Two Dimensional Position Correction Algorithm for High-Squint Synthetic Aperture Radar in Wavenumber Domain Algorithm

Abstract

In the traditional high squint angle $\omega -k$ imaging algorithm, the impact of a high squint angle on azimuth and range positioning is not considered but does include two aspects: the azimuth position shift caused by a high squint angle and the impact of Stolt interpolation on range positioning under a high squint angle. From the viewpoint of the geometric features of data acquisition in high-squint SAR and the characteristics of the $\omega -k$ imaging algorithm, this paper analyzes the causes of azimuth position offsets and range position offsets. According to the causes, closed-form mathematical expressions quantifying these coupled spatial distortions are derived. The $\omega -k$ imaging algorithm process is adjusted, and the correction factor is embedded into the imaging process to achieve offset-free, high-resolution imaging in the case of high-squint SAR.

1. Introduction

Synthetic aperture radar (SAR) is an active remote sensing detection sensor with a high resolution [1,2]. It is a coherent detection radar [3] which uses synthetic aperture technology to achieve a large equivalent antenna diameter through movement of the platform. Thus, high-resolution capabilities in the azimuth direction are obtained through synthetic aperture technology, where a high resolution in the range direction is achieved by transmitting large-bandwidth signals [4]. Compared with passive sensors, such as optical and infrared sensors, SAR as a microwave active sensor has a longer working wavelength and does not rely on external signal sources [5]. It can overcome interference from harsh environments such as clouds, smoke and dust and achieve the capabilities of all-weather, all-day work [6,7].

In general, SAR working in the low-squint case can basically meet the needs of reconnaissance and surveillance. However, SAR needs to work at a high squint angle to obtain battlefield intelligence information and predict situations in the battlefield in advance [3]. The high squint angle case has the ability to ensure that reconnaissance aircrafts have more maneuvering time to collect intelligence safely. This is of great significance in the reconnaissance of local wars in the battlefield under modern conditions [8,9].

When the beam’s squint angle is higher, the cross-coupling between the range and azimuth is more serious. The $\omega -k$ algorithm has less approximation and higher accuracy and is usually used for high-squint SAR. For high-squint SAR, accurate Doppler centroid estimation is crucial, impacting essential processes such as azimuth positioning, range cell migration correction (RCMC) and azimuth compression. Errors in estimation can degrade image quality, leading to azimuth position offsets, ambiguities and signal-to-noise ratio (SNR) loss. In practical data imaging processing, for scenarios with varying squint angles or continuously changing azimuth scanning angles, the data are often divided into multiple sub-blocks for separate processing to enable position correction throughout the imaging process dynamically. Each sub-block can be considered to have an equivalent squint angle, corresponding to an SAR sub-image. By combining the SAR sub-images with their equivalent squint angles, dynamic correction of the imaging results is achieved using calibration methods, followed by the synthesis of all sub-images into a complete large-scale image. In practice, mechanical or measurement errors often lead to insufficient accuracy in the acquired squint angles, and it is necessary to estimate squint angles based on echo signals. Liu integrated azimuth compression, spectral distortion analysis and circular convolution to mitigate interference from partially exposed bright targets. Its non-iterative design reduces computational complexity, making it highly efficient for non-homogeneous scenes with strong scatters [10]. Yang removed dominant point targets by downsampling the azimuth spectra and using Gaussian curve fitting. Linear fitting across range cells further minimized random errors. While well suited for high-contrast environments, this requires careful parameter tuning [11]. Li designed a scheme for bi-static forward-looking SAR. This iterative scheme estimates the Doppler centroid via a range migration slope and minimum waveform entropy. It effectively resolved Doppler ambiguity without relying on pulse repetition frequency (PRF) adjustments, making it ideal for bi-static geometries [12]. Lee focused on real-time processing. This method evaluates estimation quality using spectrum symmetry and distortion metrics. Weighted least squares and selective windowing enhance ambiguity resolution, striking a balance between accuracy and computational efficiency for onboard systems [13]. Madsen constructed a sign-Doppler estimator (SDE), which employs time-domain sign correlations and offers low computational complexity. While robust in non-homogeneous scenes, it struggles in low-SNR conditions [14]. Long used a threshold angle to determine the need for range compression to address chirp coupling in high-squint SAR. By mitigating spectral distortion caused by overlapping echoes, it is particularly beneficial for airborne and spaceborne systems with large squint angles [15].

In the traditional high-squint imaging algorithm, the impact of high-squint imaging on the azimuth and range positioning is not considered [8,16,17], specifically the azimuth position shift caused by a high squint angle and the impact of Stolt interpolation on range positioning under a high squint angle. Firstly, the influence of a high squint angle on the azimuth positioning is analyzed, and then the influence of frequency domain interpolation on range positioning is analyzed. When Stolt interpolation is performed, there is a change in the frequency range of the mapped domain compared with the effective frequency range of the original domain. This means that the equivalent sampling rate is different from the original one. This variation was not considered in previous processing algorithms, and it requires resampling the range frequency axis to match the sampling distribution after mapping. Otherwise, it will cause displacements in the range position. This kind of deviation has a relatively small impact when SAR operates at low squint angles, but as the squint angle increases, it becomes more and more severe. Therefore, during the imaging process for a high-squint SAR, it is essential to correct these deviations to ensure the accuracy of two-dimensional positioning.

On the basis of previous studies [17,18,19], this paper provides the constraint relationships between working parameters such as the observation scene range, squint angle, incidence angle and beam width in the context of high-squint scenarios. These relationships are essential for achieving effective ground observation area information acquisition via radar. The causes of azimuth and range position offsets in the acquisition and imaging processes of high-squint SAR data are analyzed. The analytic expressions of the offset in the azimuth and range are provided. The workflow of the $\omega -k$ imaging algorithm is adjusted to achieve high-resolution imaging without position offset under high-squint conditions.

