An Efficient Signal Reconstruction Algorithm for Stepped Frequency MIMO-SAR in the Spotlight and Sliding Spotlight Modes

Multiple-input multiple-output (MIMO) synthetic aperture radar (SAR) using stepped frequency (SF) waveforms enables a high two-dimensional (2D) resolution with wider imaging swath at relatively low cost. However, only the stripmap mode has been discussed for SFMIMO-SAR.This paper presents an efficient algorithm to reconstruct the signal of SFMIMO-SAR in the spotlight and sliding spotlightmodes, which includesDoppler ambiguity resolving algorithmbased on subaperture division and an improved frequency-domain bandwidth synthesis (FBS) method. Both simulated and constructed data are used to validate the effectiveness of the proposed algorithm.


Introduction
According to the minimum antenna area constraint, high resolution and wide swath (HRWS) pose contradicting requirements on synthetic aperture radar (SAR) system design [1].This system-inherent limitation can be overcome by singleinput multiple-output (SIMO) SAR system [2], which has been extended to multiple SAR modes, such as spotlight and sliding spotlight, to fulfill the different requirements of spatial resolution and coverage in future SAR missions [3][4][5].In high-resolution SAR, wide bandwidth signals are needed to obtain a high range resolution.However, the direct transmission of a wide bandwidth signal may lead to a high cost of the hardware [6][7][8].A solution to this problem is the transmission of stepped frequency (SF) waveforms in combination with bandwidth synthesis technologies [6][7][8].
The SF multiple-input multiple-output (MIMO)-SAR, which is defined as the MIMO-SAR system transmitting and receiving a class of SF subband signals simultaneously by multiple azimuth channels, respectively, combines the advantages of the SIMO system and SF waveforms and enables HRWS imaging at relatively low cost [9,10].However, only the stripmap mode SF MIMO-SAR has been discussed.
In the spotlight and sliding spotlight modes, the steering of the antenna beams leads to an increase of the azimuth bandwidth and thus raises difficulties for both azimuth reconstruction and bandwidth synthesis.Furthermore, the HRWS leads to a great amount of data, which may cause high computational burden, so an efficient signal reconstruction algorithm is required.
In this paper, an efficient algorithm is presented to reconstruct SF MIMO-SAR signal for the spotlight and sliding spotlight modes.Firstly, a Doppler ambiguity resolving algorithm based on the azimuth subaperture division is presented for azimuth reconstruction of each SF subband signal.Then, an improved frequency-domain bandwidth synthesis method without upsampling in range is proposed to efficiently synthesize the subband signals.
This paper is organized as follows.In Section 2, the signal model of SF MIMO-SAR is introduced.In Section 3, the Doppler ambiguity resolving algorithm based on subaperture division is described.An improved FBS method without upsampling in range is proposed and the flowchart of the signal reconstruction algorithm is shown in Section 4. In Section 5, simulation and real data processing results are presented.Finally, the conclusion is presented in Section 6.

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International Journal of Antennas and Propagation

Signal Model
where   is the center frequency of the whole synthesized bandwidth signal and Δ , denotes the center frequency interval between the th subband signal and the whole bandwidth signal.
Figure 1 shows the imaging geometry of sliding spotlight SF MIMO-SAR with two channels as an example.The radar moves along the -axis with a constant velocity V, and the main beams of  transmit/receive (Tx/Rx) channels in azimuth are steered from forward to backward, respectively, pointing to the rotation center of the corresponding mode.The echoes are received by all the  channels simultaneously, so  channels data are achieved for each subband signal.If the azimuth multichannel samples of each subband signal do not coincide, the PRF will be improved  times after azimuth reconstruction, so the equivalent PRF after azimuth reconstruction is  ⋅ PRF.In order to ensure relative wide range coverage, the PRF is set to ensure the equivalent PRF after azimuth reconstruction to be lower than the total Doppler bandwidth  azi and slightly greater than the instantaneous Doppler bandwidth  ins [4,5].
Suppose that   and   , respectively, denote the relative positions in azimuth of the th transmit channel (Tx) and the th ( = 1, 2, . . ., ) receive channel (Rx).The multichannel data of each subband signal can be converted into equivalent self-transmitting and self-receiving data corresponding to the effective phase centers (EPCs) by compensating the phase errors [8].The compensating function for the subband  signal received by Rx is denoted by  1, ().After the compensation of the phase errors,  EPCs signal can be achieved for subband  signal, whose azimuth coordinate can be given as The EPC corresponding to Tx1 and Rx1 is defined as the reference EPC, whose azimuth coordinate is set to be zero.It is shown in (2) that, due to different positions of transmit channels, the EPCs of different subband signals corresponding to a same Rx channel are of different azimuth coordinates.These geometry differences between subband signals should be removed for the bandwidth synthesis.
After range compression, the signal of a point target (,   ) transmitted by Tx and received by Rx can be expressed as where 2 is the slant range from point  to the EPC corresponding to Tx and Rx,  is the speed of light,  is the fast time,   is the slow time, and   (⋅) and  azi (⋅) denote the range window function and the azimuth window function, respectively.

