Rotating Parabolic-Reflector Antenna Target in SAR Data: Model, Characteristics, and Parameter Estimation

Abstract

Parabolic-reflector antennas (PRAs), usually possessing rotation, are a particular type of targets of potential interest to the synthetic aperture radar (SAR) community. This paper is aimed to investigate PRA’s scattering characteristics and then to extract PRA’s parameters from SAR returns, for supporting image interpretation and target recognition. We at first obtain both closed-form and numeric solutions to PRA’s backscattering by geometrical optics (GO), physical optics, and graphical electromagnetic computation, respectively. Based on the GO solution, a migratory scattering center model is at first presented for representing the movement of the specular point with aspect angle, and then a hybrid model, named the migratory/micromotion scattering center (MMSC) model, is proposed for characterizing a rotating PRA in the SAR geometry, which incorporates PRA’s rotation into its migratory scattering center model. Additionally, we in detail analyze PRA’s radar characteristics on radar cross-section, high-resolution range profiles, time-frequency distribution, and 2D images, which also confirm the models proposed. A maximal likelihood estimator is developed for jointly solving the MMSC model for PRA’s multiple parameters by optimization. By exploiting the aforementioned characteristics, the coarse parameter estimation guarantees convergency upon global minima. The signatures recovered can be favorably utilized for SAR image interpretation and target recognition.

1. Introduction

Parabolic-reflector antennas (PRAs), due to their excellent beam directionality, are widely used in radar surveillance, navigation, microwave communication, and radio astronomy. Although they are antennas in themselves, PRAs are also present as a particular type of targets, gaining growing interest in some scenarios to the synthetic aperture radar (SAR) community [1]. Their scattering and rotation characteristics, which are obtainable with SAR, may be of valuable aid for target recognition. Therefore, it is desirable to detect and recognize them from SAR returns or imagery. However, this work is difficult because of two reasons. On one hand, the backscattering of PRAs is so complicated that they cannot be characterized by point scattering centers, and the directionality, favorable as it is for PRAs as antennas, results in aspect sensitivity and weak backscattering at most aspect angles when they are present as SAR targets. On the other hand, their rotation, due to mechanical scanning, changes the Doppler history and therefore makes their SAR images blurred and enigmatic. And unfortunately, rotation cannot be processed by the mature technique, that is, SAR ground moving target indication (SAR/GMTI), though it is a valid tool for uniformly moving targets [2].

Conventionally, rotating PRAs (RPRAs) were investigated as micromotion targets. In 2009, the SAR/micromotion target indication concept was presented to generalize the SAR/GMTI technique and to intensively examine the detection and imaging problems for micromotion targets [3]. Actually, early in 2004, the picket fence shape was revealed to be representative of a rotating radar dish in the SAR azimuth image [4], but the dish (viz. PRA) was simplified as a combination of point scatterers, which are not qualified for characterizing PRA’s complex scattering. In 2005 and 2010, wavelet decomposition and chirplet decomposition were used to successfully separate the return of an RPRA from that of stationary parts. The returns were collected by the US Navy APY-6 radar [5, 6]. After separation, time-frequency distribution (TFD) was utilized to analyze the phase modulation and autocorrelation to estimate the rotating frequency. The RPRA detected was annotated in the SAR image [1]. However, the RPRA’s signature blurred in SAR imagery was not explained, and the aforementioned approaches are susceptible to range cell migration (RCM). TFD also usually suffers from cross-term interference. In 2007, RPRA data, collected by a microwave SAR system in University of Zurich, were analyzed [7]. A point-symmetrical image of the RPRA was obtained, and the time-frequency image showed a small fraction of a sinusoidal phase modulation. The conclusions, however, are only valid for slow rotating frequency and short coherent processing interval (CPI). PRA’s radar characteristics have not been deeply studied either.

In this paper, we focus on characterization and non-TFD-based estimation of PRAs, both stationary and rotating, in spotlight SAR data. There are different types of PRAs, but we restrict our scope to a rotation-symmetrical PRA with a single reflector. Instead of processing returns in a single range cell, we use the phase history in the wavenumber domain, that is, the 2D Fourier transform (FT) of complex imagery, to extract PRAs and their parameters. We build for a PRA a new non-point-scattering center model able to describe the aspect sensitivity and the sliding of scatterers and then incorporate the motion model into it to obtain a hybrid scattering/motion model. Radar characteristics of PRAs are then analyzed. Afterwards, an estimation algorithm for this model is proposed. This idea fully exploits a priori information on target models, implements joint estimation of scattering and motion parameters, and circumvents TFD and other tough issues, such as clutter suppression, detection, RCM correction, and parameter estimation which have to be conducted separately and explicitly in SAR/GMTI.

