Performance Evaluation of Frequency-Domain ... - IEEE Xplore

IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 7, NO. 2, FEBRUARY 2014

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Performance Evaluation of Frequency-Domain Algorithms for Chirped Low Frequency UWB SAR Data Processing Daoxiang An, Member, IEEE, Yanghuan Li, Xiaotao Huang, Member, IEEE, Xiangyang Li, and Zhimin Zhou

Abstract—This paper studies the performance of the frequencydomain algorithms (FDAs) for low-frequency ultra-wideband synthetic aperture radar (UWB SAR) data processing. First, a generalized theoretical derivation of the FDAs is presented from the viewpoint of SAR signal processing. The derivation not only provides a deeper understanding to the imaging principle of the extended Omega-K algorithm (EOKA), but also makes it compatible and comparable with the other FDAs. Second, the performance comparison on different FDAs is made based on theoretical analysis, simulation and experimental data. The comparison results show that the Omega-K algorithm (ωKA) has the highest imaging precision in the ideal case (i.e, no motion error), but its application is limited by the poor ability of compensating motion errors. In contrast, the EOKA and nonlinear chirp scaling algorithm (NCSA) have excellent performance on dealing with the motion error, but they can only be applied under specific preconditions. Besides, as cetner frequency gets lower, the fractional bandwidth and integration angle get larger, the imaging precision of NCSA greatly decreases, while the ωKA and EOKA still keep high precision. Index Terms—Frequency-domain algorithms (FDAs), low frequency, Synthetic aperture radar (SAR), ultra-wideband (UWB).

I. INTRODUCTION

T

HE low-frequency ultra-wideband synthetic aperture radar (UWB SAR) has excellent foliage or ground penetrating capability to detect concealed targets [1]–[6]. However, the UWB signal and large integration angle used in low-frequency ( 1 GHz) UWB SAR bring new complexities and challenges to the traditional SAR image formation processing. In the past years, many algorithms were proposed for SAR data processing, but only three kinds of algorithms have been widely used in the low-frequency UWB SAR. They are the kinds of back projection (BP) algorithms [1], [7], the nonlinear chirp scaling algorithm (NCSA) [8]–[11] and the extended Omega-K algorithm (EOKA) [12]. The BP class algorithms belong to the time-domain algorithm (TDA), while the other two kinds of algorithms belong to the frequency-domain algo-

Manuscript received November 29, 2012; revised March 14, 2013 and May 12, 2013; accepted May 15, 2013. Date of publication June 18, 2013; date of current version February 03, 2014. This work was supported by the National Natural Science Foundation of China: Research on the key techniques of low frequency ultrawide band synthetic aperture radar interferometry (61201329). The authors are with the College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China, 410073 (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSTARS.2013.2265272

rithm (FDA). The TDAs have two main drawbacks. The first is the heavy computational load. To overcome this problem, some effectively modifications [1], [13], [14] have been proposed, which can keep the accuracy and robust of TDAs but with a reduced computational load. The second drawback is the poor ability to integrate accurate autofocus algorithm into its imaging process [15], and this shortcoming limits the real application of TDAs for SAR systems equipped with the low/moderate accuracy Inertial Navigation System (INS)/Global Positioning System (GPS). Dissimilarly, the NCSA and EOKA can implement the MOCO perfectly based on the raw data. So although they have some approximations in the imaging processing, they are still widely used due to the excellent performance on integrating with the data-based MOCO strategy[16] and the lower computational load. The NCSA and EOKA are the modifications of the chirp scaling algorithm (CSA) [17] and the Omega-K algorithm (ωKA) [18], [19], respectively. Due to the approximations exist in the standard CSA, it can not provide enough accuracy in the highly squint or low-frequency UWB SAR cases. To resolve this problem, the NCSA [8] is proposed, which have better performance on processing the raw data acquired by highly squint SAR systems. Compared to the kind of CS algorithms, the ωKA has higher accuracy in processing the highly squint or low-frequency UWB SAR data. However, in the ωKA, only the first-order motion error correction can be applied directly [12], which is the range-independent part of the real MOCO, and is applied directly before or after range compression. However, in the case of high-resolution SAR systems, a precise MOCO of the range-dependent components of the motion error is essential [15]. Similar to the Doppler-based MOCO approach, the range-dependent MOCO processing step has to be implemented after range compression and after the range cell migration correction (RCMC), is not possible in traditional ωKA processing. In order to resolve this problem, the EOKA is proposed [12], which can integrate the high precise MOCO correction [15], [16] in the general formulation of the ωKA. Until now, nearly all the other FDAs are proposed and derived from the viewpoint of SAR signal processing. In contrast, the ωKA and EOKA are originally derived from the principle of wave equation [12], [18] or from the viewpoint of time domain signal processing [19]. Due to the poor relationship with SAR image generation, above derivations are inconvenient to understand. Besides, the different derivations of different frequencydomain algorithms make it inconvenient to evaluate their focusing performance in a uniform coordinate. To overcome this