2. Observation Geometry

The observation geometry of the squint SAR is illustrated in Figure 1. In Figure 1, the platform flies along the x direction, with an altitude h and flying speed $v_a$. The observation squint angle is ${\theta }_s$, which refers to the beam pointing direction of the antenna. The projection of the squint angle in the horizontal plane is $\beta$, and the beam width of the antenna in the azimuth is ${\theta }_{az}$. R is the shortest range in the side-looking SAR, $\alpha$ is the downward-looking angle at a certain range gate, $P(x_p,y_p)$ is the target in the observed scene, $R_{sp}$ is the slant range when the beam center passes through the target P at a squint angle ${\theta }_s$, and M and $D(x_0,y_0)$ are the intersection points of the beam’s center and the horizontal plane in the side-looking and squint SAR modes, respectively.

3. $\mathbf{\omega }$-k Imaging Algorithm of the Squint SAR

3.1. The Two-Dimensional Constraints of the Squint Angle on the Antenna Beam

3.1.1. Constraints of the Beam Width at a Certain Elevation and Squint Angle on the Observation Range in the Range Direction

From the observation geometry in Figure 1, it can be seen that the shortest slant range in the side-looking mode is the function of the platform flight’s altitude, with $R={\displaystyle \frac{h}{cos\alpha }}$.

Due to the presence of squint angles, the intersection point between the beam’s center ray and the horizontal plane is not on the y axis, resulting in a certain distance:

(1)$MD=2htan\left(\alpha \right)sin\left({\displaystyle \frac{\beta }{2}}\right)=2{\displaystyle \frac{h}{cos\alpha }}sin\left({\displaystyle \frac{{\theta }_s}{2}}\right)$

According to the expression of R and $MD$, the projection of a squint angle ${\theta }_s$ in the horizontal plane is $\beta$:

(2)$\beta =2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}\right)$

According to the limitations of actual flight and the observation conditions, in the forward squint mode, both ${\theta }_s$ and $\alpha$ are within the interval $\left[0,{\displaystyle \frac{\pi }{2}}\right]$. The above equation is only meaningful when $sin{\displaystyle \frac{{\theta }_s}{2}}<sin\alpha$ is satisfied; that is, the constraint conditions of ${\theta }_s<2\alpha$ must be met. According to the above geometric relationship, under the constraint of a downward-looking angle $\alpha$ and squint angle ${\theta }_s$, the cross flight coordinates of the beam’s center on the horizontal ground are

(3)$\begin{array}{c}{y}_0=htan\left(\alpha \right)cos\left(\beta \right)\hfill \\ =htan\left(\alpha \right)cos\left(2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}\right)\right)\hfill \end{array}$

In general, the platform’s flight altitude, squint angle, and incident angle are all limited to a certain range. Once the ranges of the flight altitude h, squint angle ${\theta }_s$, and downward-looking angle $\alpha \in \left[{\alpha }_c-{\displaystyle \frac{{\theta }_r}{2}},{\alpha }_c+{\displaystyle \frac{{\theta }_r}{2}}\right]$ are given, the elevation range which the radar beam can cover can also be determined, where ${\alpha }_c$ is the center of the antenna’s beam at a certain elevation.

Equation (3) describes the range coordinate values observed under a given platform flight altitude, squint angle, and radar downward-looking angle. According to the properties of composite functions and the monotonicity of the trigonometric function, $y_0$ is a monotonic increasing function in the interval. Based on this property, the boundary values of $y_0$ can be determined, which are

(4)$\left\{\begin{array}{c}{y}_{0max}=htan\left({\alpha }_{max}\right)cos\left(2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin{\alpha }_{max}}}\right)\right)\\ y_{0min}=htan\left({\alpha }_{min}\right)cos\left(2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin{\alpha }_{min}}}\right)\right)\end{array}\right.$

To determine the observation range in the elevation direction under a squint angle, the monotonicity of Equation (3) with respect to $\alpha$ is analyzed. Here, $tan\left(\alpha \right)$ is increasing monotonically within the interval. Meanwhile, ${\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}$ is decreasing monotonically in the interval, and $arcsin\left(x\right)$ is increasing monotonically in the interval with respect to x. Based on the properties of the composite functions, $arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}\right)$ decreases monotonically. Additionally, since $cos\left(x\right)$ is a monotonic decreasing function in the interval, it follows that $cos\left(2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}\right)\right)$ is monotonically increasing within the interval.

In accordance with the aforementioned boundary constraints, when specifying the y-axis coordinates of the target point within the scene during the simulation design, it is imperative that these coordinates do not exceed the prescribed constraint range. Conversely, given a particular observation scene and squint angle, these constraints can also serve as a valuable guide for radar system design. To ensure comprehensive coverage of the scene, it is essential to utilize these constraints to ascertain the antenna’s downward-looking angle and elevation beam width.

3.1.2. Constraints of Azimuth Beam Width and Squint Angle on Observing Range in Flight

In general, the constraints on the observed scene are two-dimensional. The previous section mainly analyzes the limitations of the parameters on the observation width in the range direction. To achieve complete coverage of the scene of interest, it is necessary to design the flight range in the azimuth direction. The complete squint observation process of a single point target P is illustrated in Figure 2. The moments when the front edge of the beam, the center of the beam, and the back edge of the beam pass through the target P are expressed as $t_{as}$, $t_{ac}$, and $t_{ae}$, respectively.

