A Non-Reconstruction Multi-Coset Sampling-Based Algorithm for Frequency Estimation with FMCW Lidar

Abstract

Frequency-Modulated Continuous Wave (FMCW) lidar for long-distance measurements face challenges in signal acquisition and frequency estimation due to the high sampling rates required, leading to increased processing load, cost, and power consumption. Although sub-Nyquist sampling can alleviate the burden of high sampling rates, it requires a complex reconstruction process that degrades real-time performance. In this study, we propose a frequency estimation algorithm based on multi-coset sampling (MCS) that not only reduces the sampling rate but also avoids reconstructing the original signal. This algorithm performs a preliminary frequency estimation by exploiting the relationship among the signal support set, the measured sequences by sampling spectrum, and the original signal spectrum, and then refines the spectrum to obtain an accurate frequency estimate. Since the algorithm relies solely on the sampled sequences for estimation, frequency ambiguity may occur during the calculation. We analyze the causes of ambiguity and propose a support set determination method to eliminate this issue. Simulation results demonstrate that the proposed algorithm attains the Cramér–Rao lower bound (CRLB) at low signal-to-noise ratios. It achieves a 10-fold improvement over Nyquist method and a 35 dB SNR reduction compared with the original MCS, while maintaining stable performance down to −20 dB.

1. Introduction

Frequency-modulated continuous-wave (FMCW) lidar has become an important area of research in modern sensing systems due to its low cost, high measurement accuracy, and robust noise immunity [1,2]. It is widely used in fields such as autonomous driving systems [3,4], robotics [5], and non-contact shape measurement [6,7,8].

In FMCW lidar, mixing the echo signal with the transmitted signal generates a beat signal, whose frequency encodes the target distance [9]. With increasing measurement distance, the beat signal changes from low to high frequency. However, as a key lidar parameter, frequency resolution—determined by the ratio of sampling rate to number of samples—must be paid for with a steep rise in sample count under the Nyquist criterion if higher spectral discrimination is to be achieved, imposing heavy demands on the front-end analog-to-digital converter (ADC) and on processing and storage resources. This reduces real-time performance and increases energy consumption, which significantly impacts applications such as autonomous driving.

Traditionally, lidar signal processing has mainly relied on spectral estimation to calculate frequency and obtain measurement distance. Frequency refinement algorithms mainly include the Rife algorithm [10], the Quinn algorithm [11], and the chirp Z transform (CZT) algorithm [12]. The CZT algorithm enables analysis of signals across different frequency ranges and offers advantages such as low computational cost and high accuracy. It is widely used for spectral refinement. Ref. [13] introduces repeated use of the CZT algorithm to improve frequency estimation accuracy. Ref. [14] proposes using CZT to construct nonlinear equations that describe the relationship between frequency, amplitude, phase, and the observed spectrum. Ref. [15] proposed an algorithm that combines CZT refinement of frequency and phase estimation to achieve accurate frequency estimation. The accuracy of the above algorithm, nonetheless, remains limited by the Nyquist rate.

Recently, compressed sensing (CS) [16,17,18,19] theory has been found to be able to accurately restore sparse signals with a small number of samples, which has attracted great attention. Prof. Y. C. Eldar proposed a sub-Nyquist sampling technique for sparse signal based on compressed sensing [20,21,22]. This technique includes sampling structures such as random demodulator (RD), multi-coset sampling (MCS), modulated wideband converter (MWC). RD is primarily suited for multitone signals consisting of discrete harmonic components, while the physical channel structure of the MWC system is more complex, requiring prefabricated pseudorandom sequences. In contrast, the MCS system features a simpler architecture, consisting of multiple analog-to-digital converters (ADCs) synchronized to a common clock frequency to form a non-uniform sampling system. Each ADC operates with distinct phase shifts, thereby constructing a matrix for compressive sampling [23,24,25]. Due to its simple structure and low cost, it is widely employed in sparse signal reconstruction research [26]. Ref. [27] proposed a sub-Nyquist sampling frequency estimation method, which uses a configurable sparse multi-set (SMC) sampling method to achieve better reconstruction accuracy by configuring the delays between different sampling channels. Ref. [28] proposed a sampling pattern selection algorithm to minimize redundant reconstruction of the signal model and optimize the sensing matrix. The reconstruction process in MCS is referred to as the multiple measurement vector (MMV) problem. Ref. [22] introduces the continuous to finite (CTF) module, which transforms the theoretical infinite measurement vector (IMV) model into an equivalent finite-dimensional MMV model. The reconstruction algorithms primarily utilize greedy algorithms [29] to identify the sparsest solution, followed by the application of the least squares (LS) method for signal reconstruction. Ref. [30] proposes a tensor-based signal processing framework for FMCW signals, revealing their spectral sparsity and demonstrating the feasibility of incorporating CS. Ref. [31] proposes a meshless method for compression-aware frequency estimation. All the aforementioned algorithms must reconstruct the original signal before proceeding to signal processing. The reconstructed signal undergoes DFT for initial estimation, followed by spectral refinement to achieve accurate frequency estimation.

