Off-Grid RFI Suppression Method in Synthetic Aperture Radar Based on Mismatch Reconstruction and Dictionary Rotation

In the process of radio frequency interference (RFI) suppression for synthetic aperture radar (SAR), when frequencies of RFIs are not located on discrete grids of the frequency spectrum, i.e., RFIs are off-grid, energies of RFIs will spread out, degrading suppression performances of lots of existing methods. To this end, aiming at the off-grid RFI suppression problem, in this article, a mismatch reconstruction and dictionary rotation-based RFI suppression method is proposed. First, using the defined support set, an optimization model is constructed to reconstruct mismatches of off-grid RFIs. Then, these mismatch variables are utilized to rotate the original Fourier transform-based dictionary so that frequencies of RFIs can be well matched. The energies of RFIs can be refocused under the representation of rotated dictionaries. At this time, we construct and solve another optimization problem to subtract RFIs and obtain the cleaned SAR signals. The real-world SAR data and simulated RFI data-based experiments demonstrate the effectiveness of the proposed method.

Generally, RFI suppression methods can be divided into three categories: nonparametric, parametric, and semiparametric [5]. For nonparametric methods, Cazzaniga and Guarnieri proposed the frequency domain notch filter (FNF) method many years ago [6]. After that, Zhou et al. [7] make use of the Eigen-subspace projection (ESP) based method to suppress RFIs . Another kind of great nonparametric method, the adaptive filters, is suggested by Le et al. [8] and is improved by Vu et al. [9]. Note that the common basic idea of nonparametric methods is to form notches, targeting the frequency points or, equivalently, the subspaces where RFIs locate. Although these nonparametric methods can significantly suppress RFIs, some SAR signals in these notches will also be filtered simultaneously. To make it worse, when energies of RFIs are spread (off-grid case), we have to set wider notches, causing a massive loss of SAR signals.
Apart from these nonparametric methods, some parametric methods are also proposed [10], [11], [12], [13] in recent years. The basic idea of parametric methods is to obtain the key parameters, say, frequencies of RFIs, to construct subspaces of RFIs, and to project the original mixed signals into the complementary subspaces of RFIs, acquiring the cleaned SAR signals. For instance, Liu et al. [10] took advantage of the iterative adaptive approach (IAA) to obtain a high-resolution frequency spectrum and then to search RFI frequencies. Similarly, Huang and Liang proposed a gradual relaxation-based cyclic approach (G-RELAX) method [11] to achieve the same goal. One can see that the prerequisite for the success of these parametric methods is that frequencies of RFIs precisely locate on discrete grids of the frequency spectrum. Although parametric methods have perfect suppression performances in this case, it is hard to guarantee that this situation can always happen in practice. On the contrary, in the off-grid RFI case, on one hand, the energies of each RFI will be spread out from one subspace to many other subspaces; on the other hand, one cannot obtain the real frequency of each RFI but view peaks corresponded points in the frequency spectrum as frequencies of RFIs. Therefore, after projecting, still, lots of energies of RFIs will be retained.
Nowadays, sparse recovery (SR) technology is extensively used in many areas, including RFI suppression in SAR. In SR methods, some parameters exist, but they have nothing to do with RFI frequencies. Thus, we say these SR methods are semiparametric. Also, we call these parameters hyperparameters. For the first time, Nguyen and Tran utilized the sparse recovery method in RFI suppression [14], [15], [16]. Then, other scholars also do some great works [17], [18], [19], [20], [21], [22], [23], [24]. The basic idea of sparse recovery methods in RFI suppression is to find the sparsest solution under the dictionary, which is generally the inverse discrete Fourier transform (IDFT) matrix. This idea is based on the narrowband feature of RFIs, or, say, RFIs are sparse in the frequency domain. All RFI information, including amplitudes and frequencies, is contained in the sparsest solution. Specifically, in this solution, positions and the corresponding numerical values of nonzero elements represent the frequencies and complex amplitudes of RFIs, respectively. Thus, the cleaned SAR signals are acquired by subtracting RFIs, which are linear combinations of columns or atoms of the IDFT dictionary in the manner of the sparsest solution from the original mixed signals. Based on this, one can construct various optimization models derived from different properties of RFI or SAR signals. For example, Nguyen and Tran make use of the robust principle component analysis model, which is based on the low-rank property of RFI and sparse property of SAR signals [16]. Huang et al. proposed a fast algorithm utilizing the row sparse property of RFI in the frequency domain [17]. Liu et al. [21] proposed a joint sparse recovery method in time-frequency. It is shown that sparse recovery methods can effectively protect the SAR signals and suppress RFIs simultaneously. Nevertheless, in the face of off-grid RFIs, suppression performances of semiparametric methods will also degrade. An important reason is that, in this case, each RFI cannot be represented by a single atom but by an approximative linear combination of numerous atoms in the dictionary. That is, energies of each RFI are separated from one atom, or equivalently, one subspace (on-grid case), to the whole space (off-grid case). Thus, the fundamental premise, RFIs are sparse in the frequency domain, is not fully satisfied. Considering this problem, Huang et al. designed an augmented dictionary to describe atoms precisely [17]. The computational complexity, however, is much higher.
In conclusion, for successfully suppressing RFIs, it is suggested that frequencies of RFIs are located on-grid, or, in other words, RFIs corresponded Fourier vectors are in the IDFT dictionary. In real applications, however, one cannot ensure that. That is, a mismatch or bias exists between each real frequency of RFI and a frequency grid point. Existing of these mismatches degrades the whole performance of RFI suppression.
In this article, aiming at the off-grid RFI suppression problem, an off-grid RFI suppression in SAR via mismatch reconstruction and dictionary rotation method is proposed. First, we define a set called the support set. Based on the definition, we design null-space-projection (NSP) operators as constraints to reduce the interaction between different RFIs. Then, we construct a multivector optimization model to reconstruct each mismatch corresponded Fourier vectors of RFIs. Finally, we propose a dictionary rotation sparse recovery method to suppress RFIs by using these reconstructed mismatch vectors. Experiments demonstrate the effectiveness of the proposed method.
The rest of the article is organized as follows: in Section II, we introduce the on-grid and off-grid signal models. We will see the difference between on-and off-grid RFIs intuitively. The support set is also defined in this part. Then, in Section III, we introduce the proposed method, including NSP operators, constructed optimization models, and solving processes. The experiment results and performance analyses are discussed in Section IV. Finally, Section V is the conclusion.
Throughout this article, vectors are denoted by bold-face lowercase characters, while matrices and sets are denoted by bold-face uppercase characters. We use (·) T , (·) H , (·) † , and (·) * to denote the transpose, conjugate transpose, inverse operations, and the optimal variables. · 1 , · 2 , and · F are 1-norm, 2norm, and F-norm, respectively. We make use of • to denote the pointwise multiplication. We denote a positive definite matrix M as M 0.

