A New Quarter Concave Cylinder Linked Dihedral Reflector for Fully Polarimetric Calibration of Wideband Nonreciprocal Radar Systems

Traditionally, fully polarimetric calibration of a nonreciprocal radar system requires measurements of at least two passive calibrators, such as a dihedral corner reflector plus a metal plate or sphere. Interchanging measurements of multiple calibrators results in not only higher complexity but also degraded uncertainty. In this work, a new polarimetric passive calibrator is proposed, which is designed as quarter concave cylinder linked dihedral (QCCLD). The backscattering of a single QCCLD contains both depolarizing and nondepolarizing components when rotating along the radar line of sight (LOS). This unique characteristic makes it an excellent polarimetric calibrator, which allows fully polarimetric calibration of a nonreciprocal radar system by measuring just a single QCCLD. The theoretical polarimetric scattering matrix (PSM) is derived based on physical optics (PO). Using complex exponential (CE) model-based parametric representation, a novel polarimetric calibration procedure is developed to suppress undesirable scattering components, which degrade the calibration accuracy. Experimental calibration results are presented with the polarization isolation improvement of more than 15 dB over 6–18 GHz frequency band, demonstrating the usefulness of the proposed QCCLD calibrator for fully polarimetric calibration of a wideband nonreciprocal radar system.

Abstract-Traditionally, fully polarimetric calibration of a nonreciprocal radar system requires measurements of at least two passive calibrators, such as a dihedral corner reflector plus a metal plate or sphere. Interchanging measurements of multiple calibrators results in not only higher complexity but also degraded uncertainty. In this work, a new polarimetric passive calibrator is proposed, which is designed as quarter concave cylinder linked dihedral (QCCLD). The backscattering of a single QCCLD contains both depolarizing and nondepolarizing components when rotating along the radar line of sight (LOS). This unique characteristic makes it an excellent polarimetric calibrator, which allows fully polarimetric calibration of a nonreciprocal radar system by measuring just a single QCCLD. The theoretical polarimetric scattering matrix (PSM) is derived based on physical optics (PO). Using complex exponential (CE) model-based parametric representation, a novel polarimetric calibration procedure is developed to suppress undesirable scattering components, which degrade the calibration accuracy. Experimental calibration results are presented with the polarization isolation improvement of more than 15 dB over 6-18 GHz frequency band, demonstrating the usefulness of the proposed QCCLD calibrator for fully polarimetric calibration of a wideband nonreciprocal radar system.