While MCS technology reduces sampling rate requirements, it remains burdened by the reconstruction step, which introduces redundant computations and increases system energy consumption. To address this limitation, we propose an MCS-based frequency estimation method that does not require reconstruction of the original signal. Instead, it determines the signal frequency directly from the spectrum of the sampled sequences, thereby simplifying the algorithm and enhancing noise robustness. Furthermore, when the covariance matrix is used to obtain the support set in CTF, noise in the sampled sequences can cause substantial interference, making the method unsuitable for low-SNR scenarios. In response to this issue, we propose a method that extracts the spectral amplitude from the sampled sequences and performs correlation detection with the sensing matrix to obtain the support set. This method effectively avoids the extensive computations required by the CTF method and demonstrates strong robustness. The support set can be accurately obtained as long as the signal spectrum is not overwhelmed by noise. The main contributions of this paper are summarized below.

• We propose a non-reconstruction frequency estimation algorithm for FMCW lidar based on the MCS structure. The method exploits the frequency corresponding to the maximum spectral amplitude of the sampled sequences, together with the support set, to locate the frequency position of the original signal, thereby achieving frequency estimation without reconstruction. To solve the ambiguity problem in frequency estimation, we also propose a support set solution method based on the correlation between the spectrum of the sampled sequences and the sensing matrix.
• We employ a two-stage approach for beat signal frequency estimation, comprising initial frequency estimation and spectral refinement. A sampled sequence is first selected to obtain the center frequency, and ambiguity is resolved using the support set to derive a preliminary frequency estimate. High-resolution frequency is obtained by refining the spectral sampled sequence through the CZT. Compared with existing methods, the proposed approach significantly simplifies the algorithmic structure and exhibits enhanced robustness against noise.
• Extensive simulation experiments were conducted to validate the performance of the proposed algorithm. The results demonstrate that the estimation accuracy of the method outperforms other sub-Nyquist approaches, achieving good performance at an SNR of −20 dB and attaining the Cramér–Rao lower bound under certain conditions.

The remainder of the paper is organized as follows. Section 2 introduces the principle of beat signal formation, explains the MCS principle, and analyzes the conventional MCS reconstruction process. Section 3 introduces a reconstruction-free MCS algorithm and integrates it with the CZT refinement method for frequency estimation. Section 4 experimentally validates the effectiveness of the proposed algorithm. Section 5 concludes this article.

2. FMCW Lidar and Multi-Coset Sampling Principles

2.1. Principles of FMCW Lidar

The basic structure of the FMCW lidar system is illustrated schematically in Figure 1a. The laser beam is polarized, producing two identical optical signals: one serving as the local reference signal and the other as the transmitted signal. The transmitted signal (Tx) is reflected on the surface of the target object and generates an echo signal (Rx) of the same form, which has a time delay $\tau$ compared with the local reference signal. The two signals interfere and form a beat signal through a balanced detector, as shown in Figure 1b. The Magnitude spectrum of the beat signal is presented in Figure 1c. Assume the distance between the laser and the target is R, and let c represent the propagation speed of the laser in air. Then, the time delay $\tau$ can be expressed as $\tau =2R/c$. B denotes the frequency modulation (FM) bandwidth of the laser and T denotes its sweep period. The frequency of the beat signal $f_b$ is generated by the frequency difference between the local reference signal and the echo signal, and $f_b$ has a linear relationship with the delay $\tau$, expressed as:

$$f_b=k\tau ={\displaystyle \frac{4BR}{Tc}}$$

(1)

Thus, the distance of the target can be determined by

$$R={\displaystyle \frac{cT}{4B}}{f}_b.$$

(2)

Notably, T and B are predetermined, and the calculation of the target distance primarily relies on the frequency $f_b$ of the beat signal $x(t)$. The ranging process can be accomplished by identifying the maximum magnitude of the beat signal spectrum $\left|X\left(f\right)\right|$, as expressed in the following equation:

$$f_b={\arg }_f\max \left|X\left(f\right)\right|.$$

(3)

Figure 1c illustrates that the beat signal is sparse in the frequency domain. The MCS structure can reconstruct the original signal after undersampling the sparse signal. We propose using the MCS structure to replace the Nyquist structure of a single-chip ADC for signal acquisition of the beat signal generated by FMCW lidar.

