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Adaptive reconstruction for azimuth signal of multichannel HRWS SAR imaging system
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 40 (2018)
Abstract
The reconstruction of azimuth signal in multichannel synthetic aperture radar (SAR) for highresolution and wideswath (HRWS) imaging requires exact steering vectors. The information of ambiguities and system parameters are used to create the steering vectors. The knowledge of ambiguities involves ambiguity number and index; the system parameters include the pulse repetition frequency (PRF), platform velocity, and channel spacing. However, in some cases, the knowledge of ambiguities is Dopplervariant and there exist errors in the system parameters, which may degrade the reconstruction performance. In this work, an adaptive reconstruction method is developed. Firstly, we discuss the azimuth sampling of multichannel SAR and derive the equation of ambiguity index. We then utilize the azimuth crosscorrelation to determine the aliasing number. Afterwards, based on the spatial spectrum estimation methods, an equivalent system parameter is adaptively calculated. With the aliasing number and the equivalent parameter, we can obtain the ambiguity information of each Doppler bin. Therefore, the steering vectors are constructed and the azimuth ambiguities can be suppressed. Compared with stateoftheart reconstruction methods, our method achieves excellent performance even in highly nonuniform sampling case. Without relying on any system parameters, the proposed method has better practicality and applicability. We conduct extensive experiments including simulations and real data processing to verify the effectiveness and evaluate the performance of the proposed method.
Introduction
Synthetic aperture radar (SAR) is an excellent remote sensor with good capability to provide allweather and allday observation of Earth’s surface [1, 2]. In recent years, the highresolution and wideswath (HRWS) SAR imaging has gained intensive attention from the research community [3–12]. However, the high azimuth resolution and wide swath coverage contradict each other in the conventional single channel SAR [3, 4]. In the light of the Nyquist sampling theory, the finer the azimuth resolution, the larger the operational pulse repetition frequency (PRF) will be. Whereas, the wideswath coverage requires a low PRF to avoid range ambiguities. Consequently, the azimuth multichannel SAR is utilized to solve this inherent contradiction and obtain HRWS SAR images [5]. In general, multiple uniformly spaced receiver channels along track are configured in the multichannel SAR as shown in Fig. 1. The single transmitter of the multichannel SAR emits a broad beam signal with a low operational PRF to illuminate wide area. The receiver channels simultaneously collect the echo signal to compensate the temporal sampling.
Because of the low operational PRF, the Doppler spectrum for each channel data is aliased. The unambiguous recovery of the Doppler spectrum is essential for the HRWS SAR imaging. On the basis of the onetoone correspondence between the Doppler frequency and azimuth squint angle, the spectrum components within one Doppler bin can be identified as virtual signal sources from different known directions [4]. Therefore, the azimuth signal of multichannel SAR can be reconstructed by digital beamforming (DBF) techniques in rangeDoppler domain. The Krieger DBF algorithm [3] and the postDoppler spacetime adaptive processing (STAP) method [4] are two representative DBF methods for azimuth signal reconstruction [13]. The principle of Krieger DBF algorithm is to solve a system of linear equations to reconstruct the Doppler spectrum. In comparison, the postDoppler STAP method employs the Capon beamformer to suppress the azimuth ambiguities [14]. However, the nonuniform azimuth sampling will deteriorate the performance of the aforementioned DBF methods in terms of the signaltonoise ratio (SNR). From another perspective, the azimuth sampling of multichannel HRWS SAR can be regarded as the recurrent nonuniform sampling. Inspired by the theory of recurrent nonuniform sampling derived by Eldar and Margolis in [15, 16], a filterbank reconstruction algorithm is proposed in [17].
The steering vectors related to different spectrum components of one Doppler bin are the crucial point of DBF techniques for ambiguity suppression. The knowledge of Doppler ambiguities (i.e., ambiguity number and index) and system parameters consisting of the operational PRF, platform velocity and spacing between two adjacent channels are utilized to construct the steering vectors. Nevertheless, the number and index of ambiguities are considered to be constant over the baseband Doppler frequencies in the above DBF methods, causing the degradation of SNR in the nonuniform sampling cases [18]. In contrary, the knowledge of ambiguities is Dopplervariant in some sampling cases, which also complicates the construction of steering vectors. Several improved DBF methods for ambiguity suppression are presented in [18–20]. The key issues of these improved methods are the calculation of effective Doppler bandwidth (EDB) and the ambiguityindex screening (AIS) operation, by which the information of ambiguities can be obtained. The method proposed in [18], termed as IDBF, defines an equivalent sampling spacing for the computation of EDB and then utilizes the Doppler spectrum weighting. The IDBF overcomes the drawbacks of Krieger DBF algorithm, but at the cost of degradation of azimuth ambiguitytosignal ratio (AASR) and azimuth resolution. Inspired by IDBF, a method named as ImpMMSE is proposed in [20]. The ImpMMSE uses a weighting process for the calculation of EDB and the minimum mean square error (MMSE) criterion to suppress azimuth ambiguities. Correspondingly, a remarkable improvement of AASR is realized. Besides, [19] calculates the baseband Doppler centroid and the EDB based on the Capon estimation method [14].
In the previous improved DBF reconstruction algorithms, the calculation of EDB and the AIS operation are critically dependent upon the system parameters. However, the system parameters may be inaccurate or unknown in some applications, which degrades the performance of ambiguity suppression [19, 21–23]. In fact, the real value of velocity is influenced by platform perturbation [23]. Additionally, the PRF and channel spacing may be lacking or influenced by measured errors [19, 21].
