A Nonlinear Compensation Method for Enhancing the Detection Accuracy of Weak Targets in FMCW Radar

Abstract

To achieve precise detection of target geometric features, Ka/W/sub-terahertz band imaging radar systems with ultra-wide instantaneous bandwidth have been developed. Although dechirp-based receiver architectures allow for low-sampling-rate signal acquisition, they require precise linearity in chirp signals, often necessitating precompensation for nonlinear errors. While most research addresses polynomial-based error correction, periodic errors remain underexplored, despite their potential to obscure weak targets and introduce spurious ones. This paper proposes a novel software-based correction method that integrates neural networks and joint optimization strategies to correct periodic phase errors. The method first employs neural networks for frequency estimation, followed by phase-matching techniques to extract amplitude and phase data. Parameter estimation is refined using the Adaptive Moment Estimation (ADAM) algorithm and Limited-Memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) optimization. Nonlinear errors are corrected via matched Fourier transforms. Simulations and experiments demonstrate that the proposed method effectively suppresses spurious targets and enhances the detection of weak targets, demonstrating strong robustness and practical applicability, thereby significantly enhancing the target detection performance of the ultra-wideband radar system.

1. Introduction

The frequency-modulated continuous wave (FMCW) radar offers several advantages over traditional pulse radar systems, such as lightweight design, structural simplicity, low cost, and low probability of interception [1]. As a result, FMCW radar has found widespread applications in fields such as short-range security inspection, autonomous driving, deformation monitoring, and vital sign detection [2,3,4,5]. To achieve finer target range resolution, radar systems require ultra-wide instantaneous bandwidth. However, such ultra-wide bandwidth imposes stringent demands on signal acquisition systems, making analog demodulation (dechirp) a crucial solution for processing ultra-wideband signals. However, dechirp technology requires exceptionally high linearity in the chirp modulation of transmitted and received signals, especially when the signal bandwidth is large. Failure to meet these linearity requirements can result in significant degradation of range resolution and even prevent accurate imaging [6].

Unfortunately, existing transmit–receive chains inevitably introduce some degree of nonlinearity into the signal. For instance, linear frequency modulation (LFM) signals generated using digital direct synthesis (DDS) and phase-locked loops inherently exhibit certain nonlinear characteristics. Furthermore, nonlinear components such as frequency multipliers, mixers, and filters in the signal transmission path exacerbate the degradation of chirp linearity. Consequently, in ultra-wideband application scenarios requiring high-quality range resolution, nonlinear correction techniques are indispensable. These techniques play a pivotal role in determining the imaging quality of range profiles and are critical for ensuring the performance of FMCW radar systems [7,8].

Current nonlinear correction techniques can be broadly categorized into hardware-based and software-based approaches. Hardware methods aim to minimize chirp nonlinearity through real-time preprocessing by adjusting the signal generation circuitry. Common hardware techniques include predistortion voltage compensation, delay phase detection, delay phase-locked loops, and various predistortion methods addressing VCO modulation voltage distortion and DDS digital distortion [9,10,11]. Although hardware methods can output high-linearity chirp signals in real time, their parameter designs are often susceptible to environmental factors such as temperature, resulting in insufficient stability.

In contrast, software correction methods have garnered significant attention due to their strong versatility, portability, and high correction accuracy. Unlike the real-time processing of hardware approaches, software methods employ digital signal processing techniques to estimate and correct nonlinear factors in the sampled intermediate-frequency (IF) signals.

Existing software correction methods primarily focus on estimating and compensating for the nonlinearities in the transmission chain. These methods typically model nonlinear phase errors as one of three types: polynomial phase, periodic phase, and stochastic phase. In practical radar systems, all three types of phase errors coexist, albeit with varying proportions depending on the system. Most current research focuses on the estimation and correction of polynomial phase errors. For example, Anghel et al. proposed the use of the high-order ambiguity function (HAF) to estimate polynomial coefficients and correct space-variant nonlinear errors through time-domain interpolation and resampling, making it suitable for engineering applications [12]. Wang et al. employed the wavelet synchrosqueezed transform to derive the time-frequency relationship of the reference chirp signal for fitting nonlinear parameters. The residual video phase (RVP) method was then applied to compensate for nonlinearities in chirp signals. However, the WSST-based approach has strict limitations, such as requiring that the signal frequency must vary slowly [13]. Similarly, Rongyao Zheng et al. combined the complete ensemble empirical mode decomposition with HAF to estimate polynomial phase errors and corrected them via time-domain resampling. These methods successfully eliminate polynomial nonlinear errors [14].

Nevertheless, none of the aforementioned methods address the correction of periodic phase errors, and studies focusing on high-frequency periodic nonlinearities remain scarce. Ayhan et al. suggested that periodic nonlinearities may arise from various sources, such as DDS spurs or clock spurs, but no correction solution was proposed [15]. Zhao et al. extracted phase information directly from the sampled signal and used a three-component sinusoidal model for fitting, which could partially mitigate the effects of periodic nonlinearities. However, this approach often fails to meet the requirements for direct sampling under ultra-wideband conditions [16].