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AN et al.: PERFORMANCE EVALUATION OF FREQUENCY-DOMAIN ALGORITHMS FOR CHIRPED LOW FREQUENCY UWB SAR DATA PROCESSING

problem, a new explanation of the imaging principle of ωKA from SAR signal processing viewpoint is presented in [9]. The new explanation makes the ωKA more compatible with other FDAs. However, no derivation of the EOKA from SAR signal processing viewpoint was reported in public literatures. Since the different derivations, it is inconvenient to carry out a performance comparison of the EOKA and NCSA in theory for processing the low-frequency UWB SAR data. The aim of this paper is twofold. First, based on the previous work presented in [9], a generalized derivation of the different FDAs is carried out. Based on the derivation, a novel explanation for the EOKA from the SAR signal processing viewpoint is given. The novel explanation makes it easier to understand the principle of the separation of the RCMC term from the actual azimuthal focusing by adopting the modified Stolt interpolation. Second, based on the generalized derivation, performance of the EOKA and NCSA for the low-frequency UWB SAR imaging is evaluated from the aspects of limitations, phase aberrations, computational load and MOCO strategy. The comparison results show that each algorithm has some advantages and disadvantages. On one hand, the ωKA and EOKA have higher accuracy than the NCSA for processing low-frequency UWB SAR data, while the EOKA and NCSA have excellent performance on compensating motion error. On the other hand, the ωKA is suitable for processing the data acquired in the ideal case (i.e., no motion error), meanwhile, the EOKA and NCSA can only be applied in the case that the specific precondition is satisfied. Besides, as the fractional bandwidth and integration angle get larger, the imaging precision of NCSA greatly decreases, so it is not suitable for processing the high-resolution low-frequency SAR data with very high fractional bandwidth and very large integration angle. It should be noted that the ωKA is also known as the range migration algorithm (RMA) [20], and the main difference between them is that the RMA uses the dechirp processing for range compression (RC) while the ωKA uses the matched filtering for RC operation. However, the kernel processing steps of their imaging procedure are the same, i.e., the Stolt interpolation. So, the derivations and explanations of the ωKA presented in this paper are also valid for the RMA. This paper is organized as follows. Section II gives the generalized derivations of the FDAs from the SAR signal processing viewpoint. Section III makes a performance comparison on different FDAs for processing the low-frequency UWB SAR data based on theoretical analysis. In Section IV, experiments using simulated and real data are carried out to validate the correctness of the theory analyses presented in the previous sections. Finally, concluding remarks are presented in Section V.

II. GENERALIZED DERIVATIONS OF THE FREQUENCY-DOMAIN ALGORITHMS A. Generalized SAR Signal Model For our analysis, we consider only the phase terms of the SAR signal, ignoring the initial phase. Besides, the general underlying assumptions (e.g. start-stop approximation) are also as-

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sumed valid in the formula derivation of this paper. As in the development presented in [9], the phase of the demodulated baseband SAR signal can be described as (1) where is the speed of light, is the range chirp rate, is the center wavelength (i.e., the wavelength corresponding to the center frequency ), is the fast time, is the range to a given target at slow time and the range of closest approach . The first term of (1) denotes the azimuth modulation, which is a function with respect to the wavelength and the changing range to the target. The second term in (1) is the transmitted chirp delayed by the two-way travel time to the target. The derivations of most FDAs are carried out in the wavenumber or two-dimensional (2D) frequency domain, and the general SAR signal in the 2D frequency domain can be described as (2) where is the moving velocity of the radar platform, is the range frequency, and is the azimuth frequency. Equation (2) is a general and accurate expression of the phase of the SAR signal spectrum [9]. This expression allows one to derive matched filters directly in the 2D frequency domain, which makes algorithms in this domain attractive. The first term in (2) is range modulation term. The second term is the range-azimuth coupling term, which varies with the slant range. This coupling term is very important in SAR imaging, because all FDAs can be viewed as an approximate correction of the coupling term in different preconditions. The essential difference of the performances of FDAs comes from the precision of the correction of the coupling term, which will be discussed in the following deductions in detail. For the simplicity of carrying out the following derivation, we rewrite (2) as follows

where Let

(3) is the migration parameter.

(4)

denote the range-azimuth coupling term. Under the following assumption

(5)

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We can expand the square root term of (3) into a Taylor series with respect to , then (3) becomes

and perform the Stolt interpolation operation on the remainder terms. Let denotes the remainder terms, then we have

(10) (6) where

Similar with the ωKA, a bulk reference function multiplication is needed in the 2D frequency domain before implementing the Stolt interpolation operation. The reference function is , where

(7) (8) denotes the coefficient of the th term. The second term of (6), which is independent of the range frequency , carries the phase information required to focus the data in azimuth direction at range . The third term of (6), linear in , represents the range cell migration (RCM). The remainder higher order ( 3rd) terms of (6) represents the range-azimuth coupling phases. In the following subsections, derivations and explanations of the FDAs are given from the viewpoint of SAR signal processing. Remarks: For all frequency algorithms derived based on (6), the Taylor expansion must be relatively accuracy. Otherwise, we can not obtain high precise imaging result. So, inequality (5) gives the upper limit on the factional bandwidth as (9) where is the fractional bandwidth defined as the ratio of the signal bandwidth to the center frequency . The fractional bandwidth is therefore limited by . Therefore, the frequency algorithms (such as the CS class algorithms) derived based on (6) are not suitable for some specific low-frequency UWB SAR with fractional bandwidth larger than 0.8 [6], [11].