In this paper, we propose an adaptive azimuth reconstruction algorithm for multichannel SAR. The proposed method can adaptively obtain the information of ambiguities and estimate an equivalent parameter of PRF, platform velocity, and channel spacing. Firstly, concerning the uniformity of azimuth sampling, we propose to classify the azimuth sampling of multichannel SAR as uniformsampling, undersampling, and oversampling. Based on this novel classification, the aliasing number, the number of aliased copies of Doppler spectrum to be reconstructed, will be computed more simply. We can use the azimuth crosscorrelation (cf. [24–26]) is utilized to determine the aliasing number. Besides, the calculation equation of ambiguity index for each Doppler bin is deduced. Notice that the estimation of EDB is essentially equivalent to the determination of the aliasing number. Afterwards, we calculate the equivalent parameter F_{ p } by means of spatial spectrum estimation methods which have been widely used in generic imaging problems [27–29]. If there exist redundant channels, the idea of subspacebased methods can be utilized [30, 31]; otherwise, we use the Capon estimator [14, 21]. Based on the aliasing number and the equivalent parameter (denoted as F_{ p }), the ambiguity number and ambiguity indexes of each Doppler bin can be acquired. Thus, the accurate steering vectors can be adaptively constructed based on only the raw data, without using any system parameters. Consequently, our method can obtain good performance of ambiguity suppression even when the azimuth sampling is highly nonuniform.
The rest of this paper is organized as follows. Section 2 briefly introduces the signal model of multichannel SAR. Different cases of azimuth sampling in multichannel SAR are investigated in Section 3. Section 4 presents the detailed derivation of the aliasing number and the equivalent system parameter. Extensive experimental results on simulations and real data processing and discussions are given in Section 5. Section 6 concludes this paper.
Notation Vectors are denoted by lowercase boldface letters and matrices by uppercase boldface letters. The symbols (∙)^{T}, (∙)^{∗}, and (∙)^{H} denote the transpose operation, conjugate operation, and Hermitian transpose operation, respectively. \(\underset {\eta }{\mathbb {E}}\{\bullet \}\) represents the expectation operation over η. Besides, diag{∙} represents a diagonal matrix constructed by a vector.
Signal model
As illustrated in Fig. 1, a single transmitter (Tx) is combined with M uniformly spaced receiver (Rx) channels along track in the multichannel HRWS SAR system. The spacing between adjacent receiver channels is D. In terms of the data acquisition mode, the multichannel SAR signal can be regarded as transmitted and received both from the effective phase centers (EPCs) after the EPC processing [5, 12]. The EPC is in the middle position of the transmitter and its respective receiver. Therefore, the displacement from mth (m=1,⋯,M) EPC to the first one can be denoted as d_{ m }=(m−1)·d, where d=D/2. With the EPC processing, the relationship between the signal received by mth channel and the first channel in the twodimensional time domain can be expressed as
where η_{ m }=d_{ m }/v_{ r }; τ and η represents the range and azimuth time, respectively; v_{ r } is the platform velocity; x_{ m }(τ,η) is the signal recorded by mth channel. Note that x_{1}(τ,η) is generated by the multichannel SAR and its Doppler spectrum is aliased as a result of the low operational PRF. Let x_{0}(τ,η) denote the reference signal, which can be regarded as recorded by a singlechannel SAR under the condition that the Nyquist sampling criterion is satisfied.
As a result of the low operational PRF represented by f_{ p }, the Doppler spectrum of each channel data is aliased. Suppose that there exist I, denoting the ambiguity number, azimuth ambiguities at baseband Doppler frequency f_{ b }, where −f_{ p }/2≤f_{ b }<f_{ p }/2. Let x_{ m }(τ,f_{ b }) and s(τ,f_{ b }+i·f_{ p }) represent the received signal of mth channel in rangeDoppler domain and the equivalent unambiguous Doppler spectrum at frequency f_{ b }+i·f_{ p } from range gate τ, respectively. i refers to the ambiguity index ranging from i_{ min } to i_{ max }. Notice that, sometimes the variables i_{ min } and i_{ max } vary with f_{ b }, and hence the ambiguity number calculated by (2) is Dopplervariant [19].
Besides, let n_{ m }(τ,f_{ b }) be the zeromean white Gaussian noise recorded by mth channel. Thus using vector notation, the multichannel SAR signal in rangeDoppler domain can be:
The M×1 vectors x(τ,f_{ b }) and n(τ,f_{ b }) are the multichannel output and the noise vector, respectively, which can be written as
The Dopplervariant steering vector a_{ i }(f_{ b }) corresponding to the ambiguity component s(τ,f_{ b }+i·f_{ p }) is formulated as
For the azimuth signal reconstruction, the EPC spacing d, platform velocity v_{ r } and operational PRF f_{ p } are important parameters. These system parameters can be converted into an equivalent parameter F_{ p }, which can be expressed as
As a consequence, we can define another new parameter
where −F_{ p }/2≤F_{ b }<F_{ p }/2. Since the value of d/v_{ r } can be regarded as constant, we can use F_{ p } and F_{ b } to represent f_{ p } and f_{ b }, respectively. As a result, the steering vector a_{ i }(f_{ b }) can be reformulated as
The steering vector a_{ i }(F_{ b }) is equivalent to a_{ i }(f_{ b }). Therefore, we can construct the steering vector a_{ i }(f_{ b }) by (9) as long as F_{ p } and the ambiguity indexes i are known. Besides, the steering vector a_{ i }(f_{ b }) is independent of range delay. The reconstruction of azimuth signal is implemented bin by bin in range direction, which means the knowledge of range delay is unnecessary in the process of azimuth ambiguity suppression. Therefore, we can reconstruct the azimuth signal when F_{ p } and i are obtained.
Azimuth sampling of multichannel SAR
This section analyzes the azimuth multichannel SAR sampling in terms of the uniformity of azimuth sampling. In specific, we describe the proposed classification of multichannel SAR sampling and derive the calculation equation of ambiguity index.
The fundamental idea behind the multichannel HRWS SAR is that the lack of temporal sampling points are compensated by the spatial sampling points. The optimum PRF, represented by f_{ po }, fulfills a uniform spacetime sampling along the synthetic aperture and can be written as
It can be seen that f_{ po } is determined by the platform velocity, channel number, and channel spacing. Unfortunately, such a rigid selection of PRF cannot always be satisfied because of the timing diagram constraints in practical applications [5]. Any deviation from the optimum frequency f_{ po } will result in nonuniform sampling in azimuth. Accordingly, different kinds of azimuth sampling schemes of multichannel SAR are generated with the change of f_{ p } as demonstrated in Fig. 2.