2.2. Principles of MCS

The structure of MCS is shown in Figure 2, which has several similar sampling channels, each with a different delay unit, and all channels have the same sampling rate. The Nyquist sampling period of the original signal is defined as $T_{nyq}$, with the corresponding Nyquist sampling rate $f_{nyq}=1/T_{nyq}$. For a given factor $L$, the sampling rate of a single-channel ADC is set to $f_p=1/\left(L{T}_{nyq}\right)$. To ensure that the zero frequency of the original signal is not split, the division coefficient is typically chosen to be an odd number $L=2L_0+1$. The set $C={\left\{\left.c_i\right\}\right.}_{i=1}^q$ represents the delays for q sampling channels, where $q<L$. The delay for each channel is expressed as $c_i{T}_{nyq}$, and the sampled result can be expressed as follows:

$${\mathit{y}}_i[n]=\left\{\begin{array}{c}x\left(n{T}_{nyq}\right), n=mL+c_i,m\in \mathbb{Z}\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(4)

Applying DFT to (4) leads to the following equation:

$${\mathit{Y}}_i\left(f\right)=\underset{\mathit{A}}{\underbrace{{\displaystyle \frac{1}{L{T}_{nyq}}}{\displaystyle \sum _{l=-L_0}^{L_0}{e}^{\frac{j2\pi l{c}_i}{L}}}}}\underset{\mathit{Z}\left(f\right)}{\underbrace{X\left(f+l{f}_p\right)}},i\in \left[1,q\right],f\in F_p$$

(5)

where $\mathit{Y}\left(f\right)$ denotes the spectra of all sampling channels, $\mathit{Y}\left(f\right)=$${\left[Y_1\left(f\right),Y_2\left(f\right),\dots \dots ,Y_q\left(f\right)\right]}^T$, $A$ denotes a sensing matrix of dimensions $q\times L$, with the element in the i-th row and l-th column defined as ${\mathit{A}}_{i,l}=$$\exp \left(j2\mathrm{\pi }{c}_{i}l\right)/\left(L{T}_{nyq}\right)$ and $F_p=[-f_p/2,f_p/2)$.

The reconstruction process of MCS can be interpreted as solving underdetermined linear equations. Typically, the support set S, representing the positional indices of the non-zero elements in sparse vectors, is determined by constructing a new linear equation through the CTF method [22]. Once S is identified, (5) is converted into an over-determined equation $\mathit{Y}\left(f\right)=$${\mathit{A}}^S{\mathit{Z}}_S\left(f\right)$, where ${\mathit{A}}^S$ denotes the set of column vectors of matrix A indexed by S, and ${\mathit{Z}}_S\left(f\right)$ denotes the set of row vectors of matrix $\mathit{Z}\left(f\right)$ indexed by S. The original signal is reconstructed by solving ${\mathit{Z}}_S\left(f\right)$ using the least squares method. The expression for CTF is provided below:

$$\left\{\begin{array}{c}\mathit{Q}={\displaystyle \int \mathit{Y}\left(f\right)·\mathit{Y}{\left(f\right)}^H}df\\ \mathit{Q}=\mathit{V}{\mathit{V}}^H\\ \mathit{V}=\mathit{A}\mathit{U}\end{array}\right..$$

(6)

It is worth noting that, due to the sparse nature of the signal in the frequency domain, (6) has a unique sparsest solution $\overline{\mathit{U}}$, and the support set of $\overline{\mathit{U}}$ is the same as the support set of $\mathit{Z}\left(f\right)$. There are many algorithms, such as orthogonal matching pursuit (OMP) [32,33], multi-signal classification algorithm (MUSIC) [34], and basis pursuit denoising (BPDN) [35,36], etc., to find the support set of $\overline{\mathit{U}}$, which is equivalent to obtaining the support set S of the original signal.

3. Proposed MCS Frequency Estimation Algorithm

3.1. Frequency Estimation Mathematical Model

Theoretically, the reconstruction results based on MCS can be directly used for frequency estimation. For beat signals, obtaining the frequency alone is sufficient for ranging without requiring signal reconstruction. To simplify the calculation, we propose a reconstruction-free MCS-based frequency estimation system. Figure 3 shows the block diagram of the proposed system. First, the support set is obtained by using the sub-Nyquist sampling spectrum of $\mathit{y}[n]$, and the ambiguity of the signal is eliminated. Then, the frequency of the sampled signal is obtained under the action of CZT, and combined with the corresponding support set, $f_b$ is obtained without reconstruction. Next, we will discuss the analysis of each part in detail.

In [9], the simplified and normalized expression for $x(t)$ is given by

$$x(t)=E\cos \left(2\pi f_{b}t+\theta \right).$$

(7)

The Fourier transform (FT) of $x(t)$ gives

$$X\left(f\right)={\displaystyle \frac{E}{2}}\left(e^{j\theta }\delta (f-f_b)+e^{-j\theta }\delta (f+f_b)\right)$$

(8)

We define a one dimensional vector $C={\left[c_1,c_2,\dots ,c_q\right]}^{\mathrm{T}}.$ Substituting (8) into (5), the spectrum of $\mathit{y}[n]$ can be expressed as:

$$\mathit{Y}\left(f\right)={\displaystyle \frac{1}{LT}}{\displaystyle \sum _{l=1}^{2L_0+1}{e}^{\frac{j2\pi C\left(l-L_0-1\right)}{L}}}\mathit{Z}\left(f\right)$$

(9)