(11) After the reference function multiplication, the Stolt mapping [12], [21] is performed by replacing the square bracket factor of (10) with the shifted and scaled range frequency variable, , that is (12) From the above derivations, we can find that the modified Stolt interpolation of (12) is equivalent to apply the SRC and RCMC simultaneously, but without the azimuth compression (AC) operation. The extracted term is only dependent on the slant range and azimuth frequency, so it can be compensated in the range-Doppler domain after the Stolt interpolation. At this stage, the AC operation is separated from the nonseparated 2D imaging procedure of the ωKA. Applying the range IFFT on the interpolation result, the echo signal in the range-Doppler domain is obtained, and the AC can be performed in this domain by multiplying the echo signal with the azimuth filtering reference function, whose phase is given by

B. Derivation of EOKA In [9], a new derivation of the ωKA based on the expansion of (6) is presented, and a novel explanation of the imaging principle of the ωKA from the SAR signal processing viewpoint is given. The new derivation and explanation make the ωKA more compatible with other FDAs and easier for understanding. In this paper, the same approach is taken to derive and explain the EOKA from the same viewpoint. From the discussion in the previous subsection, we know that the term of (4), which is independent of the range frequency , denotes the azimuth modulation. It is well known that the Stolt interpolation performs a mapping of the range frequency axis, so it is reasonable to extract the term from the range-azimuth coupling term ,

(13) Remarks: In the derivation of EOKA, the range-azimuth coupling phase term is expanded as a power series in the range frequency domain. However, it should be noted that in the real implementation of the EOKA, it is not necessary to expand these terms, but it is very useful for explanation and understanding and make the derivation of EOKA compatible with other FDAs. The main advantage of the EOKA is that the range and azimuth focusing processes are implemented separately by the modified Stolt interpolation and without any approximation throughout the overall processing. So in principle, EOKA has the same precise as the ωKA.


C. Derivation of NCSA The kind of CS algorithms are also derived by the Taylor expansion of (6). In the derivation of the original CSA [17], only terms up to the second order of (6) are included. This approximation is valid for conventional high-frequency narrow-band SAR systems but might not valid for the low-frequency UWB SAR systems with large integration angle and high relative band ratio. In 1996, the NCSA [8] was proposed, which considers the cubic phase terms for the Taylor expansion of (6), and takes spatial variant range migration into account. These modifications make the NCSA has better performance on processing the squinted SAR data, and becomes a candidate for the low-frequency UWB SAR data. For the convenient of algorithms comparison, it is necessary to give a short review of the traditional NCSA. Combing the first term and the fourth term of (6), and manipulating the combined result as follows: (14) is the new range frequency modulation (FM) where rate, which can be viewed as the combined FM rate of the radar pulse and the SRC filter. Expanding at , and keeping up to the first order, yields

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TABLE I SAR SYSTEM PARAMETERS

At last, the AC operation is performed by multiplying with the azimuth filtering reference function of (13). Remarks: From the above derivations, we can find that the intrinsic difference of the different FDAs is the compensation of the signal spectrum phase. In the kind of ωKA algorithms, the spectrum phase is perfectly compensated by the Stolt interpolation. In the kind of CS algorithms, the nonlinear Stolt interpolation is replaced by a linear transformation, which is implemented by a complex multiplication based on the specific order approximations. III. PERFORMANCE ANALYSIS OF THE FREQUENCY-DOMAIN ALGORITHMS

(15) is the FM rate at the reference range , and is the slope of variation. A derivation that keeps higher order terms in the expansion of (15) is presented in [10]. The first step of the NCSA is to perform a cubic phase filtering operation in the 2D frequency domain. The phase of the filter is where

In this section, we present some analyses on the performance of FDAs on processing the UWB SAR data, which have different center frequency, different fractional signal bandwidth and different integration angle, but the same spatial resolutions. Table I lists the main parameters of these SAR systems. The analyses are carried out based on the simulated data, and no motion error is assumed. Also, no weighting function or apodization technique is applied. A. Analysis of the Limitations

(16) The processing steps of NCSA followed the cubic phase filtering includes nonlinear chirp scaling, reference function multiplication and residual phase correction. The phases of these reference functions corresponding to each step are given by

(17)

Compared to the ωKA, an extra precondition should be satisfied in the EOKA and NCSA. This precondition is (20) will not get complex which guarantees that the term values at the overall azimuth frequency. In general SAR systems, the pulse repetition frequency (PRF) is higher than the Doppler bandwidth, which is a function of the maximum integration angle and the signal fractional bandwidth. From a system designer’s point of view, the PRF must be larger than two times of the maximum Doppler frequency

(18)

(21)

(19)

and are the minimum signal wavelength and where the maximum integration angle. The maximum value of the quadratic term in (i.e., when ) can be represented by a function of the SAR system’s parameters as

