#### Abstract

This paper proposes a novel image formation algorithm for the bistatic synthetic aperture radar (BiSAR) with the configuration of a noncooperative transmitter and a stationary receiver in which the traditional imaging algorithm failed because the necessary imaging parameters cannot be estimated from the limited information from the noncooperative data provider. In the new algorithm, the essential parameters for imaging, such as squint angle, Doppler centroid, and Doppler chirp-rate, will be estimated by full exploration of the recorded direct signal (direct signal is the echo from satellite to stationary receiver directly) from the transmitter. The Doppler chirp-rate is retrieved by modeling the peak phase of direct signal as a quadratic polynomial. The Doppler centroid frequency and the squint angle can be derived from the image contrast optimization. Then the range focusing, the range cell migration correction (RCMC), and the azimuth focusing are implemented by secondary range compression (SRC) and the range cell migration, respectively. At last, the proposed algorithm is validated by imaging of the BiSAR experiment configured with china YAOGAN 10 SAR as the transmitter and the receiver platform located on a building at a height of 109 m in Jiangsu province. The experiment image with geometric correction shows good accordance with local Google images.

#### 1. Introduction

Recently, bistatic synthetic aperture radar has been a hot topic because of its advantage of flexible configuration, abundant information, anti-interception, anti-interference, and so forth. According to the platforms of BiSAR transmitter and receiver [1, 2], BiSAR systems include space-borne BiSAR, air-borne BiSAR, space-borne and air-borne hybrid BiSAR, ground and air-borne/space-borne hybrid BiSAR, and so forth. Nowadays, a novel low-cost and simple-configured BiSAR system is popular which employs an overpass space-borne SAR as a non-cooperative transmitter and a stationary receiver located on a hill or high building. The non-cooperative BiSAR can acquire images with high spatial resolution.

Meanwhile, high spatial resolution could be obtained by using relative motion between antennas and its target [3–7]. However, because of the physical separation between transmitter and receiver, a series of new technological problems still needs to be solved, especially those associated with synchronization [8–10].

One focus on BiSAR imaging is concerned with the synchronization errors, consisting of space synchronization, time synchronization, and phase synchronization. Space synchronization can be solved by the prior knowledge of satellite pass and image opportunity. But the time and phase synchronizations are more complicated and time-consuming.

So far, bistatic image formation algorithms have been basically developed and improved. Rocca firstly introduced Dip Move Out (DMO) method [11] into BiSAR imaging, which is derived from earthquake signal processing methods, converting bistatic echoes to monostatic echoes. Many other researchers were dedicated to approximate methods to obtain two-dimensional spectrum, including the Loffeld Bistatic Formula (LBF) [12], and the series reversion method [13], and Instantaneous Doppler Wavenumber (IDW) [14]. Additionally, for the common adaptability, Back Projection (BP) method [15] has been widely used. In the bistatic case with a fixed receiver, only the transmitter contributes to the Doppler history, which makes the different scatterers in the same range cell have different Doppler histories [16, 17]. Wong and Yeo put forward a NLCS (Nonlinear Chirp Scaling) algorithm [18], which is only suitable in cases with the small bistatic angle. After that, some extended NLCS and Range Migration Algorithm (RMA) and modified BP algorithm were proposed and gained good results [19–24]. Furthermore, some bistatic system concepts were advanced [25].

However, those synchronization errors could not be eliminated in the absence of accurate orbit data and some important system parameters information, including squint angle, orbit height, satellite velocity, off-nadir angle, and accurate sampling time, while all above imaging algorithms were based on the synchronized echo and thus could not be directly implemented to nonsynchronous echo.

This paper discusses the feasibility of focusing nonsynchronous bistatic echo and introduces a novel imaging algorithm based on the direct signal. This paper is arranged as follows, the basic geometry model and signal model are established in Section 2. Then, the focusing algorithm based on direct signal and image contrast optimization that will be derived in that Section 3. At last, the experimental results will be presented in Sections 4 and 5 will summarize the paper.