In summary, there is currently no effective method for periodic error correction. Modeling periodic errors as multi-sinusoidal models for parameter estimation poses a series of challenging problems. First, the parameter estimation of general multi-sinusoidal signals is typically based on frequency estimation, which is often realized through the Fourier transform of discrete-time samples. However, accurate frequency estimation requires a large number of samples, imposing strict demands on practical applications [17]. Although methods based on eigenvalue decomposition, such as Multiple Signal Classification (MUSIC), Prony’s method, and Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) [18,19,20], can achieve high-frequency resolution under limited data length and sampling rates, their performance degrades significantly in low signal-to-noise ratio (SNR) scenarios [21,22]. Second, periodogram-based methods and eigenvalue decomposition techniques require prior knowledge of the number of frequencies to be estimated. However, this information is typically unavailable in practical scenarios, necessitating the use of additional criteria such as the Akaike Information Criterion (AIC) [23] or the Minimum Description Length (MDL) [24] to determine the number of sinusoidal components in the signal. Finally, after obtaining the frequencies, a concise and efficient method is still needed to estimate the corresponding amplitude and phase parameters.

Building on recent advancements in frontier technologies such as deep learning [25,26], this paper employs a neural network-based approach to address the aforementioned challenges. The proposed method trains a neural network using simulated data. Experimental results demonstrate that the proposed neural network significantly outperforms existing frequency estimation methods, particularly under medium-to-high noise conditions, precisely estimating frequencies from data contaminated with noise of unknown variance. Furthermore, the paper adopts a neural network model to estimate the number of frequency components in the signal. Based on this foundation, the neural network model is first applied for the preliminary estimation of the frequencies associated with periodic sinusoidal errors. A phase-matching method is then proposed to extract amplitude and phase information, thereby determining the initial parameters of the periodic sinusoidal errors.

This paper proposes a joint estimation algorithm that combines Adaptive Moment Estimation (ADAM) [27,28] and the Limited-Memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm to achieve more accurate estimation of periodic errors [29]. In the process of periodic error estimation, the ADAM algorithm leverages its adaptive learning rate adjustment capability to provide a rapidly convergent initial solution, while the LBFGS algorithm refines the parameters through precise second-order approximations, thereby improving the accuracy of the final estimation results. This joint estimation algorithm not only performs exceptionally well in handling periodic errors but also holds significant potential for application in other complex nonlinear optimization problems.

The remainder of this paper is organized as follows. Section 2 provides a detailed introduction to the IF error model and the analysis of periodic phase distortion. Section 3 presents the estimation and correction process for periodic phase errors. Section 4 validates the effectiveness and superiority of the proposed method using real-world data. Finally, Section 5 concludes the paper with a summary of the findings.

2. Modeling and Analysis of Periodic Phase Errors

2.1. Periodic Phase Error Model

The ideal FMCW signal model is expressed as:

$$s_{\mathrm{t}}\left(t\right)=exp\left\{\mathrm{j}\left[2\pi \left(f_{c}t+{\displaystyle \frac{1}{2}}{K}_r{t}^2\right)\right]\right\},$$

(1)

where $f_c$ is the carrier frequency and $K_r$ is the chirp rate, defined as the ratio of bandwidth to the modulation period. The received signal is represented as the transmitted signal delayed by a time $\tau$ and is expressed as:

$$s_{\mathrm{r}}\left(t\right)=s_{\mathrm{t}}(t-\tau )=exp\left\{j\left[2\pi \left(f_c(t-\tau )+{\displaystyle \frac{1}{2}}{K}_r{(t-\tau )}^2\right)\right]\right\},$$

(2)

where $\tau ={\displaystyle \frac{2R}{c}}$ represents the signal delay, R denotes the target distance, and c is the speed of light.

After performing mixing (dechirp) on the received signal and the transmitted signal, the resulting ideal intermediate frequency (IF) signal can be expressed as:

$$s_{if}\left(t\right)=s_{\mathrm{t}}\left(t\right)·s_{\mathrm{r}}^{*}\left(t\right)=exp\left\{j2\pi \left(f_c\tau +K_r\tau t-{\displaystyle \frac{1}{2}}{K}_r{\tau }^2\right)\right\}$$

(3)

where $K_r\tau t$ represents the beat frequency of the target and $-{\displaystyle \frac{1}{2}}{K}_r{\tau }^2$ denotes the Residual RVP [30]. In near-field imaging, the time delay is minimal, and thus, the RVP term is ignored in subsequent processing. In this paper, the phase error of the beat signal is further modeled as the sum of a polynomial phase set and a sinusoidal function set:

$$\begin{array}{c}\epsilon \left(t\right)={\displaystyle \sum _{l=2}^L}{p}_{l}t\tau +{\displaystyle \sum _{k=1}^K}{A}_k\tau sin\left({\omega }_{k}t+{\varphi }_k\right)\\ ={\displaystyle \sum _{l=2}^L}{p}_{l}t\tau +{\displaystyle \sum _{k=1}^K}{a}_{k}sin\left({\omega }_{k}t+{\varphi }_k\right)\end{array}$$

(4)

where L denotes the order of the polynomial error, p represents the polynomial coefficients, and M indicates the number of sinusoidal signals in the periodic phase error. Additionally, $a_k$, ${\omega }_k$, and ${\varphi }_k$ correspond to the amplitude, angular frequency, and phase of the periodic phase error, respectively. Based on the nonlinear phase term analysis process in [12,13], the IF signal expression containing the nonlinear phase term is as follows:

$$\begin{array}{c}{s}_{if}\left(t\right)=exp\left\{\mathrm{j}2\pi \left[f_c{\tau }_{ref}+K_{r}t{\tau }_{ref}+{\displaystyle \sum _{l=2}^L}{p}_{l}t{\tau }_{rf}+{\displaystyle \sum _{k=1}^K}{A}_k{\tau }_{ref}sin\left({\omega }_{k}t+{\varphi }_k\right)\right]\right\}\end{array}$$

(5)

where ${\tau }_{ref}$ represents the time delay of the reference target. After removing the polynomial phase error using methods such as those in [12,13,14], the beat signal becomes as follows:

$$s_{if}\left(t\right)=exp\left\{\mathrm{j}2\pi \left[f_c{\tau }_{ref}+K_{r}t{\tau }_{ref}+\sum _{k=1}^K{a}_{k}sin\left({\omega }_{k}t+{\varphi }_k\right)\right]\right\}$$

(6)

The impact of periodic phase errors on the range profile will be discussed in detail below.