The $l$-th column of $\mathit{Z}(f)$ is given by:

$${\mathit{Z}}_l\left(f\right)={\displaystyle \frac{E}{2}}\left(e^{j\theta }\delta (f-f_b+\left(l-L_0-1\right)f_p)\right.\left.+e^{-j\theta }\delta (f+f_b+\left(l-L_0-1\right)f_p)\right)$$

(10)

It can be seen that the original signal spectrum, denoted as $X\left(f\right)$, is partitioned into L segments to form the unknown vector $\mathit{Z}\left(f\right)$ with dimensions $L\times N.$

Each segment of $X\left(f\right)$ is shifted by a multiple of the single-channel sampling rate $f_p$, the index of the spectral interval in which the $f_b$ resides forms the support set. According to (10), in the spectral interval where $f_b$ resides, there exists one $f_{\psi }\in \left[-f_p/2,f_p/2\right)$ such that the frequency of $\mathit{Z}\left(f\right)$ satisfies the following relation:

$$-f_{\psi }+\left(m_1-L_0-1\right)f_p=-f_b$$

(11)

$$f_{\psi }+\left(m_2-L_0-1\right)f_p=f_b$$

(12)

where $m_1$ and $m_2$ denote the support sets of $X\left(f\right)$. Based on the above equation, it can be concluded that the support set of $X\left(f\right)$ satisfies the following relationship:

$$m_1+m_2=2\left(L_0+1\right)$$

(13)

The symmetry of $m_1$ and $m_2$ about $L_0+1$ provides a crucial basis for addressing the frequency estimation ambiguity in subsequent analysis. It is only necessary to obtain $f_{\psi }$ and the corresponding support set $m_1$ or $m_2$, and the beat signal frequency can be obtained according to Formulas (11) and (12) without reconstructing the original signal.

We arbitrarily select one channel from the MCS structure and record its channel index as i. The spectrum of the corresponding sampled sequences is ${\mathit{y}}_i\left(f\right)$. By combining (9) and (10), ${\mathit{y}}_i\left(f\right)$ can be expressed as follows:

$$\begin{gathered} {\mathit{y}}_i\left(f\right)={\displaystyle \frac{E}{2LT}}\left(e^{{\displaystyle \frac{-j2\pi c_i{L}_0}{L}}}{e}^{{\displaystyle \frac{-j2\pi c_i\left(L_0-1\right)}{L}}}\dots e^{{\displaystyle \frac{j2\pi c_i{L}_0}{L}}}\right)\times \left(\begin{array}{c}{e}^{j\theta }\delta (f-f_b-L_0{f}_p)+e^{-j\theta }\delta (f+f_b-L_0{f}_p)\\ e^{j\theta }\delta (f-f_b-\left(L_0-1\right)f_p)+e^{-j\theta }\delta (f+f_b-\left(L_0-1\right)f_p)\\ ⋮\\ e^{j\theta }\delta (f-f_b+L_0{f}_p)+e^{-j\theta }\delta (f+f_b+L_0{f}_p)\end{array}\right) \\ \begin{array}{c}={\displaystyle \frac{E}{2LT}}\left(e^{j\left(\theta +{\displaystyle \frac{2\pi c_i\left(m_2-L_0-1\right)}{L}}\right)}\delta (f-f_b+\left(m_2-L_0-1\right)f_p)\right.\left.+e^{j\left({\displaystyle \frac{2\pi c_i\left(m_1-L_0-1\right)}{L}}-\theta \right)}\delta (f+f_b+\left(m_1-L_0-1\right)f_p)\right)\end{array} \end{gathered}$$

(14)

We conclude that $f_{\psi }$ is the center frequency of ${\mathit{y}}_i\left(f\right)$. The formulation of ${\mathit{y}}_i\left(f\right)$ can then be generalized to all channels

$$\mathit{Y}\left(f\right)={\displaystyle \frac{E}{2LT}}\left(e^{j\left(\theta +{\displaystyle \frac{2\pi C\left(m_2-L_0-1\right)}{L}}\right)}\delta (f-f_b+\left(m_2-L_0-1\right)f_p)\right.\left.+e^{j\left({\displaystyle \frac{2\pi C\left(m_1-L_0-1\right)}{L}}-\theta \right)}\delta (f+f_b+\left(m_1-L_0-1\right)f_p)\right)$$

(15)

Equation (15) shows that the center frequencies of the spectra for each channel are identical, while their phases differ. In other words, the frequencies corresponding to the maximum spectral amplitudes of the sampled sequences in each channel are approximately the same. The magnitude spectrum of $\mathit{Y}\left(f\right)$ is shown in Figure 4.

If only the center frequency of $\mathit{Y}\left(f\right)$ is known, it is not possible to directly obtain $f_b$, as this involves the problem of frequency ambiguity. Specifically, since it is unknown whether $f_{\psi }$ corresponds to the positive or negative half of the frequency axis in $\mathit{Y}\left(f\right)$, $f_b$ cannot be determined simply by identifying the frequency at which $\left|\mathit{Y}\left(f\right)\right|$ attains its maximum.