The moments when the front edge of the beam, the center of the beam, and the back edge of the beam pass through the target P are expressed by the following respective equations:

(5)$t_{as}={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{R_{sp}tan\left({\theta }_s+\frac{{\theta }_{az}}{2}\right)}{v_a}}$

(6)$t_{ac}={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{\Delta X_p}{v_a}}={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{R_{sp}sin{\theta }_s}{v_a}}$

(7)$t_{ae}={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{R_{sp}tan\left({\theta }_s-\frac{{\theta }_{az}}{2}\right)}{v_a}}$

For the point target $P\left(x_p,y_p\right)$, as shown in Figure 2, it is necessary to ensure complete coverage for the x axis during observation, and the azimuth flight time should at least include the interval $\left[t_{as},t_{ae}\right]$.

For an observed scenario, to ensure that all points in the scene can be fully observed, the azimuth observation time during observation should satisfy Equation (8):

(8)$\left\{\begin{array}{c}{X}_s=x_{p1}-R_{sp1}tan\left({\theta }_s+{\displaystyle \frac{{\theta }_{az}}{2}}\right)\\ X_e=x_{p2}-R_{sp2}tan\left({\theta }_s-{\displaystyle \frac{{\theta }_{az}}{2}}\right)\end{array}\right.$

where $(x_{p1},R_{sp1})$ and $(x_{p2},R_{sp2})$ are the coordinates of two points at the edge of the scene.

3.1.3. Constraints Among Squint Angle, Beam Width at a Certain Elevation and Beam Width for a Certain Azimuth

In order to ensure that the antenna can still perform SAR imaging of a target under high-squint conditions, it is necessary to ensure that there is an effective variation in the Doppler frequency between the target and the radar, and the slant range and the incident angle can have one-to-one correspondence, which means that the antenna beam cannot cross the flight path physically. Figure 3 shows this constraint relationship, where $\gamma$ is the projection of the edge of the antenna beam on the horizontal plane, point E is the intersection point of the beam’s front ray and the horizontal plane, point C is the intersection point of the beam’s rear ray and the horizontal plane, and $E^{\prime }$ is the intersection point of the CE extension line and the x axis.

(9)$tan\gamma ={\displaystyle \frac{DE}{OD}}={\displaystyle \frac{\frac{h}{cos\alpha }tan\left(\frac{{\theta }_{az}}{2}\right)}{htan\left(\alpha \right)}}$

(10)$DE={\displaystyle \frac{h}{cos\alpha }}tan\left({\displaystyle \frac{{\theta }_{az}}{2}}\right)$

(11)$OD=htan\left(\alpha \right)$

(12)$tan\gamma ={\displaystyle \frac{DE}{OD}}={\displaystyle \frac{\frac{h}{cos\alpha }tan\left(\frac{{\theta }_{az}}{2}\right)}{htan\left(\alpha \right)}}={\displaystyle \frac{tan\left(\frac{{\theta }_{az}}{2}\right)}{sin\left(\alpha \right)}}$

The constraint ensures that the area covered by the beam is on the one side of the platform; that is, we have

(13)$\beta +\gamma <{\displaystyle \frac{\pi }{2}}$

Combining (2) and (12), the constraint is reconstructed as

(14)$2arcsin\left({\displaystyle \frac{sin\frac{{\theta }_s}{2}}{sin\alpha }}\right)+arctan\left({\displaystyle \frac{tan\left(\frac{{\theta }_{az}}{2}\right)}{sin\left(\alpha \right)}}\right)<{\displaystyle \frac{\pi }{2}}$

In summary, it is necessary to satisfy the constraint relationships of Equations (4), (8) and (14) when selecting the working squint angle, incident angle, antenna azimuth beam width and scene target coordinates in the simulation analysis, and the above constraint relationships can also be used for parameter selection during system design.

3.2. Analysis of Signal Characteristics

The characteristics of the signal in the slant range plane are analyzed. The transmitted signal is a linear frequency modulation signal, and the received signal model is expressed as follows:

(15)$\begin{array}{c}{s}_r\left(t_a,t_r\right)=w_r\left(t_r-{\displaystyle \frac{2r\left(t_a\right)}{c}}\right)w_a\left(t_a-t_{ac}\right)\hfill \\ ·exp\left\{-j4\pi f_0{\displaystyle \frac{r\left(t_a\right)}{c}}+j\pi K_r{\left(t_r-{\displaystyle \frac{2r\left(t_a\right)}{c}}\right)}^2\right\}\hfill \end{array}$

(16)$r\left(t_a\right)=\sqrt{{\left(v_a{t}_a-x_p\right)}^2+{\left(R_{sp}cos{\theta }_s\right)}^2}$

where $w_r$ and $w_a$ are the range envelope and azimuth envelope of the received two-dimensional signal, respectively, $t_a$ is the azimuth’s slow time, which is sampled by the pulse repetition frequency (PRF), $t_r$ is the range’s fast time and c is the speed of electromagnetic waves in a vacuum. $K_r$ is the chirp rate of the transmitted linear frequency-modulated signal, $f_0$ is the operating wavelength of SAR, and r is the instantaneous slant range between the target and the radar, which is a function of the slow time $t_a$. Meanwhile, $t_{ac}={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{R_{sp}sin{\theta }_s}{v_a}}$ is the time when the beam’s center passes through the target P. Firstly, the slant range characteristics under a high squint angle are analyzed, and Taylor expansion of the slant range $r\left(t_a\right)$ at $t_{ac}$ is performed:

(17)$\begin{array}{c}r\left(t_a\right)=R_{sp}-sin{\theta }_s{v}_a\left(t_a-t_{ac}\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{v_a^2{cos}^2\left({\theta }_s\right)}{R_{sp}}}{\left(t_a-t_{ac}\right)}^2\hfill \\ +{\displaystyle \frac{1}{2}}{\displaystyle \frac{v_a^{3}sin{\theta }_s{cos}^2\left({\theta }_s\right)}{R_{sp}^2}}{\left(t_a-t_{ac}\right)}^2+\cdots \hfill \end{array}$

The first term in Equation (17) is the slant range when the beam’s center passes through the target point. The second term is the linear component, the third term is the secondary migration component, and the fourth term is the third-order component. In high-squint mode, the target has significant linear-range walking momentum in the slant range.