II. SIGNAL MODEL
This section introduces signal models of both on-grid and off-grid RFIs. To express off-grid RFIs unambiguously, we define the support vector and the support set. Then, we discuss the difference between the two kinds of RFIs. It is found that compared to on-grid RFIs, energies of off-grid RFI are spread in the frequency domain, making them more challenging to be suppressed.

A. On-Grid RFI Signal Model
Owing to the two characteristics of RFI (high energy and very narrow bandwidth), scholars model RFI as a superposition of several complex sinusoidal signals [1]. Thus, under the impact of RFIs, the total echo SAR signal for a single pulse is A n e j2πf αn kT s + s(k) + n(k), k = 1, 2, . . .Nr (1) where x represents the total echo signal in this pulse and s is for the SAR signal. n is the Gaussian white noise, and r stands for RFI, which can be modeled as the sum of N complex sine functions. A n and f αn are the complex amplitude and frequency for nth RFI, respectively. Nr is the number of samples of signals with N Nr, and T s is the sampling period. Generally, s and n can be regarded as a whole, called the useful signal u u(k) = s(k) + n(k). ( The IDFT dictionary is a square matrix with each column a Fourier vector, that is where the mth vector with frequency f m in D is Note that RFI can be expressed as a compact form where α n =[1, e j2πf αn ·1·T s , e j2πf αn ·2·T s , . . ., e j2πf αn ·(Nr−1)·T s ] T = e j2πf αn t , n = 1, 2, . . ., Nr is also a Fourier vector.
Denoting Γ the set of on-grid frequency points. Then, for any frequency f i ∈ Γ, we have d i ∈ D. Now if RFI frequencies are on-grid, that is, f αn ∈ Γ and α n ∈ D, then r can also be expressed as where w is the sparse vector (since N Nr) with N nonzero elements A n .