This paper is organized as follows. In Section 2, geometrical optics (GO), physical optics (PO), and graphical electromagnetic computation (GRECO) are used to calculate the backscattering or complex radar cross-section (RCS) of a PRA. Then, based on the calculated results, we propose a new scattering center model and a hybrid scattering/motion model in Section 3. We then in Section 4 analyze the characteristics of PRAs. A few interesting signatures of PRAs are revealed, for example, periodicity of high-resolution range profile (HRRP) sequences, bowl- and bowknot-shaped images. Sections 5 and 6 provide a maximum likelihood estimator and a few results, and conclusions are reported in Section 7. The derivation throughout the paper may be a little difficult to follow, and we will try our best to clarify the problem.

2. Complex RCS Calculation for PRA

Backscattering of an antenna consists of two parts, that is, structured scattering, whose scattering principle is the same of as common objects, and pattern scattering, which is closely related to antenna’s loads. Pattern scattering is weak for a PRA and, hence, not taken into account here. We will calculate PRA’s complex RCS via GO, PO, and GRECO. GO is simple and is particularly suitable for smooth surface. PO has high accuracy near target’s normal direction and can provide closed-form solutions in some particular aspects. And GRECO, as a tool and software, incorporates many electromagnetic methods and can efficiently provide high-precision numerical solutions.

2.1. GO

As Figure 1 shows, let 𝑃 denote the focus parameter of a PRA, 𝜏, and 𝛿 denote the radius and angle (relative to -z axis) of any point on a PRA. Other parameters encountered are defined in Figure 1 where the focus is located at the origin. 𝑂-z is the symmetrical axis. Then, the amplitude of the PRA’s complex RCS, that is, the square root of RCS, can be obtained from the GO theory [8] √𝜎_GO=√𝜋𝑃cos²𝜂. (1)

Figure 1 

PRA geometry (𝑥-𝑧 slice).

Its phase can be easily derived from the position of the specular point 𝑆. Combining the amplitude and phase, we have the complex RCS √𝜎_GO=√𝜋𝑃cos²𝜂𝑃exp−𝑗𝐾2cos𝜂, (2) where 𝐾=4𝜋𝑓/𝑐 is the wavenumber, 𝑐 is the speed of light, and 𝑓 is the frequency. When choosing PRA’s bottom center 𝑂₁ as the phase reference, we need to multiply (2) with exp(𝑗𝐾⋅𝑃/2⋅cos𝜂) and then have √𝜎_GO=√𝜋𝑃cos²𝜂exp−𝑗𝐾𝑃sin²𝜂.2cos𝜂 (3)

Equations (2) and (3) hold only when 𝜂 is chosen such that the specular point does not move outside the PRA. Specifically, when the incident field illuminates the PRA front side, 𝜂 should be subjected to 𝜂𝐷≤arctan.2𝑃 (4)

2.2. PO

To facilitate application of PO, we use the Cartesian coordinate system and the cylinder coordinate system. Then in Figure 1, for any point 𝑄 in the PRA, suppose that its position in the cylindrical coordinate system is (𝜌,𝜑,𝑧)=(𝜌,𝜑,𝜌²/(2𝑃)−𝑃/2). It can be vectorized in the Cartesian coordinate system as 𝐫^′=𝑥,𝑦,𝑧^𝑇=𝜌𝜌cos𝜑,𝜌sin𝜑,²−𝑃2𝑃2^𝑇. (5)

The normal vector at 𝑄 is 𝐧=−𝜌cos𝜑𝑃,−𝜌sin𝜑𝑃,1^𝑇/𝜌1+²𝑃². (6)

The backscattering direction can be expressed by 𝐫̂=sin𝜂,0,cos𝜂^𝑇. (7)

According to PO, the complex RCS of the PRA can be written as √𝜎_PO𝑘=−𝑗√𝜋₀^2𝜋₀^𝐷/2𝐫̂⋅𝐧exp𝑗𝐾𝐫̂⋅𝐫^′𝜌1+²𝑃²𝜌𝑑𝜌𝑑𝜑. (8)

Generally, (8) can be calculated by using numerical methods, and its closed-form solution cannot be obtained due to its complexity. However, the closed-form solution exists for some particular incident angles. When 𝜂=0, for example, the complex RCS we derive is √𝜎_PO𝜂=0=−𝑗2√𝐾𝐷𝜋𝑃sin²𝑃16𝑃exp−𝑗𝐾2−𝐷²=√16𝑃𝐷𝜋𝑃exp−𝑗𝐾²𝐷16𝑃−exp𝑗𝐾²𝑃16𝑃×exp−𝑗𝐾2−𝐷².16𝑃 (9)

2.3. GRECO

GRECO is an effective tool for calculating (complex) RCS of electrically large objects [9]. It is capable of calculating the reflected and diffracted fields with high precision and efficiency by using graphic hardware acceleration and can also perform well for a wide range of aspect angles. Therefore, we select it as a numeric method and benchmark for the GO and PO methods. The result will be given in Section 6.