We perform a Fourier transform of Equation (15) in the range and organize the phase terms to obtain the following expression:

(18)$\begin{array}{c}{\varphi }_r\left(t_r\right)=-4\pi f_0{\displaystyle \frac{r\left(t_a\right)}{c}}+\pi K_r{\left(t_r-{\displaystyle \frac{2r\left(t_a\right)}{c}}\right)}^2\hfill \\ -2\pi f_r{t}_r\hfill \end{array}$

Using the principle of stationary phase (POSP), the relationship between the range’s fast time and the range frequency can be approximately determined with the following equation:

(19)${\displaystyle \frac{d{\varphi }_r}{d{t}_r}}=2\pi K_r\left(t_r-{\displaystyle \frac{2r\left(t_a\right)}{c}}\right)-2\pi f_r=0$

The relationship between the fast time and range frequency is expressed as follows:

(20)$f_r=K_r\left(t_r-{\displaystyle \frac{2r\left(t_a\right)}{c}}\right)\to t_r={\displaystyle \frac{f_r}{K_r}}+{\displaystyle \frac{2r\left(t_a\right)}{c}}$

We can substitute Equation (20) into Equation (18) to obtain

(21)$\begin{array}{c}{S}_r\left(t_a,f_r\right)=\int s_r\left(t_a,t_r\right)exp\left(-j2\pi f_r{t}_r\right)d{t}_r\hfill \\ =W_r\left(f_r\right)w_a\left(t_a-t_{ac}\right)exp\left\{-j4\pi {\displaystyle \frac{f_0+f_r}{c}}r\left(t_a\right)-j\pi {\displaystyle \frac{f_r^2}{K_r}}\right\}\hfill \end{array}$

where $W_r=w_r\left({\displaystyle \frac{f_r}{K_r}}\right)$ is the range’s spectrum envelope, the first phase term corresponds to the phase induced by the two-way wave path difference of the echo signal at a slant range $r\left(t_a\right)$ and the second phase term is not required for focusing in range, which needs to be removed in range compression. It should be noted that the range sampling rate must satisfy the sampling theorem $f_r\in \left[-{\displaystyle \frac{f_s}{2}},{\displaystyle \frac{f_s}{2}}\right]$, where $f_s$ is the range’s sampling rate. In addition, the carrier frequency $f_0$ of radar is often much greater than the range frequency; that is, $f_0\gg f_r$.

By performing a Fourier transform of Equation (21) in the azimuth and organizing the phase terms, we obtain

(22)$\begin{array}{c}{S}_r\left(f_a,f_r\right)=\int S_r\left(t_a,f_r\right)exp\left(-j2\pi f_a{t}_a\right)d{t}_a\hfill \\ =W_r\left(f_r\right)\int w_a\left(t_a-t_{ac}\right)exp\left\{\begin{array}{c}-j4\pi {\displaystyle \frac{f_0+f_r}{c}}r\left(t_a\right)\hfill \\ -j2\pi f_a{t}_a-j\pi {\displaystyle \frac{f_r^2}{K_r}}\hfill \end{array}\right\}d{t}_a\hfill \\ =W_r\left(f_r\right)\int w_a\left(t_a-t_{ac}\right)exp\left\{j{\varphi }_a\right\}d{t}_a\hfill \end{array}$

Using the POSP, the relationship between the azimuth time and azimuth frequency satisfies the following equation:

(23)$\begin{array}{c}{\displaystyle \frac{d{\varphi }_a}{d{t}_a}}=-{\displaystyle \frac{4\pi \left(f_0+f_r\right)}{c}}{\displaystyle \frac{v_a\left(v_a{t}_a-x_p\right)}{\sqrt{{\left(R_{sp}cos{\theta }_s\right)}^2+{\left(v_a{t}_a-x_p\right)}^2}}}\hfill \\ -2\pi f_a=0\hfill \end{array}$

(24)$f_a=-{\displaystyle \frac{2\left(f_0+f_r\right)v_a\left(v_a{t}_a-x_p\right)}{c\sqrt{{\left(R_{sp}cos{\theta }_s\right)}^2+{\left(v_a{t}_a-x_p\right)}^2}}}$

(25)$t_a={\displaystyle \frac{x_p}{v_a}}-{\displaystyle \frac{c{R}_{sp}cos{\theta }_s}{2v_a^2\left(f_0+f_r\right)\sqrt{1-\frac{f_a^2{c}^2}{4{\left(f_0+f_r\right)}^2{v}_a^2}}}}{f}_a$

We can substitute Equation (25) into Equation $r\left(t_a\right)$ to obtain

(26)$\begin{array}{c}r\left(t_a\right)=\sqrt{{\left(v_a{t}_a-x_p\right)}^2+{\left(R_{sp}cos{\theta }_s\right)}^2}\hfill \\ =\sqrt{{\left({\displaystyle \frac{c{R}_{sp}cos{\theta }_s}{2v_a\left(f_0+f_r\right)\sqrt{1-\frac{f_a^2{c}^2}{4{\left(f_0+f_r\right)}^2{v}_a^2}}}}{f}_a\right)}^2+{\left(R_{sp}cos{\theta }_s\right)}^2}\hfill \\ =\sqrt{{\displaystyle \frac{c^2{\left(R_{sp}cos{\theta }_s\right)}^2{f}_a^2}{4v_a^2{\left(f_0+f_r\right)}^2-f_a^2{c}^2}}+{\left(R_{sp}cos{\theta }_s\right)}^2}\hfill \\ ={\displaystyle \frac{R_{sp}cos{\theta }_s}{\sqrt{1-\frac{f_a^2{c}^2}{4{\left(f_0+f_r\right)}^2{v}_a^2}}}}\hfill \end{array}$