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, where . At this stage, it is easy to get the limitation on the maximum integration angle by resolving (20) (22) In some specific low-frequency UWB SAR system [6], [11], the highest PRF with regard to extremely large integration angle (up to nearly 180 ) will make it become complex value, in which case the EOKA and NCSA can not be directly applied. There are two ways to overcome this problem: one way is to set the azimuth frequency outside the maximum Doppler bandwidth to zero, and then the imaging algorithms are applied. Another way is to assume a higher velocity of the SAR platform in processing to keep real at all azimuth frequencies [11]. Defocusing caused by this can be compensated in some subsequent processing. B. Analysis of the Approximations The main approximation error of the kind of ωKA class algorithms is caused by the Stolt interpolation, which has impact on the image amplitude and phase. However, real data processing results [12] show that the impact of the interpolation is very small and can be ignored even in the interferometric SAR data processing. Compared to the kind of ωKA class algorithms, there are two other main approximations exist in the imaging process of the NCSA. One of them is the specific order Taylor approximation, which is the basis for the chirp scaling processing. The other approximation is related with the linear approximation of the range FM rate. In the following subsection, the phase errors caused by these two approximations are evaluated and their impacts are analyzed in detail. 1) Higher-Order Approximation of the NCSA: In the low-frequency UWB SAR data processing, the neglected higher order ( 4th) range-azimuth coupling phases (HOP) of the NCSA have significant effect on the focused image quality. To resolve this problem, a HOP correction (HOPC) function at the reference range can be applied in the 2D frequency domain at the beginning of the NCSA. The HOPC function is given by

Fig. 1. The HOPEs for the targets at different slant ranges. (a) The original values. (b) After implementing the HOPC operation.

factor in (24), but a residual HOPE exists for the targets at other ranges, and the residual HOPE increases as the targets away from the reference range. Obviously, the residual HOPE increases as the center frequency gets lower, the signal bandwidth gets larger and/or the integration angle gets larger. Considering system 2 in Table I, three targets labeled as A, B, and C are located in the imaged scene at ranges of m, m and m, respectively. Fig. 1 shows the diagrams for the HOPE evaluation of the different targets before and after implementing the HOPC operation. It is easy to find that HOPE gets smaller after performing the HOPC, which means its influence on the focusing quality gets weaker. 2) Range Frequency Modulation Approximation of the NCSA: In the derivation of the NCSA presented in the previous section, a linear approximation of the range FM rate is adopted by using the Taylor expansion of (15). However, the Taylor expansion is feasible only when the following precondition is satisfied (25)

(24)

We refer to the slant range as the “break point”. Inequality (25) is satisfied in most high-frequency narrow-band SAR systems. However, as the wavelength increases, inequality (25) is rarely satisfied. For example, assuming the SAR systems with a following typical parameters: chirp bandwidth MHz, azimuth resolution m, radar platform velocity m/s. Fig. 2(a) shows the diagram of . It can be seen that decreases with the increases of . In Fig. 2(a), the “break point” is 9.032 km, where m. In such case, the “break point” may lie in the interval , where and denote the nearest slant range and farthest slant range, respectively. Fig. 2(b) shows the diagram of the range FM rate with respect to the slant range when the “break point” exists in the range swath. It can be seen that the approximation error of suddenly becomes large around the “break point”. The approximation error of will introduce a quadratic phase error (QPE) in the spectrum, and degrades the focusing quality of the image. Moreover, to obtain the higher accuracy Taylor expansion result, i.e., (15), the following criterion should be satisfied:

It is found that, after applying the HOPC, the residual HOPE becomes zero at the reference range because of the

(26)

(23) By subtracting (23) from (6), we can have the residual HOP error (HOPE) after applying the HOPC, which is given by


Fig. 2. Analysis of the “break point”. (a) Diagram of .

. (b) Diagram of

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Fig. 4. Diagram of the range block size versus integration angle for different UWB SAR systems.

Let denote the target range offset apart from the reference range. Using (24), the maximum residual HOPE of a target can be evaluated by

(28) Similarly, it can be deduced from (27) that the maximum QPE of a target can be evaluated by Fig. 3. The QPEs for the targets at different slant ranges. (a) The QPEs in CSA. (b) The QPEs in NCSA.

The QPE induced by the approximation error of the range FM rate can be evaluated by (29) (27) It is easy to find that the QPE increases as the center frequency gets lower, the signal bandwidth and/or the integration angle gets larger. Using the same parameters presented in the previous subsection, Fig. 3 shows the QPEs of targets A, B and C in the CSA and NCSA, respectively. It is found that the QPE increases as the target away from the reference range. Moreover, for targets at the same range, the QPE in the NCSA is much smaller than that in the CSA, and this is one of the reasons that why the NCSA has higher precision than the CSA. 3) Range Block Processing of the NCSA for Large Scene: Obviously, the HOPE and QPE of the NCSA presented in the last subsection will limit the range swath width that can be well focused. Therefore, when applying the NCSA on processing large scene low-frequency UWB SAR data, a range block processing method [22] is usually used at the tradeoff of imaging efficiency. Then, the NCSA is used for the block data focusing. At last, the final combined image is obtained by mosaicing all the focused subimages together. The size of the range block is determined by analyzing the allowable HOPE and QPE, and the invariance block size in the NCSA depends on the specific radar parameters. In real applications, the size of the invariance region is limited by a number of factors. Generally, the main factors are the residual HOPE and the QPE. The following discussion is focused to these two factors only.