B. Off-Grid RFI Signal Model
An off-grid frequency f i means that f i / ∈ Γ. However, it can still be seen as a frequency point in Γ plus a bias. Therefore, the off-grid RFI model can be expressed as where δ αn is a frequency bias and δ αn = e j2πδ αn t is called the mismatch vector. Nevertheless, the above expression is ambiguous since the chosen f n is freedom. That is, a frequency of RFI can be decomposed into an arbitrary f n ∈ Γ plus a corresponded bias. A great way to choose f n and the corresponded d n is as follows. For a real RFI frequency f αn , we can always find F s means the frequency sampling rate. Then, we can decide d n in (6) by Thus, under this definition, (6) is turned into We call d sup −αn the support vector for the nth RFI. Next, we combine all d sup −αn to form a set κ, say, the support set. One of the advantages of such definitions is removing ambiguous of (6) while another is constructing the NSP operators, which will be discussed later.
At this moment, the total echo signal model can be expressed as follows: The frequency spectrums of on-grid and off-grid RFI are shown in Fig. 1(a) and (c), respectively. We can see that the energies of the off-grid RFI are spread out from a single frequency point compared to the on-grid RFI. To explain this phenomenon, let us consider the two diagrammatic drawings shown in Fig. 1(b) and (d), in which blue arrows are Fourier vectors in D; red arrows are RFIs. It is clear that, we see, an off-grid RFI cannot be represented by a single Fourier vector, but a linear combination of many Fourier vectors. Therefore, its energy not be concentrated but distributed in the frequency domain. As described before, many RFI suppression performances of methods will be unsatisfactory in this case. Take the FNF method as an example. We can suppress the RFI in Fig. 1(a) by setting a filter with a tiny notch (from the 350th frequency bin to the 352nd frequency bin); on the contrary, in Fig. 1(c), the notch has to be much larger (about from 300th frequency bin to 400th frequency bin). Therefore, more SAR signals will be lost in the latter case compared to the former.

III. PROPOSED METHOD
For now, we have no information about vectors on the righthand side of (9). However, an important information, the support set κ can be obtained in the first place. Suppose somehow κ is achieved. At this time, (9) can be turned into a multivector optimization problem for each δ αn , which contains mismatch information. We then construct an optimization model and solve it for getting each δ αn . After that, each δ αn is utilized to "rotate" the IDFT dictionary for offsetting the mismatch between each off-grid RFI and its support vector. Next, we construct another optimization model to obtain the cleaned SAR signals by using these rotated dictionaries. We will introduce the detailed procedure in this section.
A. Obtaining Mismatch Vector δ αn 1) Step 1: Principle Component Extraction: In this section, our goal is to obtain mismatch vectors δ αn for RFIs. Since mismatch vectors are only related to RFIs, we need to remove useful signals as much as possible to reduce the influence of them in the first place. Here we make use of the Eigensubspace method introduced in [7] to extract RFIs. We divide x into Nr − L + 1 subvectors x 1 , x 2 , . . ., x Nr−L+1 by a sliding window with length L, and then use them as samples to estimate the where Λ is a diagonal matrix with elements (eigenvalues) in the descending order, and V is the matrix consisting of corresponding eigenvectors. Again, since RFIs have high energy, we can see big eigenvalue corresponded eigenvectors as RFI subspaces. To this end, one can obtain RFI data by projecting each subvector onto the RFI subspaces where q means the number of big eigenvalues of R x and v j is the jth big eigenvalue corresponded eigenvector. Rearranging the obtained matrix [h 1 , h 2 , . . ., h Nr−L+1 ], one can obtain the vector x mainly composed by RFIs.

2)
Step 2: Peak Searching: After the above processing, a few peaks standing for RFIs still exist. To this end, we do a local maximum search for each peak to find the corresponding frequency point f n ∈ Γ and the Fourier vector d n , which is also a support vector d sup −αn according to the above definition. At this time, number of RFIs N , each d sup −αn as well as the support set κ are obtained.

3) Step 3: Optimization Model Construction:
We first define the null-subspace-projection (NSP) operator, denoting as Θ n , for the nth RFI r n as follows: where \ means we remove d sup −αn from the set κ.
Then, the proposed optimization model is as follows: where σ is a very small number. The meaning of (13) is to decompose x into N vectors. The purpose of using NSP operators, which take part in constraints in (13), we see, is to limit the mutual interferences among RFIs.