3. Scattering Center Model and Hybrid Model

3.1. Migratory Scattering Model for a Stationary PRA

The scattering center models have developed from the point scattering model, damped exponential/Prony model [10], through the geometric theory of diffraction (GTD) model [11], localized scattering model [12], and attributed scattering model [13], to the polynomial model [14] and migratory scattering model [15]. Their difference lies in that they use different models to characterize the dependency of complex RCS, including amplitude and/or phase, on frequency and/or aspect angle. We can find from (2) that, for a PRA, the complex RCS amplitude depends on aspect angle 𝜃 (𝜃=𝜂 when aspect angle is relative to PRA’s axis of symmetry), resulting in aspect sensitivity, while the phase also depends on aspect angle but the dependency is not of the form of sin𝜃 or cos𝜃 as ideal point scattering is (1/cos𝜂 instead), which means that the scattering center will move when the aspect angle varies. By (2), we have in effect proven that a PRA can be approximated by the migratory scattering model containing a single scattering center, and the approximation has sound physical grounds. However, the amplitude term of complex RCS in (2), enough as it is for coarse analysis, is not an accurate model intended for parameter estimation, since GO does not take into account more scattering components, for example, edge diffraction. Therefore, we use a function, 𝑠(𝜂_𝜃), to describe arbitrary aspect dependency. Then, from (3), we obtain the migratory scattering model for a stationary PRA 𝑔(𝐾,𝜃)=𝑠(𝜃)exp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃+𝑃sin²𝜂_𝜃2cos𝜂_𝜃, (10) where (𝑥,𝑦) are the coordinates of the PRA’s bottom center 𝑂₁, and 𝜂_𝜃 is the 𝜃-dependent incident angle.

For instance, When choosing PRA’s geometrical center as the phase reference (in order for agreement with the GRECO geometry in Figure 3 and Section 4), we have 𝑔𝐾,𝜃^′=𝑠𝜃^′exp−𝑗𝐾sin𝜃^′4+𝑃sin²𝜋/2−𝜃^′2cos𝜋/2−𝜃^′=𝑠𝜃^′exp−𝑗𝐾sin𝜃^′4+𝑃cos²𝜃^′2sin𝜃^′. (11)

When 𝜃^′=𝜔𝑡 is around 90^∘ (PRA’s front against radar), where 𝜔 is the angular frequency and 𝑡 is the slow time, the instantaneous Doppler can be obtained by taking the derivative of the phase in (11) 𝑓_𝑑≈17.5𝜆cos(𝜔𝑡)≈8.75𝜋𝜆−17.5𝜔𝜆𝑡. (12)

Equation (12) indicates that the PRA’s return in azimuth is a sinusoidal frequency-modulated (SFM) signal or approximately a linear frequency-modulated (LFM) signal when the PRA is illuminated broadside.

3.2. Hybrid Model for a RPRA in the SAR Geometry

It is well known that a PRA usually possesses rotation in operation. Thus, we need to establish its motion model by finding the range from SAR to the specular point 𝑆. For the SAR geometry shown in Figure 2, 𝑂 is the center of the scenario illuminated by SAR. Assume that a PRA is rotating around its vertical axis, and the pivot is the top of the vertical axis, identical to 𝑂₁. The angle between the 𝑂-𝑧^′ axis and the PRA’s axis of symmetry, 𝜙_𝑐, remains constant during the rotation around 𝑂₁𝑂₂. 𝜃_𝑐=−arctan(𝑌_𝑐/𝑋_𝑐) is the slant angle. When 𝑌_𝑐=0, SAR is side-looking, and otherwise it operates in squint-looking mode. 𝜃 is the angle of light of sight (LOS) in 𝑂-𝑥𝑦 (i.e., slant plane). The relationship between 𝜃 and 𝑡 is 𝑋_𝑐tan𝜃=𝑉_𝑎𝑡−𝑌_𝑐. (13)

Figure 2 

SAR geometry with a RPRA.

Figure 3 

PRA geometry for GO, PO and GRECO.

When 𝜃_𝑐=0 and the extent of aspect angle 𝜃 is small, tan𝜃≈𝜃, and then (13) reduces to 𝑉𝜃=^𝑎𝑋_𝑐𝑡. (14)

Clearly, 𝜃 also has the meaning of slow time.

For deriving the range model, we at first make the following definitions.

In 𝑂-𝑥𝑦 at slow time 𝑡, 𝑃_𝑡 is at (−𝑋_𝑐,𝑌_𝑐−𝑉_𝑎𝑡) (SAR’s position), 𝑂₁ is at (𝑥,𝑦) (bottom center of the PRA).

In 𝑂-𝑥^′𝑦^′𝑧^′ at slow time 𝑡, the normalized vectors of 𝑂₁𝐹 and OH are 𝐧_𝑂1𝐹= (cos γ sin ϕ_c, sin γ sin ϕ_c, cos ϕ_c), 𝐧_OH = (cos α₀ sin β₀, sin α₀ sin β₀, cos β₀),where 𝛾=𝜔𝑡+𝛾₀, 𝜔=2𝜋𝑓_𝑚 is the rotating angular frequency and 𝛾₀ is the initial phase, both unknown, and 𝛼₀ as well as 𝛽₀ are known for parameter estimation in Section 5.

