Scattering vectors

In comparison with full polarimetric mode, the number of values which have to be measured can be reduced, leading to the name compact polarimetry. The scattering vector of full polarimetric systems can be written as

(1)$$ {\vec {k}}_L = [S_{HH};S_{HV};S_{VH};S_{VV}], $$

where $S_{ab}$ represent the complex backscattering terms. These can be transformed into a circular basis using

(2)$$\left[ {\matrix{ {S_{LL}} & {S_{LR}} \cr {S_{RL}} & {S_{RR}} \cr } } \right] = 0.5\left( {\left[ {\matrix{ 1 & {-i} \cr 1 & i \cr } } \right]\left[ {\matrix{ {S_{HH}} & {S_{HV}} \cr {S_{VH}} & {S_{VV}} \cr } } \right]\left[ {1\,\quad 1-i\,\quad i} \right]} \right),$$

according to [Reference Knott, Tuley and Shaeffer16], while the indices H, V and L, R represent linear horizontal or vertical and circular left- or right-handed polarizations.

The scattering vector of DCP is reduced to just two values. If a right-handed circularly polarized antenna is used to transmit, the target vector will contain the values

(3)$$ {\vec {k}}_{DCP} = [S_{RR};S_{RL}]. $$

If a left-handed corresponding antenna transmits, all subscripts have to be inverted. The target vector of the hybrid mode CTLR, which will be used in this contribution, can be declared as

(4)$$ {\vec {k}}_{CTLR} = [S_{RH};S_{RV}], $$

if a right-handed circularly polarized antenna is used. For left-handed transmission, the subscript R has to be replaced by L.

Sensor system

The presented sensor system, which can be seen assembled in Fig. 1(a), has been designed to compare many different possibilities of several classification issues in automotive applications especially for solutions concerning compact polarimetry. Detailed information about the basic set-up and the circuit of the RF-fronted ofFig. 1(b) can be found in a previous contribution [Reference Erhart, Lutz, Walter, Mantz, Hügler and Weigel17]. The generated waveform can be programmed to CW or FMCW. Measurements in the following sections are conducted using frequency ramps of 1 GHz bandwidth and 1 ms duration. The transmitted power deviation over frequency can be seen plotted in (Fig. 2).

Fig. 1.

Presented sensor system. (a) Capsuled sensor system (b) RF-frontend.

Fig. 2.

Deviation of transmitted power during frequency chirp.

The two types of antennas will be explained in the subsequent subsection. Both have been rotated or mirrored to achieve linear and circular polarizations in both alignments or directions of rotation respectively. Each antenna is placed twice on the RF-PCB so that all combinations can be measured in bi-static mode and direct comparisons are possible. This array can be seen in the middle of Fig. 1(b) and sketched and magnified in Fig. 3. The amplification, as well as the control of the system via SPI, is done on separate PCBs. The A/D conversion is done externally, thus compact geometrical dimensions and a modular setup are guaranteed. During operation, the power amplifiers of the TRX modules can be controlled using a USB connection.

Fig. 3.

Patch array of the microstrip antennas with different polarizations.

Microstrip patches

Linearly polarized patch

The linearly polarized microstrip antenna, which is represented in Fig. 4, excited in ${\rm TM}_{10}$ mode exclusively, radiates a wave that is polarized linearly and parallel to the direction of the feed. Its impedance matching is adjusted to a frequency bandwidth from about 76–78 GHz. The quality of this matching is shown in Fig. 5(a). To measure the polarization which is transmitted by the microstrip patch, a pyramidal horn was used as a receiving component placed in and turned around boresight direction of the microstrip antenna which is transmitting a constant frequency of 77 GHz [Reference Golio18, p. 6–127]. The axial ratio, which is defined as the ratio between the minor and major axis of the polarization ellipse, can be gathered from the diagram in Fig. 5(b) by dividing greatest by smallest value. While ideal linearly polarized antennas would exhibit infinite axial ratio values, the measured value of about 35 dB is sufficient for the kind of technical usage in this contribution.

Fig. 4.

Sketch of a linearly polarized microstrip antenna.

Fig. 5.

Impedance bandwidth and axial ratio measurement results of a linearly polarized microstrip antenna. (a) Impedance beamwidth (b) Meas. axial ratio.

Circularly Polarized Patch

In general, there are two different ways to radiate circular polarization using microstrip antennas [Reference Richards, Lo and Harrison19]. Firstly, there are dual-feed antennas which rely on feeding networks that accomplish equal amplitudes and a $90^\circ $ phase difference between two orthogonal excitation points. Secondly, there are antennas with just one feed which are dependent on the existence of two modes orthogonal to each other. In comparison, the dual-feed antennas offer a higher impedance and axial ratio bandwidth but a lower 3 dB axial ratio beamwidth and they are in need of more space due to the feeding network [Reference Lo and Lee20, p. 10–57]. Because the two latter mentioned properties are important criteria for a good performance of our sensor system, microstrip antennas with one feed have been chosen.