4)
Step 4: Optimization Model Solution: Let us go back to (13). This is a convex optimization problem with multioptimal vectors. We solve it by iteration. Note that the initial values of iterative variables are set to zeros.
Denoting y αn = diag(d sup −αn ) · δ αn and using the Lagrangian with penalty method, (13) is turned into an unconstrained problem where c n 0 are penalty coefficients. Next we make use of the famous alternative direction method of multiplier (ADMM) [25] to (14), that is in the kth iteration [this is denoted as superscript (k)]. By taking the derivative of (15) for y α1 , the minimum point can be achieved as where I is the Nr × Nr identity matrix. The penalty coefficient c 1 guarantees (2I + c 1 Θ 1 · Θ H 1 ) 0. We can use the same way to obtain y αN . Repeat these steps (15)-(16) until the following stop criterion is satisfied: where τ 1 is a preset numerical value, or the maximum iteration number is reached. The optimal solutions of (13) are Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. Recall (9), we know that each δ * αn in (18) contains an amplitude A n which is not needed. Thus, we do an extra process, that is where δ * αn(l) means the lth element in δ * αn . At this time, we have obtained mismatch vectors δ * αn for RFIs.

B. Constructing Rotated Dictionaries and Suppressing RFIs 1) Step 1: Rotated Dictionaries Construction:
In the above procedure, we have obtained each mismatch vector. Here we use them to construct the rotated dictionaries, that is where RoD αn is the rotated dictionary for the nth RFI and 1 is an all ones vector with dimension Nr × 1. Also, it is quite easy to prove that each RoD αn is a unitary matrix. The rotated dictionary is the dot product between the original IDFT dictionary and another matrix related to δ * αn . All Fourier vectors in the IDFT dictionary are given an extra mismatch phase. We say for an off-grid RFI with mismatch vector δ * αn , it is more sparse in RoD αn than in D since mismatch is approximately offset in advance by the extra mismatch phase. That is, more energies of off-grid RFIs are concentrated. Thus, RFI suppression performance can be improved when using the sparse recovery method, not under the IDFT dictionary but the rotated dictionary. We show the geometrical explanation in Fig. 2. Intuitively, we rotate the IDFT dictionary, like a turntable, to align the support vector with corresponded off-grid RFI.
2) Step 2: Optimization Model Construction: Using the rotated dictionaries we obtained above, we can conduct the following optimization model for RFI suppression: where ε is a preset hyperparameter and w n are weightings satisfying N n=1 w n = 1. Each z αn is an optimization variable to be obtained.
Apparently, when all RFIs are on-grid, (21) is exactly the 1-norm-based SR model. Thus, we say the above model can be seen as an extension of the SR method under different rotated dictionaries in the off-grid case.

3) Step 3: Optimization Model Solution:
For (21), we use the Lagrangian multiplier method, that is Then, (22) can be turned into It is easy to prove that the second term in (23) satisfies the condition of Lipschitz gradient [26]. Thus, next we can take advantage of the proximal gradient method [26] for (23), obtaining αn . At this time, the shrinkage operator [27] can be utilized to (24), and we have where T ρ is for the shrinkage operator with is acquired. We can use the same way to obtain z The Lagrange variable λ is updated as follows: where μ (k) > 0 is the step size. Repeat these steps (24)- (26) until the stop criterion is satisfied where τ 2 is a preset numerical value or the maximum iteration number is reached. In this model, initial values of iterative variables are also set to zeros. The final cleaned signals are RoD αn · z * αn (28) where z * αn is the obtained nth optimal variable. After processing all RFI-contained pulses, the final cleaned SAR data is acquired.

C. Computational Complexity Analysis
In this section, we make a discussion about the computational complexity of the proposed method. Only main procedures are in consideration. In the mismatch obtaining process, from (16) we see that in each iteration, complexity for updating y α1 is O(N × Nr 2 + Nr 3 ). Thus, the total complexity in each iteration is O(N 2 × Nr 2 + N × Nr 3 ). In the RFI suppression process, from (23)- (26) we see that in each iteration, complexity for updating all z αn plus λ is O(N × Nr 3 + (2N 2 + 1) × Nr 2 ). Therefore, the total computational complexity for the proposed method is O[itr1 × (N × Nr 2 + Nr 3 ) + itr2 × (N × Nr 3 + (2N 2 + 1) × Nr 2 )]. itr1 and itr2 are iteration numbers in the two processes, respectively. According to our experience, itr1 is about 5 ∼ 7 and itr2 is about 13 ∼ 20.
In this section, we introduce and discuss the proposed method. The procedure of model solving is also represented in detail. The computational complexity of the proposed method is analyzed. The experiment results are shown in the following section, and detailed performance analyses are discussed.