3.2. Support Set Estimation

To address the problem of frequency ambiguity, we propose an efficient support set computation method. The proposed approach resolves frequency estimation ambiguity while maintaining strong robustness to noise.

We denote $\left|f_{\psi }\right|$ as the frequency corresponding to the maximum spectral amplitude on the positive frequency axis of $\left|{\mathit{y}}_i\left(f\right)\right|$. Thus, Equation (5) can be rewritten as:

$${\mathit{y}}_i\left(\left|f_{\psi }\right|\right)={\mathit{a}}_i\mathit{Z}\left(\left|f_{\psi }\right|\right),\left|f_{\psi }\right|\in \left[0,f_p/2\right]$$

(16)

where ${\mathit{a}}_i$ is the i-th row of the sensing matrix A. The vector $\mathit{Z}\left(\left|f_{\psi }\right|\right)$ with dimensions $L\times 1$ can then be expressed as:

$${\mathit{Z}}_l\left(\left|f_{\psi }\right|\right)=\left\{\begin{array}{l}{\displaystyle \frac{E}{2}}{e}^{-j\theta },l=m_1\\ 0,\mathrm{otherwise}\end{array}\right.\mathrm{or}{\mathit{Z}}_l\left(\left|f_{\psi }\right|\right)=\left\{\begin{array}{l}{\displaystyle \frac{E}{2}}{e}^{j\theta },l=m_2\\ 0,\mathrm{otherwise}\end{array}\right..$$

(17)

The product of the s-th column element of the ${\mathit{a}}_i$ and $\mathit{Z}\left(f\right)$ can be written as:

$${\mathit{y}}_i\left(\left|f_{\psi }\right|\right)={\mathit{a}}_i^s{\mathit{Z}}_{m_1}\left(\left|f_{\psi }\right|\right)={\mathit{a}}_i^s{\displaystyle \frac{E}{2}}{e}^{-j\theta }$$

(18)

$${\mathit{y}}_i\left(\left|f_{\psi }\right|\right)={\mathit{a}}_i^s{\mathit{Z}}_{m_2}\left(\left|f_{\psi }\right|\right)={\mathit{a}}_i^s{\displaystyle \frac{E}{2}}{e}^{j\theta }$$

(19)

Extending (18) and (19) to all channels gives:

$${\mathit{Y}}_i\left(\left|f_{\psi }\right|\right)={\mathit{a}}^s{\mathit{Z}}_{m_1}\left(\left|f_{\psi }\right|\right)={\mathit{a}}^s{\displaystyle \frac{E}{2}}{e}^{-j\theta }$$

(20)

$${\mathit{Y}}_i\left(\left|f_{\psi }\right|\right)={\mathit{a}}^s{\mathit{Z}}_{m_2}\left(\left|f_{\psi }\right|\right)={\mathit{a}}^s{\displaystyle \frac{E}{2}}{e}^{j\theta }$$

(21)

It follows that $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ is linearly correlated only with the values in the s-th column of the sensing matrix A, and that $s=m_1$ or $s=m_2$. As a result, it is only necessary to detect the correlation between $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ and the sensing matrix $\mathit{A}$ to identify the columns with the highest linear correlation, thereby determining the support set S.

Considering the influence of noise, the spectrum of the sampled sequences can be expressed as $\mathit{Y}’\left(f\right)=\mathit{Y}\left(f\right)+\lambda \left(f\right)$, where $\lambda \left(f\right)$ denotes a white Gaussian noise vector with zero mean and variance ${\sigma }^2$. When the support set is obtained using the CTF module, Q is expressed as:

$$\begin{array}{c}\mathit{Q}’={\displaystyle \int \left(\mathit{Y}\left(f\right)+\lambda \left(f\right)\right)·{\left(\mathit{Y}\left(f\right)+\lambda \left(f\right)\right)}^H}df\hfill \\ ={\displaystyle \int \mathit{Y}\left(f\right)·\mathit{Y}{\left(f\right)}^{H}df}+{\displaystyle \int \mathit{Y}\left(f\right)·{\lambda }^H\left(f\right)df}+{\displaystyle \int \mathit{Y}{\left(f\right)}^H·\lambda \left(f\right)df}+{\displaystyle \int \lambda \left(f\right)·{\lambda }^H\left(f\right)df}\end{array}$$

(22)

where is the ideal matrix for computing the support set. However, for beat signal frequency estimation, when the noise $\lambda \left(f\right)$ increases beyond a certain level, $\mathit{Q}’$ contains a substantial amount of noise. During the singular value decomposition of $\mathit{Q}’$, the matrix V becomes significantly distorted, resulting in low accuracy in support set acquisition. The proposed algorithm obtains the support set by exploiting the correlation between the sensing matrix and the spectrum of the sampled sequences. As long as the frequency $\left|f_{\psi }\right|$ and the corresponding spectrum $\mathit{Y}\left(f\right)$ can be correctly extracted from the sampled sequence spectrum, the support set can be accurately determined. Compared with the noise limitation of the CTF method, the proposed approach only requires that the maximum amplitude of $\mathit{Y}\left(f\right)$ not be submerged by noise (i.e., $E/\left(2LT\right)>\lambda \left(f\right)$), which is significantly less restrictive than that of CTF.