Many different shapes of radiating devices with a single feed have been investigated, yet[Reference Gao, Luo and Zhu21, p. 1–9]. The presented sensor system is equipped with truncated corner square microstrip patches, which are designed to obtain a resonance frequency of 77 GHz in both diagonal directions. The truncation of two corners, which is shown in Fig. 6, leads to the presence of a higher mode with a phase difference of $90^\circ $. These two radiating electric fields are not parallel to the linearly polarized microstrip patches resulting in low crosstalk. By truncation of the other two remaining corners, the sense of direction can be reversed. The impedance bandwidth is shown in Fig. 7(a). The axial ratio measurements have been performed using a pyramidal horn antenna as the receiving component and turning around boresight direction in the same way as before. The two extremes can be found at the expected angles around $\pm 45^\circ $ and lead to an axial ratio value of about 1 dB.

Fig. 6.

Sketch of a truncated corner square microstrip antenna transceiving right-handed circular polarization.

Fig. 7.

Impedance matching and axial ratio measurement results of a circularly polarized truncated corner square microstrip antenna. (a) Impedance bandwidth (b) Meas. axial ratio.

Dielectric Lens

The shape of the dielectric lens, as shown in Fig. 1(a), is used to focus the linearly and circularly polarized waves. Because of the recognition of double bounces and the characterization of surface textures are of primary interest, focusing is requested in elevation direction only. Hence, a planoconvex cylindrical lens, which is made of polyetherimide, has been chosen as the beam-forming component. By optimizing this lens two aims have to be achieved. Firstly, the focal plane has to be adapted to the distance between the lens and the before mentioned mircrostrip antennas. The simulated results regarding the distribution of the electric field are represented in the contour plot in Fig. 8. The microstrip antennas are placed at the origin of this Cartesian coordinate system. Secondly, the geometry of the lens has to be adjusted until the desired emission attunes. As described before, a narrow focusing in the direction of elevation is required while the electromagnetic waves should spread widely orthogonal thereto. A one-way antenna diagram of the dielectric lens fed by one of the circular microstrip antennas is plotted in Fig. 9. Dashed lines represent results from simulations, solid ones are based on measurements.

Fig. 8.

Simulated distribution of the E-Field effected by a plane wave approaching from top.

Fig. 9.

One-way antenna diagram of the dielectric lens fed by a microstrip antenna.

Evaluation measurements

A perfect object to check or calibrate a polarimetric radar system is a square angled dihedral. The polarimetric backscattering depends on the angle between the orientation of the polarization plane of the antenna and the axis of the dihedral. The scattering matrix can be evaluated by

(5)$$\left[ {\matrix{ {S_{HH}} & {S_{HV}} \cr {S_{VH}} & {S_{VV}} \cr } } \right] = \left[ {\matrix{ {\cos (2\psi )} & {\sin (2\psi )} \cr {\sin (2\psi )} & {-\cos (2\psi )} \cr } } \right],$$

in which ψ represents the rotation angle of the dihedral in boresight direction [Reference Huynen4]. The results from several measurements conducted with linearly polarized antennas and varying angle ψ can be depicted descriptively using the Pauli decomposition [Reference Jong-Sen Lee22, p. 214f]. Here the scatter matrix is divided into the Pauli basis and the corresponding coefficients α, β, and γ.

(6)$$\left[ {\matrix{ {S_{HH}} & {S_{HV}} \cr {S_{VH}} & {S_{VV}} \cr } } \right]{\rm } = \displaystyle{1 \over {\sqrt 2 }}\left( {\alpha \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right] + \beta \left[ {\matrix{ 1 & 0 \cr 0 & {-1} \cr } } \right] + \gamma \left[ {\matrix{ 0 & 1 \cr 1 & 0 \cr } } \right]} \right).$$

Furthermore, to check the circularly polarized results, they can be transformed into a linear basis using

(7)$$\left[ {\matrix{ {S_{HH}} & {S_{HV}} \cr {S_{VH}} & {S_{VV}} \cr } } \right] = 2\left( {\left[ {\matrix{ 1 & 1 \cr i & {-i} \cr } } \right]\left[ {\matrix{ {S_{LL}} & {S_{LR}} \cr {S_{RL}} & {S_{RR}} \cr } } \right]\left[ {\matrix{ 1 & i \cr 1 & {-i} \cr } } \right]} \right),$$

before the Pauli decomposition can be applied [Reference Knott, Tuley and Shaeffer16, p. 74]. The coefficients can then be calculated by the following equations

(8)$$\eqalign{\alpha = & S_{HH} + S_{VV} \cr \beta = & S_{HH}-S_{VV} \cr \gamma = & S_{HV} + S_{VH}.} $$

These coefficients are squared and plotted in its common color code red, blue, and green in Fig. 10. As expected, the values of the first Pauli coefficient, α, which corresponds to single and odd bouncing, are very low compared with the other ones in both diagrams. Values ofthe second and third coefficient, β and γ represent double- and even-bouncing with the difference of having their maximum at $0^\circ $ and $\pm 90^\circ $ or at $\pm 45^\circ $, respectively. Theoretical values, calculated according to equations (5) to (7), confirm the reliability of the sensor system and are plotted in the same figure in dashed lines of the same color. Simulated coefficient α is $-\inf $ at all angles and has been omitted for clarity.