IV. EXPERIMENTS AND ANALYSIS
In Section IV, the real-world airborne SAR data experiments are conducted to prove the effectiveness of the proposed method. The main SAR parameters are shown in Table I.

A. RFI Suppression Experiments With Real SAR Data
In this experiment, Nr = 1000 and two off-grid RFIs have frequencies (−9+0.01)MHz and (13.5+0.02)MHz, respectively. Note that the second terms in brackets are frequency biases. The energy of each RFIs is 90 dB (note that the energy of SAR signals in each pulse is about 75 dB). The energy of the noise is set to 60 dB. The original polluted SAR image is shown in Fig. 3. Under the impact of RFIs, one can obtain no ground information. Targets are hidden behind a large number of bright lines formed by RFIs.
To prove the effectiveness of the proposed method, we choose some other representative methods: FNF [6], ESP [7], IAA [10], and SR [23], as comparisons. The final SAR images after suppressing RFIs by these methods are shown in Fig. 4. In addition, to see more details, we choose two small regions (red rectangle and green rectangle) with large and small radar cross sections, respectively, in the SAR image. The partially detailed drawings of the above small region images are shown in Figs. 5 and 6.
To better understand the obtained results, meanwhile, let us investigate the range frequency spectrums (from the 500th pulse) after RFI suppression by the above methods shown in Fig. 7. Fig. 7(a) shows the frequency spectrum of RFIs plus SAR signals. Fig. 7(b) is the frequency spectrum after RFI suppression by the FNF method. We see two notches appear. Although in notches RFIs are significantly suppressed, SAR signals are also lost. Therefore, the corresponding final SAR images are defocused in range, see Fig. 5(b). This is the same explanation for the ESP method; see Figs. 7(c) and 5(c). Fig.  7(d) is the frequency spectrum after RFI suppression by the IAA method. Note that RFI energies in the 351st and 726th frequency cells are significantly suppressed. Nevertheless, since RFIs are off-grid, spread RFI energies still exist. Therefore, in Figs. 5(d), 6(d), and 7(d), we still can see some strong RFI energies. For the SR method, we can see from Fig. 7(e) that although SAR signals are well protected (no notch), some RFI energies are retained (see the partial figure in Fig. 7(e), 345th to 360th, and     715th to 740th frequency cells). This is also reflected in Fig.  6(e), where one can see the residual energies of RFIs.
From Fig. 7(f), we can see that the proposed method can effectively protect SAR signals; compared to the SR method, the frequency spectrum obtained by the proposed method is closer to the frequency spectrum of SAR without RFIs. Therefore, on one hand, the SAR image obtained by the proposed method is well-focused, see To further analyze the performances of the above different methods, we conduct a quantitative experiment based on rootmean-square error (RMSE) and signal-noise-ratio (SNR) [28], which are defined as follows: where P and P are the original SAR echo matrix without RFIs and the signal matrix after RFI suppression, respectively. O is a null matrix. Na is the total number of pulses. Furthermore, since RFIs are additive signals, we can also use the Suppression Ratio to investigate the capabilities of RFI suppression of methods Suppression Ratio = 20log 10 where subscripts out and in represent for output and input signals, respectively. Quantitative results are shown in Table II. The quantitative indices show that the proposed method has the minimum RMSE as well as the maximum SNR. Note that the Suppression Ratio index of the proposed method is lower than the FNF method. This is because the FNF method places emphasis on RFI suppression while the proposed method aims at balancing RFI suppression and SAR signals protection. Note that apart from the FNF method, the proposed method also has the maximum suppression ratio index. Therefore, we conclude that the above experiment results support our analyses and prove the effectiveness of the proposed method.

