Two-Dimensional Imaging Algorithm Based on Linear Prognosis for Space Target in Bistatic ISAR System

In bistatic inverse synthetic aperture radar (Bi-ISAR) system, its image resolution is lower than monostatic ISAR system. In order to solve this problem, the linear prognosis algorithm is adopted in the imaging process and the imaging algorithm based on linear prognosis is proposed. Space target Bi-ISAR imaging is taken as example in the research. The one-dimensional range profile is created through pulse compression mehod. Before the azimuth compression, burg entropy maximum algorithm in Levions recursive method is used to estimate the prognosis coefficients and the azimuth echo data. Then Fourier transformation is used to compress the azimuth data in order to get the high resolution azimuth image. This imaging method can obtain the two-dimensional image with the resolution equal to the monostatic ISAR or even higher than it. Simulation experiments have verified the effectiveness and availability of the algorithm.


Introduction
Due to the separated transmitting and receiving station, the bistatic Inverse Synthetic Aperture Radar (ISAR) can still get the 2--D (two-dimensional) object image even when the relative radar line of sight has no rotational movement, which is the better characteristic than monostatic ISAR.So, bistatic ISAR has the higher imaging probability than monostatic ISAR [1], [2].Otherwise, when the receiving station is located in the front, the imaging range augments at the same time, which could improve the detection and imaging ability for stealth targets [3]- [5].In recent years, the main research of bistatic ISAR is focused on imaging principle and various compensation algorithms [6]- [9].
During the bistatic ISAR imaging process, the time-variant bistatic angle can lead to a time-variant bistatic imaging resolution, which makes the 2-D image become blurred [10].The movement of space targets is complicated, so it needs to image the object in a short integration time and avoid blurring phenomenon of 2D image.
Because of the short integration time, the bistatic angle and image resolution can be looked as the constant during imaging time, which could effectively avoid phenomenon of image blurring.At the same time, the target characteristics have relatively small changes when imaging during small rotation angel, which can decrease the difficulty of movement compensation.While the coherent integration time becomes shorter, the resolution is definitely decreased.So it is the prerequisite to research the ISAR imaging algorithm when the rotation angle is small.
The ISAR imaging becomes spectrum estimation problem when the movement compensation is accomplished as researched in the paper [11].So, the imaging problem in small integration time can be changed into a high-resolution spectrum estimation problem with short data sequence.As for the problem of high-resolution spectrum estimation with short data sequence, there are two kinds of processing methods as introduced in the papers [12], [13].The first kind is spectrum estimation method based on spectrum functions such as MUSIC, ESPRIT and so on.This method could get the higher frequency resolution but it has the fake amplitude information and it is also sensitive to noise.The second kind is spectrum estimation method based on DFT data extrapolation.This method complies with some principles which extrapolates new data based on the existent data.Then all the data are used to accomplish spectrum estimation based on DFT method which will equivalently increase data length and improves spectrum resolution.The second method has attracted more attentions because it has  ISSN: 1693-6930 TELKOMNIKA Vol. 12, No. 4, December 2014: 933 -941 934 high resolution, small calculation works and the higher possibility of realization.So the second method is adopted in this paper to research the 2-D imaging algorithm for bistatic ISAR according to space targets.