Doppler Ambiguity Resolving Based on Subaperture Division
In the spotlight and sliding spotlight modes, the time-variant squint angle inevitably leads to an extension of the azimuth spectrum [4,5].The time-frequency distribution (TFD) of the sliding spotlight mode is shown in Figure 2(a).The azimuth reconstruction processing based on the subaperture division for the sliding spotlight SIMO SAR [4] can be extended to SF MIMO-SAR in the spotlight and sliding spotlight modes to resolve Doppler ambiguity of each subband signal.The subaperture division of subband signals should ensure that the processed Doppler bandwidth of the whole bandwidth signal is smaller than the equivalent PRF.Since the whole bandwidth is very wide to obtain a high resolution, the influence of range frequency on the processed Doppler bandwidth should be taken into account, which is different from the case in [4].
The beam central frequency varying with slow time   and range frequency can be expressed as where  ∈ [  − Δ  /2,   + Δ  /2] is the range frequency, which covers all the subbands and  denotes the timevariant azimuth squint angle.From Figure 2(b), the processed azimuth bandwidth  sub of the whole bandwidth signal can be approximately expressed as where  rot = 2V 2   / rot denotes the slope rate of the variation of the beam central Doppler frequency,  rot is the rotational distance, and  , denotes the Doppler center of the th subaperture, which varies with the range frequency.
To reconstruct the Doppler spectrum of the whole bandwidth signal, the duration of each slow time block  sub is chosen to ensure that the processed bandwidth  sub in each block is less than  ⋅ PRF.After the subaperture division, the signal described by ( 5) is multiplied by a Doppler shift function  2, (  , ) = exp(−2   , (  )) to move the Doppler spectrum to the Doppler baseband.Then, the Doppler ambiguity can be resolved by spatial filtering [3][4][5] for each subband signal.The spatial filtering is a weighting operation of a multichannel signal for each subband in the Doppler domain.Considering the great amount of SF MIMO-SAR data, in order to reduce the computational burden of weight vectors of spatial filtering, the signals are multiplied by compensating functions to make the weight vectors independent of specific Doppler frequency [5], and the compensating functions can be constructed as where   ∈ (− ⋅ PRF/2, −( − 1) ⋅ PRF/2) is the Doppler frequency.Then the weight vectors of the spatial filter can be obtained by where Then, the Doppler spectrum can be reconstructed by where S 1, = [ 1, (,   , 1), . . .,  1, (,   , )] is the multichannel signal vector of subband .It can be observed from (9) that a  () is independent of specific Doppler bin, and thus w  () does not need to be updated with azimuth Doppler bins, which can sufficiently reduce the computation burden.
After the Doppler ambiguity resolving, the Doppler spectrum of subband  signal can be expressed as where   ∈ (− ⋅ PRF/2,  ⋅ PRF/2) denotes the Doppler frequency.From (11), we can see that each subband signal is equivalent to a single channel signal, and the geometry differences between different subband signals are removed.Then the bandwidth synthesis can be performed in Doppler domain.

Improved FBS Method in Doppler Domain
In general, there are two popular methods for the bandwidth synthetic of SAR signal: the synthetic the time-domain bandwidth synthesis (TBS) method and the FBS method.The TBS method is accurate, but the long duration of the synthetic pulse causes a high computation burden making it inefficient [6].The FBS method is also accurate and more efficient, so it is widely used for bandwidth synthesis of SAR signal [7,8].
The FBS method proposed in [7] transforms the subband signals to range frequency domain and then shifts the spectrum of subband  signal by Δ , and performs coherent summation to obtain the whole bandwidth signal.However, considering the signals are discrete, the spectrum can only be shifted by an integer number of frequency bins in frequency domain.So this method is precise only if the residual fractional part of Δ , is zero.A solution to this problem is  times upsampling and then multiplying the signal of subband  by a frequency shift function in time domain to shift the spectrum by Δ , [8].However, the processing of upsampling adds an extra FFT,  times zeropadding in frequency domain, and an extra IFFT to the processing of bandwidth synthesis, which is inefficient with respect to both the use of data space and processing time.
To avoid signal upsampling in range, an improved FBS method in Doppler domain is proposed.Firstly, a slight shift of the range frequency spectrum is performed.The shift Whole band Subband 1 amount is the value of the fractional part of Δ , , so the frequency shift functions can be constructed as where Δ  denotes the frequency interval of a frequency bin and   is the integer part of Δ , /Δ  .Note that the frequency shifting amount is smaller than the value of a frequency bin, so that the range frequency spectrums do not alias.
Then, after the spectrum shift, the signals are transformed into range frequency domain.The center frequency distance between subband  signal and the whole bandwidth signal spans over   frequency bins, so spectrum shift and precise coherent summation can be performed as shown in Figure 3. Therefore, in comparison to the FBS method used in [8], the proposed improved FBS method can perform bandwidth synthesis precisely without upsampling of signal and thus lowers the computation load.After the coherent summation, the whole bandwidth signal in 2D frequency domain is obtained, which can be processed by conventional SAR imaging algorithms for corresponding modes [4,11].
Figure 4 shows the flowchart of the proposed algorithm in the case of  = 2.It consists of three parts: subaperture division, Doppler ambiguity resolving, and bandwidth synthesis.
Suppose   and   denote the range and azimuth sampling numbers of each subband signal, respectively.According to the flow of the proposed algorithm, the computational load of improved FBS method can be written as (1/2)  log 2   +     , while the computational load of FBS method presented in [8] is (1/2)  log 2   +   log 2   +  2     .Therefore, the proposed improved FBS method can sufficiently lower the computational load of bandwidth synthesis.