3.3. Elimination of Frequency Estimation Ambiguity

Since the number of spectral segments L is an odd integer, and the spectrum $X\left(f\right)$ is symmetric about zero, the range of the k-th spectral interval can be expressed as $\left[k{f}_p-f_p/2,k{f}_p+f_p/2\right]$. When the spectrum $X\left(f\right)$ of the beat signal is divided into L sub-bands for combination, the relationship between the values of frequency $f_b$ and $k{f}_p$ results in two possible scenarios, as illustrated in Figure 5. The fundamental cause of the frequency estimation ambiguity lies in the indeterminacy of the sign of $k{f}_p$. According to (12), when $f_b>$$k{f}_p$, the sign of $f_{\Psi }$ is positive, and when $f_b<$$k{f}_p$, the sign of $f_{\Psi }$ is negative. Therefore, the ambiguity can be eliminated simply by determining the relationship between $f_b$ and $k{f}_p$ without reconstructing the original signal. We establish a relationship between the ambiguity in frequency estimation and the support set. By combining (14) and (15), the support set satisfies $0<m_1\le L_0+1$ and $L_0+1\le m_2<2\left(L_0+1\right)$. When the positive frequency axis of $\mathit{Y}\left(f\right)$ is selected, the support set can be classified into two cases:

Case I: When the beat frequency $f_b$ is located to the left of the midpoint of the k-th spectral interval, i.e., as illustrated in Case I of Figure 5. According to (17), (18) and (20), the negative frequency axis of $X\left(f\right)$ is involved in the support set calculation. By using the positive frequency axis of $\mathit{Y}\left(f\right)$, the frequency $\left|f_{\psi }\right|$ is obtained. Then, the correlation between $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ and the sensing matrix A is computed to obtain the support set $s=m_1$. The inverse DFT is applied to the spectrum $\mathit{Y}\left(f\right)$ of the sampled sequences to obtain:

$$\mathit{y}[n]={\displaystyle \frac{E}{LT}}\cos \left(-2\mathrm{\pi }\left(\left(m_1-L_0-1\right)f_p+f_b\right)\left(nL{T}_{nyq}\right)\right.\left.-\theta +{\displaystyle \frac{2\mathrm{\pi }C\left(m_1-L_0-1\right)}{L}}\right)$$

(23)

From the above equation, it follows that $f_{\Psi }<0$. Combining this with (11), the estimate of the beat signal frequency is

$${\mathit{f}}_b=\mathit{k}{\mathit{f}}_p-\left|{\mathit{f}}_{\psi }\right|=\left(L_0-m_1+1\right){\mathit{f}}_p-\left|{\mathit{f}}_{\psi }\right|.$$

(24)

Case II: For a beat signal whose frequency $f_b$ lies to the right of the midpoint of the k-th spectral interval (i.e., f_b > kf_p), the situation corresponds to Case II in Figure 5. Based on (17), (19) and (21), the positive frequency axis of $X\left(f\right)$ is involved in the support set calculation. The correlation between $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ and the sensing matrix A is computed to obtain the support set $s=m_2$. The inverse DFT is applied to the spectrum $\mathit{Y}\left(f\right)$ of the sampled sequences to obtain:

$$\mathit{y}[n]={\displaystyle \frac{E}{LT}}\cos \left(2\mathrm{\pi }\left(f_b-\left(m_2-L_0-1\right)f_p\right)\left(nL{T}_{nyq}\right)\right.\left.+\theta +{\displaystyle \frac{2\mathrm{\pi }C\left(m_2-L_0-1\right)}{L}}\right)$$

(25)

From the above equation, it can be seen that $f_{\Psi }>0$. Together with (12), his yields the estimate of the beat signal frequency as

$$f_b=k{f}_p+\left|f_{\psi }\right|=\left(m_2-L_0-1\right)f_p+\left|f_{\psi }\right|$$

(26)

By computing the correlation between the magnitude spectrum of the sampled sequences and the sensing matrix A, the corresponding support set s is identified. When the support set satisfies $s<L_0+1$, it corresponds to Case I, and the frequency can be derived from (24). Conversely, when the support set satisfies $s>L_0+1$, it corresponds to Case II, and the frequency can likewise be obtained from (26). This procedure effectively resolves the ambiguity in frequency estimation.

3.4. Spectral Refinement

For the unknown matrix $\mathit{Z}\left(f\right)$, the support set has been determined in this paper, and the preliminary frequency estimate of the beat signal is obtained accordingly. However, the frequency accuracy is still insufficient to meet the requirements of FMCW lidar, necessitating further refinement of $\left|f_{\psi }\right|$.