Echo modeling of space target in bistatic ISAR system
The imaging geometry relationship of moving target in bistatic ISAR system is showed in Figure 1. is the rotation angle of bisector during imaging process.If we make the assumption that the bistatic radar is ideally synchro and LFM signal is transmitted with the echo which is sampled in intermediate frequency and desample it to baseband frequency, the echo of scatter i c can be denoted through (1) as follows.
Where t is the time-in-pulse; m t is the pulse transmitting time; 0 f is carrier frequency;  is frequency slope; p T is pulse width; ci  is the non-back scattering intensity of scatter i c in the th i range bin.In the paper [6], it gets the research result that whether the target moving track is in the same plane with baseline of bistatic radar, the range transformation of scatter can be expressed as follows (2). Where is target time-varying bistatic angle during the imaging process; is the range course between target phase center and transmitting-receiving station, which is the translational movement item of target; R t is the rotational range item of scatter i c .It can be found from type (2) that in bistatic ISAR imaging, the range transformation can still be divided into two parts, i.e, translational movement item and rotational movement item So through bistatic ISAR system also can get the target image through range compression, range alignment, initial phase correction, azimuth compression just as monostatic ISAR system researched in the paper [6].
When the rotational angle is small which is equal to integration of short time, the bistatic angle ( ) m t  can be thought as the constant approximately.It is assumed that the constant bistatic angel is thought to be 0  .If the translational range item has been totally compensated, the range transformation item of scatter i c can be simplified as type ( 3) At this time, target range resolution of and azimuth resolution can be expressed as types ( 4), ( 5) respectively, the expressions are accordance with the research work in the paper [14].
In the equations mentioned above, w B is the transmitting signal bandwidth; rot   is the rotational angle of bisector during imaging process.From types (4) and ( 5), it can be found that when the transmitting signal bandwidth is fixed, the target range resolution in small rotation angle imaging condition is the constant.But the azimuth resolution is totally determinded by rotational angle rot   . When the imaging rotational angle rot   is small, the azimuth resolution is also small as well, which could not satisfy with the demands of imaging system.So the imaging algorithm in small rotational angle condition should to be researched and improved.

Analysis of Bistatic ISAR resolution
As transmitting station and receiving station are separated from each other in bistatic ISAR system, thus the imaging resolution is related with bistatic angle.If the imaging bistatic angle is  , then the target range resolution can be expressed as following type (6) [14], [15].
Where _ 0 rc  is the target range resolution; w B is the transmitting signal bandwidth;  is the imaging bistatic angle; c is the transmitting speed of electromagnetic wave in free space.It can be found from type (6) that as there exists bistatic angel, the bistatic ISRA range resolution is lower than the monostatic ISAR range resolution 2 w c B with the same signal bandwidth.For the convenience of comparison, bistatic ISRA system is thought to be an equivalent monostatic ISAR system in the bisector.If the transmitting carrier frequency is 0 f , signal bandwidth is w B , bistatic angle is  , then the equivalent carrier frequency of monostatic radar is . So the resolution also could be expressed as type (6) reported in the paper [15].This equivalent model can explain why the range resolution is decreased with bistatic angel, so this model has been widely used during the existent bistatic ISAR research work of the papers [13]- [15].The frequency of echo could not be changed except Doppler effects of target.And the target scattering echo frequency and its bandwidth could not be changed as the existence of bistatic angle.So, there must be some causes that make the bistatic range resolution different from that of monostatic ISAR system.In the Figure .1, it is shown that the wave distance is different from scatter 1 c and 2 c in different range areas.

cos 2
In the demonstration of scatter wave distance difference in the Figure 1, no matter where the scatter is, the main lobe width of echo after pulse compression is constant because the echo signal width is constant, if it only judges from the echo characteristics index.Otherwise, the essence of radar range resolving ability is the echo time-delay resolving.When the matching filter based on FFT is used to accomplish range profile compression, whether the radar is monostatic or bistatic, if the echoes of two scatters are needed to be distinguished, and the time-delay of the two scatters echo after pulse compression is 3dB width at least.And the round wave distance difference d  could be denoted as below type (8).
The range resolving ability is the width of the smallest resolving range in the targets.The bistatic range-bin width is determinded by the distance between the two intersection dots, which are the dots of bistatic bisector intersected by the two rang-same lines with the same focus which is also verfied by the paper [16], [17].The distance between the two dots is also the distance between the two range-same lines.So, when the target round wave distance difference in the type (7) is the same as showed in type ( 8), scatter 1 c and scatter 2 c can be separated in range axis by the square.And now, the distance difference of the two scatters in range axis is , which is expressed in the type (6).The resolution is lower than the monostatic ISAR system with the same signal bandwidth as 2