Then, we can easily have cos𝜂_𝜃=𝐧_𝑂1𝐹⋅𝐧_{𝑂1𝑃𝑡}≈𝐧_𝑂1𝐹⋅𝐧_𝑂𝑃𝑡≈𝐧_𝑂1𝐹⋅𝐧_OH=sin𝜙_𝑐sin𝛽₀cos𝛾−𝛼₀+cos𝜙_𝑐cos𝛽₀, (15) where the approximation is because 𝑂₁𝑃_𝑡 is approximately parallel to 𝑂𝑃_𝑡 and even to OH when the extent of 𝜃 is not large.

We can also easily have 𝑃_𝑡𝑂₁=𝑉_𝑎𝑡−𝑌_𝑐+𝑦²+𝑋_𝑐+𝑥². (16)

From (3) or (10), we have 𝑂₁𝑆⋅𝐧_{𝑃𝑡𝑂1}=𝑃sin²𝜂_𝜃2cos𝜂_𝜃. (17)

Then using the far-field plane wave assumption and (16) as well as (17), we can obtain the range from 𝑃_𝑡 to 𝑆 at slow time 𝑡𝑅(𝑡)=^def𝑅_𝜃=𝑃_𝑡𝑂₁+𝑂₁𝑆⋅𝐧_{𝑃𝑡𝑂1}=𝑉_𝑎𝑡−𝑌_𝑐+𝑦²+𝑋_𝑐+𝑥²+𝑃sin²𝜂_𝜃2cos𝜂_𝜃≈𝑉_𝑎𝑡−𝑌_𝑐²+𝑋²_𝑐+𝑥cos𝜃+𝑦sin𝜃+𝑃sin²𝜂_𝜃2cos𝜂_𝜃, (18) where 𝜂_𝜃 can be calculated by (15).

After range compression and discarding insignificant factors representing the beam pattern and window functions, SAR returns can be written in the frequency slow time domain as 𝐺(𝐾,𝜃)=𝑠(𝜃)exp−𝑗𝐾𝑅(𝑡). (19)

When processing spotlight SAR data with algorithms such as the polar format algorithm (PFA), one usually compensates the returns with the scenario center’s phase, that is, to multiply (19) with exp(𝑗𝐾(𝑉_𝑎𝑡−𝑌_𝑐)²+𝑋²_𝑐). Hence, we obtain the resulting returns 𝑔(𝐾,𝜃)=𝑠(𝜃)exp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃+𝑃sin²𝜂_𝜃2cos𝜂_𝜃(𝜃)+𝑃=𝑠exp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃21cos𝜂_𝜃−cos𝜂_𝜃=^𝐼_𝑖=0𝑏(𝑖)𝜆_𝑖(𝜃)+𝑃×exp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃21cos𝜂_𝜃−cos𝜂_𝜃, (20) which is named the hybrid scattering/motion model, or migratory/micromotion scattering center (MMSC) model, where 𝜆_𝑖(𝜃) are orthogonal basis functions. It is of the same form as (10) derived based on a turntable, but herein it is derived from the SAR geometry and its 𝜂_𝜃 is determined by (15). This model has incorporated motion parameters into the scattering center model, providing the rationale for joint estimation of scattering-motion parameters. It will be seen in next section that the model provides a very good representation of a PRA. By the way, the 2D inverse FT of (20), interpolation probably needed, will produce an SAR complex image.

Assume that 𝜃 is minor and near 0^∘, 𝜃_𝑐=0^∘, and 𝜂_𝜃 is around 0^∘ or 180^∘. Then, we have 1cos𝜂_𝜃≈2−cos𝜂_𝜃,when𝜂_𝜃isaround0^∘,−2−cos𝜂_𝜃,when𝜂_𝜃isaround180^∘. (21)

Applying (14), (15), and (21) to (20) leads to𝑔()(𝜃)𝐾,𝜃≈𝑠exp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃−𝑃cos𝜂_𝜃+𝑃signcos𝜂_𝜃(𝜃)×𝑉≈𝐶⋅𝑠×exp−𝑗𝐾_𝑎𝑦𝑡𝑋_𝑐−𝑃sin𝜙_𝑐sin𝛽₀cos𝜔𝑡+𝛾₀−𝛼₀, (22) where 𝐶 is an insignificant unknown constant factor. The instantaneous Doppler, or micro-Doppler, can be obtained by taking the derivative of the phase 𝑓_𝑑≈−2𝜔𝑃sin𝜙_𝑐sin𝛽₀𝜆sin𝜔𝑡+𝛾₀−𝛼₀−2𝑉_𝑎𝑦𝜆𝑋_𝑐. (23)

Both (22) and (23) indicate that the RPRA’s azimuth return is an SFM signal, but in effect, it can be approximated by an LFM signal due to broadside flash.

4. Radar Characteristics of a PRA and Validation of the MMSC Model

The analysis proceeds with radar characteristics of a PRA, both stationary and rotating, in terms of RCS, HRRPs, TFD, and SAR images. The results from GO, PO, and GRECO are all based on the following geometry and parameters.

