Terahertz Squint SAR Imaging Based on Decoupled Frequency Scaling Algorithm

Highlights

What are the main findings?

• In the terahertz band, the two-dimensional coupling effect under high-squint-angle conditions is aggravated, making phase errors non-negligible and causing traditional frequency-domain algorithms to fail to achieve precise focusing. Therefore, this paper proposes a terahertz squint SAR imaging algorithm based on decoupled frequency scaling.
• Following time-domain decoupling, the proposed algorithm combined range frequency scaling with azimuth nonlinear chirp scaling, which can achieve better imaging results.

What is the implication of the main finding?

• The experimental results show that the proposed algorithm can achieve better imaging results and provide an effective technical approach to squint SAR imaging in the terahertz band.
• This result has application value in airborne terahertz radar detection, cooperative observation of UAV formation, and other fields.

Abstract

Terahertz synthetic aperture radar (SAR) can achieve high-resolution imaging of the target area through a large bandwidth, while squint imaging can flexibly detect the target area by adjusting the beam direction. However, the two-dimensional coupling effect intensifies under squint conditions, making it challenging for traditional frequency domain algorithms for high-resolution imaging. This paper analyzes the Doppler variations and proposes a two-dimensional decoupling algorithm for squint SAR imaging in the terahertz band. The proposed algorithm decouples in the time domain and combines the improved frequency scaling operation with the azimuthal nonlinear frequency scaling operation to obtain the focused SAR image. Compared to the Range Doppler algorithm and nonlinear frequency scaling algorithm, the simulation and experimental results verified the effectiveness of the proposed algorithm, which demonstrates the application potential for squint SAR imaging in the terahertz band.

1. Introduction

Synthetic Aperture Radar (SAR) forms a large virtual aperture through platform motion to achieve two-dimensional imaging [1]. In the terahertz band (0.1–10 THz), its short wavelength helps generate wideband signals and high-gain narrow beams, enabling high-resolution imaging in both range and azimuth directions [2,3,4]. The shorter synthetic aperture time also allows for rapid imaging [5,6,7,8]. Squint SAR imaging, as an important operational mode, uses flexible beam steering to image targets ahead of the platform, providing richer target information and diverse observation perspectives [9,10].

However, efficient imaging for high-squint SAR data remains challenging [11,12,13]. Conventional SAR imaging techniques face significant challenges under high-squint conditions due to severe two-dimensional coupling effects, which prevent accurate phase modeling in the frequency domain [14]. While algorithms such as the chirp scaling algorithm (CSA) and the frequency scaling algorithm (FSA) offer solutions for small squint angles, their performance degrades substantially with increasing squint angles [15,16]. Even with their improved algorithms, the chirp scaling algorithm (ECSA) and the nonlinear frequency scaling algorithm (NFSA) are unable to meet the precise imaging requirements under conditions of large squint and large bandwidth in the terahertz band [17,18].Although the modified Range Doppler algorithm (MRDA) partially mitigates coupling through range cell migration correction, it remains restricted to small scene sizes [19,20]. Frequency domain approaches such as the range migration algorithm (RMA) avoid explicit analytical expressions and accommodate larger squint angles in the terahertz band, yet their efficiency is limited by Stolt mapping operations and inability to adapt the parameters range-dependently [21,22]. Similarly, the Polar Format Algorithm (PFA) suffers from scene size restrictions due to wavefront curvature effects, and though extendable via multi-step processing, this exacerbates computational costs [23]. Recent advances have incorporated motion compensation, sub-aperture processing, and advanced interpolation to enhance scalability, though balancing precision and computational efficiency remains a constant research challenge [24,25].

The back projection algorithm (BPA) and its variants represent the most widely used time domain approaches for high-squint SAR imaging, renowned for their principled simplicity and adaptability to various radar systems and scene configurations [26]. By processing echoes pixel by pixel based on precise time delays in the slant range, BPA achieves accurate imaging without being constrained by severe squint angles or complex geometries [27,28]. However, since it does not exploit the azimuth invariance property of SAR signals, it suffers from high computational complexity, which severely limits its real-time application on high mobility platforms [29]. Especially in the terahertz frequency band, the amount of echo data increases significantly, and it is difficult for the efficiency of the BP algorithm to meet the requirements. Consequently, despite its robustness, the inefficiency of BPA underscores the continued importance of developing frequency domain fast imaging algorithms for high-squint SAR data.

This paper proposes an improved decoupled frequency scaling algorithm for squint SAR imaging in the terahertz band. By analyzing variations in Doppler parameters and combining two-dimensional time-domain decoupling, range frequency scaling, and azimuth nonlinear frequency scaling methods, the processing flow of the proposed algorithm is derived. Simulations of multiple point targets and airborne experiments demonstrate that, in the terahertz band, this algorithm offers significant advantages over RDA and NFSA.

The structure of this paper is organized as follows. Section 2 develops the terahertz squint imaging echo model and analyzes the Doppler characteristics of terahertz radar echoes. Section 3 proposes an improved imaging algorithm to address the two-dimensional coupling problem under terahertz squint conditions. In Section 4, multipoint target simulations are conducted to compare and analyze the proposed algorithm with traditional algorithms, demonstrating its superior performance. Section 5 analyzes the imaging quality based on airborne high-squint terahertz radar data utilizing both the proposed and conventional algorithms. Finally, Section 6 concludes this paper with a summary of the research findings.