Fig. 10.

Measured and simulated Pauli values calculated from linearly and circularly polarized sets of the antenna. (a) linearly polarized (b) circularly polarized.

If single or double bounce is the only information which is desired, compact polarimetric approaches can be taken in consideration, too. Using DCP, the interpretation of measured values is straightforward regarding the differentiation of single and double bounces. Because circularly polarized waves change their sense of rotation at each bounce, it is sufficient to compare the amplitudes of the two receiving channels. The magnitude at the cross-polarized configuration $S_{RL}$ represents the amount of single and odd bounces while the co-polarized channel $S_{RR}$ receives the power of double and even bounces, so no further calculations have to be executed. The two values of the target vector from equation (3) are plotted in Fig. 11(b) using the same measurement set-up as before.

Fig. 11.

Compact polarimetric approaches to distinguish single from double bounces. (a) Meas. DCP (b) Meas. CTLR.

This simple analysis is in contrast to the evaluation process of CTLR-mode. Using this compact polarimetric approach, desired ratios of even and odd bounces have to be calculated. In this contribution, the three-component compact decomposition proposed by Cloude et al. [Reference Cloude, Goodenough and Chen23] will be applied. Initially, the Stokes vector of the received signal has to be calculated. This can be done using the values of the corresponding target vector from equation (4) and the following equations.

(9)$$\eqalign{S_0 = & \vert S_{RH}\vert ^2 + \vert S_{RV}\vert ^2 \cr S_1 = & \vert S_{RH}\vert ^2-\vert S_{RV}\vert ^2 \cr S_2 = & 2 \Re(S_{RH}S_{RV}^{\ast}) \cr S_3 = & -2\Im(S_{RH}S_{RV}^{\ast}).}$$

In these equations, $S_{kl}$ denotes the complex voltage in the subscripted combination of polarization, asterisks denote the complex conjugate and ℜ and ℑ the real and imaginary part of the value in brackets, respectively. The Stokes vector can be used to calculate the amount of the dihedral power $P_D$, the surface scattering power $P_S$ and the volume power $P_V$.

(10)$$\eqalign{P_D = & \displaystyle{{1} \over {2}} \left(\sqrt{S_1^2 + S_2^2 + S_3^2} + S_3\right) \cr P_S = & \displaystyle{{1} \over {2}}\left(\sqrt{S_1^2 + S_2^2 + S_3^2}-S_3\right) \cr P_V = & S_0-\sqrt{S_1^2 + S_2^2 + S_3^2}.} $$

The values of $P_S$ and $P_D$ are plotted in Fig. 11(b) for the same measurements as before using a dihedral asa target witha varying angle in boresight direction.

Double bounce recognition

The recognition of double bounces is an issue in automotive applications which is mentioned frequently regarding false targets caused by additional reflections at crash barriers. Because these signals have been modified by a car or other moving targets, they are much harder to distinguish as signals reflected by stationary targets without a Doppler modification. Figure 12a shows a measurement set-up to create double bounce reflections and single bounce scatters. Two plates have been adjusted at $\pm 10^\circ $ from boresight direction. In Fig. 12(b), in which the results of the Pauli-decomposition are plotted against the distance, the shorter single bounce path is highlighted with blue color and the longer double bounce path is marked in red color. The first Pauli coefficient, α is plotted dashed and reaches its maximum in the blue highlighted area while the second coefficient climaxes in the red area. Thus a separation of single and double bounces using the Pauli-decomposition is possible. Subsequent diagrams in Fig. 13 show the results gathered from the two compact polarimetric modes explained in the sections before. Again, solid lines represent the measured double bounce results and dashed lines the single bounce correspondents. As a separation in these diagrams is clearly visible, both compact polarimetric modes can be utilized in this field of application.

Fig. 12.

Sketch and measurement results of feasibility set-up for double bounce recognition using PAULI decomposition. (a) Sketch of measurement set-up (b) Meas. PAULI-coefficients.

Fig. 13.

Measurement results of feasibility set-up for double bounce recognition using DCP and CTLR. (a) Meas. DCP (b) Meas. CTLR.

Conclusion

A sensor system, which is able to perform compact polarimetric as well as full polarimetric measurements has been designed and put into operation successfully. The antennas, which were used to perform the polarized transceiving have been explained and the dielectric lens as the beam-forming component has been characterized. Measurements to characterize the radar system and to separate single from double bounces have been conducted with full polarimetric as well as the two compact polarimetric approaches DCP and CTLR. These led to good and comparable results in this chosen automotive scenario and further investigations are planned. One subsequent field of application is the characterization of surface textures particularly with respect to the classification of road conditions.