We use the uniformity of azimuth sampling κ as defined in Eq. (11) to quantitatively analyze the effect of f_{ p } on the azimuth sampling.
The uniformity of azimuth sampling has crucial effect on the suppression of azimuth ambiguities. Combining (10) and (11) yields a new formation of f_{ p }:
We observe a proportional relationship between the uniformity of azimuth sampling κ and the operational PRF f_{ p }.
According to the value of the uniformity of azimuth sampling, the azimuth sampling can be classified into three categories: κ=1 refers to the uniformsampling (Fig. 2a); κ>1 indicates the oversampling (Fig. 2b); κ<1 represents the undersampling (Fig. 2c). The oversampling scheme is described that d_{0}, the displacement from Mth EPC to the first one in the next pulse, is smaller than d. If d_{0} is larger than d, the sampling is undersampling. Notice that d_{0} would reduce with the increase of κ. When the sampling position of the first channel coincides with that of Mth channel, κ becomes M/(M−1) and the received data are redundant [5]. For the sake of system complexity of the multichannel SAR, κ is assumed not to be larger than M/(M−1) in this paper.
The aliasing number denoted by N is defined as the number of aliased copies of the Doppler spectrum within the scope of [−f_{ p }/2,f_{ p }/2] and can be calculated by
where B_{ rd } represents the reconstructed EDB of multichannel SAR by ambiguity suppression. It should be noted that the maximum value of aliasing number is M. As derived in [18, 19], the uniformsampling, where B_{ rd } is Mf_{ po }, achieves optimal performance on ambiguity suppression. In this work, B_{ rd } is designed to approach Mf_{ po }. Therefore, in the cases of uniformsampling and undersampling where κ≤1, the aliasing number is M and the value of B_{ rd } can be κ·Mf_{ po } based on (12). Nevertheless, the aliasing number is less than M in the case of oversampling. When the case of coincidingsampling (κ=M/(M−1)) occurs, the aliasing number is M−1 and B_{ rd } is equivalent to M·f_{ po }. Thus, the reconstructed EDB B_{ rd } for uniformsampling and coincidingsampling can both achieve the value of M·f_{ po }. We can conclude that the aliasing number N ranges from M−1 to M for sampling schemes with M/(M−1)>κ>1. Besides, κ·Mf_{ po } should be at least larger than the 3dBDoppler bandwidth of multichannel SAR in undersampling cases. Thus, in terms of the valuing of κ, we have the criterion:
where B_{ d } is the 3dBDoppler bandwidth of multichannel SAR.
As known, except the system parameters, the reconstruction of azimuth signal involves another three parameters: the aliasing number N, the ambiguity number I, and the ambiguity index i. The aliasing number is an overall concept reflecting the relationship between B_{ rd } and f_{ p }. N may not be an integer in the oversampling scheme. The azimuth ambiguities are just the spectrum components within a specific Doppler bin f_{ b }. The number of azimuth ambiguities is referred to as ambiguity number, which must be an integer. Also, I is Dopplervariant when the oversampling scheme is generated. The ambiguity index i indicates the connection between the instantaneous Doppler frequency f_{ a } [19] being reconstructed and the baseband Doppler frequency f_{ b }, which can be expressed as
Figure 3 shows the relational diagram between f_{ a } and f_{ b }, which is the Doppler spectrum diagram (DSD) [19]. The ambiguity indexes i may vary over f_{ b } under the influence of the channel number M and the aliasing number N. When M is odd, the index i can be formulated by (16) (Fig. 3a); otherwise, i can be acquired by (17) (Fig. 3b). Obviously, the ambiguity indexes i are Dopplervariant in the case of oversampling. With the equivalence between f_{ p } and F_{ p }, we can determine the ambiguity indexes i of each Doppler bin as long as F_{ p } and N are obtained. We also observe that the ambiguity number I is less than M at some Doppler bins in the oversampling scheme, meaning the existence of redundant channels. However, there is no redundant channel in the cases of uniformsampling and undersampling.
Adaptive reconstruction method
In this section, we introduce the proposed adaptive azimuth reconstruction method for multichannel HRWS SAR. The proposed method can be separated into two principal procedures: the first one is the determination of the aliasing number N, and the other one is the estimation of the equivalent parameter F_{ p }. Then the algorithm steps are given. Finally, we analyze the computation complexity of our adaptive reconstruction method, demonstrating low computation load.
Determination of aliasing number
Here we elaborate the determination of aliasing number N by using crosscorrelation technique presented in [26]. Based on the azimuth crosscorrelation, the coherence between x_{0}(τ,η) and x_{0}(τ,η+Δη) can be
According to (1), the received data of mth channel is just the delayed version of the first channel with time shift η_{ m }=d_{ m }/v_{ r } during a pulse. While, η_{0}, termed as the time shift between Mth channel in the former pulse and the first channel in the next pulse, is η_{0}=d_{0}/v_{ r }. Recall that d_{0} denotes the distance from the Mth EPC during the former pulse to the first EPC in the latter pulse. Thus, the time shift for adjacent azimuth samples generated from the same pulse is η_{ c }=d/v_{ r }; the time shift for adjacent azimuth samples in different pulses is η_{0}. Note that we derive that in the case of oversampling, the expression η_{0}<η_{ c } can be obtained; otherwise, the expression η_{0}≥η_{ c } can be determined.
Utilizing (1) and (18), the coherence of neighboring azimuth samples in the same pulse, denoted by α, can be defined as
The coherence of adjacent samples from different pulses, namely γ, can be derived as
Combining (1), (18), (19), and (20), the approximation of α and γ can be obtained:
Considering that μ(Δη) decreases with an increasing Δη [32], the azimuth sampling of multichannel SAR is oversampling when α<γ; otherwise, the azimuth sampling is uniformsampling or undersampling. Furthermore, if the value of γ approaches 1, we can infer that η_{0} is zero and the azimuth sampling is coincidingsampling.
As discussed in Section 3, the aliasing number N is influenced by the uniformity of azimuth sampling κ, which determines the azimuth sampling scheme. The value of N can be expressed as:
Bear in mind, in the case of uniformsampling (κ=1), the aliasing number N equals M; in the coincidingsampling case (κ=M/(M−1)), N is just M−1. Hence, after the classification of azimuth sampling schemes, the aliasing number can be adaptively determined. With the description in (23), we only focus on the calculation of N for the oversampling scheme. Here, we formally present the criterion in terms of the determination of N when κ>1. Based on the above analysis, we creatively propose the following determination criterion:
Since α<γ≤1 in the oversampling scheme, (24) is a convex function related to the coherence of adjacent azimuth samples in multichannel SAR.
According to (24), N is in the range from M−1 to M. For the azimuth uniformsampling, γ is equivalent to α and hence the aliasing number is M. When the operational PRF leads to coincidingsampling, γ reaches 1 and thus N reduces to M−1. For other oversampling schemes where κ∈(1,M/(M−1)), N is larger than M−1 but less than M. Besides, as the increase of κ, the aliasing number reduces. Thus, we can conclude that the reconstructed EDB B_{ rd } approaches Mf_{ po } under the criterion formula expressed in (24).
With the coherence coefficients α and γ, we can clarify which case the azimuth sampling is. After that, the aliasing number N can be directly obtained as M or calculated by (24). Furthermore, we can adaptively decide the aliasing number without using any system parameter.
It should be noted that the coherence coefficients α and γ are not obtained in reality due to the expectation operations in (19) and (20). α and γ can be estimated by (25) and (26), respectively.
For the purpose of improving estimation accuracy, we use all of the azimuth samples of each range gate to calculate coherence coefficients.
Estimation of the equivalent system parameter
In this part, we present the estimation of the equivalent system parameter F_{ p }. The spatial spectrum estimation techniques can be utilized to adaptively estimate F_{ p } [33]. The principles of the classical algorithms, including Capon method, multiple signal classification (MUSIC) method and estimation of signal parameters via rotational invariance technique (ESPRIT), will be employed in this paper. Since there is no redundant channel in the uniformsampling or undersampling case, only the Capon method can be used. In comparison, the three spectrum estimation techniques are all suitable for some Doppler bins of the oversampling case. However, the subspace based methods have higher accuracy than Capon method. Also note that the Capon method and MUSIC method require spectral peak searching to obtain the estimation of F_{ p }, which would complicate the estimation process. There should be a good balance between the estimation accuracy and the complexity regarding which technique is selected to calculate F_{ p }.
Capon method
A new steering vector b(F_{ a }) over the instantaneous Doppler frequency f_{ a } is defined as
where F_{ a } is a new equivalent parameter expressed as
Thus the inferior and superior limits of F_{ a } for estimating F_{ p } are −κ/2 and κ/2, respectively. Moreover, based on the categorization of the azimuth sampling scheme and the inequality (14), the scope of F_{ a } can be approximately determined.
The Capon spectrum at F_{ b } (equivalent to f_{ b }) over F_{ a } is
where R(F_{ b }) is the covariance matrix of multichannel SAR signal at F_{ b }. R(F_{ b }) can be expressed as
The array manifold A(F_{ b }) at F_{ b } is given by
and the matrix R_{ ss }(F_{ b }) given by
where s(τ,f_{ b })=[s(τ,f_{ b }+i_{ min }f_{ p }), ⋯,s(τ,f_{ b }+i_{ max }f_{ p })]^{T}. Besides, \(\sigma _{n}^{2}\) denotes the variance of noise, I_{ M } is an identity matrix with size M×M. Because of the unavailability of R(F_{ b }) in practical applications, it is usually replaced with the sample covariance matrix \({\hat {\boldsymbol {R}}}(F_{b})\) which is calculated by
where L is the number of used training samples of every receiver channel and x(τ_{ l },f_{ b }) denotes the lth training sample vector in range direction from f_{ b }.
The peaks of P_{ Capon }(F_{ a },F_{ b }) correspond to the ambiguities within Doppler bin f_{ b }. Hence, the interval between adjacent peaks of the Capon spectrum P_{ Capon }(F_{ a },F_{ b }) is just F_{ p }.
MUSIC method
The eigendecomposition of the covariance matrix R(F_{ b }) as expressed in (31) can be
where \({\lambda _{1}} \ge {\lambda _{2}} \ge \cdots \ge {\lambda _{I}}\ge {\lambda _{I+1}} = \cdots = {\lambda _{M}} = \sigma _{n}^{2}\) are the eigenvalues of R(F_{ b }) sorted in descending order and e_{ m } are the corresponding eigenvectors. The eigenvalues and their corresponding eigenvectors are relevant to the baseband Doppler frequency f_{ b }. The matrix E_{ s }=[e_{1},⋯e_{ I }] and matrix E_{ n }=[e_{I+1},⋯e_{ M }] are utilized to span the signal subspace and noise subspace, respectively. The formulations of matrices Λ_{ s } and Λ_{ n } are Λ_{ s }=diag{λ_{1},⋯,λ_{ I }} and \({{\boldsymbol {\Lambda }}_{s}} = \sigma _{n}^{2}{{\boldsymbol {I}}_{M  I}}\), respectively. The dimension of the signal subspace is equivalent to the ambiguity number I and can be obtained by using (2), (16) and (17). The MUSIC pseudo spatial spectrum at F_{ b } over F_{ a } can be calculated by
The distance between the adjacent peaks in the pseudo spectrum P_{ MUSIC }(F_{ a },F_{ b }) is just F_{ p }.
ESPRIT method
Define two selection matrices with size (M−1)×M:
where 0_{M−1} indicates a zero vector with size (M−1)×1. Let us multiply the array manifold A(F_{ b }) as in (32) by J_{1} and J_{2}, respectively, two submatrices are obtained as
With the steering vector a_{ i }(F_{ b }) as formulated in (9), we can express the relationship between A_{1}(F_{ b }) and A_{2}(F_{ b }) as
where
Ψ is just the translational invariance matrix induced by the configuration of uniformly spaced channels in the multichannel HRWS SAR.
Take into account that the variable F_{ b } on the right side of (35) is omitted for brevity. Then, multiplying matrix E_{ s }(F_{ b }) as appearing in (35) by J_{1} and J_{2}, respectively, we have
According to the theory of ESPRIT [31], there exists an underlying rotational invariance between E_{s1}(F_{ b }) and E_{s2}(F_{ b }), which is expressed as
Φ(F_{ b }) just represents the rotational invariance between signal subspaces. Moreover, Φ(F_{ b }) can be formulated as
where T(F_{ b }) is a nonsingular matrix and can be written as
From (46), we know that Ψ and Φ(F_{ b }) have the same eigenvalues; the diagonal elements of Ψ are just the eigenvalues of Φ(F_{ b }). The difference value between two adjacent phases (in radian) of the eigenvalues of Φ(F_{ b }) equals 2π·F_{ p }, and hence the value of F_{ p } can be acquired. Besides, Φ(F_{ b }) can be easily obtained by using (45).
Algorithm steps
Based on the above derivation and discussion, here, we present the algorithm steps of the proposed adaptive azimuth reconstruction method as illustrated in Fig. 4. The implementation details are as follows:

Perform azimuth crosscorrelation to the multichannel SAR received data by (25) and (26), and obtain the coherence coefficients α and γ.

Through comparing α and γ, the azimuth sampling scheme is determined. In the case of uniformsampling or undersampling, the aliasing number can be directly obtained as M; otherwise, N is calculated by (24).

Implement azimuth fast Fourier transform (FFT) to each receiver channel data and obtain the multichannel output expressed in rangeDoppler domain.

Based on the oddeven character of M and the calculated aliasing number N, estimate the equivalent parameter F_{ p } by the principles of Capon, MUSIC or ESPRIT method.

Calculate the ambiguity indexes and ambiguity number of every Doppler bin by (16) or (17).

With the ambiguity index i and F_{ p }, construct the Dopplervariant steering vector a_{ i }(F_{ b }).

Suppress azimuth ambiguities by MMSE criterion proposed in [10, 20] and the idea in [21] to reduce computation load. Finally, obtain the HRWS SAR image by using conventional imaging algorithm, like the chirp scaling (CS) algorithm [1].
We can use several Doppler bins to estimate F_{ p } and then average these estimates in order to improve the estimation accuracy. Besides, the assumption that the signal is bandlimited can be relaxed due to the utilization of the MMSE criterion.
Computation complexity
Here, the computation load of the determination of aliasing number N and equivalent system parameter F_{ p } is investigated. Since the multiplications need more computation time and hardware sources, we only consider the calculation burden of multiplication. The computation complexity of the proposed adaptive reconstruction method is listed in Table 1. Assume that there are N_{ a } azimuth samples in one range gate for each channel data of multichannel SAR, and K Doppler bins are selected to improve the estimation accuracy of coherence coefficients and the equivalent parameter.
According to Eqs. (19), (20) and (24), the number of complex multiplication for determining the aliasing number N is 3·K·M·N_{ a }. Based on Eq. (34), the sample covariance matrix \({\hat {\boldsymbol {R}}}({F_{b}})\) needs L·M^{2} complex multiplications. In addition, the estimation of F_{ p }(F_{ b }) by using spatial spectrum techniques as presented in our paper requires \(\mathcal {O}({M^{3}})\) complex multiplications for each Doppler bin. Thus, the computation complexity for estimating equivalent system parameter is \(K\cdot \left (L\cdot M^{2}+\mathcal {O}({M^{3}})\right)\). Therefore, we can conclude that the total computation complexity for our method is \(3KMN_{a}+K\left (L M^{2}+\mathcal {O}({M^{3}})\right)\). With the relationship that N_{ a }≫M, Na≫K and Na≫L, we can confirm that the computation complexity of our method is not high and can be easily implemented.
Experiment results and discussion
To verify the effectiveness and evaluate the performance of our proposed adaptive azimuth reconstruction method, we conduct extensive experiments on simulated data and real multichannel SAR data. We perform evaluation of the adaptive determination of aliasing number N. Then, the validity of the estimated equivalent parameter F_{ p } and its performance over SNRs and azimuth sampling uniformity κ are demonstrated. We also analyze how the errors of F_{ p } estimated by our method affect the azimuth reconstruction. Finally, the real airborne multichannel SAR data is utilized to verify the effectiveness of our reconstruction method.
Simulations about aliasing number
A sixchannel SAR system with relevant parameters summarized in Table 2 is employed to generate the simulated data and then implement the following simulation experiments. From the perspective of the aliasing number N, we perform comparisons between the proposed adaptive reconstruction method and several other stateoftheart reconstruction methods including Krieger DBF [3], IDBF [18], and ImpMMSE [20]. Note that the weighting factor ρ is set to be 0.5 in the ImpMMSE reconstruction method. The performance of these aforementioned algorithms over the uniformity of azimuth sampling κ would be investigated. The uniformity of azimuth sampling κ is set to be from 0.9 to 1.2, where κ=1 indicates the uniformsampling and κ=1.2 just the coincidingsampling. Moreover, for the sake of comparison, the overall reference is compared to all methods. The values of overall reference are achieved by a single channel SAR with the operational PRF Mf_{ p }. Besides, for the purpose of evaluating AASR, the Doppler bandwidth B_{ D } is set to be 8165 Hz yet the 3dBDoppler Bandwidth B_{ d } is 6290.6 Hz. Independent white noise is added to the simulated multichannel SAR data such that the SNR of raw data is 20 dB.
Firstly, we conduct experiment with respect to the estimation of coherence coefficients α and γ. All of the available azimuth samples are used to calculate α and γ. Figure 5 shows the calculation results of coherence α and γ versus κ. In spite of the varying κ, the value of α is 0.27. The value of γ rises with the increasing κ when κ is in the range from 0.9 to 1.2. We see that the curves α and γ intersect at κ=1 where the azimuth sampling is uniform. Observe that α>γ when the azimuth sampling is undersampling (0.9≤κ<1) and α<γ in case of oversampling (1<κ≤1.2). We also observe that γ reaches 1 in the case of coincidingsampling (κ=1). The calculation results are in good agreement with theoretical analysis as discussed in Section 4.1. This experiment proves that the estimation of α and γ by (25) and (26) have high accuracy. Therefore we can claim that the uncertainty of estimation for coherence values α and γ is negligible, which lays a foundation for the evaluation of aliasing number N.
Essentially, a primary difference among the above reconstruction methods is the calculation of aliasing number N. The aliasing number reflects the reconstructed EDB and hence controls the azimuth resolution. Figure 6 shows the aliasing number and the relative resolution degradation (in percentage) against varying κ realized by different reconstruction methods, respectively. The value of the difference between the resolution and reference resolution divided by the reference resolution is just the relative resolution degradation. From Fig. 6a, we can see that the aliasing number determined by Krieger DBF is constant and equivalent to M. In the case of oversampling (κ>1), the aliasing numbers obtained by our method, IDBF and ImpMMSE all reduce with an increasing κ. The aliasing number of our method decreases more quickly, which infers that its EDB is closer to M·f_{ po }. Moreover, we note that the aliasing number of IDBF is slightly inferior to that of our method when κ approaches 1.2. One can also notice that the aliasing numbers of IDBF and ImpMMSE are less than M in some undersampling cases, which violates the fact that the ambiguity number is always M for any Doppler bin f_{ b } when κ<1. Figure 6b shows that the relative resolution degradations of our method, IDBF and ImpMMSE are all under 20%, which are maintained at acceptable levels. We know that the azimuth resolution of multichannel SAR is determined by the EDB. The ambiguity suppression in multichannel SAR can achieve excellent performance when EDB reaches Mf_{ po }. Although our method seems to provide the ’worst’ performance in resolution comparing with other methods in case of oversampling, its EDB is closest to Mf_{ po }. Moreover, the SNR scaling factor Φ_{ bf } and AASR could better represent the reconstruction performance. The finest azimuth resolution does not mean the best performance of ambiguity suppression. Next we will implement simulations to evaluate the reconstruction performance realized by different algorithms.