2. Signal Model

The imaging geometry of squint SAR is shown in Figure 1. The radar operates in strip-map mode. The green ellipse in the figure represents the beam coverage area, while the blue area shows the entire region scanned by the radar beam. The angle between the direction of the radar beam and the direction perpendicular to the velocity is the squint angle θ. Suppose that the time when the radar platform is located at the point A is the azimuth slow time starting point, the closest distance between the point target P and the flight trajectory is $R_B$, and the slant range when the beam center scans over the target P is $R_0$. On the line parallel to the flight track, the starting position of the beam center is the point B.

After the slow-time $t_m$, the radar platform moves to point $A^{\prime }$, with a lateral displacement of $v{t}_m$. Let the distance between the target point P and point B be $X_p$. From the geometric relationship, the instantaneous slant range $R(t_m;R_0)$ between the point target P and the radar is given as follows:

$$R(t_m;R_0)=\sqrt{{(v{t}_m-X_p)}^2+R_0^2-2R_0(v{t}_m-X_p)sin\theta }$$

(1)

When the condition $|v{t}_m-X_p| \ll R_0$ is satisfied, expanding Equation (2) as a Taylor series around $t_m={\displaystyle \frac{X_p}{v}}$ yields the following:

$$\begin{array}{cc}\hfill R(t_m;R_0)=& R_0-vsin\theta \left(t_m-\frac{X_p}{v}\right)+{\displaystyle \frac{v^2{cos}^2\theta }{2R_0}}{\left(t_m-\frac{X_p}{v}\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{v^3{cos}^2\theta sin\theta }{2R_0^2}}{\left(t_m-\frac{X_p}{v}\right)}^3\hfill \\ \hfill & +\sum _{n=4}^{\infty }{\displaystyle \frac{1}{n!}}{\displaystyle \frac{{\partial }^{n}R(t_m;R_0)}{\partial t_m^n}}{|}_{t_m=\frac{X_p}{v}}{\left(t_m-\frac{X_p}{v}\right)}^n\hfill \end{array}$$

(2)

In the expansion of the instantaneous slant range, the first term represents the nominal range between the point target and the radar. The second term is the linear range walk term. The third term is the quadratic range curvature term. The fourth term is the cubic range shift term. Higher-order terms (fourth-order and beyond) are termed higher-order range shift terms [30].

It can be observed that the magnitudes of each order of range cell migration (RCM) depend on the inherent squint angle of the moving platform and the azimuth position of the target. Specifically the RCM terms increase with the azimuth distance $X_p$, and the RCM terms decrease with the nominal slant range $R_0$.

Radar achieves high range resolution by transmitting wide-bandwidth signals in the range direction. Due to the high carrier frequency in the terahertz band, large transmission bandwidths (typically exceeding 5 GHz) can be achieved. Limited by the sampling rate of conventional analog-to-digital (AD) converters, direct sampling is generally not employed in terahertz SAR imaging systems [31]. Instead, the dechirp method is utilized for signal reception, thereby reducing the required sampling rate [32].

Typically, the transmitted radar signal is a linear frequency modulated (LFM) signal:

$$s(\tau ,t_m)=\mathrm{rect}\left({\displaystyle \frac{\tau }{T_p}}\right)\mathrm{rect}\left({\displaystyle \frac{t_m}{T_a}}\right)exp\left[j2\pi \left(f_{c}t+{\displaystyle \frac{1}{2}}\gamma {\tau }^2\right)\right]$$

(3)

where $t=t_m+\tau$ represents the full time, $\tau$ denotes the range fast time, $t_m$ is the azimuth slow time, $T_p$ is the pulse width, $\gamma$ is the chirp rate, $f_c$ is the carrier frequency, and the rectangular envelope function is defined as follows:

$$\mathrm{rect}\left(u\right)=\left\{\begin{array}{cc}1\hfill & \left|u\right|\le {\displaystyle \frac{1}{2}}\hfill \\ 0\hfill & \left|u\right|>{\displaystyle \frac{1}{2}}\hfill \end{array}\right.$$

Taking the scene center distance $R_c$ as the reference range, the reference signal at this distance is:

$$s_{\mathrm{ref}}(\tau ,t_m)=\mathrm{rect}\left({\displaystyle \frac{\tau -\frac{2R_c}{c}}{T_p}}\right)\mathrm{rect}\left({\displaystyle \frac{t_m}{T_a}}\right)exp\left\{j2\pi \left[f_c\left(t-{\displaystyle \frac{2R_c}{c}}\right)+{\displaystyle \frac{1}{2}}\gamma {\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^2\right]\right\}$$

(4)

For a point target P with instantaneous slant range $R(t_m;R_0)$, the echo signal can be expressed as:

$$\begin{array}{cc}\hfill s_r(\tau ,t_m)=& \mathrm{rect}\left({\displaystyle \frac{\tau -\frac{2R(t_m;R_0)}{c}}{T_p}}\right)\mathrm{rect}\left({\displaystyle \frac{t_m}{T_a}}\right)\hfill \\ \hfill & ·exp\left\{j2\pi \left[f_c\left(t-{\displaystyle \frac{2R(t_m;R_0)}{c}}\right)+{\displaystyle \frac{1}{2}}\gamma {\left(\tau -{\displaystyle \frac{2R(t_m;R_0)}{c}}\right)}^2\right]\right\}\hfill \end{array}$$