Within a processing block, the residual HOPE and the QPE should be limited by , where is normally set to less than 0.5. Using (28), the maximum range deviation from the reference can be obtained from (30) And using (29), the maximum range deviation from the reference is found from the following: (31) Finally, the range invariance block size is as follows: (32) Fig. 4 shows the diagrams for the range block size evaluation for the SAR system in Table I. From Fig. 4, we can find that the range block size gets smaller as the integration angle gets larger. For example, the range block size is only 73.1 m for system 1, and this size is too small in real application. In such case, the range block processing will be failed, and this leads to the poor ability of NCSA to process large range swath and high-resolution UWB SAR data with low center frequency, high fractional bandwidth and large integration angle. C. Computational Load Analysis In the following analysis of the computational loads of the frequency algorithms, the increase of the computational load induced by the MOCO operation of each algorithm is assumed the

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same, thus the MOCO strategy has no impact on the comparison result. In the real application of EOKA, there are two optional imaging flows can be chosen. One option, referred to as EOKA1, is to perform the range compression prior to the Stolt interpolation, and remove the amount of data corresponding to the range throwaway region at the beginning. Using this option, smaller data sizes are needed in the subsequent operations but at the expense of an extra pair of range FFT and IFFT. Another option, referred to as EOKA2, is to incorporate the range compression with the reference function multiply, and remove the amount of data corresponding to the range throwaway region after the Stolt interpolation. Using this option, larger data sizes are needed in the 2D frequency domain to accommodate the range matched filter throwaway region, but no extra range FFT and IFFT is needed. In [9], a quantitative comparison of the computation loads of RDA, ωKA and CSA is carried out by estimating the number of floating point operations (FLOPs). Using the same approach, a comparison of the computation loads of the EOKA and NCSA is carried out in this paper. Let denote the number of input range samples per line, denotes the number of output range samples after the range matched filter throwaway, and is the ratio of the number of the samples in the range chirp signal to the number of the input range samples. Moreover, let denote the interpolator length used in the Stolt interpolation. Then using the same definition presented in [9], the computational loads of EOKA implemented with the two different options are given by

Fig. 5 Diagram of the computation loads versus .

and vector multiplication operations, which are easy to implement in software and hardware. In contrast, due to the Stolt interpolation, the implementation of EOKA is relative complex and difficult. Another point that should be noted is that the NCSA can only process the range chirp signal, so in its imaging procedure, a large mount of data need to be stored up until the range IFFT operation. If range compression has already been performed or a dechirp-on-receive mode is used, the echo signal has to be expanded with a chirp signal. Based on the above analysis, in real applications, one should carefully choose the proper imaging algorithm according to the specific SAR system parameters and the practical requirement. IV. EXPERIMENT RESULTS

(33) (34) Similarly, the computational load of the NCSA is given by (35) Assuming the interpolator length , and the number of input range samples , then the computational loads of the EOKA and NCSA, measured in Giga-FLOPs (GFLOPs), are plotted as functions of in Fig. 5. Observing the curves shown in Fig. 5, we can get the following conclusions: First, the computational loads of the EOKA with both implementations are smaller than that of the NCSA. Second, when is small, the computational load of EOKA1 is larger than that of EOKA2. However, as the increases, the computational load of EOKA1 decreases faster than that of EOKA2. When , the computational load of EOKA1 is smaller than that of EOKA2. So in the real application, the proper processing option of the EOKA should be carefully chosen according to the specific SAR system parameters. From above analysis, we can find that the NCSA has heavier computational load than the EOKA when measured in GFLOPs. However, the implementation of NCSA only includes the FFT

A. Simulation Results The simulated data used in this study is also based on the parameters of the three SAR systems given in Table I. In the simulation, two targets are arranged in the illuminated scene along the azimuth center at the range 10 km (i.e., center slant range) and 9.6 km, respectively. In the imaging processing, the center slant range is selected as the reference range. Four approaches ωKA, EOKA, NCSA without HOPC and NCSA with HOPC are subject to be compared. To obtain a fair comparison, no weighting function or sidelobe control approach is used. Figs. 6–8 show the imaging results obtained by different algorithms. Fig. 6 shows the imaging results of System 1. It is easy to recognize that the images obtained by the ωKA and EOKA are well focused and their focusing quality are very close. Fig. 6(c) shows the imaging result obtained by using the NCSA with HOPC. We can find that the bottom image in Fig. 6(c) (Corresponding to the target at the center range) shows an improvement in focusing but still not well focused, while the top image shows seriously degraded focusing. The defocusing of the center target can be explained by the phase errors (except for the HOPE and QPE) caused by the other approximations in the imaging steps of NCSA (such as the chirp scaling, RCMC, etc.). For most cases, these phase errors are very small and their influence can be ignored. However,


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Fig. 6. Contour plots of targets obtained via different approaches for system 1. (a) ωKA; (b) EOKA; (c) NCSA with HOPC; (d) NCSA without HOPC.