#### 2. Signal Model

The acquisition geometry of non-cooperative BiSAR systems is shown in Figure 1, where the transmitter is moving along a linear trajectory and the receiver is stationary. The center of scenario is selected as coordinate origin, the - plane is tangent to the earth surface and -axis is parallel to the satellite velocity, where the position vector of the transmitter is at the azimuth time , the equivalent satellite velocity is , the incidence angle is , the incidence angle of receiver is , the fixed receiver is located at , the target position vector is , is the range between transmitter and target at azimuth time , is the range between target and stationary receiver, is the minimum range from the target to the transmitter, and represents the closest range from receiver to the flight path of transmitter. Then, its reflected echo can be expressed as where represents the fast time, is the velocity of light, is the pulse width, is the carrier frequency, is the Frequency Modulation (FM) rate, is the bistatic range at azimuth time , and is the duration time from transmitter to target to receiver at azimuth time :

Let us denote the minimum range from the target to the transmitter by and the minimum bistatic range sum by ; that is,

Transforming the echo signal into range Doppler domain based on the principle of stationary phase (detail in the appendix), we have where and are the envelopes in range time domain and azimuth frequency domain, respectively, is the azimuth frequency, is the wavelength, is the Doppler centroid, is the FM (Frequency Modulation) rate in range Doppler domain, is the FM rate of SRC filter, is the factor of range migration, and is the curvature factor:

#### 3. Two-Dimensional Focusing Algorithm

##### 3.1. Range Focusing Based on Parameters Estimation Using Direct Signal

As for non-cooperative transmitter, passive measurement for transmitted signal is needed [26, 27]. However, only little prior information could be obtained from China Space Agency (CSA), except bandwidth, wavelength, and pulse width, which could be used to estimate the chirp-rate directly.

Before range processing, the recorded SAR data must be partitioned into two-dimensional by the estimated PRF. The estimated PRF, which is coarse but accurate enough for the proposed algorithm, can be obtained by measuring the positional difference of adjacent peak [28, 29], after range compression is implemented on the one-dimensional direct signal.

Range compression is implemented in frequency domain and then converted the result into time domain. However, as the transmitter is working in the squint mode, the differences between and could not be ignored. As a result, secondary range compression is inevitable.

In case that the receiver of BiSAR is a stationary receiver, the Doppler centroid and Doppler chirp-rate could be expressed as where is the squint angle of the transmitter.

Substituting (7) and (9) into (6), we have

Equation (10) indicates that the FM rate of SRC matched filter is related to Doppler centroid, squint angle, and Doppler chirp-rate. However, these imaging parameters are not feasible for the non-cooperative BiSAR. Fortunately, the direct signal could be used to these key parameters’ estimation and the following image focusing.

###### 3.1.1. Estimate the Solution of the Optimal Doppler Chirp-Rate

After range compression of direct signal, the Doppler phase could be extracted easily because of its high SNR. The extracted peak phase of direct signal is .

Considering that is the function of azimuth time , thus it can be expanded as Taylor series at : where are the coefficients of the expanded phase.

Generally, the first two items in (11) are good enough to guarantee the expansion accuracy, and (11) could be estimated as where is the bistatic range at .

According to (9), (12) is rewritten as where is the estimated Doppler centroid and is the estimated Doppler chirp-rate.

As can be seen from (13), the peak phase of direct signal is regarded as the quadratic function composed of and . Obviously, the accurate estimation of and could be obtained through quadratic curve fitting between phase and azimuth time .

However, because of the existence of synchronization errors, the estimated Doppler centroid and Doppler chirp-rate are inaccurate. We denote and , where and are the differences between the estimated value and true value.

It is known to us all that three common problems existed in BiSAR system, including PRF partition errors, introduced phase noise, and different carrier frequency between transmitter and receiver. Of all three errors above, the PRF partition errors and differences of carrier frequency between transmitter and receiver mainly affect the first order term of in (13), while the error caused by phase noise would be guaranteed by system design and stable system capability [30–32]. As a result, the impacts introduced by these errors on can be ignored, and we have .

After obtaining the accurate estimation of , the FM rate of SRC matched filter thus could be simplified as a quadratic function composed of and . In the following, the simplified two-dimensional optimization will be applied to get optimal and .