As shown in Figure 3, the origin is now set at its geometrical center 𝑂₂ according to our GRECO software’s definition. 𝑂₁𝑂₂=𝑂₂𝑂₃=0.25 m. The PRA’s depth 𝑂₁𝑂₃=0.5 m, 𝑃=9 m, and 𝐷=6 m. The frequencies are from 9.8 GHz to 10.2 GHz stepped by 0.01 GHz with central frequency 𝑓_𝑐=10 GHz (therefore the bandwidth is 400 MHz and the range resolution is 0.375 m), and the aspect angles are from −180^∘ to 180^∘ stepped by 0.1^∘. Then, we use GRECO to calculate the complex RCS of a PRA on a turntable (equivalent to radar’s circling a stationary PRA, at an angular frequency 𝜔=𝜃^′/𝑡). We now analyze the calculated results.

4.1. Stationary PRA

4.1.1. RCS Characteristics

As shown by Figures 4(a) and 4(b), the PRA’s RCS exhibits strong dependency on aspect angle 𝜃^′. Its scattering is intense (about 250 m² or 24dB⋅m², far larger than the PRA’s aperture area 𝜋(𝐷/2)²≈28m²) when the LOS deviates from its axis of symmetry by no more than about 18^∘. Otherwise its RCS abruptly declines to zero. This phenomenon is generally called broadside flash [16]. From (4), |𝜂|≤arctan[6/(2×9)]≈18.4^∘, which agrees with GRECO results (solid line) and means that GO has correctly predicted the aspect supporting interval. Figure 4(c) shows that GO can also give a coarse prediction for the RCS-𝜃^′ function by frequency-free (1). For more accurate description, 𝑠(𝜃) is therefore used in (10).

(a)

(b)

(c)

Figure 4 

PRA RCS’s aspect dependency. (a) GRECO, at carrier frequency 10 GHz; (b) GRECO, at different frequencies; (c) GRECO and GO, at carrier frequency 10 GHz.

From Figure 4(b), we can also find that the RCS-𝜃^′ curves are almost identical for different frequencies, except when 𝜃^′=±90^∘, indicating weak frequency dependency of RCS. This is confirmed by Figure 5(a), which shows the RCS-frequency functions are valued between 22 and 26 in dBm², a small range, regardless of aspects (except 𝜃^′=±90^∘). Therefore, our MMSC model, though not considering RCS’s dependency on frequency, that is, 𝑠(𝜃) does not contain 𝑓 in (20), is judged reasonable from this viewpoint. However, when the LOS is perpendicular to the PRA’s bottom, RCS takes on obvious oscillation for PO and GRECO (Figure 5(b)). The reason may be stated in terms of multiple scatterers; that is, there may be additional scattering centers besides the specular point predicted by GO at this aspect.

(a)

(b)

Figure 5 

RCS’s frequency dependency. (a) GRECO, at different aspects; (b) GRECO, PO, and GO, 𝜃′=90∘.

4.1.2. HRRP Characteristics

Figure 6 depicts the HRRP sequence of the PRA, derived from GRECO data. Since HRRPs at some aspects are too weak to display in Figure 6(a), we modify their amplitudes for every aspect into a normalized value, as shown in Figure 6(b). We will analyze its HRRP characteristics from the following points.(1)Overall, strong HRRPs occur periodically when 𝜃^′∈[−108^∘,−72^∘]∪[72^∘,108^∘], which is consistent with our analysis of RCS. The HPPRs’ intensity is almost identical for front and back illumination.(2)When illuminated broadside (𝜃^′=±90^∘), the PRA has two scattering centers at ±0.25m with the same amplitude (two peaks in Figure 6(a); only for this aspect, the bandwidth is 2 GHz as opposed to 400 MHz to resolve them). This can be interpreted by (9) given by PO, the second square-bracketed term of which clearly shows that the positions of the two scattering centers are ±𝐷²/(16𝑃)=±0.25m. And the conclusion also confirms our conjecture on the reason of RCS oscillation in Figure 5(b). In effect, the two centers correspond to the PRA’s bottom center and edge.(3)When 𝜃^′∈[−108^∘,−72^∘]∪[72^∘,108^∘] and 𝜃^′≠±90^∘, the PRA assumes a single scattering center. When the PRA’s front side is illuminated, the center moves further with deviation of the LOS from the PRA’s symmetry axis. The case is just the opposite when the back side is illuminated. This conclusion also agrees with our model’s prediction (see (10) and (11)). More specifically, in Figure 3, when 𝜃^′=108^∘, the scattering center should move to sin𝜃^′/4+𝑃cos²𝜃^′/(2sin𝜃^′)≈0.69m, which is in good agreement with the reality in the top-right corner of Figure 6(a). Beyond this range, the center will move outside the PRA and therefore disappear.(4)When 𝜃^′∉[−108^∘,−72^∘]∪[72^∘,108^∘], the HRRPs are too weak and we have to resort to Figure 6(b). Clearly, for these aspects, the scattering is rather complicated due to disappearing of the specular point, and more scattering centers are present beyond the prediction of GO and our models. Fortunately, these HRRPs are so weak that they can be safely ignored.