(5)

After coherent demodulation through dechirping between the echo signal and the reference signal, the output signal becomes the desired dechirped echo:

$$\begin{array}{cc}\hfill s_{\mathrm{dechirp}}(\tau ,t_m)=& \mathrm{rect}\left({\displaystyle \frac{\tau -\frac{2R(t_m;R_0)}{c}}{T_p}}\right)\mathrm{rect}\left({\displaystyle \frac{t_m}{T_a}}\right)\hfill \\ \hfill & ·exp\left\{j\left[-{\displaystyle \frac{4\pi }{\lambda }}{R}_{\Delta }+{\displaystyle \frac{4\pi \gamma }{c^2}}{R}_{\Delta }^2-{\displaystyle \frac{4\pi \gamma }{c}}{R}_{\Delta }\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)\right]\right\}\hfill \end{array}$$

(6)

where $R_{\Delta }=R(t_m;R_0)-R_c$ and $\lambda$ is the wavelength of the carrier frequency. The second term in the exponential is known as the residual video phase (RVP) term, introduced by the dechirp mechanism. This term has no utility for imaging and must be removed in subsequent processing stages.

Under the dechirp receiving scheme, when a squint angle exists, the Doppler centroid of the signal is no longer zero. After removing the RVP (Residual Video Phase) term, the Doppler history derived from the echo signal expression is as follows:

$${\Phi }_a=-{\displaystyle \frac{4\pi }{\lambda }}{R}_{\Delta }$$

(7)

The azimuth Doppler frequency is then:

$$f_a={\displaystyle \frac{1}{2\pi }}·{\displaystyle \frac{d{\Phi }_a}{d{t}_m}}=-{\displaystyle \frac{2}{\lambda }}·{\displaystyle \frac{dR(t_m;R_0)}{d{t}_m}}\approx {\displaystyle \frac{2}{\lambda }}vsin\theta -{\displaystyle \frac{2v^2{cos}^2\theta }{\lambda R_0}}\left(t_m-{\displaystyle \frac{X_p}{v}}\right)$$

(8)

From this, we obtain the Doppler centroid frequency:

$$f_{ac}={\displaystyle \frac{2vsin\theta }{\lambda }}$$

(9)

and the Doppler chirp rate:

$$K_a=-{\displaystyle \frac{2v^2{cos}^2\theta }{\lambda R_0}}$$

(10)

Approximating with the slant range at scene center, the Doppler bandwidth can be derived as follows:

$$B_a=|K_a|T_a={\displaystyle \frac{2v^2{cos}^2\theta }{\lambda R_c}}·{\displaystyle \frac{1}{v}}·L_{sar}={\displaystyle \frac{2v{cos}^2\theta }{\lambda R_c}}·{\displaystyle \frac{{\theta }_{BW}{R}_c}{cos\theta }}={\displaystyle \frac{2v{\theta }_{BW}cos\theta }{\lambda }}$$

(11)

When $\theta =0$, the Doppler centroid frequency becomes zero, and the Doppler bandwidth reduces to ${\displaystyle \frac{2v{\theta }_{BW}}{\lambda }}$, where ${\theta }_{BW}$ is the azimuth beamwidth. To avoid azimuth aliasing, the pulse repetition frequency (PRF) must exceed the Doppler bandwidth. With a carrier frequency of 220 GHz, a squint angle of 60°, and a platform velocity of 60 m/s, the Doppler bandwidth is calculated as 7768.48 Hz for the simulated data (beamwidth of 1°) and 1075.87 Hz for the measured data (beamwidth of 1.4°). In both simulation and practical measurement scenarios, the selected parameters satisfy the anti-aliasing condition.

The influence of different squint angles and carrier frequencies on Doppler information is shown in Figure 2 and Figure 3. The influence of carrier frequency on phase errors of different orders is illustrated in Figure 4.

It can be observed that as the squint angle increases, the Doppler bandwidth continuously decreases, but the Doppler center frequency continuously increases, which can cause azimuthal aliasing issues. In the terahertz band, when the squint angle is the same, the phase error increases with increasing carrier frequency, and the high-order phase error cannot be ignored. The Doppler center frequency increases significantly with the decrease in wavelength, the Doppler bandwidth also increases as the wavelength decreases when the beamwidth is fixed, which poses more challenges to imaging. Moreover, as the Doppler frequency varies with the target’s range and azimuth position, the use of a reference target at the scene center may lead to incomplete phase compensation for targets located near the edges. This results in defocusing in the peripheral regions of the image.

3. Methods

The imaging method consists of two main parts: range processing followed by azimuth processing. As the squint angle increases, the range walk becomes significantly larger than the range curvature, causing severe coupling between the azimuth and range directions. The flowchart of the proposed algorithm is shown in Figure 5.

Compared to the nonlinear frequency scaling method, the method proposed in this paper first corrects the distance walk in the time domain. Then, a frequency scaling operation corrects the range curvature, followed by range compression (RC) and phase compensation to complete range compression. After compensating for the cubic phase term, an improved nonlinear chirp scaling is performed, multiplied with a reference function to complete azimuth compression, and the final two-dimensional image is formed in the distance frequency domain azimuth time domain.

