Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4047983, 10 pages

https://doi.org/10.1155/2017/4047983

## A Signal Processing Algorithm Based on 2D Matched Filtering for SSAR

School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Shouguo Yang

Received 16 April 2017; Revised 26 October 2017; Accepted 8 November 2017; Published 26 December 2017

Academic Editor: Yann Favennec

Copyright © 2017 Shouguo Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study discusses a smart radar antenna scanning mode that combines features of both the sector-scan mode used for conventional radar and the line-scan mode used for synthetic aperture radar (SAR) and achieves an application of the synthetic aperture technique in the conventional sector-scan (mechanically scanned) radar, and we refer to this mode as sector-scan synthetic aperture radar (SSAR). The mathematical model is presented based on the principle of SSAR, and a signal processing algorithm is proposed based on the idea of two-dimensional (2D) matched filtering. The influences of the line-scan range and speed on the SSAR system are analyzed, and the solution to the problem that the target velocity is very high is given. The performance of the proposed algorithm is evaluated through computer simulations. The simulation results indicate that the proposed signal processing algorithm of SSAR can gather the signal energy of targets, thereby improving the ability to detect dim targets.

#### 1. Introduction

When the radar detects the target remotely, the target is normally assumed to be a point located in a resolution cell, and the energy of the target echo is assumed to be evenly distributed in the resolution cell. The synthetic aperture technique is used to improve the resolution of a radar system by reducing the resolution cell size and concentrating the signal energy that contains information about the target. Because the energy density of the noise remains constant but the energy density of the target increases in the same resolution cell, the signal-to-noise ratio (SNR) is improved, which helps to detect a dim target. However, the majority of existing early-warning radar systems are ground-based radar (GBR). Synthetic aperture radar (SAR) requires a moving platform with a sufficiently high speed, which GBR systems lack. Supposing that a GBR system could perform a long-range fast line-scan relative to the target, this system would not meet the tactical requirements for a radar system. Therefore, it is not realistic to copy the working scene of airborne radar to that of GBR. In addition, the majority of existing early-warning radar systems use the mechanical scanning mode, which makes them lose the synthetic aperture function. Applying the synthetic aperture technique in a conventional mechanically scanned radar system remains an urgent problem.

Sector-scan synthetic aperture radar (SSAR) provides an effective approach for solving the mentioned problem above. SSAR is the improvement of ground-based sector-scan radar, which combines the advantages of the sector-scan method and the line-scan method. And the line-scan method is a beam moving transversely along the antenna surface quickly. In [1], the working mode and implementation method of SSAR are given. And it is found that when the radar is line-scanning quickly, regardless of whether the target is moving, the target has radial velocity relative to the receiving point, which is not the same as the ISAR. ISAR must wait for the plane to cross the beam transversely, so that it becomes passive. But SSAR can find target actively. It is also pointed out in [1] that SSAR can achieve wideband signals in slow-time dimension (i.e., azimuth dimension) by line-scan method, which provides the necessary conditions for compression in slow-time dimension. In this paper, we mainly study the signal processing algorithm of SSAR, so that the energy of target echo signal is obviously gathered to detect the weak point target. That is the fundamental purpose of SSAR. However, SAR and ISAR are mainly for high-resolution imaging of target. Therefore, the purpose of SSAR is different from that of SAR and ISAR.