By substituting Equation (25) into ${\varphi }_a$, the approximate phase can be expressed as follows:

(27)$\begin{array}{c}{\varphi }_a\left(f_a,f_r\right)=\hfill \\ =-{\displaystyle \frac{4\pi \left(f_0+f_r\right)R_{sp}cos{\theta }_s}{c}}\sqrt{1-{\displaystyle \frac{f_a^2{c}^2}{4{\left(f_0+f_r\right)}^2{v}_a^2}}}\hfill \\ -2\pi f_a{\displaystyle \frac{x_p}{v_a}}-\pi {\displaystyle \frac{f_r^2}{K_r}}\hfill \end{array}$

By substituting Equation (27) into Equation (22), the two-dimensional spectrum of the echo is

(28)$S_r\left(f_a,f_r\right)=W_r\left(f_r\right)W_a\left(f_a-f_{ac}\right)exp\left\{j{\varphi }_a\left(f_a,f_r\right)\right\}$

where $f_{ac}$ is the Doppler center frequency and $W_a$ is the antenna pattern in the azimuth:

(29)$f_{ac}={\displaystyle \frac{2v_a}{\lambda }}sin{\theta }_s$

Performance Analysis of Point Target

In order to evaluate the performance of the algorithm, it is necessary to quantify the imaging quality of the point targets. Compared with the side-looking imaging mode, the target will rotate in the case of a large squint, and the rotation angle is related to the SAR squint angle. The azimuth and range are no longer orthogonal. Therefore, when performing two-dimensional slice analysis of the point targets in an image, it is necessary to first rotate them to obtain the target image with an orthogonal azimuth and range. The rotation and analysis process is as follows:

• (1). Firstly, extract the point target from the two-dimensional SAR image;

• (2). Then, perform 16-fold interpolation of the image;

• (3). Rotate the interpolated point target at an angle of ${\theta }_s$. The rotation process is as follows:

• (a). Calculate the new sampling intervals in the range and azimuth after rotation. If the range and azimuth sampling intervals before rotation are $\Delta R$ and $\Delta A$, respectively, then the calculation formulas for the range and azimuth sampling intervals after rotation are [8,20]

(30)$\left\{\begin{array}{c}\Delta R^{\prime }=\Delta Rcos\left({\theta }_s\right)\\ \Delta A^{\prime }=\Delta Acos\left({\theta }_s\right)\end{array}\right.$

• (b). Rotate the coordinate system and determine the range of coordinates after rotation. If the number of sampling points in the range and azimuth directions of the captured, unrotated image is $N_r$ and $N_a$, respectively, then the coordinates in the range and azimuth directions before rotation can be expressed as R and x, respectively:

(31)$R=\left[-floor\left(N_r/2\right):floor\left(N_r/2\right)\right]\Delta R$

(32)$x=\left[-floor\left(N_a/2\right):floor\left(N_a/2\right)\right]\Delta A$

where floor refers to the function in the programming language, which means the greatest integer less than or equal to a given real number.

Then, calculate the two-dimensional coordinate systems corresponding to a clockwise rotation ${\theta }_s$, which are

(33)$\left[\begin{array}{c}{R}^{\prime }\\ x^{\prime }\end{array}\right]=\left[\begin{array}{cc}cos\left({\theta }_s\right)& sin\left({\theta }_s\right)\\ -sin\left({\theta }_s\right)& cos\left({\theta }_s\right)\end{array}\right]\left[\begin{array}{c}R\\ x\end{array}\right]$

The distribution range of the two-dimensional coordinate system after the rotation can be obtained, where $x^{\prime }\in \left[x_{min}^{\prime },x_{max}^{\prime }\right]$, $R^{\prime }\in \left[R_{min}^{\prime },R_{max}^{\prime }\right]$.

• (c). Normalize the coordinates after rotation. The matrix composed of discrete samplings before rotation is uniformly sampled, but the corresponding coordinate points after rotation may no longer be uniformly spaced. Therefore, coordinate homogenization and image interpolation need to be carried out based on the coordinate distribution after rotation. The coordinates after homogenization are $R^{″}$ and $x^{″}$, respectively.

• (i). Calculate the required number of samplings based on the new sampling interval and coordinate range after coordinate rotation:

(34) ${N_r}^{\prime }=floor\left(\left(R_{max}^{\prime }-R_{min}^{\prime }\right)/\Delta R^{\prime }\right)$

(35) ${N_a}^{\prime }=floor\left(\left(x_{max}^{\prime }-x_{min}^{\prime }\right)/\Delta A^{\prime }\right)$

• (ii). Reset the coordinates:

(36) $R^{″}=\left[-floor\left(N_r^{\prime }/2\right):floor\left(N_r^{\prime }/2\right)\right]\Delta R^{\prime }$

(37) $x^{″}=\left[-floor\left(N_a^{\prime }/2\right):floor\left(N_a^{\prime }/2\right)\right]\Delta A^{\prime }$

• (d). For image interpolation, based on the two-dimensional coordinate distribution of $R^{\prime }$ and $x^{\prime }$ before interpolation and the target image, interpolate the image values at $R^{″}$ and $x^{″}$.

• (4). Slice analysis of the rotated image in both the azimuth and distance directions should be performed.

3.3. Two-Dimensional Position Offset Analysis

3.3.1. Azimuth Position Offset and Correction

According to the imaging geometry in Figure 2, it can be seen that an additional offset $\Delta X_P=R_{sp}sin\left({\theta }_s\right)=Rtan\left({\theta }_s\right)$ was added to the azimuth position at a squint angle ${\theta }_s$ compared with the side-looking SAR. After Stolt interpolation, the offset of each range gate is corrected to the reference distance. The offset $\Delta X=R_{ref}tan\left({\theta }_s\right)$ is considered the overall offset, and it needs to be removed in the imaging process to ensure accurate azimuth position.

