1. Introduction
Synthetic aperture radar (SAR) is an all-day, all-weather active microwave remote sensing system, which has become an indispensable means for Earth observation [
1]. High resolution can provide finer feature information, and wide imaging can provide a broader observation scene. Therefore, high-resolution wide-swath (HRWS) imaging is a significant development direction for a modern SAR system [
2,
3,
4,
5,
6,
7,
8]. However, the conventional SAR system cannot realize both azimuth high-resolution and range wide-swath imaging due to the minimum antenna area constraint [
4,
9,
10,
11,
12]. At present, the basic working modes of SAR systems, such as ScanSAR [
13], Spotlight SAR [
14], TOPS [
15], and Mosaic [
16], achieve a compromise between azimuth resolution and range swath by allocating and adjusting the synthetic illumination time of ground scene. However, they cannot fundamentally solve the contradiction between imaging swath and resolution. The displaced phase center multiple azimuth beams (DPC-MAB) SAR system [
9,
10,
12,
17,
18] arranges multiple phase centers along the azimuth direction with equal spacing to receive scene echoes, which equivalently increases the sampling rate in time dimension by increasing sampling rates in the spatial dimension. Then, the alias-free full azimuth spectrum is reconstructed by the digital beamforming (DBF) technique to alleviate constraints of azimuth resolution and range swath on pulse repetition frequency (PRF) [
2,
3,
4,
6,
12,
17]. The DPC-MAB technique has become one of the effective means for realizing HRWS SAR imaging and has been successfully deployed on TerraSAR-X [
19], RADARSAT-2 [
20], ALOS-2 [
21], GF-3 [
22,
23], LT-1 [
24], and other azimuth multichannel (AMC) SAR systems.
Theoretically, the HRWS SAR image can be well focused after AMC signal reconstruction. However, due to error factors, such as temperature variations, antenna radiation pattern difference, and undesirable receiver components, there are often inconsistencies in amplitude, phase, range sampling time, and antenna phase center locations between channels. These imbalances among channels, especially amplitude and phase inconsistencies, can significantly reduce the suppression effect of spectrum ambiguity in subsequent reconstruction processing algorithms, which, in turn, introduces virtual targets on the final image and seriously degrades the quality of SAR imaging. Therefore, channel imbalance consistency calibration is a crucial step in AMC SAR data processing [
19,
25]. Amplitude error can be accurately corrected by the channel balancing technique [
26]. Generally speaking, range sampling time imbalance (RSTI) can be accurately measured and compensated by an internal calibration system. Meanwhile, the azimuth time-domain cross-correlation (ATC) method can also effectively estimate the RSTI from echo data [
27]. For the spaceborne SAR system, the position error of the antenna phase center is tiny, the position error along the track can be ignored, and the radial position error can be regarded as a tiny phase error [
28,
29,
30]. Thus, the main task of channel consistency calibration is the efficient estimation and calibration of phase imbalance between channels.
For the phase imbalance problem between channels, the available channel error calibration methods are divided into two categories in this paper: processing for signal data and processing in the image domain, both of which have various advantages and disadvantages. The approaches based on signal data processing are mainly the following: The ATC method is operationally efficient in estimating the RSTI and phase error [
27]. However, the system Doppler center frequency accuracy affects phase error estimation results in more. Although the signal subspace (SSP) method has more accurate estimation results [
31], it needs eigen decomposition of the covariance matrix to obtain signal subspace and noise subspace, so it has much computation and is affected by the SNR of echo data. The minimum variance distortionless response (MVDR) method is also an accurate phase imbalance estimation method [
32] but is affected by Doppler center frequency estimation accuracy. However, it has a lower computational complexity compared with the SSP method. Then, methods that perform processing in the image domain, including the image weighted minimum entropy (WME) method [
33,
34], the image least
-norm (LLN) method [
35], and the maximum normalized image sharpness (MNIS) method [
30], are both characterized by high estimation accuracy and robustness, but generally have high computational complexity.