The signal after RVP correction is as follows:

$$\begin{array}{c}\hfill s\left(\tau ,t_m\right)=w_r\left(\tau -{\displaystyle \frac{2R\left(t_m\right)}{c}}\right)w_a\left(t_m\right)·exp\left\{-j\left[{\displaystyle \frac{4\pi }{\lambda }}{R}_{\Delta }+{\displaystyle \frac{4\pi \gamma }{c}}{R}_{\Delta }\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)\right]\right\}\end{array}$$

(12)

The instantaneous slant range expression can be approximated as follows:

$$\begin{array}{cc}\hfill R(t_m;R_0)=& \sqrt{{(v{t}_m-X_p)}^2+R_0^2-2R_0(v{t}_m-X_p)sin\theta }\hfill \\ \hfill \approx & \sqrt{v^2{cos}^2\theta {\left(t_m-{\displaystyle \frac{X_p}{v}}\right)}^2+R_0^2}-vsin\theta \left(t_m-{\displaystyle \frac{X_p}{v}}\right)\hfill \\ \hfill & +{\displaystyle \frac{v^3{cos}^2\theta sin\theta }{2R_0^2}}{\left(t_m-{\displaystyle \frac{X_p}{v}}\right)}^3\hfill \end{array}$$

(13)

Using range walk correction to solve two-dimensional coupling, range walk correction is performed using the time-domain correction function:

$$H_{RWC}(\tau ,t_m)=exp\left\{-j\left[{\displaystyle \frac{4\pi }{\lambda }}+{\displaystyle \frac{4\pi \gamma }{c}}\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)\right]vsin\theta t_m\right\}$$

(14)

After range walk correction, frequency scaling is performed in the 2D frequency-domain for range curvature correction:

$$\begin{array}{cc}\hfill s\left(f_r,f_a\right)=& W_r{W}_a·exp\left(-j2\pi f_a{\displaystyle \frac{X_p}{v}}\right)exp\left(-j{\displaystyle \frac{\pi }{\gamma }}{f}_r^2\right)exp\left(-j{\displaystyle \frac{4\pi R_c}{c}}{f}_r\right)\hfill \\ \hfill & ·exp\left[-j{\displaystyle \frac{4\pi }{c}}{X}_{p}sin\theta \left(f_r+f_c\right)\right]exp\left[-j4\pi R_0\sqrt{{\displaystyle \frac{{\left(f_c+f_r\right)}^2}{c^2}}-{\displaystyle \frac{f_a^2}{4v^2{cos}^2\theta }}}\right]\hfill \end{array}$$

(15)

After Fourier transforming it to the distance time domain, azimuth frequency domain, the echo signal becomes:

$$\begin{array}{cc}\hfill s\left(\tau ,f_a\right)=& w_r{W}_a·\left\{exp\left(-\mathrm{j}{\displaystyle \frac{4\pi }{\lambda }}{X}_{p}sin\theta \right)\right.exp\left(-\mathrm{j}2\pi f_a{\displaystyle \frac{X_p}{v}}\right)exp\left[-\mathrm{j}{\displaystyle \frac{4\pi }{\lambda }}{R}_{0}D\left(f_a\right)\right]\hfill \\ \hfill & ·exp\left\{-\mathrm{j}{\displaystyle \frac{4\pi \gamma }{c}}\left[{\displaystyle \frac{R_0}{D\left(f_a\right)}}-R_c+X_{p}sin\theta \right]\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)\right\}\hfill \\ \hfill & ·exp\left\{j{\displaystyle \frac{2\pi {\gamma }^2{R}_{0}cos\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_{c}D{\left(f_a\right)}^3}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^2\right\}\hfill \\ \hfill & ·exp\left\{-j{\displaystyle \frac{2\pi {\gamma }^3{R}_0{cos}^2\theta sin\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_c^{2}D{\left(f_a\right)}^5}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^3\right\}\hfill \\ \hfill & ·\left.exp\left[j{\displaystyle \frac{\pi {\lambda }^{2}sin\theta R_0{f}_a^3}{4v^3{cos}^4\theta D{\left(f_a\right)}^3}}\right]\right\}\otimes exp\left[-\mathrm{j}\pi \gamma {\tau }^2\right]\hfill \end{array}$$

(16)

where the scaling factor is $D\left(f_a\right)=\sqrt{1-{\displaystyle \frac{{\lambda }^2{f}_a^2}{4v^2{cos}^2\theta }}}$, and ⊗ is the convolution symbol.

Using the frequency scaling method for calibration, in order to prevent excessive bandwidth after scaling, a new scaling function is adopted:

$$H_{FS}\left(\tau ,f_a\right)=exp\left\{j\pi cos\theta \gamma \left[1-D\left(f_a\right)\right]{\tau }^2\right\}$$

(17)