I. INTRODUCTION
P OLARIMETRIC scattering matrix (PSM) provides an exquisite description of the interaction between electromagnetic (EM) wave and radar target, which plays important roles in scattering diagnosis and target identification [1], [2]. In practice, antenna crosstalk and channel imbalance distort the received signal in polarimetric radar systems, which have to be solved through fully polarimetric calibration. There are mainly two tasks in polarimetric calibration of a radar system: 1) design an appropriate polarimetric calibrator, which has known theoretical PSM, and 2) develop measurement and processing procedures to obtain all the distortion parameters from measured data. Since the standard polarimetric calibration signal model has been proposed in the 1990s [3], [4], different polarimetric calibrators and improved calibration methods have been proposed for specific requirements [5], [6], [7], [8], [9]. Recently, in terms of the new calibrators, Monzon [10] proposed a unidirectional conducting canonical object to satisfy the requirements as a cross-polarized bistatic calibration device. Olk et al. [11] proposed a wire mesh with high cross-polarization level for the calibration of monostatic and bistatic radar cross section (RCS) facility operating at W-band. A mainlobe steered dihedral (MSD) object was proposed by Beaudoin et al. [12], which can be applied to bistatic polarimetric calibration. Kong and Xu [13] proposed a rhombus-shaped dihedral, which can be used for polarimetric calibration and background clutter extraction simultaneously. Ali and Perret [14] proposed an augmented depolarizing circular scatterer based on resonant elements, which performs well in a compact range. For improved calibration methods, Muth [15], [16] proposed a nonlinear calibration technique based on Fourier analysis, suppressing the effects of system drift and background clutter in measurement environment effectively. Wu and Xu [17] proposed an improved calibration technique for the case of quasi-monostatic polarimetric measurement system, which is very common in RCS test ranges.
For fully polarimetric calibration of a nonreciprocal radar system, there are eight distortion parameters needing to be solved, i.e., four crosstalk parameters and four polarization channel gain factors [9]. Measurements of a single traditional dihedral corner reflector cannot provide enough independent equations, resulting in requirement of a second nondepolarizing calibrator, such as a metal plate, sphere, or cylinder, and so on. In other words, the measurements of at least two different conventional calibrators are required for fully polarimetric calibration of a nonreciprocal radar system. Extra workload and complexity are added in polarimetric measurements when interchanging calibrators and concerning with the accurate position and orientation.
To simplify the measurement and calibration procedure, a new single polarimetric calibrator consisting of a quarter concave cylinder linked dihedral (QCCLD) reflector was first proposed by Wu and Xu [18], [19], which can accomplish This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ fully polarimetric calibration for nonreciprocal radar systems by measuring just itself. In this article, starting from our previous work, further in-depth studies of the QCCLD calibrator are made as follows. The scattering mechanisms are analyzed in detail. In addition, the theoretical PSM varying with rotation angle along the radar line of sight (LOS) is derived based on physical optics (PO). A novel polarimetric calibration procedure based on parametric representation using complex exponential (CE) model [20], [21], [22], [23] is presented to suppress the interference of undesirable scattering components for enhanced accuracy. Finally, experimental results are presented to validate the usefulness of the QCCLD polarimetric passive calibrator.
This article is organized as follows. The three-dimensional (3-D) geometry, theoretical PSM, and scattering mechanism of the proposed QCCLD calibrator are analyzed in Section II. In Section III, the polarimetric measurement signal model and the CE model parametric representation-based calibration procedure are presented. Experimental measurements and calibration results are illustrated in Section IV with analysis to validate the proposed QCCLD calibrator. We summarize this article in Section V.

A. Geometry and Theoretical PSM
The previous work [19] has suggested that the frequency dispersion characteristic of a concave cylindrical surface is more stable than a metal plate. Besides, less diffraction wave and interaction with the concave cylinder are excited by a triangular-shaped dihedral than a rectangular-shaped one. As illustrated in Fig. 1, the QCCLD calibrator is designed as a combination of a separated triangular-shaped dihedral and a quarter of concave cylinder, which is determined by three parameters, i.e., the height h, the radius of the concave cylinder r , and the width of the triangular plate w.
When the calibrator is rotating with an angle of θ along the radar LOS, the PSM can be expressed as  where S cyl and S dih are the scattering components of concave cylinder and dihedral at 0 • rotation, respectively, which are derived in Appendix-A based on PO. Substitute (A13) and (A18) into (1), the theoretical backscattering PSM of the QCCLD calibrator can be written as where H and V stand for horizontal and vertical polarizations, respectively.

B. Backscattering Mechanism Analysis
The backscattered field of QCCLD calibrator consists of four kinds of mechanisms, i.e., the specular reflection wave, the multiple reflection wave, the diffraction wave, and the surface wave. The scattering mechanisms are illustrated in Fig. 2, where the EM wave incidents along the y-axis. The number markers of different scattering centers (SCs) are detailed in Table I.  The ultrawideband (UWB) backscattering RCS from 100 MHz to 36 GHz with a 100 MHz frequency step of a specific QCCLD calibrator with h = 190 mm, w = 96 mm, and r = 48 mm is calculated using method of moment (MoM) code of FEKO software [24]. Fig. 3

(a) and (b)
shows the RCS and high-resolution range profile (HRRP) at 0 • rotation. A Chebyshev window function is used to suppress the sidelobe. The RCS increasing with frequency shows a periodical oscillation characteristic due to the vector addition of SC1 and SC2, corresponding with (2) and (5) of the theoretical PSM. In addition, the traveling wave of SC8 can only be excited at VV polarization at this rotation angle [25]. Fig. 4(a) and (b) shows the RCS and HRRP at 45 • rotation. It is seen that SC1 dominates in co-polarization component, while SC2 dominates in cross-polarization component. The fully polarimetric HRRP sequences of QCCLD calibrator varying with rotation angle along radar LOS are shown in Fig. 5, where Hamming window function is used. It is seen that the SC1 keeps constant, while the SC2 varies periodically with the rotation angle.

