
+1 ¡1
Z
+1
¡1
Z
+1
P(U0 , V0 )jB(U, V, U0 , V0 )j2 dU0 dV0 :
¡1
(24) Equation (24) is usefully interpreted as a superposition integral [37]. This means that ˆ V)g results from the “summation,” over all EfP(U, the (U0 , V0 ), of the weighted beampatterns, the weights being the P(U0 , V0 )s. It is also possible to find an interpretation of (24) in terms of a convolution. To do so, we introduce B0 (U0 , V0 ) = B(0, 0, U0 , V0 ) =
N¡1 M¡1 1 X X j2¼U0T ½ j2¼V0 m ne e NM n=0 m=0
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Fig. 7. Four illustrative antenna arrays. (a) Spherical array (N = 72). (b) Bowl-shaped array (N = 56). (c) Uniform circular array (N = 30). (d) ULA (N = 12).
written as
which is the beampattern if the array is steered to (U, V) = (0, 0). Noting that B(U, V, U0 , V0 ) = B0 (U0 ¡ U, V0 ¡ V), we can rewrite (24) as
8¯ ¯2 9 M¡1