(a)

(b)

Figure 6 

HRRP sequences of the PRA with (a) original and (b) normalized amplitude. GRECO data are used.

4.1.3. TFD Characteristics

We also plot the TFD according to the PRA’s complex RCS at the central frequency, only for aspects at which the PRA’s RCS is strong. The PRA is placed on a “rotatable” turntable, and thus the aspect angle has the sense of slow time. Figure 7(a) shows the TFD based on GRECO-derived data. it is clear that the PRA contains three main scattering centers. One is the specular point, related to the slant line in Figure 7(a). The slant line agrees well with (12) and its graphical presentation in Figure 7(b). Two is at the PRA’s edge, that is, points 𝐴 and 𝐵 in Figure 3, related to the top and bottom curves in Figure 7(a). Each curve is effectively a small fraction of a sine wave. We now try to explain the vertical line in the middle of Figure 7(a). When 𝜃^′=90^∘, different parts of the PRA have wide-range relative velocities, from 0 corresponding 𝑂₁ to the maximum and minimum corresponding to point 𝐴 and 𝐵 in Figure 3. Therefore the instantaneous Doppler at 𝜃^′=90^∘ contains all the components throughout between the maximal, and minimal Doppler, leading to a vertical line in Figure 7(a). In addition, it is clear that, though there are several components in Figure 7(a), the slant line has the strongest intensity. Thereby it is reasonable for our scattering center models to contain only the specular scattering component.

(a)

(b)

Figure 7 

TFD of the PRA with data from (a) GRECO and (b) our PRA scattering center model.

4.1.4. 2𝐷 Image Characteristics

The 2D images of the PRA are formulated by the convolution back-projection (CBP) algorithm, displayed in Figure 8 for narrow and wide angles, respectively. From Figure 8(a), three scattering centers are present, with the middle bright one at the PRA’s bottom center and the other darker two at its edge. This is in accordance with Figure 7(a). However, in the wide-angle case (Figure 8(b)), the PRA exhibits a bowl-shaped signature, and the bowl’s size is equal to the true size of the PRA. The two points corresponding to the PRA’s edge are still visible at the bowl’s edge. The bowl shape is due to the migration of the specular point within such a wide angle. From Figure 6 it is well known at different aspects, that the specular point is located at different positions, and therefore they cannot be coherently integrated into a single strong point in the resulting image. Instead, the energy is spread over the bowl shape. Moreover, there is an other weak energy distributed at the left and right edges of Figure 8(b). This energy results from other types of scattering from the PRA, for example, creeping wave. Again, the PRA, when its back side faces radar, will assume similar signatures to Figure 8, but the bowl’s direction is the opposite. To summarize, 2D images provide a very good representation of PRA’s structure, and the specular scattering contributes to PRA’s dominant energy.

(a)

(b)

Figure 8 

2D images of the PRA in a turntable, by CBP, using GRECO data. (a) 𝜃′∈[88∘,92∘]; (b) 𝜃′∈[0∘,180∘].

4.2. Rotating PRA

When a PRA is rotating in the SAR geometry, its radar characteristics will considerably vary due to the rotation. In what follows, we proceed with the analysis of an RPRA based on the MMSC model since the real SAR data of an RPRA are unavailable for the moment. Simulation parameters are shown in Table 1.

Table 1 

SAR and RPRA’s parameters.

4.2.1. RCS, HRRP, and TFD Characteristics

Suppose 𝑓_𝑚=0.3 Hz, which is also a typical value for real RPRAs. In Figure 9(a), as is expected, the RPRA’s RCS sequence takes on periodicity due to the rotation and the period is about 3.3 s, in good agreement with the true value 1/𝑓_𝑚. Its TFD also shows the periodicity (Figure 9(b)). In any segment, the slantline-shaped TFD represents an LFM signal, or a fraction of a SFM signal, as expected by (23). In effect, a sine wave is impossible to be displayed in the TF image owing to broadside flash. Note also that the slant lines may be blurry due to short time and the TF-resolution limit. This kind of periodicity is also characteristic of the HRRP sequence in Figure 9(c). However, a curve similar to the left one in Figure 6(a) is not present. This is because that the back of the PRA has no chance to be illuminated by SAR for the parameters in Table 1.

(a)

(b)

(c)

Figure 9 

RPRA’s characteristics. (a) RCS sequence; (b) TFD; (c) HRRP sequence.

4.2.2. SAR Image Characteristics

The presence of rotation changes the Doppler history of the PRA’s returns, thus bringing out new signatures distinct from of the stationary PRA. When the PRA rotates at a small frequency, it takes on blurred lines in the SAR image (Figure 10(a)). When it rotates more rapidly, the image takes on a bowknot shape with several intersectant lines (Figure 10(b)). The faster the PRA rotates, the more lines there are. When 𝑓_𝑚=2 Hz, we cannot distinguish these lines from each other. As a result, the image exhibits another kind of bowknots as shown in Figure 10(c). We also find that the angular extent of the bowknots is approximately equal to that of SAR, that is, 10^∘, in Figures 10(b) and 10(c), after the units of the two axes are equalized. These interesting signatures can provide new features for target recognition, notwithstanding that the underlying reason is enigmatic and remains to be explored in the future.