for the high-resolution low-frequency UWB SAR with very high fractional bandwidth and very large integration angle, the impacts of such kinds of phase error can no longer be ignored. Comparatively, due to the influence of the HOPE, in the image obtained by the NCSA without HOPC, both the two targets are seriously degraded, as shown in Fig. 6(d). Fig. 7 shows the imaging results of System 2. Observing Fig. 7(a) and (b), it is easy to recognize that the images obtained by the ωKA and EOKA are still well focused and their focusing quality are very close. However, in the image obtained by the NCSA with HOPC, the center target is well focused (See the bottom image in Fig. 7(c)), but the other target shows a little degraded focusing (See the top image in Fig. 7(c)). This can be explained as follows: as the center

frequency gets higher and the factional bandwidth as well as the integration angle gets smaller, the focusing performance of the NCSA is mainly dependent on the influence of the HOPE and QPE. For the target at the center range, the HOPE is completely removed, and it is well focused. However, the range position of the other target is very close to the upper limit of the range block size, and its focusing quality is influenced by the residual HOPE and QPE. So, in the case of that the range swath is larger than the block size, the range block processing method should be applied to obtain the well focused swath. Fig. 7(d) shows the imaging result obtained by using the NCSA with HOPC. Similarly to the previous simulation, the image is still completely defocused due to the influence of the uncompensated HOPE.

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TABLE II MEASURED PARAMETERS OF THE SIMULATED TARGETS FOR FIG. 6

TABLE III MEASURED PARAMETERS OF THE SIMULATED TARGETS FOR FIG. 7

Fig. 8 shows the imaging results of System 3. In this case, the images obtained by the ωKA and EOKA are well focused, as shown in Fig. 8(c). Meanwhile, the targets in the image obtained by the NCSA with HOPC also get well focused, as shown in Fig. 8(c). This can be explained as that the range swath is smaller than the range block size, so the residual HOPE and QPE are smaller than the tolerance value, so they have little impact on the target focusing quality. However, due to the influence of the uncompensated HOPE, the image obtained by the NCSA without HOPC is still defocused in some extent, as shown in Fig. 8(d). Furthermore, a quantitative evaluation on the focusing quality of the images shown in Figs. 6–8 is made, and the basic terms of SAR image quality are measured, such as the spatial resolution, integrated sidelobe ratio (ISLR) and peak sidelobe ratio (PSLR). Spatial resolution is considered to be the most significant parameter in SAR image quality measurements, while ISLR and

PSLR are used to evaluate quantitatively the performance of the SAR imaging algorithms. Here, the definitions of the above parameters presented in [23], [24] are used. In [23], the azimuth resolution is defined by the 3 dB width in azimuth of the impulse response function of a point target. In [24], the ISLR is defined by the ratio of the energy within the sidelobe area bounded by concentric ellipses and to the energy of mainlobe area bounded by ellipse , and the PSLR is defined by the peak magnitude found within sidelobe area to the one found within mainlobe areas. Tables II–IV show the measured results. It is well known that the ωKA allows imaging of scenes with high precision, and the measured parameters obtained by the ωKA are very close to the theoretical values [6], [10]. So, the image obtained by the ωKA is used a reference for the SAR image quality evaluation in this study. By comparing with measured results of the EOKA and NCSA with that of ωKA, it is


Fig. 9. Processed P-band UWB SAR images. (a) Obtained by the EOKA. (b) Obtained by the NCSA.

easily to evaluate the performance of the EOKA and NCSA for processing typical low-frequency UWB SAR data in terms of SAR image quality. Observing the Tables II–IV, we can find that comparative measured resolutions parameters achieved by the EOKA as ωKA for both of the targets in the three cases. However, observing the Tables II and III, the ISLRs and PSLRs achieved by the EOKA are a little worse than those achieved by the original ωKA for System1 and System2, while the comparative ISLRs and PSLRs achieved by the EOKA as ωKA for System3. This can be explained as follows: as the factional bandwidth and the integration angle get larger, the accuracy of the Taylor series of (6) decreases, and this impacts the imaging performance of EOKA in some extent. In contrast, for System 1, the range resolution achieved by the NCSA with HOPC is a little worse than the reference value,

TABLE IV MEASURED PARAMETERS OF THE SIMULATED TARGETS FOR FIG. 8

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Fig. 10. Zoom images. (a) Zoom of the region circled by the dashed rectangle in Fig. 9(a); (b) Zoom of the region circled by the dashed rectangle in Fig. 9(b); (c) Zoom of the region circled by the solid rectangle in Fig. 9(a); (d) Zoom of the region circled by the solid rectangle in Fig. 9(b).