Although existing methods have made significant progress, reducing computational complexity while estimating channel phase errors accurately and robustly remains a challenge due to the complexity of a scene and the need for fast processing in real time. In order to address this challenge, a novel method of minimizing the sum of sub-band norms (MSSBN) for reconstruction azimuth spectrums is proposed to estimate phase errors of the AMC SAR system. Specifically, the reconstructed azimuth spectrum by the DBF technique has the smallest sum of sub-band spectrum norm when channel phases are consistent. In this paper, by modeling phase imbalance estimation as an optimization problem, typical global optimization algorithms, such as optima quest of nonlinear program (OQNLP) [
36] and particle swarm algorithms (PSO) [
36] can be used to find phase imbalances by the MSSBN method. Meanwhile, the proposed algorithm limits the influence of range samples by range pulse compression, and range-variation phase errors can be fitted by estimating the phase error at different sampling locations in the range direction. The MSSBN method does not require more spatial degrees of freedom for phase error estimation and has a broader range of application scenarios than the SSP method. Meanwhile, the MSSBN method has a higher estimation accuracy than the ATC method and is unaffected by the Doppler center frequency estimation accuracy. Since there is no need to image the reconstructed signal data of each channel, the MSSBN method has higher computational efficiency than the LLN method. The phase information of all scenes is converted to the same Doppler bandwidth using pulse compression and azimuth fast Fourier transform (FFT), and computational efficiency can be further improved by downsampling for azimuth spectrum without degrading estimation accuracy.
The remainder of this paper is organized as follows:
Section 2 introduces the signal model and reconstruction algorithm of the AMC SAR system, then analyzes the effect of channel imbalance and performs precompensation for phase error estimation.
Section 3 details the derivation of the proposed algorithm.
Section 4 reports the simulation and actual experimental results with a discussion. Then, conclusions are provided in
Section 5.
3. Method
When the phase between channels is consistent, the sum of the norm of reconstructed spectrum sub-bands has a minimum value, so the proposed method will be proven and derived based on the subadditivity of the linear normed space [
40] in this section. According to the analysis in
Section 2, the phase imbalance is treated as a constant error in this paper. As is known, the phase imbalance between channels is a relative quantity, so the first channel is set as the reference channel, and the phase error of the
channel is set as
, where
. Therefore, the phase error matrix is shown as
According to the Krieger reconstruction matrix [
3], the echo signal only with a phase error of each channel is reconstructed as
where
,
, and
denote the reconstructed spectrum with phase error, the aliased echo with phase error for each channel, and the full alias-free spectrum, respectively. As shown in the second equation of Equation (26), the phase error can be considered a part of the filter matrix or echo signal. The matrix
related to the Doppler frequency does not have an effect on the analysis.
The reconstructed filter matrix and transmission matrix generate a weighted matrix , and the complete aliasing-free spectrum is mapped to the reconstructed spectrum by . At the same time, it can be seen from Equation (26) that has nothing to do with the Doppler frequency but is only related to the reconstruction filter, the system transmission matrix, and channel errors. Therefore, this paper first analyzes the weighted matrix. The phase errors from to were assigned to channel 2 and channel 3, respectively, to obtain the reconstruction filter with errors, and the weighted matrix with errors was obtained by matrix multiplication.
The simulation parameters in
Figure 4 are from
Table 1, which shows the schematic diagram of each component of the weighted matrix affected by the channel phase error. As can be seen from the figure, when the phase error of the channel is consistent with that of the reference channel, that is, when the phase error in the figure is simultaneously 0, the diagonal elements of the weighted matrix all reach the maximum value 1, and other elements reach the minimum value 0. However, the reconstruction filter is known in the actual channel phase error estimation. In contrast, the phase error is an unknown quantity present in the echo, and the phase error cannot be estimated directly by the filter matrix. According to Perceval’s theorem [
1], when the power of the ambiguous spectrum is minimized, an alias-free reconstructed spectrum can be obtained [
41]. However, the ambiguous spectrum cannot be separated from the signal spectrum, so the ambiguous spectrum is considered together with the signal spectrum in this paper. Fortunately, the sub-band of the reconstructed spectrum is obtained by the weighted summation of the sub-band of the aliasing-free full spectrum, so it is necessary to add and analyze the corresponding elements of the weight matrix.