(a)

(b)

(c)

Figure 10 

SAR images of an RPRA, formulated by CBP, when (a) 𝑓𝑚=0.2Hz, (b) 𝑓𝑚=0.3Hz, and (c) 𝑓𝑚=2Hz.

4.2.3. Effective Illumination Condition

In Figure 9(c), we have found that the back of the PRA is not illuminated by SAR. In effect, even if it is illuminated, there is not backscattering strong enough if the incident angle is too large. In such a case, the illumination is not effective. Then, what is the condition of effective illumination? Clearly, |𝜂_𝜃|≤Δ or 𝜋−Δ≤|𝜂_𝜃|≤𝜋+Δ needs and only needs to be satisfied where Δ is half the aspect supporting interval (e.g., 18^∘). Substituting it into (15) and after some simple manipulations, we find that the sufficient and necessary conditions for effective front or back illumination are, respectively, 𝛽₀−𝜙_𝑐𝛽≤Δ₀+𝜙_𝑐≥𝜋−Δ. (24)(25)

If (24) and (25) are combined, we readily obtain a necessary condition of both sides being illuminated 𝜋Δ≥3. (26)

However, in general, Δ<𝜋/4 for PRAs. Therefore, it is impossible for both sides of a PRA to be effectively illuminated.

The analyses made earlier have fully supported our scattering center models and provided deep insight into PRA’s characteristics. These characteristics can be used as feature for SAR image interpretation and target detection or recognition. They can also be used for parameter’s coarse estimation as shown in Section 5.

5. MLE of Parameters in the MMSC Model

Once the MMSC model is built, the estimation for parameters in this model can be performed from measured data, and there are various ways available, for example, MLE [14], the regularization method [17], and the sparse representation method [15], though they were not intended for our problem herein. Actually, from a unified viewpoint, all of them belong to optimization algorithms for solving inverse problems. In this paper, we develop an MLE method for the MMSC model, which is equivalent to the nonlinear least squares estimation [14]. Note that only one PRA target is assumed for simplicity, but it can be generalized for multiple targets by introducing RELAX or CLEAN techniques, thereby circumventing the estimation of the model order.

5.1. Maximal Likelihood Estimator

In (20), 𝑠(𝜃) has been parameterized in basis functions. For notational expediency, (20) can be further vectorized as 𝐠=Φ(𝜗)𝐛+𝐯, (27) where 𝐠 is the measured echo vector, 𝐯 is the noise vector, 𝜗=^def(𝑥,𝑦,𝑓_𝑚,𝛾₀,𝜙_𝑐,𝑃) is unknown, 𝐛=^def[𝑏(0),…,𝑏(𝐼)]^𝑇 is unknown, and 𝑒=_𝐾𝜃^defexp−𝑗𝐾𝑥cos𝜃+𝑦sin𝜃+𝑃sin²𝜂_𝜃2cos𝜂_𝜃,Φ=^def𝑒_𝐾1𝜃10⋱0𝑒_{𝐾1𝜃𝑀}…𝑒_{𝐾𝑁𝜃1}0⋱0𝑒_{𝐾𝑁𝜃𝑀}𝜆₀𝜃₁…𝜆_𝑂𝜃₁𝜆⋮⋮⋮₀𝜃_𝑀…𝜆_𝑂𝜃_𝑀, (28) where the derivation of Φ is not difficult but tedious, therefore not given here.

We try to solve for 𝜗 and 𝐛 from (27). Clearly, this is an inverse problem. Using MLE, we at first obtain 𝜗’s estimate 𝜗=argmin_𝜗𝐠−ΦΦ^𝐻Φ⁻¹Φ^𝐻𝐠²₂. (29) Afterwards, by substituting 𝜗 into Φ, we have the least-squares estimate of 𝐛𝐛=Φ^𝐻Φ⁻¹Φ^𝐻𝐠. (30)

Finally, by use of 𝐛, the reflectivity function can be found as 𝑠̂(𝜃)=^𝐼_𝑖=0𝑏̂(𝑖)𝜆_𝑖(𝜃). (31)

5.2. Determination of the Initial Value of 𝜗

Equation (29) is an optimization function, and it is easy to cause local minima. One may either try to find efficient global optimization algorithms, which have not been found to date, or to determine initial values with higher precision. The latter line of thought entails a thorough understanding of the target characteristics, which, fortunately, has been analyzed in detail in Section 4. Therefore, we give the coarse estimation method for 𝜗.

(𝑥,𝑦): From Figure 10, the center of an RPRA’s blurred image indicates the position of the scattering center. So, (𝑥,𝑦) can be estimated, respectively, by 𝑥=∬𝑥^′⋅𝐼𝑥^′,𝑦^′𝑑𝑥^′𝑑𝑦^′∬𝐼𝑥^′,𝑦^′𝑑𝑥^′𝑑𝑦^′𝑦=∬𝑦^′⋅𝐼𝑥^′,𝑦^′𝑑𝑥^′𝑑𝑦^′∬𝐼𝑥^′,𝑦^′𝑑𝑥^′𝑑𝑦^′, (32) where 𝐼(𝑥^′,𝑦^′) is the gray-level image.