3.3.2. Range Position Offset and Correction

In the $\omega -k$ algorithm, Stolt interpolation needs to be performed in the two-dimensional frequency domain. The implementation of interpolation can be divided into two methods. One is to keep the number of frequency sampling points unchanged while the frequency sampling interval changes. The other approach is to keep the frequency sampling interval unchanged, but the number of frequency sampling points will increase. In the first case, it was found that shifts in the range position occurred. The shift phenomenon was centered around $R_{ref}$, and the overall effect was that the image was compressed.

The underlying causes of the aforementioned phenomena have been meticulously analyzed. According to the setting process of the range frequency $f_{rdd}$ in interpolation, the equivalent sampling rate after mapping is $F_{r_{new}}$. The frequency range of $f_{rdd}$ is then forcibly allocated across $N_r$ points, resulting in a frequency sampling interval of $F_{r_{new}}/N_r$($F_{r_{new}}>F_r$), whereas the original frequency sampling interval is $F_r/N_r$. The corresponding range frequency distributions are depicted in Figure 4. The parallelogram shape indicates the change in the range frequency corresponding to each echo. It can be seen that the change in the range frequency is modulated by the azimuth frequency. Due to the modulation effect of the azimuth frequency, the range frequencies corresponding to each pulse echo datum are not completely consistent, forming the aforementioned parallelogram region. The red curve illustrates the frequency distribution with an interval of $F_{r_{new}}/N_r$, and the green curve represents the distribution with an interval of $F_r/N_r$. This allocation effectively stretches the spectral sampling interval by a factor of $F_{r_{new}}/F_r$. Consequently, stretching in the frequency domain translates to compression in the time domain, which aligns with the observed phenomenon.

Due to the fact that the time-domain image obtained by frequency domain interpolation is compressed around the reference range $R_{ref}$, the compression amount of the interpolated image relative to $R_{ref}$ is

(38)$\Delta r={\displaystyle \frac{F_r}{F_{r_{new}}}}(r_s-R_{ref})={\displaystyle \frac{F_r}{F_{r_{new}}}}({\displaystyle \frac{n}{F_r}}{\displaystyle \frac{c}{2}}-R_{ref})$

where n is the range index. Therefore, after the imaging is completed, the slant range is allocated by the new sampling rate, which should be

(39)$r_s^{\prime }=R_{ref}+\Delta r$

In the aforementioned interpolation process, the interpolated range frequency $f_{rdd}$ is constrained to $N_r$ points, thereby altering the frequency interval. This alteration in the frequency interval subsequently impacts the positioning of the range. To mitigate this issue, $f_{rdd}$ is adjusted to keep the frequency interval unchanged, where $\Delta f_r=F_r/N_r$, $f_{rdd}=f_{r{d}_{min}}:\Delta f_r:f_{r{d}_{max}}$. Then, interpolation is performed, which causes the range size of the interpolated image to change, but the resulting image is not compressed. The upward scale of the range needs to be modified.

To mitigate this issue, $f_{rdd}$ is adjusted to maintain a constant frequency interval, specifically $\Delta f_r=F_r/N_r$, and it is redefined as $f_{rdd}=f_{r{d}_{min}}:\Delta f_r:f_{r{d}_{max}}$. Subsequent interpolation is then carried out. Although this process results in a change in the range size of the interpolated image, it ensures that the image is not compressed. Consequently, the scaling factor for the range dimension needs to be modified accordingly:

(40)$r_s=R_{s_{min}}+\left(0:N_r^{\prime }-1\right)/F_{r_{new}}c/2+c{T}_p/4$

3.4. Diagram of the Algorithm

The commonly used imaging process of the $\omega -k$ algorithm is shown in Figure 5, which mainly includes Doppler centroid estimation and compensation, bulk compression, differential compression, azimuth position correction and range position correction.

The Doppler centroid determines the alignment of phase histories during azimuth compression. Errors in its estimation lead to misregistration of targets, blurring and loss of resolution. Accurate Doppler centroid values are also essential for range cell migration correction (RCMC), ensuring energy from a single target remains coherent across the synthetic aperture. Incorrect centroid estimates exacerbate ambiguities caused by the periodic nature of the Doppler spectrum, particularly in high-squint or wide-swath configurations. The classic algorithm based on the azimuth spectrum is used for Doppler centroid estimation. By estimating the azimuth antenna pattern and extracting the peak point position in the range-Doppler domain, the Doppler centroid $f_{ac}$ can be obtained. Then, we can compensate for the Doppler frequency offset through multiplication by a factor $exp\left(-j2\pi f_{ac}{t}_a\right)$.

The bulk compression and differential compression are achieved by multiplying the two-dimensional frequency reference function and Stolt interpolation, respectively. Bulk and differential compression are pivotal in addressing the RCMC and storage demands of SAR $\omega -k$ imaging, demonstrating superior capability in addressing two-dimensional coupling and focusing challenges inherent in SAR data processing and achieving precise phase preservation across both the range and azimuth dimensions. Bulk compression provides initial phase correction, and differential compression ensures fine-range correction, particularly for wide-band and high-squint SAR systems.

Azimuth position correction is achieved through multiplication by the phase correction factor $exp\left(j2\pi f_a{R}_{ref}tan\left({\theta }_s\right)/v_a\right)$ after Stolt interpolation in the range-Doppler domain, and range position correction is achieved by reconstructing the range sampling grid according to Equation (39). Alternatively, the modified Stolt interpolation incorporates two critical adaptations; the equation $f_{rdd}=f_{r{d}_{min}}:\Delta f_r:f_{r{d}_{max}}$ governs the interpolation coverage adjustment in the range-frequency domain, while Equation (40) regulates the geometric recalibration of the range axis.

