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In this work, a 2-D subaperture polar format algorithm (PFA) based on stepped-chirp signal is proposed. Instead of traditional pulse synthesis preprocessing, the presented method integrates the pulse synthesis process into the range subaperture processing. Meanwhile, due to the multi-resolution property of subaperture processing, this algorithm is able to compensate the space-variant phase error caused by the radar motion during the period of a pulse cluster. Point target simulation has validated the presented algorithm.

Synthetic aperture radar (SAR) becomes an important tool in modern remote sensing for its all-weather, day and night capability to provide high-resolution maps of scene of interest. The demand for radar images is constantly pushing for finer resolutions. This quest for the resolving power has two major consequences [

Due to technical limitations, particularly the limited sampling rate of the analog to digital converters, synthetic bandwidth technique [

With respect to the azimuth dimension, high resolution is obtained by coherent integration over a large aperture. The generally linear radar flight trajectory assumption, which is the basis of frequency domain image formation algorithm, is deviated, especially when nonplanar motion (NPM) occurs. Polar format algorithm (PFA) [

In this paper, a new image formation algorithm which incorporates the synthetic bandwidth technique with subaperture processing is proposed. Instead of traditional pulse synthesis preprocessing, it integrates the pulse synthesis into range subaperture processing. Meanwhile, it is able to compensate the space-variant phase errors caused by the radar motion during the pulse cluster.

Consider a spotlight SAR operating in the geometry of _{n}_{n}_{0} and _{0} at the center of the aperture. The distance from the APC to the scene center is _{cn}_{0} = 0. A target scatter is located at (_{x}_{y}_{sn}_{Δ} = (_{cn}_{sn}_{0} is the carrier frequency of the transmitted signal, _{s}

The differential range _{Δ} can be expressed as [_{0} + _{1}_{2}^{2} when the cubic and higher order terms are ignored. Since the coefficients of this polynomial are dependent on the target position, this error is space-variant.

Inserting

Performing range resampling formulated by

For the purpose of clearness and simplicity, we still use

If the space sampling position _{n}

First, we divide the azimuth and range aperture into subapertures, respectively, by making
_{1} is the azimuth intra-subaperture index limited within −_{1} /2≤ _{1} ≤ _{1} /2 − 1, _{2} is the azimuth inter-subaperture index limited within −_{2}/2≤_{2}≤_{2}/2 − 1, Δ_{2} is the azimuth data decimation factor, _{1} is the range intra-subaperture index limited within −_{1} /2 ≤ _{1} ≤ _{1} / 2 − 1, _{2} is the range inter-subaperture index limited within −_{2}/2≤_{2} ≤_{2}/2−1, and _{2} is the range data decimation factor. Using

Next, applying the quadratic order approximation of _{1}, _{1}, _{2}, _{2} sequentially, we get
_{1} and _{1} in such a way that the phase error terms caused by wavefront curvature in the subaperture can be neglected. It is helpful to note that error terms in the last two terms can be compensated due to space position information extracted from the coarse resolution images. Now to facilitate the analysis, we rewrite _{e}_{1} = _{0}_{0}_{x}dα_{2}_{2}_{1}, _{e}_{2} = _{0} (1 + _{0}_{2}_{2})(_{1}Δ_{2}_{2}+_{2} (Δ_{2}_{2})^{2}), and _{e}_{3} = _{0}_{0}_{2}_{0}_{2} are the undesired terms, which should be compensated in this algorithm.

From _{1}, following by phase correction _{e}_{1}, and then perform a DFT across _{1} to get the coarse resolution images. After the second phase error compensation _{e}_{2}, we perform a CZT across _{2} to get azimuth fine resolution. Finally, after compensating the third phase error term _{e}_{3}, a DFT across _{2} results in the final fine resolution image.

To reduce the transmission bandwidth, and meanwhile to achieve the high range resolution, it is possible to transmit series of narrow-band signals centered at different carrier frequencies. For example, an equivalent wideband LFM chirp can be assembled from lesser-bandwidth chirp segments in the data processing stage. These subchirp signals, which are referred to as a pulse cluster, are transmitted as separate pulses, each with their own carrier frequencies. The carrier frequencies distribute sequentially to keep the spectrums covering the desired bandwidth.

Now assume that each pulse cluster has _{2} chirp segments each with bandwidth _{s}_{0}, and the step carrier frequency is Δ_{s}_{2} th (−_{2}/2≤_{2} ≤ _{2}/2−1) subchirp is _{0} + _{2}Δ_{2}th backscattered echo signal in the _{1} is the range sample index in each chirp segment, _{2} is the chirp segment index, and _{n,k}_{2} and squint angle _{n,k}_{2} vary not only with index _{2}, which is undesirable. The latter change of _{n,k}_{2} and _{n,k}_{2} is resulted from the radar motion during the pulse cluster. Neglecting this variation will introduce space-variant phase errors which limit the focused scene size. But in

The key to analysis of the characteristics of the _{n,k}_{2} cos_{n,k}_{2} in terms of tan _{n,k}_{2}. From the geometry in _{0,0} is the grazing angle at the aperture center corresponding to _{n,k}_{2} = 0.

