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The airborne downward looking sparse linear array three dimensional synthetic aperture radar (DLSLA 3-D SAR) operates nadir observation with the along-track synthetic aperture formulated by platform movement and the cross-track synthetic aperture formulated by physical sparse linear array. Considering the lack of DLSLA 3-D SAR data in the current preliminary study stage, it is very important and essential to develop DLSLA 3-D SAR simulation (echo generation simulation and image reconstruction simulation, including point targets simulation and 3-D distributed scene simulation). In this paper, DLSLA 3-D SAR imaging geometry, the echo signal model and the heterogeneous parallel technique are discussed first. Then, heterogeneous parallel echo generation simulation with time domain correlation and the frequency domain correlation method is described. In the following, heterogeneous parallel image reconstruction simulation with two imaging algorithms, e.g., 3-D polar format algorithm, polar formatting and L1 regularization algorithm is discussed. Finally, the point targets and the 3-D distributed scene simulation are demonstrated to validate the effectiveness and performance of our proposed heterogeneous parallel simulation technique. The 3-D distributed scene employs airborne X-band DEM and P-band Circular SAR image of the same area as simulation scene input.

The airborne downward looking sparse linear array three dimensional synthetic aperture radar (DLSLA 3-D SAR) can be placed on small and mobile platforms to acquire high resolution full 3-D microwave images. The 3-D resolution is acquired by wave propagation dimensional pulse compression with wide band chirp signal, along-track aperture synthesis with flying platform movement, and cross-track aperture synthesis with physical sparse linear array [

Downward Looking Imaging Radar (DLIR) was first introduced by Gierull in 1999 [

In order to assess the key technology and validate the imaging algorithms, it is very important and essential to develop DLSLA 3-D SAR simulation. DLSLA 3-D SAR echo generation simulation with airborne platform motion error can be used for DLSLA 3-D SAR motion compensation research. DLSLA 3-D SAR echo generation simulation with array channel amplitude and phase inconsistency error can be used for DLSLA 3-D SAR array channel amplitude and phase inconsistency compensation research. Furthermore, the effectiveness and performance of the DLSLA 3-D SAR imaging algorithms can be demonstrated with image reconstruction simulation.

Considering the heavy computation complexity and computation load in 3-D circumstances, heterogeneous parallel simulation technique for DLSLA 3-D SAR simulation is demonstrated in this paper [

The remainder of the paper is organized as follows. In Section 2, the DLSLA 3-D SAR imaging geometry, the echo signal model and the heterogeneous parallel technique are illustrated. In Section 3, DLSLA 3-D SAR heterogeneous parallel echo generation simulation with time domain correlation and frequency domain correlation method is discussed. In Section 4, DLSLA 3-D SAR heterogeneous parallel image reconstruction simulation with 3-D polar format algorithm, polar formatting and L1 regularization algorithm is described. The effectiveness and performance of the proposed DLSLA 3-D SAR heterogeneous simulation is demonstrated with point targets and 3-D distributed scene simulation in Section 5. Finally, the conclusion and the current focus of our research are outlined in Section 6.

Airborne DLSLA 3-D SAR operates nadir observation and its working geometry is shown in _{1} is the angle between QP and QX′ called the along-track Doppler cone angle. γ_{2} is the angle between QP and QY′ called the cross-track Doppler cone angle. ϕ is the angle between OP and OP′, θ is the angle between OP′ and OZ. The coordinates of the target P in Cartesian coordinate can be written as (ρsinϕ, ρcosϕsinθ, ρcosϕcosθ).

For conventional side-looking 2-D SAR echo signal acquisition, only the along-track Doppler cone angle γ_{1} is obtained with platform movement, which means only along-track resolution is obtained. The targets with different cross-track Doppler cone angle located in the same range and along-track bin cannot be distinguished in 2-D SAR. For airborne DLSLA 3-D SAR echo signal acquisition, along-track Doppler cone angle γ_{1} is obtained with platform movement and cross-track Doppler cone angle γ_{2} is obtained with sparse array, which means the scatters can be differentiated in the along-track and the cross-track dimension. Shadowing and lay over effects cannot be avoided in the side-looking 2-D SAR as the grazing angle is small, especially in the far range gate units. Airborne DLSLA 3-D SAR owns large grazing angle as it operates nadir observation, therefore, shadowing and lay over effects can be overcome [

A chirp signal with carrier frequency _{c}_{r}_{P}

The APC coordinate is (_{m}_{n}

In this paper, we apply a heterogeneous parallel technique on the Central Processing Unit (CPU) [