3.5. Simulation Experiments

In order to verify the effectiveness of the analysis and imaging algorithm mentioned, a simulation experiment is carried out in this section. The SAR system and simulation parameters are shown in Table 1.

The distribution of target points in the simulation scenario is shown in Figure 6 and Table 2.

The simulation scenario was composed of multiple discrete isolated target points, as illustrated in Figure 6. The imaging results for the original algorithm are illustrated in Figure 7. And the imaging results using the modified algorithm are shown in Figure 8. Compared with Figure 7, the imaging results for the modified algorithm show that the two-dimensional position was corrected well. The point analysis and performance comparisons between the original and proposed algorithms are shown in Table 3. The edge point P1 (3549 m, 600 m) and the center P5 (4242.6 m, 0 m) were chosen as examples. The positions of the targets after two-dimensional position correction are listed in the third line of Table 3.

Compared with the original algorithm, the results obtained by the proposed algorithm were corrected well and basically consistent with the set values. The standard deviation of the azimuth’s estimated position via statistical analysis was 0.12 m, and that of the range position positioning was 0.21 m. The performances of the imaging results were analyzed quantitatively using the peak-to-sidelobe ratio (PSLR), integrated sidelobe ratio (ISLR) and resolution. Firstly, a point in the scene was extracted. Then, rotation and interpolation were performed. The rotated and original images are illustrated in Figure 9 and Figure 10, respectively. Finally, slice analysis in the azimuth and range was performed for the rotated images. The analysis and comparison results are shown in the last three rows of Table 3. The performance of the proposed algorithm was consistent with the original algorithm except for the position offsets, proving that the proposed algorithm achieves two-dimensional correction of the target positions while ensuring image performance.

Additionally, a comparison was made with the method of azimuth position correction using the Doppler centroid and Doppler rate. The imaging results using the Doppler centroid and Doppler chirp rate are illustrated in Figure 11. Compared with the uncorrected image, the positions of the targets were corrected. The positions of the point targets in the corrected scene were counted, and the results are shown in the table below. The standard deviation of the azimuth’s estimated position via statistical analysis was 1.00 m.

The azimuth position deviation of the edge point P1 after positioning was 1.2 m, and the azimuth position deviation of the center point P5 after positioning was 1.32 m. Compared with the correction results of the proposed algorithm, the positioning deviation was relatively large. This is because $K_a$ is the azimuth Doppler chirp rate in the range-Doppler algorithm, which approximates the azimuth signal as a linear, frequency-modulated signal. This approximation introduces errors, as it involves a Taylor expansion of the slant range, retaining only the second-order term and neglecting higher-order terms. Moreover, the approximation error became more severe as the squint angle increased. Using $K_a$ for position correction naturally introduced a certain amount of error.

An analysis of the imaging performance of the edge point P1 and the center point P5 was conducted. Then, rotation and interpolation were performed. The rotated and original images of edge point P1 are illustrated in Figure 12a and Figure 12b, respectively. The azimuth resolution was 0.42 m, the PLSR was −13.27 dB, and the ISLR was −11.04 dB. The range resolution was 0.48 m, the PLSR was −13.26 dB, and the ISLR was −11.05 dB. The rotated and original image of center point P5 are illustrated in Figure 13a and Figure 13b, respectively. The azimuth resolution was 0.42 m, the PLSR was −13.27 dB, and the ISLR was −11.04 dB. The range resolution was 0.48 m, the PLSR was −13.26 dB, and the ISLR was −11.05 dB.

In summary, compared with the original $\omega -k$ algorithm and the positioning algorithm combining the Doppler centroid and Doppler chirp rate, the proposed algorithm better corrected the target position while ensuring imaging quality.

3.6. Experiment Results

In order to better verify the effectiveness of the algorithm, L-band airborne squint flight SAR data were processed. The radar system parameters were as shown in Table 4.

The proposed algorithm was used to process SAR data to obtain a two-dimensional focused image, as shown in Figure 14. Points were selected from the scene for performance slice analysis. First, the target point was extracted from the large image scene and interpolated by a factor of 16, as shown in Figure 15a. Then, the point target image was rotated according to the squint angle, as shown in Figure 15b. The rotated point targets were analyzed by azimuth and range slicing, as shown in Figure 16a and Figure 16b, respectively. The PSLR in the azimuth was −11.62 dB, the ISLR was −8.66 dB, and the azimuth resolution was 3.4 m. In the range direction, the PSLR was −26.61 dB, the ISLR was −15.40 dB, and the resolution was 3.1 m. The actual processing results were consistent with the simulation values.

Traditional algorithms were used for processing, and the imaging results are shown in Figure 17. Points were selected from the scene for positioning and performance slice analysis. First, the target point was extracted from the large image scene and interpolated by a factor of 16, as shown in Figure 18a. Then, the point target image was rotated according to the squint angle, as shown in Figure 18b. The rotated point target was analyzed using azimuth and range slicing, as shown in Figure 19a and Figure 19b, respectively. The PSLR in the azimuth was −11.62 dB, the ISLR was −8.66 dB, and the resolution was 3.4 m. In the range direction, the PSLR was −21.57 dB, the ISLR was −15.40 dB, and the resolution was 3.1 m. From the above performance metrics, it can be seen that the imaging performance of the proposed method was comparable to that of traditional algorithms.

The position correction capability of the proposed algorithm was analyzed. Three target regions, namely S1, S2 and S3, were selected, and the coordinates of their central areas were calculated. These coordinates were then mapped to the slant range plane in the radar coordinate system. The reference coordinate values in the azimuth and range are shown in the second and third rows of Table 5, respectively. The fourth and fifth rows display the azimuth and range coordinates after correction using the proposed algorithm, respectively. The sixth and seventh rows show the azimuth and range coordinates obtained using the traditional algorithm without position correction, respectively. The statistical comparison in Table 5 demonstrates that the proposed algorithm effectively achieved two-dimensional image position correction, which was consistent with the simulation results.