2.2. Analysis of Periodic Phase Distortion

The frequency modulation linearity of an LFM signal is typically defined as:

$$\eta \stackrel{\Delta }{=}{\displaystyle \frac{{\left|f_{\mathrm{e}}\left(t\right)\right|}_{max}}{B}}$$

(7)

where $\eta$ represents the frequency modulation linearity of the LFM signal, B is the modulation bandwidth of the LFM signal, and $f_e\left(t\right)$ denotes the beat frequency. According to [31], the analysis of periodic phase errors can be carried out as follows: for frequency offsets, a Fourier analysis approach can be adopted to decompose the offset into an infinite sum of sinusoidal harmonics. To simplify the analysis, a single-frequency deviation is introduced.

$$f_{\mathrm{e}}\left(t\right)=\eta Bcos\left(2\pi f_{\mathrm{m}}t\right)$$

(8)

where $f_m$ represents the frequency of the nonlinear single-frequency deviation. Thus, the frequency domain function of the LFM signal containing periodic phase errors is expressed as:

$$f\left(t\right)=f_c+K_{r}t+\eta Bcos\left(2\pi f_{\mathrm{m}}t\right)$$

(9)

where $f_c$ represents the carrier frequency, $K_r$ denotes the chirp rate, $K_r=B/T$, and T is the pulse width. The LFM phase signal under periodic frequency deviation distortion can be expressed as:

$$\begin{array}{c}\varphi \left(t\right)=2\pi {\int }_0^{t}f\left(t\right)\mathrm{d}t\\ =2\pi {\int }_0^t\left[f_c+K_{r}t+\eta Bcos\left(2\pi f_{\mathrm{m}}t\right)\right]\mathrm{d}t\\ =2\pi \left(f_{c}t+{\displaystyle \frac{1}{2}}{K}_r{t}^2\right)+{\displaystyle \frac{\eta B}{f_{\mathrm{m}}}}sin\left(2\pi f_{\mathrm{m}}t\right)\end{array}$$

(10)

Let $\beta ={\displaystyle \frac{\eta B}{f_{\mathrm{m}}}}$, then the LFM signal with periodic phase errors can be expressed as:

$$s_{if}\left(t\right)=exp\left\{\mathrm{j}2\pi \left(f_{c}t+{\displaystyle \frac{1}{2}}{K}_r{t}^2\right)\right\}·exp\left\{\mathrm{j}\beta sin\left(2\pi {\mathrm{f}}_{\mathrm{m}}\mathrm{t}\right)\right\}$$

(11)

Assuming the output signal of an ideal FMCW signal after passing through a matched filter is $s_o\left(t\right)$, the output signal in the presence of single-frequency periodic distortion after matched filtering is expressed as:

$$\begin{array}{c}{S}_{em}\left(t\right)=J_0\left(\beta \right)E_{nv}\left[s_o\left(t\right)\right]e^{j2\pi f_{c}t}\hfill \\ +{\displaystyle \sum _{n=1}^{\infty }}{J}_n\left(\beta \right)\left\{\begin{array}{c}{E}_{nv}[s_o(t+{\displaystyle \frac{n{f}_m}{K_r}})]e^{j2\pi (f_c+{\displaystyle \frac{n{f}_m}{2}})t}\hfill \\ +{(-1)}^n{E}_{nv}[s_o(t-{\displaystyle \frac{n{f}_m}{K_r}})]e^{j2\pi (f_c-{\displaystyle \frac{n{f}_m}{2}})t}\hfill \end{array}\right\}\hfill \end{array}$$

(12)

where $J_n\left(\beta \right)$ represents the n-th order Bessel function of the first kind and $E_{nv}(•)$ denotes the envelope removal function.

From the above equation, it can be seen that when periodic phase errors are present, the output signal after pulse compression primarily consists of the main signal with an amplitude of $J_0\left(\beta \right)$ and paired echoes with an amplitude of $J_n\left(\beta \right)$. The time interval and frequency interval between the paired echoes and the main signal are $\pm n{f}_m/K_r$ and $\pm n{f}_m/2$, respectively. Based on the theoretical derivation above, the following conclusions can be drawn:

1.

The impact of periodic phase errors on radar performance is primarily caused by paired echoes;

2.

Periodic phase errors affect the detection of weak targets near the main lobe and targets in the far range;

3.

Periodic phase errors do not affect the width or position of the main lobe.

3. Estimation and Correction of Periodic Phase Errors

Based on the analysis in the previous section, periodic errors can generate paired echoes near the target, severely degrading the detection performance for nearby weak targets and far-range targets. To address this issue, periodic nonlinear correction is essential. Since nonlinearity is an intrinsic modulation characteristic of the system and independent of the target, this section first employs ideal single targets, such as delay lines or corner reflectors, to accurately estimate the system’s nonlinearity. The estimated parameters are then used to construct a matched Fourier transform, enabling space-variant nonlinear correction for multiple targets.