After frequency scaling, the echo signal becomes:

$$\begin{array}{cc}\hfill s\left(\tau ,f_a\right)=& w_r{W}_a·exp\left(-\mathrm{j}{\displaystyle \frac{4\pi }{\lambda }}{X}_{p}sin\theta \right)exp\left(-j2\pi f_a{\displaystyle \frac{X_p}{v}}\right)exp\left[-j{\displaystyle \frac{4\pi R_0}{\lambda }}D\left(f_a\right)\right]\hfill \\ \hfill & ·exp\left[j{\displaystyle \frac{\pi {\lambda }^{2}sin\theta R_0{f}_a^3}{4v^3{cos}^4\theta D{\left(f_a\right)}^3}}\right]\hfill \\ \hfill & ·exp\left\{j{\displaystyle \frac{2\pi {\gamma }^2{R}_{0}cos\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_{c}D{\left(f_a\right)}^3}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^2\right\}\hfill \\ \hfill & ·exp\left\{-j{\displaystyle \frac{2\pi {\gamma }^3{R}_0{cos}^2\theta sin\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_c^{2}D{\left(f_a\right)}^5}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^3\right\}\hfill \end{array}$$

(18)

Range compression is performed in Range-Doppler domain. The range compression function is as follows:

$$\begin{array}{cc}\hfill H_{RC}\left(\tau ,f_a\right)=& exp\left\{-j{\displaystyle \frac{2\pi {\gamma }^2{R}_{0}cos\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_{c}D{\left(f_a\right)}^3}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^2\right\}\hfill \\ \hfill & ·exp\left\{j{\displaystyle \frac{2\pi {\gamma }^3{R}_0{cos}^2\theta sin\theta \left[1-D{\left(f_a\right)}^2\right]}{c{f}_c^{2}D{\left(f_a\right)}^5}}{\left(\tau -{\displaystyle \frac{2R_c}{c}}\right)}^3\right\}\hfill \end{array}$$

(19)

After range compression, the signal becomes as follows:

$$\begin{array}{cc}\hfill s\left(\tau ,f_a\right)=& w_r{W}_a·exp\left(-j2\pi f_a{\displaystyle \frac{X_p}{v}}\right)exp\left[-j{\displaystyle \frac{4\pi R_0}{\lambda }}D\left(f_a\right)\right]exp\left[j{\displaystyle \frac{\pi {\lambda }^{2}sin\theta R_0{f}_a^3}{4v^3{cos}^4\theta D{\left(f_a\right)}^3}}\right]\hfill \end{array}$$

(20)

Residual phase compensation is performed using the following:

$$H_{RPC}\left(\tau ,f_a\right)=exp\left[-j{\displaystyle \frac{\pi {\lambda }^{2}sin\theta R_0{f}_a^3}{4v^3{cos}^4\theta D{\left(f_a\right)}^3}}\right]$$

(21)

After phase compensation, the compressed signal is:

$$\begin{array}{c}\hfill s\left(\tau ,f_a\right)=w_r{W}_a·exp\left(-j2\pi f_a{\displaystyle \frac{X_p}{v}}\right)exp\left[-j{\displaystyle \frac{4\pi R_0}{\lambda }}D\left(f_a\right)\right]\end{array}$$

(22)

After completing range compression processing, azimuth processing must be performed. The azimuth Doppler frequency exhibits spatial variability and couples with the range direction. While time-domain walk correction partially decouples these effects (sufficient for low-precision applications), the terahertz band’s higher resolution demands more precise compensation of higher-order phase terms. Particularly with increasing squint angles, these higher-order effects become non-negligible. We perform Taylor expansion on $D\left(f_a\right)$ at $f_a=0$ and approximate it to a cubic term. The echo signal becomes as follows:

$$\begin{array}{cc}\hfill s\left(f_r,f_a\right)=& W_r{W}_a·exp\left(-\mathrm{j}{\displaystyle \frac{4\pi }{\lambda }}{X}_{p}sin\theta \right)exp\left(-j2\pi f_a{\displaystyle \frac{X_p}{v}}\right)\hfill \\ \hfill & ·exp\left[-j{\displaystyle \frac{4\pi R_0}{\lambda }}+j{\displaystyle \frac{\pi \left(R_0-X_{p}sin\theta \right)\lambda }{2v^2{cos}^2\theta }}{f}_a^2+{\displaystyle \frac{\pi {\lambda }^2\left(R_0-X_{p}sin\theta \right)sin\theta }{4v^3{cos}^4\theta }}{f}_a^3\right]\hfill \end{array}$$

(23)

First, we filter higher-order terms using the following:

$$\begin{array}{cc}\hfill H_{HOF}\left(f_r,f_a\right)=& exp\left[j\pi {\displaystyle \frac{\left(2{sin}^2\theta -cos\theta \right)sin\theta {\lambda }^2{R}_0}{12\left(sin{\theta }^2-cos\theta \right)v^3{cos}^4\theta }}{f}_a^3\right]\hfill \end{array}$$

(24)

After high-order phase filtering, the echo signal becomes as follows:

$$\begin{array}{cc}\hfill s\left(f_r,t_m\right)=& W_r{w}_a·exp\left[-\mathrm{j}{\displaystyle \frac{2\pi v^2{cos}^2\theta }{\lambda \left(R_0-X_{p}sin\theta \right)}}{\left(t_m-{\displaystyle \frac{X_p}{v}}\right)}^2\right]\hfill \\ \hfill & ·exp\left[-\mathrm{j}{\displaystyle \frac{2\pi R_0{v}^3{cos}^2\theta sin\theta \left(2{sin}^2\theta -cos\theta \right)}{3\lambda {\left(R_0-X_{p}sin\theta \right)}^3\left({sin}^2\theta -cos\theta \right)}}{\left(t_m-{\displaystyle \frac{X_p}{v}}\right)}^3\right]\hfill \end{array}$$