Simulation and Raw Data Results
In this section, point targets simulation and constructed data processing results are carried out to verify the validity of the proposed algorithm.After the subaperture division, the subband 1 signal in the 2D frequency domain before and after the spatial filtering is shown in Figures 5(a) and 5(b), respectively.It can be observed that the Doppler ambiguity is resolved by the spatial filtering, and the Doppler spectrum without aliasing is reconstructed for each subband signal.Then the improved FBS method is performed to combine the two subband signals, and the whole bandwidth signal in the 2D frequency domain is shown in Figure 6(a).Figure 6(b) shows the range images of a point target before and after bandwidth synthesis, from which one can see that the high range resolution is achieved after bandwidth synthesis.Then the imaging algorithm proposed in [11] is used to process the 2D reconstructed signal and the imaging result is obtained.The interpolated contour plots of three of the point targets are shown in Figure 7.These three point targets are denoted by P 1 , P 2, and P 3 , respectively, and their 2D coordinates are (−1 km, 616 km), (0 km, 617 km), and (1 km, 618 km).The plots show that the point targets are well focused.The key image quality parameters, namely, the integrated sidelobe ratios (IRWs), the peak sidelobe ratios (PSLRs), and the integrated sidelobe ratios (ISLRs), are measured and shown in Table 2, which further verify the performance of the proposed algorithm.

Constructed Data Experiment.
Since there are no SF MIMO-SAR data available for us, a single channel airborne Spotlight SAR data is used to construct SF MIMO-SAR data.The airborne Spotlight SAR data was collected on July, 2008, the main system parameters of which are given in Table 3.
Firstly, the original data is transformed into range frequency domain and the whole range bandwidth is divided into two subbands, and then the subband 2 data is transformed into Doppler domain and multiplied by a slow time delay function to simulate an EPC difference from subband 1.Finally, the two subband data are downsampled by a factor of two into equivalent two channel undersampled data, respectively.Then a two-input two-output MIMO-SAR data in spotlight mode with two subbands is obtained.The main   parameters of the constructed SF MIMO-SAR data are given in Table 4. Figure 8(a) shows the focused image of subband 1 data after Doppler ambiguity resolving, from which one can see that no ghost exists in azimuth.The focused image of the synthesized bandwidth signal is shown in Figure 8(b), and the comparison between the range images of a point target in the amplified area before and after bandwidth synthesis is shown in Figure 8(c), from which one can see that the range resolution is improved by the proposed improved FBS method.Therefore, the result of constructed real data processing also proves the effectiveness of the proposed algorithm.

Conclusion
This paper presents an efficient algorithm to reconstruct SF MIMO-SAR signal in the spotlight and sliding spotlight modes.A Doppler ambiguity resolving method based on the subaperture division is presented to reconstruct the Doppler spectrum of each subband signal.Then, considering the high computation load of SF MIMO-SAR signal synthesis, an improved FBS method is proposed to avoid upsampling of data and thus lower the computation load of bandwidth synthesis.Results of point targets simulation and constructed International Journal of Antennas and Propagation

Figure 2 :
Figure 2: The subaperture division.(a) The TFD of sliding spotlight.(b) 2D spectrum of the whole bandwidth signal.

Figure 6 :Figure 7 :
Figure 6: (a) 2D spectrum of the whole bandwidth signal.(b) Range images of a point target before and after bandwidth synthesis.

Table 1 :
Main system parameters of point target simulation.The main parameters of a sliding spotlight SF MIMO-SAR system are given in Table1.A 3 × 3 point target array, of which the targets are uniformly distributed in a scene of 2 km × 2 km in azimuth and range directions, is set in the illustrated scene.

Table 2 :
Image quality of point targets.

Table 3 :
Main system parameters of original data.

Table 4 :
Main system parameters of constructed data.