In this paper, the CZT algorithm is applied to refine the spectrum of $\mathit{Y}\left(f\right)$. The CZT is a specialized type of DFT, based on the principle of calculating the Z-transform of a finite-length sequence of sampled values at equally spaced sampling points on the unit circle of the Z-plane. Dense interpolation of the frequencies on the spectrum of the sampled signal, which is actually computed only for a specific segment of the spectrum on the unit circle, enhances the computational resolution. It is characterized by high efficiency and allows for refinement in arbitrary frequency bands.

As shown in Figure 4, the frequencies corresponding to the maxima of each channel in $\mathit{Y}\left(f\right)$ are all located at the same position. Therefore, frequency estimation can be achieved by refining only a single sampled channel. According to the location of $\left|f_{\psi }\right|$, one frequency $f_1$ on its left and $f_2$ on its right is selected as the start and end points of the refinement. The range of spectrum refinement is twice the resolution of the frequency at this point, so the values of $f_1$ and $f_2$ are determined as follows:

$$\begin{array}{l}{f}_1=\left|f_{\psi }\right|-\Delta f\\ f_2=\left|f_{\psi }\right|+\Delta f\end{array}$$

(27)

When the resolution is $\Delta f=f_p/N$, the refined spectrum is obtained through M-fold interpolation on the unit circle. In the M-point CZT algorithm, we can first obtain the discrete spectrum of ${\mathit{y}}_i[n]$

$${\mathit{y}}_i\left(z\right)={\displaystyle \sum _{n=0}^{N-1}{\mathit{y}}_i[n]}\exp \left(-j2\pi {\displaystyle \frac{f_1}{f_p}}n\right)\exp \left(-j2\pi {\displaystyle \frac{2\Delta f}{M{f}_p}}nz\right).$$

(28)

The frequency corresponding to the maximum value of $\left|{\mathit{y}}_i\left(z\right)\right|$ is the estimated frequency. Using (24) and (26), along with the frequency corresponding to the maximum value of the amplitude of the refined spectrum, the frequency estimate ${\tilde{f}}_b$ is determined.

3.5. Algorithm and Complexity Analysis

The operational pseudo-code of the algorithm is shown in Algorithm 1. The algorithm consists of three parts: preprocessing, support set estimation, and spectral refinement. First, the sampling sequence is preprocessed according to the delay factor C and segmentation coefficient L, as described in [22], to obtain ${\mathit{y}}_i[n]$. All subsequent data processing is carried out on $\mathit{y}[n]$. A DFT is performed on $\mathit{y}[n]$ to obtain the maximum magnitude on the positive frequency axis. Next, support set estimation is conducted by computing the correlation between $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ and each column of the sensing matrix A, and the column with the highest correlation is identified as the support set $s_1$. By evaluating the sign of $s_1$, the ambiguity can be resolved according to the two cases described above, and a preliminary frequency estimate is obtained. The same procedure is then applied to the negative frequency axis to acquire another support set $s_2$. Finally, a sampled sequence from the sampling value matrix is selected for spectral refinement. The refined frequency $\left|{\mathit{f}}_{\psi }\right|$ is substituted into (24) and (26), thereby obtaining the final frequency estimate ${\tilde{f}}_b$.

The complexity of the proposed algorithm is analyzed. After preprocessing the sampling points, performing the DFT on the resulting LN data points requires $LN{\log }_2\left(LN\right)$ multiplications. Since there are q sampled sequences in total, the overall computational complexity is $O\left\{qLN{\log }_2\left(LN\right)\right\}$. For support set acquisition, L correlations need to be computed, and each correlation requires q multiplications, resulting in a total complexity of $O\left\{Lq\right\}$. In the case of the CZT, only a single sampled sequence needs to be refined with a magnification factor of M, yielding a computational complexity of $O\left\{M{\log }_{2}M\right\}$. Therefore, the overall computational complexity of the scheme is $O\left\{qLN{\log }_2\left(LN\right)+Lq\right.$$\left.+M{\log }_{2}M\right\}$. The computational complexity of CTF is $O\left\{\left.q^{2}N+q^3+qLN\right\}\right.$. The results clearly indicate that the proposed method has a lower computational complexity than CTF.

Algorithm 1: Frequency estimation method based on MCS

Inputs: N-point sampled sequences $x\left((nL+C)T_{nyq}\right),$ number of channels q, refinement multiplier M

Output: preliminary frequency estimate $f_a$, support set S, frequency estimates ${\tilde{f}}_b$