(25)

The nonlinear chirp scaling operation in the range-frequency/azimuth-time domain gives the following:

$$\begin{array}{cc}\hfill H_{NLCS}\left(f_r,t_m\right)=& W_r{W}_a·exp\left[j\pi {\displaystyle \frac{2v^2\left(cos\theta -{sin}^2\theta \right)cos\theta }{\lambda R_0}}{t}_m^2\right]\hfill \\ \hfill & ·exp\left[j\pi {\displaystyle \frac{2v^{3}cos\theta sin\theta \left({sin}^2\theta -cos\theta \right)}{3\lambda R_0^2}}{t}_m^3\right]\hfill \end{array}$$

(26)

After nonlinear frequency scaling, the echo signal becomes as follows:

$$\begin{array}{cc}\hfill s\left(f_r,f_a\right)=& W_r{W}_a·exp\left[-\mathrm{j}{\displaystyle \frac{2\pi X_{p}cos\theta }{v{sin}^2\theta }}{f}_a\right]\hfill \\ \hfill & ·exp\left[\mathrm{j}{\displaystyle \frac{\pi \lambda R_0}{2v^{2}cos\theta {sin}^2\theta }}{f}_a^2+\mathrm{j}{\displaystyle \frac{\pi {\lambda }^2{R}_0}{12v^3{cos}^2\theta sin\theta \left({sin}^2\theta -cos\theta \right)}}{f}_a^3\right]\hfill \\ \hfill & ·exp\left[-\mathrm{j}{\displaystyle \frac{2\pi {cos}^2\theta \left({sin}^2\theta -cos\theta \right)}{\lambda R_0{sin}^2\theta }}{X}_p^2+\mathrm{j}{\displaystyle \frac{4\pi {cos}^2\theta \left({sin}^2\theta -cos\theta \right)}{3\lambda R_0^{2}sin\theta }}{X}_p^3\right]\hfill \end{array}$$

(27)

After nonlinear frequency scaling operation, further azimuth compression processing can be performed:

$$H_{AC}\left(f_r,f_a\right)=exp\left(-j{\displaystyle \frac{\pi \lambda R_0}{2v^2{sin}^2\theta cos\theta }}{f}_a^2\right)exp\left[j{\displaystyle \frac{\pi {\lambda }^2{R}_0}{12sin\theta \left({sin}^2\theta -cos\theta \right)v^3{cos}^2\theta }}{f}_a^3\right]$$

(28)

Finally, ignoring high-order terms, the compressed signal is:

$$s\left(f_r,t_m\right)=\sin \mathrm{c}\left\{B_r\left[f_r+{\displaystyle \frac{2\gamma }{c}}\left(R_0+X_{p}sin\theta -R_c\right)\right]\right\}·\sin \mathrm{c}\left[B_a\left(t_m-{\displaystyle \frac{X_p}{cos\theta v}}\right)\right]$$

(29)

where $B_r$ is the bandwidth of the transmitted signal. After geometric correction, the final imaging result can be obtained.

4. Simulations and Experimental Results

4.1. Simulations

The point target simulation is carried out in this section using the parameters given in

Table 1. Simulation parameters.

Description	Value
Carrier frequency	220 GHz
Bandwidth	5 GHz
Pulse width	26 µs
Pulse repetition frequency	10 kHz
Reference distance	3 km
Height	1 km
Radar velocity	60 m/s
Squint angle	60°

. The layout of the targets is shown in Figure 6. A uniform grid of 49 point targets is distributed across the scene, arranged in 7 rows and 7 columns, covering a 20 m × 20 m area with an inter-target spacing of 3.33 m. The imaging results after geometric correction of the NFSA and the proposed algorithm are shown in Figure 7.

To better compare the imaging performance of different algorithms, three representative points were magnified for detailed analysis. We select the points in the center, upper right corner, and lower right corner of the scene for analysis, which are points A, B, and C, respectively. The imaging results and the corresponding 2D profiles of different algorithms are presented in Figure 8 and Figure 9, respectively.

Observing the imaging results of multi-point targets in Figure 8 and Figure 9, it can be seen that at a 60° squint angle, the NFSA exhibits defocusing issues. This occurs because the terahertz band demands a higher phase compensation accuracy, while NFSA fails to adequately account for the two-dimensional coupling induced by the squint angle. The algorithm neglects the space variation of the compression kernel and exceeds the effective depth of focus, resulting in moderate range defocusing and severe azimuth defocusing, ultimately degrading image quality. Particularly for point targets B and C at the scene edges, the effects of chirp rate mismatch and higher-order phase terms become non-negligible, causing significant 2D defocusing.

In contrast, the proposed algorithm addresses the range–azimuth coupling through decoupling processing. Although it partially ignores the space-variance of the compression kernel and higher-order residual phase terms during subsequent operations, these simplifications do not substantially affect the final results. The azimuth mainlobe shows no noticeable broadening, and both range and azimuth impulse responses maintain ideal profiles. All point targets in the scene achieve well-focused signatures, yielding high-quality final imagery. As evidenced by the two-dimensional profiles in Figure 8 and Figure 9 and the data in

Table 2. Performance metrics comparison of simulated imaging results.