The phase of the beat echo signal is first obtained by phase unwrapping as follows:

$$y\left(t\right)=\zeta +\sum _{k=1}^K{a}_{k}sin\left({\omega }_{k}t+{\varphi }_k\right)$$

(13)

In the equation, it should be noted that $a_k=2\pi A_k{\tau }_{ref}$; $\zeta =2\pi f_c{\tau }_{ref}$ and $\zeta$ are unknowns, but $\zeta$ can be treated as a constant, which does not affect the subsequent estimation of sinusoidal parameters. Prior to fitting, a DC removal process can be applied to eliminate $\zeta$. The complete process of periodic error correction is illustrated in Figure 1. First, the phase information of the beat signal is extracted using a corner reflector or delay line. Next, initial estimates of the periodic error are obtained using a deep learning model combined with a phase-matching method. Subsequently, precise optimization is performed by integrating the ADAM and LBFGS algorithms. Finally, the phase error is corrected using Fourier matching techniques based on the estimated parameters, yielding the fully corrected beat signal.

3.1. Initial Estimation of Periodic Errors Based on a Neural Network Model

3.1.1. Frequency Estimation Model

In this paper, a model suitable for estimating periodic phase errors was trained using simulated data. As shown in Figure 2, the frequency estimation module is designed to transform the input signal into a frequency representation, enabling the extraction of frequency features from the signal. This module consists of an input layer, multiple convolutional layers, and an output layer, with ideal frequency representations generated using a narrow Gaussian kernel.

Input Layer: The input signal undergoes an initial transformation through a linear layer. This transformation projects the signal into a higher-dimensional internal representation space, enabling the model to extract a richer set of features for subsequent operations. The primary purpose of this transformation is to enhance feature representation, facilitating more accurate feature extraction during subsequent convolution operations. The use of a fully connected layer in this step allows the network to capture complex relationships within the input signal, ensuring that the initial features are adequately extracted for further processing in the network.

Convolutional Layers: The output from the input layer is processed through multiple one-dimensional convolutional layers. The design of the convolutional layers is inspired by the success of convolutional neural networks in image processing, with the primary advantage of effectively capturing local features within the signal. Each convolutional layer employs multiple filters, followed by a batch normalization layer and a ReLU activation function after the convolution operation. The convolutional kernels slide over the signal, generating feature maps that capture frequency information at various scales. This process enables the convolutional layers to extract local frequency characteristics from the signal. Batch normalization helps accelerate the training process, stabilize the model, and prevent issues such as gradient vanishing or explosion. Through this structure, the convolutional layers achieve efficient computation while extracting rich frequency features. By stacking multiple layers, the network progressively refines high-level features of the signal, enhancing its ability to represent complex frequency patterns.

Output Layer: A deconvolution layer (transposed convolution layer) is employed to upsample the internal representation to the desired frequency representation dimensions. The deconvolution layer, a specialized operation in convolutional neural networks, performs the reverse of the convolution operation. While convolution is typically used for downsampling—extracting features and reducing data dimensions using a sliding window—deconvolution is used for upsampling, restoring data dimensions through interpolation and kernel expansion. By upsampling through the deconvolution layer, the output frequency representation retains higher levels of detail, which facilitates subsequent tasks such as frequency estimation and counting. The high-resolution frequency representation produced by the deconvolution layer captures finer frequency variations within the signal, thereby enhancing the accuracy of frequency estimation.

Application of Narrowband Gaussian Kernels: Narrowband Gaussian kernels are utilized to generate the ideal frequency information during model training. The specific steps are as follows:

• Define the Gaussian Kernel: A narrowband Gaussian kernel is mathematically defined with a mean frequency and a narrow standard deviation, ensuring it focuses on a specific frequency range.
• Generate Ideal Frequency Representation: Using the Gaussian kernel, an ideal frequency representation is constructed by centering it on the target frequencies of the input signal. This representation serves as the ground truth for training the frequency estimation model.
• Apply to Training Data: For each input signal, the corresponding ideal frequency representation is computed using the Gaussian kernel, creating the supervised labels required for model training.
• Train the Model: The model learns to map the input signal to the ideal frequency representation by minimizing the error between its predictions and the Gaussian-kernel-generated frequency labels. This ensures the model is optimized for precise frequency estimation, even in complex or noisy environments.

This approach leverages the smooth and localized nature of Gaussian kernels, which not only provide a high-resolution reference for training but also enhance the network’s resilience to noise. The kernel function is designed to create an ideal, noise-resilient feature map that aids in more accurate frequency estimation during training.

Loss Function: The loss function utilizes the Mean Squared Error (MSE) to quantify the difference between the predicted frequency representation and the ground truth frequency representation.

$${\mathcal{L}}_{\mathrm{FR}}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{∥{\widehat{y}}_i-y_i∥}^2$$

(14)

The Mean Squared Error (MSE) loss function is an effective metric for quantifying the average deviation between predicted and true frequency values, making it particularly suitable for regression tasks. By minimizing the discrepancy between estimated frequencies and the ground truth, the MSE loss ensures that the network can consistently predict frequencies with outstanding performance. The training data consist of simulated signals with known frequency characteristics, which serve as a foundation for guiding network training.