III. POLARIMETRIC MEASUREMENT SIGNAL MODEL AND CALIBRATION PROCEDURE A. Polarimetric Signal Model Based on QCCLD Calibrator
The measurement signal model for polarimetric calibration of a nonreciprocal radar system can be described in the form of matrices as [9] where M is the measurement signal matrix, S denotes the true PSM of target, and T and R represent the transmitting and receiving distortion matrices, respectively. The undesirable signals B and N stand for the clutter and noise matrices, respectively.
In practical measurements, the signal-to-noise ratio (SNR) can be high enough by pulse accumulation, so the effect of N can be ignored. The background clutter B of measurement environment can usually be well suppressed by vector subtraction or Fourier analysis method proposed by Muth [15], [16]. Then, the polarimetric measurement signal model can be simplified as with Normalize the receiving and transmitting distortion matrices as and ε T V = T HV /T VV refer to the crosstalk parameters of receiving and transmitting polarization channels.
The polarization channel gain matrix can be constructed by pointwise product as where g HH , g HV , g VH , and g VV represent the polarization channel gain factors. Therefore, the polarimetric calibration model with eight distortion parameters of a nonreciprocal radar system is expressed as where the operator ⊙ stands for a Hadamard product. Substitute the theoretical PSM of QCCLD into (12), we have

B. Polarimetric Calibration Procedure
It is seen that the fully polarimetric measurement signals in (13) and (16) have the general form of M pq = a pq + c pq cos 2θ + s pq sin 2θ (17) where the subscripts p and q stand for the electric polarization vectors either H or V for the receiver and transmitter. The Fourier analysis method can be used to obtain all coefficients a pq , c pq , and s pq from measurement data to suppress the effect of background clutter [15], [16]. The signal can be described as the Fourier series M = a 0 + c 1 cos θ + s 1 sin θ + c 2 cos 2θ + s 2 sin 2θ + · · · (18) Mathematically, the coefficients a pq , c pq , and s pq of QCCLD calibrator correspond to a 0 , c 2 , and s 2 of each polarization [19]. Take HH polarization for example, we have Assuming the calibrator rotates from 0 to 2π with 2N angle samples in measurement, the coefficients for any polarization can be calculated from measured data as In fact, the coefficients a The scattering components separated by Fourier analysis method from the rotated QCCLD MoM data shown in Fig. 5 are illustrated in Fig. 6. A Hamming window function is used to suppress the sidelobe in HRRPs.
It is seen from Fig. 6 that there are undesirable components excited by interaction, diffraction, and surface waves, resulting in RCS deviation deviating from the theoretical PSM derived from PO. For example, the interferential component in Fig. 6(b) at about 0 cm is caused by the double-bounce interaction between concave cylinder and dihedral, i.e., SC3 in Fig. 2. To suppress the interference of undesirable components, a CE model-based parametric representation approach [20], [21], [22], [23] is used to extract the main SC.
The CE model expression of target scattering function is  where M is the model order or the number of scattering components;a i , α i , and r i are the complex amplitude, the frequency dispersion factor, and the distance from phase center of the ith SCs, respectively, which can be estimated by state space approach (SSA) [22]; f is the radar frequency vector.
Assuming that the kth SC can be reconstructed by the m 1 th to m 2 th scattering components, the scattering function of the kth SC can be written as where m 1 and m 2 of the kth SC can be determined by the boundary locations r m1 and r m2 of the main lobe in the HRRP domain, respectively. In practice, considering the widening of the main lobe caused by the distortion parameters (crosstalk parameters and channel gain factors), the locations r m1 and r m2 are obtained by the following mathematical optimization as: where r m0 is the peak value location of the main lobe in HRRP. The objective function is minimized by gradually increasing the distance between r m1 and r m2 . The SC extraction result from Fig. 6 is shown in Fig. 7. It is seen that after SC extraction based on CE model parametric representation, the undesirable components are well suppressed and the main SCs agree well with the theoretical value of PO. In addition, the RCS uncertainty of CE model representation is shown in Appendix-B. After separation and extraction of the three scattering components for each polarization, the distortion parameters can be solved by the polarimetric calibration model described above.
The polarimetric calibration procedure for nonreciprocal radar systems using the QCCLD calibrator is summarized in four steps.
Step 1: Obtain the fully polarimetric measurement data from a nonreciprocal radar system of the QCCLD calibrator rotated along the radar LOS.
Step 2: Separate the scattering components using the Fourier analysis method. Then, extract the main SCs using CE modelbased parametric representation approach.
Step 3: Solve the polarimetric calibration model to obtain the eight distortion parameters of the nonreciprocal radar system.
Step 4: Use the obtained distortion parameters for fully polarimetric calibration of the radar target under test.
The flowchart of the polarimetric calibration procedure is shown in Fig. 8.