Because SSAR is derived from the traditional SAR approach, the SSAR signal processing procedure is developed based on the basic SAR algorithm, the range-Doppler algorithm [2], to achieve bidirectional (radial and transverse) pulse compression and to maximize the SNR. The main difficulty in signal processing lies in the range migration problem in the slow-time dimension for targets with high acceleration and a low SNR, that is, the fast- and slow-time dimension coupling problem. Existing methods for correcting range migration (caused by the maneuvering target) are generally to compensate the estimated parameters of the target [3–5]. These methods are highly effective for targets with high SNR but are ineffective for targets with low SNR. In addition, some documents have also proposed methods for targets with low SNR. For example, Chen et al. [6] proposed a method that eliminates the linear range migration caused by the target velocity using the keystone transform and increases the SNR by accumulating adjacent range profiles using the envelope-correlation algorithm [7] or the phase gradient autofocus (PGA) algorithm [8] to compensate for higher-order motion. These methods are effective when the acceleration is not particularly high and the SNR is not particularly low. However, to obtain a high transverse compression ratio, a relatively high acceleration is artificially generated by the SSAR method. In addition, dim targets have extremely low SNR. In this case, the method of Chen et al. cannot significantly increase the SNR by accumulating adjacent range profiles, and therefore, this method is limited to low-acceleration cases and is less effective for SSAR signal processing. Li et al. [9] proposed a method for correcting range migration (caused by the maneuvering target) for low-SNR targets. This method also involves parameter estimation, and a transverse correlation function is used to increase the SNR prior to the parameter estimation step. Therefore, this method applies to only cases in which the SNR is not too low. Tian et al. [10] and Dai and Zhang [11] proposed the generalized keystone transform, which can eliminate range migration caused by target motion of the second order and above (acceleration and jerk). Li et al. [12] proposed a fast algorithm to process the result of the generalized keystone transform. This algorithm performs a fast Fourier transform (FFT) in the transverse direction rather than pulse compression to eliminate range migration, and thus, this method does not increase the SNR. Wu et al. [13] and Zhu et al. [14, 15] proposed a two-dimensional (2D) matched-filter method that simultaneously performs radial and transverse compression and then estimates the target parameters. This method requires estimation of only the acceleration and is very effective.

For targets with a low SNR, it is not applicable for SSAR to use ordinary parameter estimation and compensation methods. Inspired by the idea of 2D matched filtering [13–15], the present study proposes a signal processing algorithm based on the principles of SSAR. The performance of this algorithm is verified through computer simulations, and this study lays a solid foundation for the practical application of SSAR.

#### 2. Echo Signal Model

Zhang [16] noted that there are usually three types of radar echo models: the accurate model, the first-order approximate model, and the “stop-go” model. In the present study, the “stop-go” model is used. The “stop-go” model is also known as the “stop-and-hop” model and is obtained by further simplifying the first-order approximate model of the radar echo.

Supposing that the transmitting signal of the radar is , the receiving signal is ( is the echo attenuation coefficient). In addition, let and denote the range and the velocity of the target at time 0, respectively, and assume that the target is moving toward the radar system along the normal direction. Supposing that the transmitting signal is , the echo signal received by the radar can be expressed as follows:where and indicate fast-time and slow-time, respectively, represents the carrier frequency of the signal, represents the range of the target at time , and represents the speed of light.

When we use the slow-time dimension information of the target, we assume a total of pulse repetition intervals (PRIs). The slow-time of the th transmitting pulse is ( represents the PRI of the radar system). The relation between the fast-time and slow-time is as follows:

Zhang [16] showed that if LFM signals are used, the “stop-go” radar echo model must satisfy the following conditions to replace the accurate model:where represents the maximum velocity of the target, represents the maximum acceleration of the target, represents the correlation time in the slow-time dimension, represents the pulse width of the signal transmitted by the radar system, and represents the wavelength of the signal.

The “stop-go” model has a broad range of applications, and this model is almost always effective for radar systems, but it cannot be used for sonar systems [9].

#### 3. SSAR Signal Processing Algorithm

Supposing that the transmitting signal is expressed aswhere , the receiving signal can be expressed as

Here, only the signal is considered, and noise is ignored. In this case, the “stop-go” model is used as the signal model. Because the amplitude of the receiving signal is a constant, normalization is performed.

After the receiving signal is mixed, (5) can be expressed as

Considering the effect of the beam width, (6) can be expressed aswhere represents the location of the target in the slow-time dimension, that is, the slow-time location of the target when the target is aligned with the normal axis of the antenna, and represents the duration of the target echo in the slow-time dimension (, where represents the beam width of the antenna and represents the rotational speed of the antenna).