###### 3.1.2. The Solution of the Optimal Doppler Centroid and Squint Angle

Image contrast refers to the dynamic range of an image, while focusing generally means higher spatial resolution. However, focusing quality could directly be reflected by image contrast [33–35], and the higher the contrast, the better the focused SAR image. The image contrast can be defined as the = standard deviation of the image energy/mean of the image energy.

The optimal and can be obtained by two-dimensional optimization, But the two-dimensional searching is not a good choice. For example, if the loop times in domain and domain are and respectively, obviously, the corresponding , computation capacity is heavy and time-consuming. However, for the low earth orbit, the maximum squint angle of transmitter is around −4° even though yaw steering is not applied. Because of introduced synchronization errors, the relationship between estimated Doppler centroid and squint angle could be expressed as , where . It indicates that the relationship between and is approximately linear. As a result, the complicated two-dimensional optimization could be simplified as multi-one-dimensional optimization. Meanwhile, the computation capacity would be reduced to , where are the times of iteration.

The searching steps are shown as below.(i)Firstly, a squint angle and a group of possible Doppler centroids, by which a group of SRC matched filters could be assembled according to (10), should be given. Since the maximum squint angle of transmitter is around −4°, we set at first in order to decrease the iteration times and improve the searching efficiency.(ii)Secondly, SRC processing as well as final SAR image could be accomplished based on all assembled matched filters, and we can obtain the image contrast of each picture , where : . And then, the optimal matched filter could be found based on image contrast optimization, which consists of optimal Doppler centroid and .(iii)Thirdly, the optimal squint angle can also be searched by applying the same searching method above based on obtained and a majority of given possible squint angles. The image contrast of each picture could also be obtained, where : .(iv)Finally, the loop ends when and satisfy , where is the threshold. Otherwise, let and repeat the loop. Generally, is selected in the optimization processing [33–35].

After deriving , , and , the optimal SRC matched filter could be assembled with the expression shown below: where is the range frequency.

According to the searching steps above, the nonsynchronous echo of YAOGAN 10 is processed, and the searching results of optimal Doppler centroid and squint angle are shown in Figure 2.

**(a) Doppler centroid**

**(b) Squint angle**

Figure 2 shows that the searching results of optimal Doppler centroid and squint angle are −3811 Hz and −2.4°, respectively, with which the optimal SRC matched filter could be constructed.

##### 3.2. RCMC and Focusing in Azimuth Direction

It is known to us all that range in bistatic configuration depends on radar incidence angle and receiver angle [36, 37]. However, for the lack of the incidence angle of transmitter and receiver and the satellite height based on non-cooperative transmitter, the traditional imaging algorithms could not be applied to Range Cell Migration Correction (RCMC).

However, considering the peculiarity of low receiver, narrow beam, and small scene in our real BiSAR system (parameters are shown in Table 1), the spatial variation of RCM could be ignored, and we can realize the Range Cell Migration Correction (RCMC) by compensating the RCM of echo with that of direct signal.

As can be seen from (4), the range history of a random target in range Doppler domain is expressed as

According to (15), we denote the RCM of a random target as

It could be seen from (16) that RCM varies with range time and azimuth frequency, and the difference of RCM between a random target and stationary receiver is

Obviously, the spatial variation of RCM cannot be ignored, until (17) is smaller than half of a range cell.

In order to extract accurate RCM of direct signal, curve fitting is needed. In normal cases, quadratic curve fitting is enough to satisfy the precision. (The processing result of real data is shown in Figure 5).

For focus in azimuth direction, the echo in range Doppler domain should be multiplied by the conjugate of the peak phase of direct signal in each range gate, while it is easy to extract the peak phase in range Doppler domain for the high energy of direct signal.

The expression of azimuth matched filter could be written as

According to (4) and (18), the difference of the peak phase between a random target and direct signal in range Doppler domain is expressed as

In order to focus on the whole scene, the maximum quadratic phase error of echo should satisfy . By all appearances, from (17) and (19), the size of imaging scene is limited. Using the parameters in Table 1, the largest scene that can be focused is 2 km 2 km, and it meets our demands.

After the azimuth compression, the slant range image in two-dimensional time domain can be obtained with an azimuth IFFT.

The flow chart of the proposed algorithm is shown in Figure 3.