IV. EXPERIMENTAL RESULTS
The fully polarimetric measurement experiments based on the proposed QCCLD calibrator are carried out in an indoor test range. Fig. 9 illustrates the manufactured QCCLD calibrator and the nonreciprocal polarimetric radar system. The  size of the QCCLD is the same as described in Section II-B with h = 190 mm, w = 96 mm, and r = 48 mm. The radar system consists of two wideband dual-polarized horn antennas and a vector network analyzer (VNA). The size of the two antennas is 85 mm and the distance between them is 100 mm, resulting in a quasi-monostatic angle of 1.06 • . The configuration of the experiment is illustrated in Fig. 10. The targets under calibration are a square metal plate sized 150 mm in width and a triangular-shaped dihedral whose height and width are 300 and 150 mm, respectively. The calibrator and targets under test are mounted on a metal pylon coated with radar absorbing material (RAM). The measurement parameters are listed in Table II. Fig. 11 shows the measured fully polarimetric magnitude of the QCCLD calibrator at the center frequency of 12 GHz. It is seen that the mean values of the co-polarization signals are not zero due to the concave cylinder scattering component.   Fig. 12 illustrates the measured and fully polarimetric calibrated data for a target square metal plate. It can be seen that, before calibration, the polarization isolation of the measured data is about 30 dB. On the other hand, after polarimetric calibration, it becomes more than 45 dB in most cases over the 6-18 GHz frequency band, 15 dB better than the raw data.
In Fig. 13, the results of a target triangular-shaped dihedral with 0 • rotation before and after polarimetric calibration are presented, respectively. The dihedral is a dominant copolarization target under this rotation. It is found that for most cases, the cross-polarization component is about 50 dB down from the co-polarization component after calibration, 20 dB better than the uncalibrated data.

V. CONCLUSION
In this work, a new QCCLD calibrator is proposed for fully polarimetric calibration of nonreciprocal radar systems. The PSM of the calibrator consists of both depolarizing and nondepolarizing scattering components as rotating along the radar LOS. Therefore, fully polarimetric calibration for nonreciprocal radar systems can be accomplished through measurements of just a single QCCLD calibrator, greatly simplifying the measurement procedure. With the using of a CE model-based parametric representation, the interference of undesirable components is well suppressed during the fully polarimetric calibration process. Theoretical analysis and experimental results demonstrate that the QCCLD can be an excellent candidate for either indoor or outdoor uses.

A. Theoretical PSM of QCCLD Based on PO Solution
The relationship between the target scattering matrix and RCS is shown as [26] where S pq denotes the element of the PSM, and σ pq represents RCS.
According to the PO theory [25], the element of the PSM can be expressed as where k wavenumber; n s unit normal of the illuminated surface; e r unit vector along scattering polarization; h i unit vector of incident magnetic field; ⃗ r position vector from origin to the surface patch da; i unit vector of incident wave; s unit vector of scattering wave; A illuminated region of the surface. For the separated dihedral component, considering the location of double-bounce scattering as the phase reference center, the Cartesian coordinate system xyz is established in Fig. 14.
Suppose that the incident EM wave first illuminates the left plate, then reflect to the right plate and finally back to the receiver, the scattering component can be expressed as where dm and dn are the integral elements along the direction of width and height, respectively. In Fig. 14 Considering the scattering component that the incident EM wave first illuminates the right plate, we have S 21 = S 12 . Therefore, the separated dihedral component of QCCLD can be derived as The concave cylinder component is a quarter of cylindrical surface from a right circular cylinder, whose height and radius are h and r , respectively, shown in Fig. 15.
The PO integral of the quarter concave cylinder component can be expressed as In the case of backscattering, considering the angle limit of the integral, (A14) can be simplified as Stationary phase method is used to solve the integral in (A15). The phase function can be expanded in a Taylor series, and all terms beyond the second derivative are ignored. The stationary phase method can be described as [25] The integral in (A15) can be solved as Substitute (A17) into (A15), the concave cylinder component of QCCLD can be written as When the QCCLD is rotating along the radar LOS, the PSM can be expressed as (2)- (5) in Section II-A.

