The FDHM antenna and the corresponding CI system were numerically investigated in our previous work (Han et al., 2021). The FDHM antenna is designed with frequency diversity characteristics. Similar to the concept of frequency diversity, polarization diversity in optics is also an interesting direction (Hu et al., 2021). The structure of the metasurface element is shown in Figure 1A. The top layer of the element is a square patch that is printed on a Rogers 3006 substrate backed by a ground plane. The gap g is modified in the operating band to change the surface impedance of the element. The surface impedance of the element can be calculated as Z = j Z 0 φ c ω p 2 − 1 , (1) where Z ₀ is the impedance of the free space wave, p = 2.2 mm is the period of elements, φ is the phase difference of elements, ω is the angular frequency, and c is the speed of light. The relationship between surface impedance and frequency corresponding to different g values is shown in Figure 1B. As can be seen, for a fixed gap g, the surface impedance of the element changes along with the frequency. On the other hand, for a fixed frequency, the gap g can also modulate the surface impedance of the element. Therefore, by randomly designing the gap g for each element, one could realize a frequency-diverse metasurface antenna, which has been validated through full-wave simulation (Han et al., 2021). The fabricated FDHM antenna used in this paper is shown in Figure 2A. The panel size of the FDHM is 70 mm × 70 mm, operating from 18 GHz to 26 GHz. The gap g of each element is randomly generated within the working frequency band. A monopole antenna is designed at the center to excite the surface waves. When the antenna is working, various radiation patterns at different frequencies are formed. The measured |S ₁₁| is in accord with the simulated one, as plotted in Figure 2B, suggesting that the proposed FDHM antenna can operate in broadband from 18 to 26 GHz.

The frequency-diverse 2-D pattern of the FDHM antenna is measured by planar near-field scanning in a microwave anechoic chamber. The measurements range from 18 to 26 GHz, with a frequency step of 100 MHz. The scanning grid along the horizontal and vertical directions is 6 mm by 6 mm. The center of the scanning plane is aligned with the center of the entire array at a distance of 200 mm. The patterns at 18, 20, 21, 22, 24z, and 26 GHz are shown in Figure 3A. To more clearly see the correlation between different frequency patterns, correlation analysis was conducted on the above 81 groups of patterns. The correlation results are shown in Figure 3B. From the results, it can be clearly seen that two-dimensional patterns of different frequencies differ greatly from each other, which is exactly the requirement of the CI system (Zhao et al., 2020; Gollub et al., 2017).

The mathematical model of CI imaging is briefly introduced. Consider a 2 × 2 MIMO metasurface antenna using the fabricated FDHM antenna as the sub-aperture. Among the four sub-apertures, two FDHM antennas act as transmitters while the other two antennas act as receivers. According to the first Born approximation condition, the mathematical model of the 2 × 2 MIMO metasurface antenna can be expressed as g = H f + n , (2) where g is a column vector of echo signals that are radiated from the two transmitters and acquired by the two receivers, H is the measurement matrix, f is the objects column vector to be imaged, and n represents the noise vector. Because the 2 × 2 MIMO metasurface antenna is utilized, for each frequency measurement, we could obtain four equations in Eq. 2, which are g m i , j = ∑ n = 1 N E m i r → n · E m j r → n · f n ,   i , j = 1,2 , (3) where m denotes the frequency domain sensing index, i denotes the transmitter, j denotes the receiver, n denotes the pixel of the scene, E m i r → n and E m j r → n denote the transmitted and received electric fields. Actually, we could drop the i and j indices of Eq. 3 because one can readily swap the row vectors in a linear equation formulated by Eq. 2. Thus, the element in the measurement matrix can be denoted as H m , n T ⋅ H m , n R , which is the multiplication of the transmitted and received patterns. We note that these patterns had been measured in an anechoic chamber as mentioned before.