4. Discussion

This paper proposed a novel two-dimensional position correction algorithm for the high-squint SAR mode. The computational load was analyzed and discussed here. The algorithm mainly includes Doppler centroid estimation and compensation, a two-dimensional FFT, multiplying the reference phase, Stolt interpolation, azimuth position correction and range position correction. Doppler centroid compensation includes complex multiplication of $N_a{N}_r$ times. The computation load of the two-dimensional FFT includes $N_a$ times $N_r$ points’ FFT operations and $N_r$ times $N_a$ points’ FFT operations. Stolt interpolation requires interpolation of the points for each of the $N_a{N}_r^{\prime }$ points, in which M is the interpolation kernel length. Azimuth position correction and range correction include complex multiplication of $2N_a{N}_r^{\prime }$ times, $N_r^{\prime }$ times $N_a$ points’ FFT operations and $N_a$ times $N_r^{\prime }$ points’ FFT operations. Therefore, the computational load of the proposed algorithm is

(41)$C=2N_a{N}_r+{\displaystyle \frac{N_a{N}_r}{2}}{log}_2\left(N_a{N}_r\right)+\left(M+2\right)N_a{N}_r^{\prime }+{\displaystyle \frac{N_a{N}_r^{\prime }}{2}}{log}_2\left(N_a{N}_r^{\prime }\right)$

5. Conclusions

To ensure optimal imaging performance in high-squint SAR systems, this study established quantitative constraint relationships between the operational parameters, which include the pulse repetition frequency and platform velocity, and the observational parameters, which include the scene coverage, squint angle, incidence angle and beam width. These formulas serve as a theoretical foundation for achieving accurate data acquisition under high squint-angle configurations. To address the substantial displacement in the azimuth and range dimensions induced by large squint angles, closed-form analytical expressions for geometric distortions were derived. The proposed compensation methodology was systematically integrated into the $\omega -k$ imaging algorithm’s processing chain, enabling precise 2D spatial correction and ultimately producing high-resolution SAR images free from positional distortion.

Author Contributions

S.W. and B.W. conceptualized the study and contributed to the article’s organization; C.S., L.Y., J.C. and Z.H. contributed to the discussion of the simulation results; S.W. drafted the manuscript, which was revised by all authors. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Observation geometry.

Figure 2. Observation history of a single point target.

Figure 3. The constraint relationship between the squint angle, beam width at a certain elevation, and beam width at a certain azimuth.

Figure 4. Range frequency distribution.

Figure 5. Processing diagram of [Forumla omitted. See PDF.] algorithm.

Figure 6. Distribution of the simulation scenarios.

Figure 7. Imaging results before position correction.

Figure 8. Imaging results after position correction.

Figure 9. Point target before rotation.

Figure 10. Point target after rotation.

Figure 11. Imaging results using Doppler centroid and Doppler chirp rate.

Figure 12. Edge point target P1. (a) Point target before rotation. (b) Point target after rotation.

Figure 13. Center point target P1. (a) Point target before rotation. (b) Point target after rotation.

Figure 14. The processing results of the proposed algorithm with real data.

Figure 15. Point target in the proposed algorithm. (a) Point target before rotation. (b) Point target after rotation.

Figure 16. Slice analysis of point target. (a) Azimuth slice. (b) Range slice.

Figure 17. SAR image processed using the traditional algorithm.

Figure 18. Point target in traditional algorithm. (a) Point target before rotation. (b) Point target after rotation.

Figure 19. Slice analysis of point target. (a) Azimuth slice. (b) Range slice.

Table 1

Simulation system parameters.

Parameter	Value	Parameter	Value
Radar center frequency (GHz)	9.6	PRF (Hz)	470
Scene center slant range (km)	4.24	Azimuth beam (°)	2
Scene center slant range (km)	4.24	Azimuth beam (°)	2
Radar initial position (km)	−3.36	Velocity (m/s)	150
Transmitted bandwidth (MHz)	280	Pulse width (μs)	5

Table 2

Coordinates of points in the simulation scenario.

	P1	P2	P3	P4	P5	P6	P7	P8	P9
Range (m)	3549	4242.6	4459.8	4036.1	4242.6	4459.8	4036.1	4242.61	4459.8
Azimuth (m)	600	300	300	0	0	0	−300	−300	−600

Table 3

Point analysis and performance comparisons between the original and proposed algorithms.

	Original Algorithm		Proposed Algorithm
(Range, Azimuth)	P1 (3549, 600)	P5 (4242.6, 0)	P1	P5
Position (m)	(2959, −1469)	(4320, −2068)	(3549, 600.3)	(4243, 0)
ISLR (dB)	(−11.01, −11.05)	(−11.01, −11.05)	(−11.05, −11.04)	(−11.05, −11.04)
PSLR (dB)	(−13.27, −13.27)	(−13.27, −13.27)	(−13.26, −13.27)	(−13.26, −13.27)
Resolution (m)	(0.48, 0.42)	(0.48, 0.42)	(0.48, 0.42)	(0.48, 0.42)

Table 4

Radar system parameters.

Parameter Name	Value
Radar center frequency (GHz)	1.6
Effective radar velocity (m/s)	150
PRF (Hz)	70
Transmitted bandwidth (MHz)	50
Squint angle (°)	30

Table 5

Center coordinates of the selected areas in a real scenario.

Coordinates/Area	S1	S2	S3
Reference Range/m	3644	3822	3765
Reference Azimuth/m	187.2	27.2	400.5
Corrected Range/m	3643	3822	3764
Corrected Azimuth/m	187	28.7	401.8
Uncorrected Range/m	3228	3551	3443
Uncorrected Azimuth/m	30.3	−130.8	245.2

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