Since the wavefront curvature error term _{2}) does not play an important role in this development, it is neglected. Then inserting

As before, we assume that the space sampling position _{n,k}_{2} = _{2}+_{2}) then

Phase term Φ_{basic}_{err}_{x}_{y}

For

Comparing with _{n,k}_{2} and _{2}) varies with index _{2}, while in

Insert tan_{n,k}_{2} = _{2} + _{2}) into _{err}_{1}, _{2}) = _{o}_{0}(_{1} + _{2}_{2})] _{x}dαk_{2} is the phase error term resulted from radar motion during pulse cluster but after range resampling. Compared with phase error term Φ_{err}

Analogous to _{e}_{1} = _{0}_{0}_{x}dαK_{2}Δ_{2}_{2}_{1}, _{e}_{2} = _{0} (1+_{0}_{2}_{2}) [_{1} Δ_{2}_{2} + _{2}_{2}_{2})^{2}], and _{e}_{3} = _{0}_{0}_{2}_{0}_{2} are undesired terms just like those in PFOSA which are introduced by wavefront curvature. While error phase terms _{err}_{1}) = _{o}β_{0}_{x}dαk_{2}_{1} and
_{err}_{1},_{2}), are due to radar motion during the pulse cluster. These error terms are space-variant due to the dependence on azimuth position (range dependence is eliminated owing to range resampling). If these phase errors are not compensated, as a consequence, they result in displacement and defocus in range. Since the coarse location information can be extracted from the coarse resolution images, it is possible to compensate these errors by modifying the classical PFOSA. The new algorithm (we call it SCPFOSA) can be derived from

Step1: Perform a CZT across _{1}, get the azimuth coarse resolution estimate _{x}

Step2: Use the estimate of _{x}_{e}_{1} and _{err}_{1}), and then perform a FFT across _{1} to obtain the range coarse resolution estimate _{y}

Step3: Use the estimate of _{x}_{y}_{e}_{2}, then perform a CZT across _{2} to get the azimuth fine resolution estimate _{x}

Step4: Use the fine resolution estimation of _{x}_{y}_{e}_{3} and _{err}_{2}), and then perform a FFT across _{2} to get the range fine resolution estimate _{y}

In this section, point target simulation is employed to validate the presented algorithm. The waveform parameters are chosen as: _{s}_{2} =15. The other parameters are listed as follow: standoff range is 10km, azimuth resolution is 0.1m, and radar forward velocity is 150m/s. Two point targets are simulated, the first one is the scene center point, and the other one is located at azimuth 150m away from the scene center. The new algorithm is evaluated with respect to the classical PFOSA which doesn't compensate the error terms resulted from the use of stepped-chirp signals. In subaperture algorithm, subapertures are overlapped to control the sidelobes; in particular, they are overlapped to control the amplitude to grating lobes due to data decimation. The degree of allowable overlap will depend on the window functions employed, and sidelobe toleration limits. In our paper, the overlap rate is not the problem we are concerned, so we do not employ window function in the simulation. As the phase error term _{err}_{1}, _{2}) results in distortion and defocus only in the range, we show the range profiles of impulse response function (IRF) for the two simulated targets (_{err}_{1}) and _{err}_{2}) are both zeros, the two algorithms have almost the same response. However, for the azimuth displaced point target, the mainlobe of range profile is broadening for PFOSA, since the phase errors caused by motion of radar during pulse cluster are not compensated. While using SCPFOSA, due to the correction of these phase errors, its range profile has improved significantly (mainlobe reduce 12% and peak sidelobe ratio (PSR) reduces about 2.5dB).

In this paper, a 2-D subaperture algorithm based on stepped-chirp signal is presented. It integrates the pulse synthesis process into range subaperture processing without traditional pulse synthesis preprocessing. Meanwhile, due to the multi-resolution property of subaperture processing, this algorithm is able to compensate the space-variant phase error resulted from the motion of radar during a pulse cluster. SCPFOSA has almost the same processing flow chart with PFOSA, only the additional phase error term are added, it has the comparable computation complexity with PFOSA. Furthermore, due to its repetitive architecture in subaperture processing, the SCPFOSA is very suitable for parallel and pipeline hardware architectures.

SAR geometry.

Flow chart of SCPFOSA.

Comparison of SCPFOSA and PFOSA.