Assume that the target is moving toward the radar in the radial direction at a uniform speed and its absolute velocity is . Let the line-scan speed resulting from SSAR be denoted by and the midpoint of the slow-time duration of the echo signal be used as the reference point. Thus, the slow-time instantaneous radial velocity of the target at the reference point for the SSAR system is . According to the SSAR theory, the corresponding slow-time instantaneous range of the target iswhere represents the reference distance, that is, the slant range of the target at the slow-time instant .

According to the stationary phase principle, the Fourier transform corresponding to is .

Performing a Fourier transform on (7) with respect to fast-time , we can obtain

Applying matched filtering and substituting into (9), we can obtain

Performing an inverse Fourier transform (IFT) on , we can obtain

The expression for is also the result generated after using a matched filter.

Performing a Fourier transform with respect to fast-time on again, we can obtain

Substituting (8) into (12), we can obtain

Substituting into (13), we can obtain

Because , we have where .

Performing a Fourier transform with respect to slow-time () on , we can obtain

Performing a Fourier transform on with respect to slow-time () and substituting the result into , we can obtain

Compensating (17) by , we can obtain

In (18), the term , which is denoted by (), affects the final compression.

Clearly, the () term is related to the velocity of the target, the bandwidth of the transmitting signal, the line-scan speed of the system, and the beam width. The line-scan speed and the beam width each have a positive effect on the final compression, that is, increasing the line-scan speed of the system and the beam width can improve the compression. Increasing the line-scan speed is limited by the antenna size, and the beam width has an upper limit of approximately 38° according to the SSAR theory. Thus, the improving space of the line-scan speed is larger than that of the beam width. The velocity of the target and the bandwidth of the transmitting signal have a negative effect on the final compression (under the same condition as the condition described in the following sections). For different fast-time frequencies, the slow-time frequencies are distributed in different centers (which are modulated by the velocity of the target) and different widths (which are modulated by the bandwidth of the transmitting signal). The more discrete the distribution centers are and the greater the variations in the distribution widths are, the greater their effect on the final compression is.

Here we definewhere and .

When , the () term is expressed as follows:

Thus, we can obtain

Performing an IFT on (21) with respect to slow-time and fast-time, we can obtain

In the final compression, the amplitude of the echo signal improves times, where is the bandwidth of the transmitting signal and is the bandwidth of the LFM signal formed by the line-scan in the azimuth dimension. Thus, the capability of the radar system to detect dim targets is significantly improved.

From (22), the location of the target obtained following compression is in the fast-time dimension, which gives the range, and in the slow-time dimension, which gives the azimuth. However, the actual location is , indicating that there is a large difference, especially in the azimuth dimension, and this difference is caused by the velocity of the target.

From (22), we can obtain

In (23), there are three unknown variables but only two equations. Therefore, the precise location of the target cannot be determined. Another equation that contains the three unknown variables is required to determine the location of the target. The difference in the line-scan speed provides two additional equations, so that we can obtain

These four equations contain the three unknown variables. Solving (24) for the unknowns, we can obtain

It is recommended that two slow-time dimension equations be used. First, the velocity of the target can be determined. Then, the location of the target in the slow-time dimension is determined. The range of the target can be determined using the equation with a relatively high line-scan speed. If the difference between the two line-scan speeds is excessively high, the target detection will be affected. To improve the solution accuracy, it is recommended that the fast-time dimension equation with a relatively high line-scan speed be used to determine the range of the target. Simulations showed that this algorithm has greater accuracy when the velocity of the target is within a certain range. The performance of this algorithm decreases when the velocity of the target is very high. To address this issue, the velocity range of the target is divided into intervals, and the interval in which the velocity of the target is the lowest is selected as the basic interval for processing. When searching in other intervals, phase compensation is performed with the maximum velocity in the basic interval as the baseline value (the velocity in the middle of the other interval is times the maximum velocity in the basic interval). The detection results obtained from all of the intervals are integrated (because of the difference in the line-scan speed) based on the minimum distance criterion.

By summing up the above algorithm, the SSAR signal processing procedure is summarized in Figure 1.