but the other measured parameters achieved by the NCSA with HOPC and all the measured parameters achieved by the NCSA without HOPC are much worse than the reference values, as shown in Table II. However, as the center frequency gets higher and the fractional bandwidth as well as the integration angle gets smaller, the measured parameters achieved by the NCSA with HOPC gets better and the invariance range block size also gets larger, as shown in Tables III and IV. Although the imaging performance of the NCSA without HOPC also becomes better as the center frequency gets higher and the fractional bandwidth as well as the integration angle gets smaller, but the measured parameters are still worse than the other algorithms. Therefore, the NCSA without HOPC may be not suitable for high-resolution low-frequency UWB SAR imaging. B. Real Data Results In order to demonstrate the theory analysis presented in this paper further, a real data processing experiment was carried out. The raw data with resolution of 1.0 m 1.2 m (Range Azimuth) is collected by a stripmap mode UWB SAR. The UWB SAR system operates in P-band. The slant range of the scene center is 10 km, and the integration angle corresponding to the center range is 17.2 degrees. The radar platform moves at a height of 5.4 km, with a speed of 106 m/s. The raw data was respectively processed by the EOKA and NCSA with the same MOCO strategy. No range block processing was used in the imaging procedure. The size of the final images is 1.26 km 2.02 km (Range Azimuth). From the two

images shown in Fig. 9, we find that the roads, farmland, and trucks in forestry are clearly distinguished in the images, and the contrasts of the images are very intense. In order to give a comparison of the two focused images, Fig. 10 shows the zooms of the regions circled by the dashed and solid rectangles. From the zoom regions shown in Fig. 10(a) and (b), which correspond to the center range region, the focusing quality of the two images obtained by EOKA and NCSA is nearly the same. However, from the zoom regions shown in Fig. 10(c) and (d), which corresponding to the near range region, we can find that the texture character of the range region, we can find that the texture character of the image obtained by EOKA is clearer than that obtained by the NCSA. The reason is that the focusing performance of NCSA is affected by its approximations processing, which was discussed in Section III. V. CONCLUSIONS A generalized derivation of the FDAs is presented in this paper, and a novel explanation of the imaging principle of the EOKA is given from the viewpoint of the SAR signal processing. From the derivation, it is easy to find that by using the Stolt interpolation instead of the Taylor series approximation, the ωKA and EOKA algorithms have higher accuracy than the other frequency domain algorithms. Based on the generalized derivation, an evaluation of the performances of the EOKA and the NCSA on processing low-frequency UWB SAR data are


carried out based on theoretical analysis, simulation and experimental data. Assumptions and approximations are different in each algorithm, and the errors due to these approximations are analyzed. Based on the analysis carried out in this paper, we can get the following conclusions: 1) In the ideal case (i.e., no motion error), the ωKA is the best candidate for UWB SAR processing, because it has the highest precision and the fewest limitations. 2) In the non-ideal case (i.e., there is motion error), the EOKA is an excellent candidate for the UWB SAR processing, especially for high-resolution ( 1.0 m), low-frequency ( 300 MHz), high fractional bandwidth ( 0.6) and large integration angles 30 . However, the application of the EOKA is limited by the precondition of inequality (22). Besides, the implementation of the EOKA is more complicate than the NCSA. 3) The analysis shows that the NCSA may not be suitable for the above-mentioned high-resolution low-frequency UWB SAR processing. However, for the other UWB SAR systems which have the relative lower resolution, higher center frequency, the smaller fractional bandwidth, and/or the integration angle, the NCSA with HOPC is a good candidate for the real data processing, especially in the UWB SAR real-time imaging. This is because the NCSA has excellent performance on compensating the motion error, and its simple imaging procedure is more suitable for hardware parallel implementation and with fewer code lines. Although many detailed considerations must be examined in order to select the best algorithm for the specific UWB SAR parameters and practical requirements, the readers can begin with the guidelines presented in this paper. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful comments and suggestions which improved this paper. REFERENCES [1] L. M. H. Ulander, H. Hellsten, and G. Stenstrom, “Synthetic-aperture radar processing using fast factorized back-projection,” IEEE Trans. Aerosp. Electron. Syst., vol. 39, no. 3, pp. 243–249, Jul. 2003. [2] V. T. Vu, T. K. Sjögren, M. I. Pettersson, A. Gustavsson, and L. M. H. Ulander, “Detection of moving targets by focusing in UWB SAR—Theory and experimental results,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 10, pp. 3799–3815, Oct. 2010. [3] L. M. H. Ulander, B. Flood, P.-O. Frölind, T. Jonsson, A. Gustavsson, J. Rasmusson, G. Stenström, A. Barmettler, and E. Meier, “Bistatic experiment with Ultra-wideband VHF-band synthetic-aperture radar,” in Proc.EUSAR, Friedrichshafen, Germany, Jun. 2008, pp. 131–134. [4] Y. Na, H. Sun, Y. H. Lee, L. C. Tai, and H. L. Chan, “Performance evaluation of back-projection and range migration algorithms in foliage penetration radar imaging,” in Proc. ICIP, Singapore, Oct. 2004, pp. 21–24. [5] A. Potsis, A. Reigber, E. Alivizatos, A. Moreira, and N. K. Uzunoglou, “Comparison of chirp scaling and wavenumber domain algorithms for airborne low frequency SAR,” in Proc. SPIE—SAR Image Analysis, Modeling, and Techniques V, F. Posa, Ed., Mar. 2002, vol. 4883, pp. 25–36.