The curves of mean squared values of reconstruction filter obtained by different methods are plotted in Fig. 7. In the uniformsampling scheme, we observe that the mean values of the reconstruction filter over the instantaneous Doppler frequency f_{ a } are all equivalent to 0 dB (see Fig. 7a), and hence all the four reconstruction methods have similar performance. Whereas, the mean squared values over the outmost Doppler band are much higher in the case of oversampling as illustrated in Fig. 7b. Bear in mind that the uniformity of the oversampling case is κ=1.195, which makes f_{ p } approach the operational singular PRF 1414.1 Hz (generate coincidingsampling). The rising mean squared values of the oversampling case can be explained by the increasing condition number of the array manifold A(F_{ b }) [34]. Furthermore, the aliasing number and AIS operation, to a certain extent, are beneficial to reduce the value of condition number of A(F_{ b }). As a consequence, we can see that the IDBF and our reconstruction method have less range of rising value than the ImpMMSE and Krieger DBF method. The rising mean squared value dominates the deterioration of SNR, indicating an increasing SNR scaling factor denoted by Φ_{ bf } [5]. Although the reduction of N can enhance the SNR of the reconstructed azimuth signal, it might result in the drop of AASR [5, 20].
We further study the performance of these algorithms about SNR scaling factor Φ_{ bf } and AASR versus κ (0.9≤κ≤1.2). Figure 8a shows the curves of SNR scaling factors obtained by different methods. We note that when κ is close to 1, all methods realize the best performance in terms of Φ_{ bf }. Too large deviation from κ=1 will magnify the SNR scaling factors of all methods, implying the degradation of SNR for the reconstructed signal. Obviously, the proposed method outperforms its competitors regarding SNR scaling factor. This is because our method has the minimum aliasing number (see Fig. 6) and carries out the AIS operation. However, the reduce of aliasing number may degrade the AASR.
Figure 8b presents the AASR achieved by different methods. Clearly, when the azimuth sampling is highly oversampling, the degradation of AASR occurs in our proposed method, the IDBF method and the ImpMMSE method. The proposed method has optimum performance, even better than the reference case, when the uniformity of azimuth sampling κ is close to 1. Although the AASR of our method is slightly inferior to those of other methods in the case of highly oversampling, its AASR is still under − 30 dB. Considering that there should be a good balance between the SNR and AASR for the reconstructed signal, our adaptive method can be an excellent method to suppress azimuth ambiguities especially for nonuniform sampling cases. Moreover, our method dose not require any system parameters.
Simulations on the equivalent system parameter
We firstly use the simulated data with parameters listed in Table 2 to verify the effectiveness of the adaptive estimation of the equivalent system parameter F_{ p }, which is inspired by spatial spectrum estimation techniques. Then we evaluate the estimation accuracy of F_{ p } in terms of the average rootmeansquared error (ARMSE) versus SNR and the uniformity of azimuth sampling κ, respectively. We also investigate how the estimation errors of F_{ p } affect the azimuth reconstruction of multichannel SAR signal.
Here the uniformity of azimuth sampling is set to be κ=1.1 for the convenience of using MUSIC and ESPRIT techniques. Thus, the number of ambiguities in frequency bins around zero Doppler are 5 (cf. (17)). Besides, the range of F_{ a } is from − 0.55 to 0.55 and the SNR is 20 dB in this simulation. The spectrum envelopes at zero Doppler bin generated by different estimation techniques are plotted in Fig. 9. Observe that the peaks are uniformly spaced; the peaks of MUSIC and ESPRIT spectrums are sharper than Capon spectrum. The distance between adjacent spectrum peaks just equals the equivalent parameter F_{ p }. Therefore, we can confirm that F_{ p } can be estimated without using any system parameters, which makes our algorithm more practically applicable.
We further investigate the performance of estimating F_{ p } versus SNR under conditions with various κ (0.95, 1, and 1.15). Note that, in the case of κ=1.15, the three spectrum estimation techniques all can be utilized. Based on 200 trails, the empirical results regarding the ARMSE of F_{ p } (in percentage) are shown in Fig. 10a. From an overall perspective, the estimation accuracy is fairly good. One can see that the estimation accuracy improves with the increasing SNR. Compared with the undersampling scheme, the oversampling scheme achieves better accuracy. This experiment implies that the estimation accuracy is sensitive to the uniformity of azimuth sampling κ. Next, we explore the effect of κ on the estimation of F_{ p }.
Figure 10b plots the curves of ARMSE versus κ in distinct cases where SNR is 0 and 20 dB, respectively. Notice that the MUSIC and ESPRIT techniques are only utilized in the oversampling schemes where κ≥1.025. Clearly, the estimation accuracy of F_{ p } improves with a rising κ. Furthermore, we see that the ARMSE achieves the minimum value when κ is close to 1.05, but not 1. This can be explained that a high PRF is beneficial to enhance the estimation accuracy of F_{ p } under condition that the azimuth sampling is quasiuniform.
We can confirm that in terms of estimating F_{ p } the subspace based methods have higher accuracy than Capon method from Fig. 10. The MUSIC method achieves slightly better performance than the ESPRIT method. Whereas, the Capon method and MUSIC method involves the spectral peak searching.