For each input signal, the corresponding ideal frequency representation is computed using a Gaussian kernel. This process involves applying the kernel with a mean frequency and a narrow standard deviation, thus generating an idealized frequency map that accurately reflects the true frequency characteristics of the signal. These generated frequency labels are then utilized during the training process. The training data are specifically designed to cover a wide range of frequency values and signal-to-noise ratios (SNRs), thereby enabling the network to generalize reliably across various scenarios. To simulate real-world conditions, the dataset includes signals with varying numbers of frequency components and diverse noise levels.

Incorporating the structural design outlined previously, the frequency representation module efficiently extracts relevant frequency features from the input signals, generating high-resolution frequency maps. This architecture is capable of capturing the essential frequency components while also adapting to varying SNRs, ensuring optimal performance across different environments. The use of narrow Gaussian kernels enhances the precision of frequency estimation and significantly improves noise resilience. As a result, this approach delivers superior performance in frequency estimation tasks, particularly in scenarios involving high noise levels and complex signal conditions, where traditional methods may struggle.

3.1.2. Frequency Counting Model

The frequency counting module is designed similarly to the frequency estimation module, with the goal of determining the number of frequencies present in the signal based on its frequency representation. Like the frequency estimation module, it consists of an input layer, multiple convolutional layers, and an output layer. The input layer and convolutional layers share the same structure as those in the frequency estimation module. However, the output layer in the frequency counting module uses a linear layer to convert the internal representation into the final frequency count.

Additionally, when the frequency count is a discrete value, with a finite and predefined set of possible frequency counts, the model adjusts the loss function to the cross-entropy loss function:

$${\mathcal{L}}_{\mathrm{FC}}=-{\displaystyle \frac{1}{N}}\sum _{i=1}^N{y}_{i}log{\widehat{y}}_i$$

(15)

3.1.3. Phase Match

Through the frequency estimation and frequency counting models, the initial frequency values ${\widehat{\omega }}_k$ and the frequency count k are obtained. The estimated frequency parameters are then used to construct the phase-matching function.

$$\widehat{\varphi }=sin({\widehat{\omega }}_{k}t+{\widehat{\varphi }}_k),k=1,2,\cdots ,K$$

(16)

This is obtained by multiplying the phase matching function with the phase error signal.

$$\begin{array}{cc}\hfill {\delta }_k& ={\phi }_k\left(t\right)·{\widehat{\phi }}_k\left(t\right)\hfill \\ & =\sum _{k=1}^K{B}_{k}sin\left({\omega }_{k}t+{\varphi }_k\right)·sin\left({\widehat{\omega }}_{k}t+{\widehat{\varphi }}_k\right)\hfill \\ & =\sum _{k=1}^K{\displaystyle \frac{B_k}{2}}\left\{cos\left[\left({\omega }_k-{\widehat{\omega }}_k\right)t+\left({\varphi }_k-{\widehat{\varphi }}_k\right)\right]-cos\left[\left({\omega }_k+{\widehat{\omega }}_k\right)t+\left({\varphi }_k+{\widehat{\varphi }}_k\right)\right]\right\}\hfill \end{array}$$

(17)

As shown above, when ${\omega }_k={\widehat{\omega }}_k$, the DC component is ${\displaystyle \frac{B_k}{2}}cos({\varphi }_k-{\widehat{\varphi }}_k)$. By traversing $\left(-\pi ,\pi \right)$ over $\theta$, the phase parameter is found at the maximum DC component, while half of the DC component gives the amplitude parameter ${\widehat{a}}_k$. A one-dimensional search is sufficient to obtain ${\widehat{\varphi }}_k,{\widehat{a}}_k=B_k/2$. Let $k=k+1$ and update the error phase.

$${\varphi }_k={\varphi }_{k-1}\left(t\right)-B_{k-1}sin\left({\widehat{\omega }}_{k-1}t+{\widehat{\varphi }}_{k-1}\right)$$

(18)

Repeat the phase-matching process to obtain the parameter values of the K sinusoidal components. The above process yields the initialization values $\widehat{a}, \widehat{\omega }, \widehat{\varphi }$, serving as the starting point for the subsequent optimization algorithm.

3.1.4. Optimization Algorithm

Since our objective function is nonconvex, we used the ADAM optimization algorithm during the coarse optimization phase of model training to update parameters. Compared to traditional gradient descent methods like SGD, ADAM is more efficient and robust, often achieving better performance in fewer iterations. However, it does not guarantee finding the global optimum. In practice, combining multiple optimizers often yields better results. Therefore, this study employs L-BFGS for fine optimization.

3.1.5. Coarse Optimization—ADAM

ADAM is widely used in deep learning optimization and is an efficient optimization algorithm. Based on the concept of stochastic gradient descent, it ensures relatively low computational cost by utilizing the first-order derivatives of the objective function. ADAM dynamically adjusts the learning rate for each parameter using first- and second-moment estimates, making it highly effective in handling sparse gradients and noisy problems.

The loss function for ADAM is the MSE:

$$\mathcal{L}\left(\theta \right)={\displaystyle \frac{1}{N}}\sum _{j=1}^N{\left(y_{ture}\left(t_j\right)-y_{pred}\left(t_j\right)\right)}^2$$

(19)

$$y_{pred}=\sum _{i=1}^K{a}_{i}sin\left({\omega }_{i}t+{\varphi }_i\right)$$

(20)

where $\theta$ represents the optimization parameter $a_i, {\omega }_i, {\varphi }_i$ and $y_{ture}$ is the input data to be fitted. The detailed optimization steps are shown below:

1.