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[6] V. T. Vu, T. K. Sjögren, and M. I. Pettersson, “A comparison between fast factorized backprojection and frequency-domain algorithms in UWB low frequency SAR,” in Proc. IGARSS, Boston, MA, USA, Jul. 2008, pp. 1293–1296. [7] P.-O. Frölind and L. M. H. Ulander, “Evaluation of angular interpolation kernels in fast back-projection SAR processing,” Proc. Inst. Elect. Eng.—Radar, Sonar Navig., vol. 153, no. 3, pp. 243–249, Jun. 2006. [8] G. W. Davidson, I. G. Cumming, and M. R. Ito, “A chirp scaling approach for processing squint mode SAR data,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 1, pp. 121–133, Jan. 1996. [9] I. G. Cumming and F. H. Wong, Digital Processing of Synthetic Aperture Radar Data Algorithm and Implementation. Norwood, MA, USA: Artech House, 2005. [10] E. C. Zaugg and D. G. Long, “Generalized frequency-domain SAR processing,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 11, pp. 3761–3773, Nov. 2009. [11] V. T. Vu, T. K. Sjögren, and M. I. Pettersson, “Ultrawideband chirp scaling algorithm,” IEEE Geosci. Remote Sens. Lett., vol. 7, no. 2, pp. 281–285, Apr. 2010. [12] A. Reigber, E. Alivizatos, A. Potsis, and A. Moreira, “Extended wavenumber-domain synthetic aperture radar focusing with integrated motion compensation,” Proc. Inst. Elect. Eng.—Radar, Sonar Navig., vol. 153, no. 3, pp. 301–310, Jun. 2006. backprojector [13] J. McCorkle and M. Rofheart, “An order algorithm for focusing wide-angle wide-bandwidth arbitrary-motion synthetic aperture radar,” in Proc. SPIE AeroSense Conf., Orlando, FL, USA, Apr. 1996, pp. 25–36. [14] A. F. Yegulalp, “Fast backprojection allgorithm for synthetic aperture radar,” in Proc. IEEE Radar Conf., Waltham, MA, USA, Apr. 1999, pp. 60–65. [15] A. Moreira and Y. H. Huang, “Airborne SAR processing of highly squinted data using a chirp scaling approach with integrated motion compensation,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 5, pp. 1029–1040, Sep. 1994. [16] M. D. Xing, X. W. Jiang, R. B. Wu, F. Zhou, and Z. Bao, “Motion compensation for UAV SAR based on raw radar data,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 8, pp. 2870–2883, Aug. 2009. [17] R. K. Raney, H. Runge, R. Bamler, I. G. Cumming, and F. H. Wong, “Precision SAR processing using chirp scaling,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 4, pp. 786–799, Jul. 1994. [18] C. Cafforio, C. Prati, and F. Rocca, “SAR data focusing using seismic migration and techniques,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 2, pp. 194–207, Mar. 1991. [19] R. Bamler, “A comparison of range-doppler and wavenumber domain SAR focusing algorithm,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 4, pp. 706–713, Jul. 1992. [20] W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms. Norwood, MA, USA: Artech House, 1995. [21] D. X. An, X. T. Huang, T. Jin, and Z. M. Zhou, “Extended two-step focusing approach for squinted spotlight SAR imaging,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 7, pp. 2889–2900, Jul. 2012. [22] D. X. An, X. T. Huang, T. Jin, and Z. M. Zhou, “Extended nonlinear chirp scaling algorithm for high-resolution highly squint SAR data focusing,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 9, pp. 3595–3609, Sep. 2012. [23] V. T. Vu, T. K. Sjögren, and M. I. Pettersson, “On synthetic aperture radar azimuth and range resolution equations,” IEEE Trans. Aerop. Electron. Syst., vol. 48, no. 2, pp. 1764–1769, Apr. 2012. [24] V. T. Vu, T. K. Sjögren, M. I. Pettersson, and A. Gustavsson, “Defination on SAR image quality measurements for UWB SAR,” in Proc. SPIE Image and Signal Process. Remote Sens. XIV, Cardiff, U.K., Sep. 2008, vol. 7109A, pp. 1–9. Daoxiang An (S’10–M’11) received the B.S., M.S., and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2004, 2006, and 2011, respectively. Currently, he is a Lecturer of the National University of Defense Technology. His research interests include ultra-wideband SAR image formation, ultrawideband SAR motion compensation, and high-resolution SAR image formation.

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Yanghuan Li received the B.S., M.S., and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2004, 2006, and 2010, respectively. He is currently a Lecturer with the National University of Defense Technology. His research interests include ultra-wideband SAR imaging, SAR motion compensation, and GPS/INS navigation.

Xiangyang Li received the B.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 1993 and 2000, respectively. He is currently an Associate Professor of the National University of Defense Technology. His fields of interest include ultra-wideband radar system design and signal processing.

Xiaotao Huang (M’02) received the B.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 1993 and 1999, respectively. He is currently a Professor of the National University of Defense Technology. His fields of interest include radar theory, signal processing and radio frequency signal suppression.

Zhimin Zhou received the B.S. degree in aeronautical radio measurement and control and the M.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 1982, 1989, and 2002, respectively. He is currently a Professor of the National University of Defense Technology. His is a senior member of the Chinese Institute of Electronics. His fields of interest include ultra-wideband radar system and real-time signal processing.