B. RCS Uncertainty of CE Model Parametric Representation
In Section III-B, the CE model-based parametric representation approach is used to extract SCs of the three scattering Take HH polarization as an example for the constant and the second-order cosinusoidal components and HV polarization as an example for the second-order sinusoidal component, the RCS error caused by CE model parametric representation is illustrated in Fig. 16. It is seen that the RCS uncertainty is within ±0.01 dB, which is accurate enough for polarimetric calibration.

C. Calibration Sensitivity Analysis of a Rotating QCCLD
In practical applications of the rotating QCCLD, the calibration accuracy is related to various factors, including the location and posture of the calibrator, and the bistatic angle between the transmitting and receiving antennas, and so on. For instance, the incident EM wave can deviate from the expected orientation due to the imprecise placement of the calibrator. In this section, the calibration sensitivity of the rotating QCCLD is analyzed considering the RCS error and polarization purity affected by the monostatic, bistatic, and rotation angular errors.

1) Monostatic Angular Sensitivity:
The radar EM wave with a monostatic angular error of α is illustrated in Fig. 17(a), where the QCCLD is rotating along the expected LOS with an angle of θ . The RCS characteristics are calculated using the MoM code of FEKO software [24] at the central frequency of 12 GHz of our experiment for HH and HV polarizations. The HV polarization RCS versus rotation angle with α = 2 o is shown in Fig. 17(b), demonstrating the RCS curves of the MoM result, the Fourier analysis result, and the difference between them. Using Fourier analysis method, the RCS error and the polarization purity versus monostatic angular error of constant, second-order cosinusoidal, and secondorder sinusoidal components are illustrated, respectively, in Fig. 17(c) and (d).
It is seen from Fig. 17 that, with the angular error, an extra fourth-order sinusoidal component occurs [17]. Besides, the monostatic angular error of up to 0.7 • is allowed for the RCS errors of all the three components less than 0.1 dB. In our experiment, the accessible accuracy can be better than ±0.1 • using an electronic total station, so this error can be neglected. In addition, as seen from Fig. 17(d), the monostatic angular error of up to 2 • makes no influence on the polarization purity.
2) Bistatic Angular Sensitivity: A symmetrical bistatic angle of β between incident and scattering EM waves is illustrated in Fig. 18(a). The bistatic angle-sensitive RCS characteristics of QCCLD are shown in Fig. 18(b)-(d).
From Fig. 18, the bistatic angular error of less than 1 • is allowed for the RCS error no more than about 0.1 dB. In the experiment, a bistatic angle of 1.06 • is formed because of the separation between transmitter and receiver. In practice, either increasing measurement distance or closer installation of the two antennas can reduce the impact of bistatic angular error. In addition, it seems that a bistatic angular error of  up to 2 • makes no noticeable influence on the polarization purity.
3) Rotation Angular Sensitivity: A misaligned rotation angle error of θ on the rotating QCCLD is shown in Fig. 19(a). The RCS characteristics in this case are shown in Fig. 19(b)-(d).
In Fig. 19(c), it is seen that the RCS error caused by the rotation angular error of up to 2 • impacts little on the calibration by using the Fourier analysis method. However, this angular error makes a severe impact on the polarization purity, as seen from Fig. 19(d), where it is found that a rotation angular error of less than 0.1 • is allowable for a polarization purity better than 50 dB. The accessible mechanical accuracy of rotation angle can be about ±0.1 • by using a digital gradienter in our experiment.
In general, the quality of polarimetric calibration depends on the exact position, orientation, and alignment of the rotating QCCLD. In practical applications, the calibration error can be effectively reduced by a careful installation with highprecision instruments and using the Fourier analysis method.