**(a) Main part**

**(b) Core part**

It could be seen from Figure 3(a) that the proposed algorithm is mainly composed of four parts. Including, range compression, optimal SRC, Range Cell Migration Correction (RCMC), and azimuth compression. The core part shown in Figure 3(b), which consists of multi-one-dimensional optimization, can search the optimal and accurately and efficiently. As for each optimization, searching intervals, including and , should be defined based on precision requirements.

#### 4. BiSAR Experiment

To testify the validity of the proposed algorithm, BiSAR experiments based on the opportunity illuminator are carried out in JiangSu province of China, and the nonsynchronous echo of China YAOGAN 10 is used.

The BiSAR experimental hardware adopted in the bistatic experiments is shown in Figure 4, where different components have been labeled, while the stationary receiver is located on the building with the height of 109 m.

**(a) After SRC**

**(b) After RCMC**

There are two kinds of channels in the receiver, which are radar channel to collect the reflected echo from target area and reference channel to collect directly illuminated signal from the transmitter for synchronization processing.

At the bottom of Figure 4, an optic picture of the target area is given. It is a very good target area because different targets including roads, river, and factory rooms, were included and could be used to test the different bistatic responses from the target area. The bistatic experiments were carried out in 2012.

When the optimal Doppler centroid and squint angle are obtained based on image contrast optimization (shown in Figure 2), the optimal SRC matched filter can be applied to perform the SRC perfectly. After that, the RCM of the direct signal could be extracted to compensate the echo in two-dimensional frequency domain. The results after SRC and RCMC are shown, respectively, in Figure 5, while the extracted RCM before and after curve fitting is shown in Figure 6.

After RCMC is finished, the peak phase of the direct signal can be extracted in range Doppler domain and used to match the echo in azimuth.

For a better understanding, a classic NLCS algorithm [22] is also used to process the nonsynchronous echo of YAOGAN 10 and the comparison is shown in Figure 7.

**(a) The result of the proposed algorithm**

**(b) The result of the classical NLCS algorithm**

It can be seen from Figure 7 that (a) is obviously better focused than (b), and the river, roads, rooms, and so forth can be clearly seen in the red circles. It is suggested from the above results that the proposed algorithm has the ability to process nonsynchronous echo.

The slant range image after geometry correction and the Google optic image are also shown, respectively, in Figure 8, where the coincidence between the bistatic and optical images is found.

**(a) Image after geometry correction**

**(b) Google image**

It is suggested from Figure 8(b) that the places we are interested in are not influenced, the main targets, such as factory rooms, roads, and river in the bistatic image have a very good accordance with the optical one.

#### 5. Conclusion

Based on searching of the optimal SRC matched filter from the information of direct signal, this paper presented a novel imaging method for the nosynchronous echo based on the special mode of BiSAR with a stationary receiver. Then, The RCM and peak of direct signal were extracted for the azimuth focusing. The experiment data with china’s YAOGAN 10 SAR as the transmitter are processed by the proposed method. The images are also geometrically corrected and then visually compared with a Google image of the corresponding region. The good accordance shows the imaging algorithm’s effectiveness and the fascinating potential application of this non-cooperative BiSAR system in the future.

#### Appendix

Based on (1), the expression after range FFT could be expressed as where is the range frequency.

The phase in the integral is

According to the principle of stationary phase, the stationary phase point is

Substituting (A.3) into (A.1), and we get that

After an azimuth FFT operation, the expression of (A.4) could be written as

Obviously, the phase in the integral is In the meantime, (A.6) could be rewritten as

Based on the principle of stationary phase, the stationary phase point of (A.7) is

Substituting (A.8) into (A.7), the expression of echo in 2D frequency domain could be shown as

In order to get the expression of echo in range Doppler domain, we have

Let us denote and expand as the polynomial function of by Taylor expansion; that is,

Consequently, the phase of (A.9) could be simplified as

Similarly, we could get the stationary phase point of (A.12) where

The expression of echo in range Doppler domain could be obtained by combining (A.9), (A.10), (A.13); and (A.14):

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported by the key project of National Natural Science Foundation of China (Grant nos. 61225005, 61032009, 61172177, 61120106004, and 60890073).