Set the first moment estimate variable $m_0=0$, second moment estimate variable $v_0=0$, learning rate $\alpha$, first moment exponential decay rate ${\beta }_1$, and second moment exponential decay rate ${\beta }_2$;

2.

Calculating the gradient:

$$\begin{array}{c}{g}_{a_i,t}={\displaystyle \frac{\partial \mathcal{L}\left(\theta \right)}{\partial a_i}}\hfill \\ g_{{\omega }_i,t}={\displaystyle \frac{\partial \mathcal{L}\left(\theta \right)}{\partial {\omega }_i}}\hfill \\ g_{{\varphi }_i,t}={\displaystyle \frac{\partial \mathcal{L}\left(\theta \right)}{\partial {\varphi }_i}},\hfill \end{array}$$

(21)

where t represents the current iteration number;

3.

Update first-order and second-order moment estimates and compute bias corrections:

$$\begin{array}{c}{m}_t={\beta }_1{m}_{t-1}+\left(1-{\beta }_1\right)g_t\hfill \\ v_t={\beta }_2{v}_{t-1}+\left(1-{\beta }_2\right)g_t^2\hfill \end{array}$$

(22)

$$\begin{array}{c}{\widehat{m}}_t={\displaystyle \frac{m_t}{1-{\beta }_1^t}}\hfill \\ {\widehat{v}}_t={\displaystyle \frac{v_t}{1-{\beta }_2^t}};\hfill \end{array}$$

(23)

4.

The parameters are updated using the corrected moment estimates:

$${\theta }_t={\theta }_{t-1}-\alpha {\displaystyle \frac{{\widehat{m}}_t}{\sqrt{{\widehat{v}}_t}+ϵ}}.$$

(24)

After iterating for a certain number of times, the coarse-optimized periodic phase error parameters are obtained. These parameters will then be used for local optimization to obtain more accurate error parameters.

3.2. Precision Optimization—L-BFGS

In the second stage of optimization, the L-BFGS optimization algorithm is employed to further fine-tune the model parameters and improve the fitting accuracy. L-BFGS is a quasi-Newton method suitable for optimization problems requiring high-precision convergence. This method approximates the inverse of the Hessian matrix using limited memory, making it more efficient for large-scale optimization problems. In this paper, the L-BFGS algorithm is used to optimize the amplitude, frequency, and phase parameters, ensuring that the generated signal matches the real signal as closely as possible.

The algorithm steps are as follows:

1.

The initialized fitting parameters ${\theta }_0$ are set to the final results of the ADAM optimization. A history matrix $\left\{S_j,G_j\right\}$ is initialized to store the gradient and position vectors, with the history matrix size set to 20.

2.

Calculate the gradient g using a method similar to the ADAM algorithm, which will not be elaborated here. Then, update the position vector difference and the gradient difference. These updates are used to record the historical information, which is subsequently employed to approximate the inverse of the Hessian matrix:

$$\begin{array}{c}{S}_i={\theta }_{j+1}-{\theta }_j\hfill \\ G_j=g_{j+1}-g_j\hfill \end{array}$$

(25)

Let $q=g_j$, The first loop starts from the most recent historical record and iterates backwards, calculating the scalar ${\eta }_j$ and updating the vector q:

$$\begin{array}{c}{\eta }_j={\rho }_j{S}_j^{T}q\\ q=q-{\eta }_j{G}_j\end{array}$$

(26)

Calculate the scaling factor ${\gamma }_j$ for the initial matrix and initialize the vector r:

$$\begin{array}{c}{\gamma }_j={\displaystyle \frac{S_{j-1}^T{G}_{j-1}}{G_{j-1}^T{G}_{j-1}}}\\ r={\gamma }_{j}q\end{array}$$

(27)

The second loop starts from the furthest historical record and iterates forwards, calculating the scalar ${\chi }_j$ and updating the vector r:

$$\begin{array}{c}{\chi }_j={\rho }_j{G}_j^{T}r\hfill \\ r=r+S_j\left({\eta }_j-{\chi }_j\right)\hfill \end{array}$$

(28)

Determine the search direction $p_j$, and refine the parameters.

$$\begin{array}{c}{p}_j=-r\\ {\theta }_{j+1}={\theta }_j+{\chi }_j{p}_j.\end{array}$$

(29)

3.

If the norm of the gradient $\left|\right|g_j\left|\right|$ is smaller than the predefined tolerance $\zeta$ or if the maximum number of iterations is reached, the iteration stops. This completes the precise optimization process. In Section 4, this paper will validate the algorithm through both simulations and measured data.

3.3. Correction Using Matched Fourier Transform

Existing nonlinear correction methods typically use the time-domain interpolation resampling method to eliminate polynomial nonlinearity and achieve multi-target focusing. However, under periodic nonlinearity, solving transcendental equations to determine interpolation points becomes challenging. To address this, this paper adopts the more versatile matched Fourier transform to achieve multi-target nonlinear correction. With an analytical expression of nonlinearity, multi-target focusing can be effectively performed.

$$F_{\mathrm{MFT}}={\int }_0^{T}f\left(t\right){\mathrm{e}}^{-\mathrm{j}2\pi f\psi \left(t\right)}\mathrm{d}\psi \left(t\right)$$

(30)