It is assumed that the total number of frequency measurement modes is M and the scene is discrete with N pixels. Generally, the number of measurement modes is much smaller than scene pixels (M < N), which means Eq. 2 is a sick problem and cannot be directly analyzed. Based on the CS theory, we can solve this problem when the object is a compressible image with randomly distributed patterns. Researchers have proposed many algorithms from different angles to deal with Eq. 2 defined in the CS reconstruction problem (Bioucas-Dias and Figueiredo, 2007; Zhu and Bamler, 2012; Rao et al., 2013), and the imaging system in this paper is solved by TwIST algorithm.

The complex matrix H in the imaging system is not a full-rank matrix, so it is necessary to evaluate the performance of matrix H. As presented above, the designed FDHM antenna can generate multiple distinct radiation patterns in the frequency domain. It is important to evaluate the correlation between various radiation patterns. In this paper, we use µ (H) to describe the mutual interference characteristics of the measurement matrix H, which can be defined as μ H = ∑ m ≠ n H ∼ m , n T ⋅ H ∼ m , n 2 M M − 1 , (4) where H ∼ is the sensing matrix with all columns normalized and T denotes the transpose symbol (Duarte-Carvajalino and Sapiro, 2009). The smaller the coherence is, the better imaging reconstruction becomes. At the same time, the singular value analysis of the matrix is introduced to characterize the coherence of the measurement modes.

As mentioned above, the measurement matrix H in this paper is constructed by the discrete sampling of radiation field patterns of various frequency points in broadband. Generally speaking, the imaging performance depends on the orthogonal measurement modes, and this conclusion can be obtained from Eq. 3. It is worth noting that the orthogonal characteristics between the measurement matrix of the FDHM antenna are essentially the pseudo-random characteristics of the radiation modes in the target field, namely the dispersion characteristics. FDHM antenna has the characteristic of frequency diversity, and the number of different measurement modes in the target scene is determined by the system bandwidth. If the frequency sampling interval is dense, it will lead to different degrees of radiation mode overlap, increase data redundancy, and weaken the performance of the system. This characteristic also limits the performance of the frequency diversity antenna imaging system.

In this paper, ART is adopted, which can obtain more radiation modes and significantly improve the imaging performance of the system. In this section, the ART of the MIMO system is mainly discussed. The ART increases the system degree of freedom from two aspects: 1) The spatial position of multiple aperture antennas is different; 2) as the aperture antenna rotates around its panel axis, the field pattern in the target scene changes. In theory, the aperture antenna can be rotated around the axis at any angle, and a new measurement matrix can be obtained with each rotation while maintaining a low correlation with each other. For the imaging scene, the radiation pattern of the target scene changes with the rotation operation. The designed antenna has the characteristic of polarization insensitivity, which is conducive to the MIMO aperture rotation technology. According to the method, the number of measurement modes can be effectively increased. Considering the complexity of calculation, only four special angles 0°, 90°, 180°, and 270° are discussed in this paper. The rotation diagram of the 2 × 2 MIMO metasurface antenna integrating four FDHM antennas using ART is presented, including the states of 0°, 90°, 180°, and 270° as shown in Figures 4A–D. The two transmitters are labeled as Tx1 and Tx2, while the two receivers are labeled as Rx1 and Rx2.

The correlation of the measurement matrix generated by different rotation operations is shown in Figure 5. During this process, the whole aperture is rotated at 0°, 90°, 180°, and 270° four modes. Note that the premise of this process is that the four antennas in Figure 4 rotate 0°, 90°, 180° and 270° around their panel axes respectively. It can be seen from Figure 5A that the average correlation value µ (H) gradually decreases with the rotation operation. The singular value curve flattens with the increase of measurement modes in Figure 5B, which further verifies the effectiveness of our proposed method. We found that this method can reduce the correlation values, produce more efficient radiation patterns, and also greatly increase the number of measurement patterns. The imaging performance of the proposed method will be verified in the next section.