𝑓_𝑚: It can be estimated by finding the rotating period from an HRRP sequence (see Figure 10) and then taking its reciprocal.

𝜙_𝑐 and 𝑃: We have not found any effective method for estimating them separately. However, from (23), after obtaining 𝑓_𝑚 and 𝜔, we can find 𝑃sin𝜙_𝑐 as a whole by estimating the Doppler slope. 𝑃 can be set as a mean value from a priori knowledge, at the order of meters. 𝜙_𝑐 can then be determined accordingly.

𝛾₀: It is hard to be estimated. We just take a set of values of it as the coarse estimates; so, the objective function will be optimized with different initial values, resulting in multiple local minima, the minimum of which is finally taken as the global minimum. We find through simulation that the optimization is not sensitive to the choice of 𝛾₀’s initial values.

6. Simulation Results

In this section, we present some estimation results by the proposed method in Section 5. 𝑓_𝑚=0.3Hz, 𝛼₀=𝜋, 𝛽₀=𝜋/4, the signal-to-noise ratio (SNR) is −3 dB, and other parameters are shown in Table 1. The Dirac function is chosen as the basis function. The optimization algorithm we use is sequential quadratic programming (SQP), which has proven successful in solving our problem.

The RPRA’s SAR image is shown in Figure 11, which is similar to Figure 10(b) except for being contaminated by noise. The coarse estimates for initial values are shown in Table 2. Using the estimator proposed, we finally obtain the estimates of the PRA’s parameters, which, obviously, agree well with their true values. To our best knowledge, there is not any published work on extracting the overall information of an RPRA in SAR and so, overall comparison cannot be performed. Once the six parameters are obtained, the coefficients of the basis functions can be found and therefore the reflectivity function 𝑠(̂𝜃) results (Figure 12). Note that 𝜃 is proportional to slow time by (14). Figure 12 indicates that the reflectivity is successfully recovered, and the periodicity implies the broadside flash signature which can be favorably used in target recognition. Also, the square of the reflectivity is accordant with the RCS sequence in Figure 9(a). Although the conventional method also gives estimates the reflectivity by evaluating the mean of the returns’ amplitude sequence, it has poorer agreement with the reality, and more problematically, this method needs finding range cells containing micromotion targets at first, which is in itself a tough job but is now circumvented by our approach through building the MMSC model in the phase domain.

Table 2 

Final estimated results.

Figure 11 

PRA’s image under SNR=−3dB, by CBP.

Figure 12 

Recovered reflectivity with respect to slow time.

Again, we can see through the slices of the objective function shown by Figure 13 that improper initial values of 𝑓_𝑚 will lead to local minima, but the coarsely estimating precision for it is able to guarantee the convexity of the objective function within a necessary interval. it is similar to other parameters except 𝑃 which tolerates larger error of initial estimates (Figure 13(b)).

(a)

(b)

Figure 13 

Objective function slices along (a) the rotating frequency and (b) the focus parameter.

7. Conclusion

As a special type of targets in SAR, RPRAs have a few interesting characteristics which can be exploited for understanding SAR images. In this paper, we have at first obtained the closed-form and numeric solutions for PRA’s backscattering, via GO, PO, and GRECO, respectively. From its GO solution, a new scattering center model has been proposed for characterizing the migration of the specular point on the PRA, and then the rotating parameters have been incorporated into the model to obtain a hybrid model or MMSC model in SAR geometry. Using the models and the scattering data calculated, we have in detailed investigated the radar characteristics of a PRA and RPRA alike. Their RCS, HRRP sequences, and TFD all show that a PRA exhibits broadside flash as well as periodicity and contains a single migratory scattering center for limited aspects, whose energy is dominant. The bowl-shaped image of a stationary PRA reflects its structure perfectly, while the SAR image of an RPRA assumes distinct signatures. All these characteristics have confirmed the MMSC model, and they also provide novel information on targets’ structures and motions. Then, using this information for coarse estimation of PRA’s parameters, we have developed a maximal likelihood estimator to solve the MMSC model. Simulation results at −3 dB of SNR demonstrate the estimator’s performance. The parameters estimated, for example, the rotating frequency and focus parameter, can faithfully record the information on PRA’s motion and size and therefore can be used advantageously as features for target recognition. However, we have concentrated only on a PRA in a homogeneous scenario which can be characterized by white Gaussian noise and lacked consideration of clutters stronger and more complex. Additionally, selection of basis functions has not been discussed, either. Notwithstanding its limitation, this study provides a basis for interpretation of SAR images containing PRAs, and the idea of hybrid modeling is also instrumental for other fields such as SAR/GMTI.

Acknowledgment

The authors acknowledge the support from Provincial Science Foundation for Distinguished Young Scholars in Hunan (no. 11JJ1010) and National Science Fund of China (no. 61171133, no. 61101182, and no. 61025006).