The multi-target differential frequency signal with sinusoidal period phase error can be expressed as follows:

$$\begin{array}{c}{s}_{if}\left(t\right)={\displaystyle \sum _{q=1}^Q}exp\left\{\mathrm{j}2\pi \left[f_c{\tau }_q+K_{r}t{\tau }_q+{\displaystyle \sum _{k=1}^K}{A}_k{\tau }_{q}sin\left({\omega }_{k}t+{\varphi }_k\right)\right]\right\}\\ ={\displaystyle \sum _{q=1}^Q}exp〈\mathrm{j}2\pi \left\{f_c{\tau }_q+K_r{\tau }_q\left[t+{\displaystyle \sum _{k=1}^K}{\displaystyle \frac{A_k}{K_r}}sin\left({\omega }_{k}t+{\varphi }_k\right)\right]\right\}〉\end{array}$$

(31)

where Q is the target number and ${\tau }_q$ is the echo delay of the q target. Its Fourier transform is as follows:

$$S_{\mathrm{FT}}\left(f\right)={\int }_0^T{s}_{if}\left(t\right){\mathrm{e}}^{-\mathrm{j}2\pi ft}dt$$

(32)

Let $\psi \left(t\right)=t+{\displaystyle \sum _{k=1}^K}{\displaystyle \frac{A_k}{K}}sin\left({\omega }_{k}t+{\varphi }_k\right)$, then the MFT of ${s_i}_f\left(t\right)$ is as follows:

$$\begin{array}{cc}\hfill S_{MFT}\left(f\right)& =\int s_{if}\left(t\right)e^{-j2\pi ft}dt\hfill \\ & ={\int }_0^T\sum _{q=1}^Q{e}^{j2\pi \left[f_c{\tau }_q+K_r{\tau }_q\psi \left(t\right)\right]}{e}^{-j2\pi f\psi \left(t\right)}{\psi }^{\prime }\left(t\right)dt\hfill \\ & =\sum _{q=1}^Q[\psi \left(t\right)-\psi \left(0\right)]·exp\left(j2\pi f_c{\tau }_q\right)·sinc\left\{[\psi \left(t\right)-\psi \left(0\right)]\left(f-K_r{\tau }_q\right)\right\}\hfill \end{array}$$

(33)

where ${\psi }^{\prime }\left(t\right)$ is the derivative of $\psi \left(t\right)$, and its expression is as follows:

$${\psi }^{\prime }\left(t\right)=1+\sum _{k=1}^K{\displaystyle \frac{A_k{\omega }_k}{K_r}}cos\left({\omega }_{k}t+{\varphi }_k\right)$$

(34)

The targets at different distances are focused into the ideal $sinc$ function, and the frequency $f_p=K_r{\tau }_p$ of the target reflects the true distance of the target, thereby correcting the system’s nonlinearity.

The periodic amplitude of the new time axis $\psi \left(t\right)$ is as follows:

$${\displaystyle \frac{A_k}{K_r}}={\displaystyle \frac{a_k}{2\pi K_r{\tau }_{ref}}}$$

(35)

The transformation from the time axis t to the new time axis $\psi \left(t\right)$ is a resampling process. The MFT of the intermediate frequency signal ${s_i}_f\left(t\right)$ is equivalent to the Fourier transform on the new time axis $\psi \left(t\right)$. The one-dimensional range profile compression on $\psi \left(t\right)$ is the result after correcting the periodic phase errors.

4. Algorithm Validation

The proposed algorithm’s effectiveness and superiority are validated using both simulated and experimental data in this chapter. The first section focuses on validating the algorithm’s effectiveness. The algorithm is applied to process simulated echo signals, demonstrating its ability to correct periodic errors and detect weak real targets within error-contaminated signals. The second section highlights the algorithm’s superiority by comparing its performance with existing frequency estimation methods, such as MUSIC, FFT, and the periodogram. The results show the algorithm’s advantages at medium to high signal-to-noise ratios. In the third section, validation using experimental data further confirms the algorithm’s effectiveness, showcasing its reliability and practical advantages. To highlight the results of periodic error correction, all subsequent experiments are windowed; this step is not repeated in the later sections.

4.1. Validity Verification

As shown in Figure 3, after introducing single-frequency sine errors and third-order sine errors, one and three pairs of echoes, respectively, appear near the main lobe in the one-dimensional range profile of the echo signal. This outcome aligns with the analysis in Section 2.2 regarding the impact of periodic errors on radar signals.

Next, Gaussian noise is added to the beat signal with periodic errors, as depicted in Figure 4a. It is evident that the random noise significantly raises the noise floor and amplifies the side lobe levels. The beat signals, both with and without noise, are then processed using the proposed method, with the results shown in Figure 4b,c. The results demonstrate that the proposed algorithm successfully corrects the periodic errors in both scenarios. Notably, in Figure 4c, after periodic error correction, the processed result is comparable to the noise floor level, illustrating the algorithm’s robustness and remarkable performance in noisy environments.

Figure 5a presents the estimation results of the periodic error parameters obtained using the proposed algorithm and the FFT-based method. As seen in the figure, the parameters estimated by the FFT-based method contain some errors, while the proposed algorithm achieves a more accurate estimation. The parameters estimated by both methods are then used for periodic error correction, with the results shown in Figure 5b. Due to the bias in the estimation error, the FFT-based method fails to completely eliminate the periodic error, whereas the proposed algorithm successfully achieves the desired correction.

The paper next presents simulation experiments for weak target detection. As shown in Figure 6a, the beat signal without errors clearly detects a weak target located near the main lobe. However, after introducing third-order periodic errors, the echo of the weak target is obscured by paired echoes resulting from the periodic error. As demonstrated in Figure 6b, after applying the proposed method to correct the periodic errors, the weak target detection signal is successfully restored, leading to a significant improvement in the accuracy of target detection.