		Imaging Result of NFSA (A/B/C)	Imaging Result of the Proposed Algorithm (A/B/C)
IRW (cm)	Range	6.48/6.72/7.21	6.03/6.19/6.23
	Azimuth	6.63/7.91/13.15	6.34/6.72/6.89
PSLR (dB)	Range	−12.03/−11.14/−10.65	−13.56/−13.93/−13.63
	Azimuth	−11.56/−10.98/−10.78	−13.87/−13.06/−13.42
ISLR (dB)	Range	−10.35/−9.13/−9.01	−11.56/−11.04/−11.32
	Azimuth	−9.62/−8.03/−6.47	−11.23/−10.92/−10.39

, when the squint angle is 60°, the proposed algorithm achieves a reduction in range PSLR and ISLR compared to NFSA, along with an improvement in range resolution. Moreover, the azimuth PSLR and ISLR is lower, and the azimuth resolution is enhanced, approaching the theoretical value. These results demonstrate that the proposed algorithm offers superior focusing performance.

4.2. Experimental Results

To validate the effectiveness of the proposed algorithm, a terahertz SAR imaging experiment was performed under high-squint conditions using an airborne platform, and different algorithms were used to process the airborne measured data. The optical satellite image of the actual imaging scenario is presented in Figure 10. The radar was mounted under a small aircraft from Shaanxi Tianying Aviation Company, and the system parameters of the THz radar used for experimental testing are detailed in

Table 3. Terahertz imaging system parameters.

Description	Value
Carrier frequency	220 GHz
Bandwidth	900 MHz
Pulse repetition frequency	15 kHz
Reference distance	3 km
Radar velocity	60 m/s
Squint angle	60°
Azimuth beamwidth	1.4°
Flying height	1 km

. The radar platform passes along the light blue trajectory at a velocity of 60 m/s, with the imaging area demarcated by the red rectangular frame. Key radar parameters include carrier frequency of 220 GHz, bandwidth of 900 MHz, pulse repetition frequency (PRF) of 15 kHz, azimuth beamwidth of 1.4 degrees, and squint angle of 60 degrees, with the scene center located at a slant range of 3 km and the platform operating at an altitude of 1 km. The data were acquired using strip-map SAR mode, with the observation target being a town on the ground. According to the system parameters, the theoretical range and azimuth resolutions are 0.34 and 0.35 m, respectively.

For partial scene imaging, the corresponding optical satellite image is presented in Figure 11a. RDA, NFSA, and the proposed algorithm are applied to process this segment of echo data, and the imaging results are compared in Figure 11.

Figure 11 compares the imaging results of the RDA, NFSA, and the proposed algorithm. It can be observed that the RDA algorithm exhibits noticeable defocusing in the image due to inaccurate phase compensation. The NFSA algorithm has improved the imaging quality, but due to the lack of compensation for high-order phases, there is severe azimuthal defocusing. The algorithm proposed in this article has achieved precise phase compensation, resulting in the better preservation of the target contour and detailed features in the imaging results. The image quality metrics in

Table 4. Image quality metrics comparison of real-data imaging results.

	RDA	NFSA	The Proposed Algorithm
Image entropy	5.21	4.93	4.58
Image contrast	5.43	5.52	5.91

further confirm these observations: the proposed algorithm demonstrates the lowest image entropy (4.58), indicating a higher energy concentration, and the highest contrast (5.91), reflecting better sharpness. We select two strong scattering points (A and B) in the imaging scene for profile analysis, as shown in Figure 12 and

Table 5. Performance metrics comparison of real-data imaging results.

		Imaging Result of RDA (A/B)	Imaging Result of NFSA (A/B)	Imaging Result of the Proposed Algorithm (A/B)
IRW (m)	Range	0.42/0.41	0.37/0.36	0.36/0.35
	Azimuth	1.41/1.12	0.42/0.43	0.41/0.42
PSLR (dB)	Range	−7.65/−8.23	−9.90/−10.58	−11.22/−11.90
	Azimuth	−3.40/−3.23	−6.91/−5.79	−9.47/−9.99
ISLR (dB)	Range	−8.54/−8.93	−9.22/−9.36	−9.13/−9.64
	Azimuth	−3.81/−3.47	−6.23/−6.92	−7.29/−7.13

. By observing the two-dimensional profile, it can be seen that compared to the other two algorithms, the proposed algorithm achieves a lower azimuth peak sidelobe ratio and narrower mainlobe width for the imaging results of prominent points in the scene. The airborne measured data experiments verify that the proposed algorithm improves the focusing quality in terahertz squint SAR imaging.

5. Discussion

While the superior focusing quality validates the imaging accuracy of the proposed method, its practical value, especially for real-time processing on resource-constrained platforms, hinges on computational efficiency. Therefore, a detailed analysis of the computational complexity is presented below.