Heterogeneous parallel model is shown in

For an imaging scene, the echo signal can be generated according to _{s}_{P} (_{s}_{d}

Heterogeneous parallel echo generation simulation with the time domain correlation method is shown in

For 3-D distributed imaging scene, it will be very time consuming to generate the DLSLA 3-D SAR echo signal with time domain correlation method. This is caused by the sample depth computation and judgment for so many targets in the imaging scene. We perform wave propagation dimensional FFT to _{c}_{k}_{k}_{k}

The heterogeneous parallel echo generation simulation with frequency domain correlation method is shown in

Heterogeneous parallel echo generation simulation with time domain correlation method is time consuming while quite precise. Heterogeneous parallel echo generation simulation with frequency domain correlation method eliminates sample depth computation for every target. There exists truncated error in the frequency domain correlation method caused by use of the windowing operation with finite window length. We recommend point targets simulation to be used for imaging index (e.g., PSLR and ISLR) computation, with the heterogeneous parallel time domain correlation echo generation method, although the truncated error with heterogeneous parallel frequency domain correlation is very small and can be ignored in most instances. For large scale 3-D distributed imaging scene simulation, heterogeneous parallel frequency domain correlation echo generation method is the best choice [

After wave propagation dimensional frequency domain matched filtering, the signal can be written as

The signal in _{t}_{P}_{k}

The basic imaging term can be polar formatted in the along-track dimension and cross-track dimension from sample _{m}_{n}_{m}_{n}_{m}_{c}_{k}_{m}f_{c}_{n}_{c}_{k}_{n}f_{c}_{a}_{e}_{a}_{e}

This algorithm should be applied to uniform along-track and cross-track sample circumstances. The flow diagram of the heterogeneous parallel image reconstruction simulation with 3-D polar format algorithm is shown in

In the real system, the cross-track phase centers are usually non-uniformly and sparsely distributed due to the array element installation positions restricted by the airborne platform and the airborne wing tremor effect. Then the cross-track FFT in _{n}_{e}_{n×l}_{n}γ_{l}_{e}_{l}_{e}_{k}_{e}

The polar formatting and L1 regularization algorithm can be applied to non-uniform and sparse cross-track sample circumstances. The flow diagram of the heterogeneous parallel image reconstruction simulation with polar formatting and L1 regularization algorithm is shown in

In this section, we present two DLSLA 3-D SAR numerical simulation experiments to illustrate the performance of our proposed heterogeneous parallel simulation method. The sparse linear array we used is composed of 8 transmitting array elements and 32 receiving array elements [

The point targets simulation is used to illustrate the capability of the reconstructed image on different height with the proposed imaging algorithm. The measurement parameters used in point targets simulation are shown in

The heterogeneous parallel echo generation simulation with frequency domain correlation method possesses ringing effect after wave propagation dimensional matched filtering. This may reduce the echo generation precision. We compare the wave propagation dimensional matched filtering profile and the imaging result profile with the two echo generation methods in

The image reconstructed by 3-D PFA without wave front curvature error compensation is shown in

By comparing

After wave front curvature error compensation, the reconstructed image is in polar coordinate grid, we need to transform it to Cartesian coordinate grid according to

In order to make the numerical simulation more precise and consistent with the real 3-D distributed imaging scene, we choose an airborne X-band DEM surface and airborne P-band Circular SAR image (the image is formulated on the DEM surface with Back-projection algorithm) of the same area for distributed scene simulation [

The 3-D reconstructed image in polar coordinates without wave front curvature error compensation is shown in

The 3-D reconstructed image in Cartesian coordinates after wave front curvature error compensation is shown in

According to the ARTINO sparse array configuration principle, the equivalent phase centers are uniformly distributed. In this simulation, we pick out 50% phase centers randomly in order to simulate non-uniform and sparse cross-track sampling circumstance. The reconstructed image in polar coordinates with 50% random cross-track virtual phase centers by using the proposed polar formatting and L1 regularization method is shown in

We compare the reconstructed image with the proposed polar formatting and L1 regularization method tailored for non-uniform and sparse cross-track virtual phase centers and the reconstructed image with 3-D polar format algorithm tailored for uniform cross-track virtual phase centers. The comparison of the 3-D Cartesian image is shown in