4.2. Superiority Verification

This paper validates, through Monte Carlo simulations, that the neural network-based model outperforms existing methods in error coefficient estimation. The accuracy of the estimate is measured by computing the False Negative Rate (FNR), which is defined as the number of true frequencies that go undetected, i.e., when no estimated frequency falls within a radius of ${\left(2N\right)}^{-1}$. As shown in Figure 7a, it is evident that within the SNR range of 0–30 dB, the proposed frequency estimation module exhibits significant superiority over both the MUSIC algorithm [18] and the periodogram [17]. In this SNR range, the proposed method achieves lower estimation errors, demonstrating higher precision and stability.

As shown in Figure 7b, for the frequency counting module, the proposed method also exhibits exceptional performance. Compared to algorithms such as AIC [23], MDL [24,32], and SORTE, the proposed frequency counting module shows markedly better accuracy and robustness. These results demonstrate that the proposed method provides more precise and reliable estimates when dealing with complex signals and noise interference, highlighting its potential and superiority in practical applications.

$$\begin{array}{cc}\hfill FNR& ={\displaystyle \frac{N_e}{N_r}}\times 100\%\hfill \\ \hfill \mathrm{Error}& ={\displaystyle \frac{M_s-M_r}{M_r}}\times 100\%\hfill \end{array}$$

(36)

where $N_e$ represents the number of true undetected frequencies, $N_r$ denotes the number of true frequencies, $M_s$ refers to the estimated number of sinusoidal components, and $M_r$ indicates the number of true sinusoidal components.

4.3. Experimental Data Processing

The parameters of the equipment used in the actual measurement experiments are shown in

Table 1. Radar parameters.

Parameter/Specification	Value
Operating frequency	90–96 GHz
Sweep duration	2 ms
IF frequency	20 KHz–5 MHz
Transmitted power	15 dBm
Range	1–50 m

. The specific measurement scenario is depicted in Figure 8a, while Figure 8b shows the physical radar system. All subsequent experimental data are derived from this FMCW radar system.

This paper further validates the performance of the proposed algorithm using experimental data. As shown in Figure 9a, the phase of the real-world data are fitted using the proposed algorithm, and the corrected result is presented in Figure 9b. After correction, the proposed algorithm successfully suppresses the paired echoes caused by periodic errors to the noise level. This result demonstrates that the proposed algorithm can effectively correct periodic errors in real-world data, significantly improving signal quality and detection accuracy. The experiment confirms the algorithm’s effectiveness and robustness in practical applications.

To validate the effectiveness of the proposed algorithm in improving weak target detection capability, a series of real-world comparison experiments were conducted. First, FMCW radar was used to collect a set of strong reflection single-target echo data, as shown in Figure 10a. Due to the presence of periodic errors, false targets appeared near the main lobe. After applying the proposed algorithm for correction, as shown in Figure 10b, the false targets were successfully eliminated. Figure 10c presents a comparison of the one-dimensional range profiles before and after correction, demonstrating that the proposed algorithm significantly reduces the impact of periodic errors.

Next, in the same experimental scenario, a weak target was placed near the strong reflecting target. The resulting echo image is shown in Figure 10d. Due to the influence of periodic errors, it was difficult to discern the presence of the weak target in the scene. However, after eliminating the periodic errors using the proposed algorithm, as shown in Figure 10e, the weak target was clearly restored in the area marked by the red circle. The one-dimensional range profile in Figure 10f further illustrates this significant improvement.

In summary, the experimental results fully validate the superiority of the proposed algorithm in suppressing periodic errors and enhancing weak target detection capability, thereby significantly improving the accuracy and reliability of target detection.

To further validate the reliability of the proposed algorithm in complex multi-target environments, we applied the algorithm to multi-target correction. As shown in Figure 11a, multiple targets in the range direction generate similar false targets. We extracted the target within the red box and displayed its 1D range profile in Figure 11c. Due to the presence of periodic errors, paired echoes appear near the main lobe, leading to the formation of false targets. By applying the proposed algorithm to correct the periodic errors, the result is shown in Figure 11b. After correction, the false targets are successfully eliminated. The 1D range profile within the red box further demonstrates that the side lobes caused by periodic errors are significantly suppressed. This result further confirms the effectiveness and reliability of the proposed algorithm in complex multi-target environments, as it accurately corrects periodic errors and significantly improves target detection precision.

5. Conclusions

Periodic phase errors in ultra-wideband FMCW radar systems often result in the generation of spurious targets and the masking of genuine weak targets. This paper proposes a novel software-based correction method that integrates neural networks, phase-matching techniques, and advanced optimization algorithms. The proposed method significantly mitigates the impact of periodic errors, reliably suppresses spurious targets, and substantially enhances the detection capability for genuine weak targets. The effectiveness and robustness of the proposed method have been validated through simulations and experimental data. The results demonstrate that the method significantly improves the accuracy of one-dimensional range profiles, successfully eliminates spurious targets caused by periodic errors, and effectively enhances the radar system’s ability to detect weak targets.

In summary, the proposed periodic error correction method provides a reliable solution for high-precision target detection and performance optimization in ultra-wideband radar imaging systems. This research contributes valuable theoretical foundations for the further development of radar imaging technology and offers strong support for the design of next-generation high-precision radar systems.