The simulation parameters in
Figure 5 are shown in
Table 1. It can be seen from
Figure 5 that the shift in the position of the reconstructed filter null caused by phase imbalance leads to a change in the amplitude of each spectrum. However, a minimum value exists for their sum of the norm, and the minimum value is obtained when there is no channel phase imbalance. The spectrum of different sub-bands is different, so this paper will mathematically derive the relationship between the sub-band spectrum norm (SBN) and the channel phase imbalance.
According to the subadditivity of linear normed space, the SBN satisfies the following equation:
From the above equation, it can be seen that there is a constant lower bound for the sum of the sub-band norm (SSBN), but the minimum cannot be reached because the argument of the sub-band spectral cannot be the same. Again, according to the subadditivity of linear normed space, SBN with error can be written as
where
denotes a constant number; the detailed duction can be found in
Appendix A. There exists
when, and only when, the phase error between channels is consistent, at which time the norm of the sub-band 1 spectrum reaches a minimum value [
40]. All sub-band spectral norms satisfy the above equation, so the SSBN with phase errors can be written as
Next, the above equation is verified by simulation experiments of three channels. The simulation parameters of
Figure 6 are shown in
Table 1. As shown in
Figure 6a, the sum of three sub-band spectral norms reaches a minimum value, and the minimum value is equal to the SSBN without phase errors when the phase errors of channels 2 and 3 are 0 at the same time, i.e., when they are in phase with channel 1. As shown in
Figure 6b, the norm of the sum of sub-band spectral (NSSB) with errors is also a constant and is less than the SSBN.
Now the optimization function is established to estimate channel phase error by the MSSBN. The phase error estimates are shown as
According to Equation (26), the reconstructed spectrum after channel phase error compensation is shown as
where
. Then the norm of the
sub-band spectrum can be expressed as
Since the number of channels M is equal to the number of spectrum ambiguities
N, the SSBN can be further written in the following form:
In this case, the phase error can be estimated by the MSSBN. The estimation model for the MSSBN is developed as
where
. Many mature global optimization algorithms, such as OQNLP and PSO, can solve the optimization problem in Equation (34).
The estimation algorithm proposed in this paper is performed in the range-Doppler domain, where the phase information of all sampling points is transformed to the same Doppler frequency band by the azimuth FFT and range pulse compression operations. The phase information of all sampling points is then converted to the same Doppler band. Furthermore, after the range pulse compression, the Doppler spectrum is aggregated into a few range bins, which can significantly reduce the influence of the range dimension on the computational complexity. In order to improve the computational efficiency, downsampling must be uniformly sampled in the whole Doppler band so as not to affect the estimation results.
After estimating the channel phase error, channel error compensation is carried out for the multichannel signal with error reconstruction, and the compensation results are as
The complete process of the algorithm proposed in this paper is shown in
Figure 7. The combined flow chart summarizes the main steps of the channel error estimation and calibration process as follows:
Step 1: Estimate and calibrate the amplitude imbalance between channels using the channel balancing method.
Step 2: Estimate and calibrate the RSTI between channels by using the ATC technique.
Step 3: Perform the spectrum reconstruction of each channel’s echo signal by the reconstruction filters .
Step 4: Transform each channel’s constructed signal into the range-Doppler domain utilizing range pulse compression and azimuth FFT.
Step 5: Uniformly sample the reconstructed azimuth spectrum of each channel echo signal and search for phase error by the MNSSB.
Step 6: The reconstructed azimuth spectrum of each channel echo signal is calibrated for phase error and then summed to obtain the alias-free full azimuth spectrum.
Step 7: Image the reconstructed echo signal after imbalance calibration.