Imaging algorithm based on levions prognosis 4.1 Imaging flow of 2-D image
The main effect factor of 2-D imaging in small rotational angle is the azimuth resolution.So in the following research, it is assumed that one-dimensional range profile is obtained through digital matching filter pulse compression and the translational compensation has been accomplished.
The ISAR imaging problem can be regarded as spectrum estimation problem just same as the paper [11].The basic step of linear prognosis is firstly to estimate the filter coefficients and then extrapolate the data to the needed extent.The prognosis coefficients are estimated based on the least mean square error rule, which can ensure the difference value between actual echo and the extrapolated data is the least.In order to improve the azimuth spectrum estimation resolution, the range echo is estimated to accord with the linear prognosis model and the azimuth data integration time is extended , so the azimuth resolution is advanced.
If some range aligned range echo could be expressed as   1 ( ), , ( )  , so the extrapolated echo at time n t of this range could be expressed as In the equation ( 9), ( ) m a i is the ith coefficient of the m-order filter.With the same analysis, the echo at time 1 n m t   after extrapolated of this range echo could be expressed by (10).
In the equation(10), * ( ) m a i is the conjugate complex of ( ) If the former-latter prognosis is prosecuted by p times, then the echo after prognosis could be expressed as   1 1 ( ), , ( ), , ( ), , ( ) . If the pulse repeat cycle is T , then the azimuth resolution could be improved from 1 mT to 1 ( 2 ) T m p  .Combined with rang pulse compression, 2-D imaging algorithm flow based on linear prognosis when the rotational angel is small which is demonstrated as in the Figure 3.

Construction of prognosis filter coefficients and its order
The larger the linear prognosis filter is, the more accurate the prognosis result is.But the calculation amount is increased rapidly also, so the balance between calculation work of algorithm and prognosis error is destroied.In the paper [17], it shows that when the Auto Regressive (AR) model is used to extrapolate data, the most accurate prognosis results could be obtained when the filter order is chosen to be 1/3 of the data length.The Levinson prognosis of Burg entropy is used to estimate the filter order and its coefficients.The filter coefficients of different filter order are estimated firstly, then the prognosis error power with different filter is compared and it chooses the filter with the smallest prognosis error power.
The basic steps to get the prognosis order and its coefficients are summarized as follows.
Step1： Initial the prognosis power error and its back-and-forth prognosis error; Step2： Count the reflection coefficient m K that is used to extrapolate data; Step3： Count the back-and-forth prognosis filter coefficients via (11); Step4： Count the prognosis error power of the filter with m-order via (12) (1 ) Step5： Count the output of the filter with m -order, repeat Step2 to Step5 until the prognosis error power m P hardly become smaller.And the order and coefficients of prognosis filter are got at the same time.

Data extrapolation length
The data extrapolation length is also determinded by prognosis error , and the longer the data extrapolation length is the larger the prognosis error will be.So far, there is no quantitative choose standard concerned on this problem.What could be decided up is that the length of data extrapolation is related with SNR ( signal-noise-rate), and the higher the SNR is the longer the data extrapolation length will be.Otherwise the shorter data extrapolation length has to be chosen.In this paper, space target is chosen as the research object and the SNR of space target is relatively low compared to research result of paper [17] , it adopts the 1/2 length of original data length as the data extrapolation length thus it can improve the azimuth resolution by two times .In the simulation, the smallest receiving SNR is set to be 14dB and the target RCS is set as 4m 2 .Because the scatters are separate, respectively, so the RCS of every concrete scatter is 0.5m 2 .Simulation parameters of the radar system are shown in the Table 1.According to the parameters in Table 1, the range resolution and azimuth resolution can be expressed as  From the Figure 6 we can find that, before the azimuth data extrapolation range resolution is smaller than the interval of scatters so the scatters could be totally differentiated from range profile.But the azimuth resolution is not small enough, so the scatters could not be differentiated clearly form azimuth direction.Figure 6 is the 2-D image after data extrapolation and because the azimuth doppler resolution is improved, so the scatters could be totally differentiated from azimuth direction,and the correct 2-D image is obtained.

Conclusion
ISAR imaging could be seen as high-resolution spectrum estimation problem after the movement compensation is completed.The space target bistatic 2-D imaging problem in small rotation angle is researched in the paper.Simulation results prove that this algorithm can improve resolution in azimuth direction and can be used in practical environment.The research on how the algorithm can be used in lower SNR is also performed.

Table 1 .
Simulation radar parameters