The original SAR echo data has a size of $N_a\times N_r$, where $N_a$ and $N_r$ represent the number of samples in the azimuth and range directions, respectively. As illustrated in Figure 5, the proposed algorithm involves only FFTs and complex multiplications. Specifically, the entire procedure includes seven complex multiplications, four azimuth FFTs/IFFTs, and one range FFT. According to standard computational principles, one complex multiplication requires six FLOPs, while a 1D FFT (or IFFT) of length N requires approximately $5N{log}_{2}N$ FLOPs. Thus, for a dataset of size $N_a\times N_r$, the total complexity of the complex multiplication steps is $O\left(6N_a{N}_r\right)$, and the total complexity of FFT is $O\left(5N_a{N}_r{log}_2\left(N_{a/r}\right)\right)$ when considering both dimensions. The overall computational complexity of the proposed method can be summarized by Equation (30).

$$O\left(7\times 6N_a{N}_r+4\times 5N_a{N}_r{log}_2{N}_a+5N_a{N}_r{log}_2{N}_r\right)$$

(30)

When both Na and Nr are large, the logarithmic term dominates, leading to the simplified expression:

$$O\left(N_a{N}_r\left({log}_2{N}_a+{log}_2{N}_r\right)\right)=O\left(N_a{N}_r{log}_2\left(N_a{N}_r\right)\right)$$

(31)

In contrast, the BP algorithm consists of one range FFT and numerous complex multiplications per azimuth sample. Assuming that the range interpolation and phase application require four FLOPs per pixel per azimuth position, its complexity is:

$$O\left(4\times 6N_a^2{N}_r+5N_a{N}_r{log}_2{N}_r\right)$$

(32)

When both Na and Nr are large, this leads to the simplified expression:

$$O\left(N_a{N}_r\left(N_a+{log}_2{N}_r\right)\right)=O\left(N_a^2{N}_r\right)$$

(33)

Assuming $N_a=4096$ and $N_r=8192$, the total FLOPs required by the proposed algorithm are approximately $1.16\times {10}^{10}$, while the BP algorithm requires roughly $3.30\times {10}^{12}$. Despite incorporating various high-order compensation functions, the proposed method achieves an estimated 283-fold speed up compared to the time-domain BP algorithm. It should be noted that while constant factors may vary in practical implementations depending on hardware and optimization, the order-of-magnitude advantage remains unchanged.

The efficiency of the proposed algorithm stems mainly from its frequency-domain processing (using FFTs), which yields a complexity of $O\left(N_a{N}_r{log}_2\left(N_a{N}_r\right)\right)$ compared to the quadratic complexity $O\left(N_a^2{N}_r\right)$ of the BP algorithm. Although BP is highly accurate, its high computational cost makes the proposed method more suitable for real-time processing and large-scale data scenarios.

Regarding the processing of terahertz real-data, the azimuth data volume is exceptionally large. Utilizing the full aperture would yield very high azimuth resolution, which is often unnecessary; instead, a balanced resolution between azimuth and range is generally preferred. Therefore, a complete synthetic aperture is typically not formed, and the raw data are processed immediately after acquisition. However, it is essential to determine the minimum synthetic aperture length in advance to achieve the desired imaging resolution.

The azimuth resolution ${\rho }_a$ can be expressed in terms of the Doppler bandwidth $B_a$ and the platform velocity as ${\rho }_a={\displaystyle \frac{v}{B_a}}\approx {\displaystyle \frac{\lambda R_c}{2vcos\theta t_a}}$, where the Doppler bandwidth is given by $B_a={\displaystyle \frac{2vsin\left(\theta +{\theta }_a/2\right)}{\lambda }}-{\displaystyle \frac{2vsin\left(\theta -{\theta }_a/2\right)}{\lambda }}={\displaystyle \frac{4vcos\theta sin\left({\theta }_a/2\right)}{\lambda }}\approx {\displaystyle \frac{2vcos\theta }{\lambda }}{\theta }_a$ Here, ${\theta }_a$ denotes the synthetic aperture angle. Substituting the real-data parameters, the theoretical range resolution is approximately 0.33 m, and the azimuth resolution is also 0.33 m, ensuring a well-matched resolution in both dimensions.

In video SAR (ViSAR), the frame update strategy is often categorized as either true-frame (when the data overlap ratio $\alpha =0\%$) or pseudo-frame (when $\alpha >0\%$), with the frame rate determined by:

$$\mathrm{Frame}\mathrm{rate}={\displaystyle \frac{1}{1-\alpha }}{\displaystyle \frac{2vcos\theta }{\lambda R_c}}{\rho }_a$$

(34)

Under the pseudo-frame condition ($\alpha =80\%$), using a carrier frequency of 35 GHz for traditional Ka-band and 220 GHz for the terahertz band, the resulting frame rates are approximately 2.33 Hz for Ka-band and 14.67 Hz for THz, the latter being close to real-time video rates. Even in the true-frame case ($\alpha =0\%$), the terahertz system achieves a frame rate of about 2.93 Hz under the same experimental parameters, whereas the Ka-band only reaches 0.46 Hz. These results clearly demonstrate the significant advantage of the terahertz band in high-squint video SAR imaging, particularly in achieving high frame rates under matching resolution conditions.

6. Conclusions

This paper proposes a terahertz squint SAR imaging algorithm based on decoupled frequency scaling. Following time-domain decoupling, the proposed algorithm combined range frequency scaling with nonlinear azimuthal chirp scaling, which can achieve better imaging results. The proposed algorithm was validated through multipoint target simulations and measured data processing. It achieved better imaging results, as indicated by improvements in IRW and PSLR, and it outperformed other algorithms in high-resolution imaging of airborne terahertz squint SAR data. Furthermore, analyses of computational efficiency and imaging frame rate verified its operational advantages. These results collectively confirm the method’s value for applications such as airborne terahertz detection and UAV formation cooperative observation.