According to Fig. 10, we know that the estimation errors of F_{ p } are inevitable. These errors may degrade the performance of ambiguity suppression. Next, we conduct experiments about reconstructing azimuth signal under circumstance that errors are added to F_{ p }. F_{ p } is set to be 1.195 and the aliasing number is adaptively decided by our method. Figure 11a plots the profiles of azimuth point spread functions (PSFs) when different errors (0, 1, 5, and 10% of the original F_{ p }) are added to the equivalent system parameter F_{ p }. Figure 11b demonstrates the enlargement of the ambiguities located at the green rectangle of Fig. 11a. The PSF results show that the azimuth ambiguities can still be suppressed to − 26.32 dB even when the error of F_{ p } is 10%. Furthermore, in general, the estimation error is considerably inferior to 10% as illustrated in Fig. 10. Therefore, we can conclude that our method is effective and efficiency. On the other hand, the deterioration of ambiguity suppression indicates the importance and necessity of the exact system parameters. This experiment also verifies that our reconstruction method obtain good performance in highly nonuniform sampling case.
Demonstration with real airborne fourchannel SAR data
Real data collected by an airborne multichannel SAR system is utilized to validate the effectiveness of our adaptive azimuth reconstruction method. The airborne multichannel SAR is developed by the Institute of Electronics, Chinese Academy of Sciences (IECAS) and its main parameters are summarized in Table 3. The channel number of the airborne multichannel SAR system is 4 and hence the optimum PRF f_{ po } is 564.44 Hz. Whereas, the operational PRF of the multichannel SAR is 749.76 Hz and the uniformity of azimuth sampling κ is 1.3283. Consequently, the multichannel SAR generates a highly nonuniform azimuth sampling considering that κ=1.3333 would lead to the coincidingsampling scheme. In addition, before the process of azimuth reconstruction, we employ the approach based on the azimuth crosscorrelation to calibrate the unavoidable channel errors [32, 35].
The aliasing number N adaptively determined by our method is 3.1542. In contrast, the aliasing number is 3.0242 by IDBF method. When the weighting factor ρ is 0.5, the calculated aliasing number is 3.5121 via the ImpMMSE method. Regarding N, our method is just equivalent to the ImpMMSE with ρ=0.8668 and the IDBF equivalent to the ImpMMSE with ρ=1 for the real data. This infers that our method is effective in terms of the calculation of aliasing number and can offer a good tradeoff between AASR and SNR.
Then the equivalent parameter F_{ p } is adaptively estimated by our method. Figure 12 plots the spatial spectrums at zero Doppler bin achieved by means of Capon, MUSIC and ESPRIT techniques, respectively. Observe that there are 3 uniformly spaced peaks in every spatial spectrum, which is consistent with the fact that there exist 3 spectrum components at zero Doppler bin. The values of F_{ p } calculated from Capon, MUSIC and ESPRIT spectrum are 0.3357, 0.3350, and 0.3355, respectively. Considering that the nominal F_{ p } is 0.3321, the estimation errors for the given F_{ p } are 1.0840, 0.8797, and 1.0303%, respectively. With the results shown in Figs. 10 and 11, we can believe that the estimated F_{ p } is fairly accurate, manifesting the efficiency of our method in estimating F_{ p }.
Afterwards, the azimuth signal is reconstructed by different algorithms, i.e., the Krieger DBF, IDBF, ImpMMSE and the adaptive reconstruction method. It should be noted that our reconstruction method utilizes the adaptively estimated F_{ p } via the MUSIC technique while other reconstruction methods employ the given F_{ p }. With the reconstructed signal, the CS algorithm is employed to perform the imaging procedure [1]. Figure 13 presents the imaging results processed by different reconstruction methods. Observation of the SAR images shows that high level of residual ambiguities are still remained in Fig. 13a which is processed by the Krieger DBF method. In contrast, from Fig. 13b, c, d, we see the azimuth ambiguities can be effectively suppressed by using IDBF, ImpMMSE, and our adaptive reconstruction method, indicating an acceptable balance between the SNR and AASR.
Furthermore, we calculate the relative ambiguityplusnoise (AN) levels of the yellow dotted rectangles in Fig. 13 to quantitatively evaluate the performance of ambiguity suppression [20]. The area marked by the yellow dotted rectangle is selected for the reason that its image background is relatively pure and weak. In order to make the maximum value of the AN level for the marked zone of Fig. 13a be 0 dB, we normalize the image data achieved by the four methods. To better demonstrate the AN levels, Fig. 14 presents the enlargements of the yellow dotted areas in Fig. 13. The mean AL levels on the marked zone for the Krieger DBF, IDBF, ImpMMSE and our adaptive method are − 31.82, − 42.38, − 41.23, and − 44.75 dB, respectively. It can be seen that our proposed adaptive method outperforms its competitors in terms of the real data processing. Therefore we can verify that our adaptive method is a promising candidate to reconstruct the azimuth signal of multichannel SAR even in highly nonuniform sampling scheme. Furthermore, compared with its competitors, another great advantage of our adaptive method is that it does not need any system parameters.
Conclusions
We have proposed an adaptive azimuth reconstruction algorithm for the multichannel HRWS SAR signal. To this end, we proposed to classify the azimuth sampling of multichannel SAR and derived the calculation formula of ambiguity indexes. Then we presented the two crucial procedures of our adaptive method: the determination of aliasing number N and the estimation of the equivalent system parameter F_{ p }. Utilizing the azimuth crosscorrelation, the aliasing number can be adaptively calculated. The principles of these spatial spectrum estimation techniques are employed to estimate F_{ p }. Then, with N and F_{ p }, the ambiguity number and ambiguity indexes of each Doppler bin can be obtained. Consequently, we can adaptively create the steering vectors and then reconstruct the azimuth signal without requiring any system parameters. Comprehensive experiments on simulated data and real data of the airborne multichannel SAR developed by IECAS validate that our adaptive algorithm can effectively suppress azimuth ambiguities and offer an excellent balance between AASR and SNR. Therefore, we can claim that our method is a promising method to reconstruct the azimuth signal of the multichannel HRWS SAR even when the azimuth sampling is highly nonuniform.
Abbreviations
 AASR:

Azimuth ambiguitytosignal ratio
 AIS:

Ambiguity index screening
 AN:

Ambiguityplusnoise
 ARMSE:

Average rootmeansquared error
 CS:

Chirp scaling
 DBF:

Digital beamforming
 DSD:

Doppler spectrum diagram
 EDB:

Effective Doppler bandwidth
 EPC:

Effective phase center
 ESPRIT:

Estimation of signal parameters via rotational invariance technique
 FFT:

fast Fourier transform
 HRWS:

highresolution and wideswath
 IECAS:

Institute of electronics, Chinese academy of sciences
 MMSE:

Minimum mean square error
 MUSIC:

Multiple signal classification
 PRF:

Pulse repetition frequency
 Rx:

Receiver
 SAR:

Synthetic aperture radar
 SNR:

Signaltonoise ratio
 STAP:

Spacetime adaptive processing
 Tx:

Transmitter
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Acknowledgements
The authors also thank the anonymous reviewers for their comments and suggestions, which will improve the quality of this manuscript. The authors also thank IECAS for providing the multichannel airborne SAR raw data to this manuscript.
Funding
This work was supported by the National Natural Science Foundation of China under Grant 61601285.
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LZ conceived the main idea, conducted the experiment, discussed the results, and wrote the manuscript. YG and XL supervised the main idea and provided many valuable suggestions to the study. All authors read and approved the final manuscript.
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Zhang, L., Gao, Y. & Liu, X. Adaptive reconstruction for azimuth signal of multichannel HRWS SAR imaging system. EURASIP J. Adv. Signal Process. 2018, 40 (2018). https://doi.org/10.1186/s1363401805628
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Keywords
 Azimuth signal reconstruction
 Crosscorrelation
 Highresolution and wideswath (HRWS)
 Multichannel synthetic aperture radar (SAR)