Downward looking sparse linear array three dimensional synthetic aperture radar (DLSLA 3-D SAR) imaging geometry and echo signal model are illustrated as the foundation of the simulation. Heterogeneous parallel technique is employed to accelerate the heavy computations in the echo generation simulation and the image reconstruction simulation. The time domain correlation method and frequency domain correlation method can be selected for heterogeneous parallel echo generation simulation, according to a computation and precision balance (time domain correlation method for point targets echo generation, frequency domain correlation method for large scale 3-D distributed imaging scene echo generation). Space domain image reconstruction methods, e.g., 3-D polar format algorithm, polar formatting and L1 regularization algorithm, are preferred for cross-track imaging to reduce the memory cost as the cross-track aperture is much shorter than the cross-track swath (3-D polar format algorithm for uniform cross-track virtual antenna phase center circumstance, polar formatting and L1 regularization algorithm for non-uniform cross-track virtual antenna phase center circumstance). The effectiveness and performance of our proposed heterogeneous parallel simulation technique are validated with the point targets scene and the 3-D distributed scene simulation.

This work is supported by the National Natural Science Foundation of China Surface Program (Grant No. 61072112), and the National Natural Science Foundation of China Key Program (Grant No. 60890071). The authors would also like to thank the reviewers for their constructive comments.

The authors declare no conflict of interest.

Downward looking sparse linear array three dimensional synthetic aperture radar (DLSLA 3-D SAR) imaging geometry.

(

Heterogeneous parallel model.

Heterogeneous parallel echo generation simulation with time domain correlation method.

Heterogeneous parallel echo generation simulation with frequency domain correlation method.

Heterogeneous parallel image reconstruction simulation with 3-D polar format algorithm.

Heterogeneous parallel image reconstruction simulation with polar formatting and L1 regularization algorithm.

DLSLA 3-D SAR sparse array configuration.

(

(

Imaging result in polar coordinate grid without wave front curvature error compensation: (

Imaging result in polar coordinate grid with wave front curvature error compensation: (

Profile analysis: (

Imaging result in Cartesian coordinate grid: (

3-D distributed simulation input: (

Imaging result in (β, γ, α) coordinates: (

Imaging result in Cartesian coordinates: (

(

3-D image: (

XY projection image: (

Cross-track coordinates of array elements.

Transmitted Array | Ti = −1.34 + i × 0.02, i = 1, 2,…,4. |

Transmitted Array | Ti = −1.14 + i × 0.02, i = 5, 6,…,8. |

Receiving Array | Ri = −1.32 + i × 0.08, i = 1, 2,…,32. |

Equivalent Array | TRi = −1.28 + (i − 1) × 0.01, i = 1, 2,…,256. |

Measurement parameters used in the simulation.

Center Frequency | 37.5 GHz |

Transmitting Signal Bandwidth | 300 MHz |

A/D Sampling Frequency | 360 MHz |

Platform Fly Height | 1,000 m |

Platform Fly Velocity | 50 m/s |

Transmitting Signal Pulse Width | 1.0 us |

A/D Sampling Range Gate | [750.0, 1026.7 m] |

Range Sample Number | 1024 |

PRF (Pulse Repetition Frequency) | 5,000 Hz |

Along-track Dimension Sampling Interval | 0.01 m |

Along-track Dimension Sampling Number | 256 |

Transmitting Array Elements | 8 |

Receiving Array Elements | 32 |

Beam width of T/R Array | 14 × 14 |

Equivalent Phase Center Number | 256 |

Cross-track Dimension Sampling Interval | 0.01 m |

The measured Peak Side Lobe Rates (PSLR) and Integrated Side Lobe Rates (ISLR) without wave front curvature error compensation.

PSLR (dB) | −9.3 | −9.4 | −16.7 |

ISLR (dB) | −10.5 | −10.3 | −12.7 |

The measured PSLR and ISLR with wave front curvature error compensation.

PSLR (dB) | −13.3 | −13.3 | −13.4 |

ISLR (dB) | −14.9 | −14.3 | −12.7 |

Measurement parameters used in the simulation.

Center Frequency | 37.5 GHz |

Transmitting Signal Bandwidth | 300 MHz |

A/D Sampling Frequency | 360 MHz |

Platform Fly Height | 1,000 m |

Platform Fly Velocity | 50 m/s |

Transmitting Signal Pulse Width | 3.85 us |

A/D Sampling Range Gate | [974.0, 1026.5 m] |

Range Sample Number | 1024 |

PRF (Pulse Repetition Frequency) | 5,000 Hz |

Along-track Dimension Sampling Interval | 0.01 m |

Along-track Dimension Sampling Number | 256 |

Transmitting Array Elements | 8 |

Receiving Array Elements | 32 |

Beam width of T/R Array | 14 × 14 |

Equivalent Phase Center Number | 256 |

Cross-track Dimension Sampling Interval | 0.01 m |