B. RFI Supression Performances Under Different RFI Energies
Because in real detection scenes, SAR may be interfered by RFIs with different energies. Thus, comparison experiments under different RFI energies are also conducted. The RFI energies are set from 85 to 100 dB at regular intervals while the other conditions are unchanged. The corresponding RMSE, SNR, and suppression ratio results for methods are shown in Fig. 8. As shown in Fig. 8(a) and (b), the RMSE and SNR indices of the ESP method are relatively stable. For the IAA method, with the increase of RFI energies, more RFI energies are spread to subspaces of useful signals. Thus, even after suppression, considerable RFI energies remain in cleaned signals. This is the same reason for the FNF and SR methods. For the proposed method, RMSE and SNR indices are superior to other methods. Nevertheless, we see that SNR is also decreasing slowly with RFI energies increasing. Note that off-grid RFI corresponded Fourier vectors are not strictly orthogonal. Even if constraints in (13) work, with RFI energies growing, mutual energy leakage among different RFIs are becoming severe. Thus, in this case, errors of the obtained mismatch vectors increase, leading to the final suppression performance degradation. In Fig. 8(c), we observe that the IAA method has relatively stable capability of RFI suppression; while for the rest of the methods, suppression ratio indices increase with the increase of RFI energies. Note that when RFIs own relatively high energy, the proposed method has a better suppression ratio index, i.e., it holds a better RFI suppression capability than the other methods.

C. RFI Supression Performances Under Different Biases
Apart from RFI energies, suppression performances of methods under different RFI frequency biases should also be considered. We will discuss that in this section.
We consider two on-grid frequency points −9 and 13.5 MHz. From radar parameters in Table I, we know the frequency interval is 60 kHz. Thus, we do 11 experiments with RFI biases in the scope from −30 to +30 kHz to investigate the RFI suppression performances of the above methods. We still choose the above quantitative indices. The energies of the two RFI are both set to 90 dB. The results are shown in Fig. 9.
From Fig. 9, we see that with the bias tending to zero, suppression performances of the FNF, ESP, and IAA methods increase. This is easy to understand since off-grid RFIs tend to be on-grid, i.e., energies of RFI are more focused. Note that indices of the ESP method are not sensitive to the RFI frequency bias. This is because the ESP is not a grid but a principle component analysis-based method. Note that for the proposed method, although in the on-grid case, RMSE and SNR indices show that the suppression performance of the proposed method is worse than the SR and IAA methods, in the case of off-grid, it has the best performance. This is because off-grid RFIs can always be turned into on-grid, no matter what the biases are.

D. RFI Supression Performances Under Different Frequency Resolutions
Note that one of the keywords in this article is the grid. The width of two adjacent grid points in the frequency spectrum represents the frequency resolution. Different grid widths, or frequency resolutions, may affect the suppression performances of methods. Therefore, we investigate the RFI suppression performances under different frequency resolutions for methods in this section. In this experiment, we fix Nr = 1000. We do ten experiments with the frequency sampling rate in the scope of 40 to 80 MHz. We set 1 RFI with energy 90 dB and with frequency (1 × 10 4 + π) (Hz), which can always make sure that the RFI is off-grid. Here we still choose RMSE, SNR, and suppression ratio as performance indices. The results of them for each method are shown in Fig. 10.
From Fig. 10, one can see that with the increase of the frequency sampling rate, the suppression performances of FNF, IAA, and SR methods are improving; while for the proposed method, no matter what the frequency resolution is, we can always find the optimal rotated dictionary for each RFI. Since we know that, the mismatch vector constructs the rotated dictionary; while the mismatch vector is obtained by "squeezing" RFI energies into the support vector (this is the meaning of the objective function in (13). Note that the support vector we defined is on-grid, which is nothing to do with the frequency resolution. On the other hand, under the rotated dictionary, energies of RFI are refocused, i.e., the original off-grid RFI is turned into an on-grid RFI, which also has no relationship with the frequency resolution. To this end, one can see in Fig.  10 that the suppression performance of the proposed method is stable.

V. CONCLUSION
This article proposes an off-grid RFI suppression method in SAR via mismatch reconstruction and dictionary rotation. Specifically, we define the support set and NSP operators for off-grid RFIs. Under these definitions, we construct and solve a multivector optimization model to obtain RFI mismatch vectors. Then, we use mismatch vectors to rotate IDFT dictionaries so that energies of off-grid RFIs can be refocused under the representations of them. Finally, we suppress RFIs by constructing and solving a rotated dictionary-based optimization model. The experiments show that the RFI suppression performances of the proposed method are effective and stable.
In future work, on one hand, we will consider the fast algorithm under this framework. Iterations and running times of the proposed method are unsatisfactory, especially when N > 3. On the other hand, we will extend the proposed method to wideband interference suppression, which is more challenging.