Processing: Preprocessing the sampled sequences based on (4) produces the sequences $\mathrm{Perform}\mathrm{DFT}\mathrm{on}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{processed}\mathrm{sequences}\mathrm{individually}\mathrm{to}\mathrm{obtain}\mathrm{the}\mathrm{spectrum}\mathit{Y}\left(f\right)$ and determine $\left|{\mathit{f}}_{\psi }\right|$ and $\mathit{Y}\left(\left|f_{\psi }\right|\right)$ Support set estimation: 3. The sensing matrix A is calculated by (5) 4. $\mathrm{Calculate}\mathrm{the}\mathrm{correlation}\mathrm{between}\mathrm{each}\mathrm{column}\mathrm{of}\mathrm{the}\mathrm{sensing}\mathrm{matrix}\mathit{A}\mathrm{and}\mathrm{the}\mathrm{positive}-\mathrm{frequency}\mathrm{components}\mathrm{of}\mathit{Y}\left(\left|f_{\psi }\right|\right)$, and determine the most correlated columns $O=\underset{l\in \left[1,L\right]}{\cup }{\displaystyle \frac{\mathit{Y}{\left(\left|f_{\psi }\right|\right)}^T\times {\mathit{A}}^l}{‖\mathit{Y}{\left(\left|f_{\psi }\right|\right)}^T‖·‖{\mathit{A}}^l‖}}$ 5. $\mathrm{Find}\mathrm{the}\mathrm{most}\mathrm{relevant}\mathrm{column}:s={\arg }_l\max (O)$ 6. $\mathrm{Denote}\mathrm{the}\mathrm{most}\mathrm{relevant}\mathrm{column}\mathrm{above}\mathrm{as}{s}_1$, Select the negative frequency axis of $\mathit{Y}\left(f\right)$ and repeat steps 2, 3, 4, and 5 to obtain $s_2$ 7. $\mathrm{Obtain}\mathrm{the}\mathrm{support}\mathrm{set}S=s_1\cup s_2$ Based on (24) and (26), one can obtain a preliminary estimate of the frequency $f_a$ Refinement: 8. $\mathrm{Set}\mathrm{the}\mathrm{start}\mathrm{and}\mathrm{end}\mathrm{points}\mathrm{for}:f_1=\left|f_{\psi }\right|-f_p/N,f_2=\left|f_{\psi }\right|+f_p/N$ 9. Set the CZT parameters for spectrum refinement: $A_0=e^{-j2\mathrm{\pi }{f}_1/f_p},W=e^{-j2\mathrm{\pi }(f_1-f_2)/(f_{p}M)}$ 10. $\mathrm{Parameters}\mathrm{are}\mathrm{integrated}\mathrm{into}\mathrm{the}\mathrm{CZT}\mathrm{to}\mathrm{achieve}\mathrm{a}\mathrm{more}\mathrm{refined}\mathrm{spectrum}Y\left(z\right)$: $Y\left(z\right)={\displaystyle \sum _{n=0}^{LN-1}{y}_i\left(n\right){A_0}^{-n}{W}^{nz}},z=0,1,\dots ,M-1,i\in \left[0,q\right]$ 11. $\mathrm{Find}\mathrm{the}\mathrm{frequency}\mathrm{corresponding}\mathrm{to}\mathrm{the}\mathrm{position}\mathrm{of}\mathrm{the}\mathrm{maximum}\mathrm{value}\mathrm{in}Y\left(z\right)$: $\left|{\tilde{f}}_{\psi }\right|=\left|\max (|Y\left(z\right)|)-1\right|·(f_2-f_1)/M+f_1$ 12. $\mathrm{Substituting}\left|{\tilde{f}}_{\psi }\right|$ in (24) and (26) to obtain the final estimate of the frequency ${\tilde{f}}_b$ 13. $\mathrm{Return}{f}_a$, S, ${\tilde{f}}_b$

4. Experimental Results and Analysis

To validate the proposed method, simulation experiments were carried out to assess the frequency estimation performance.

4.1. Setting

To emulate the beat signal generated by an FMCW lidar system, we conduct experiments using narrowband signals with near-sinusoidal characteristics. Given the sparsity of the signal and the presence of noise in practical applications, the following $x(t)$ is selected to simulate the beat signal

$$x(t)=E\cos \left(2\pi ft\right)+n(t)$$

(29)

where $n(t)$ represents noise. In subsequent experiments, Gaussian white noise was used to model actual sampling noise.

In this study, the sample rate $f_{nyq}$ is configured as 200 MHz. Considering that MCS is a physically realizable system, this paper sets L to be an odd integer, $L=25$. The sampling frequency of each channel is $f_p=8\mathrm{MHz}$ C represents the sampling delay mode, and q integers are randomly selected from $[0,L]$ as the delay units, which remain unchanged during the experiment. Each channel ADC only differs in delay, while other parameters remain identical unless otherwise specified in subsequent experiments. The CZT spectrum refinement multiplier is set to $M=512$. The beat signal frequency ranges from 4 MHz to 100 MHz, with 5000 test cases randomly selected. To compare the performance of the algorithms, the root mean square error (RMSE) measures the accuracy of the frequency estimation, and the estimate set (ES) score evaluates the accuracy of the support set

$$\mathrm{RMSE}=10\lg {\left({\displaystyle \frac{1}{P}}{\displaystyle \sum _{i=1}^P{\left({\tilde{f}}_i-f_i\right)}^2}\right)}^{{\displaystyle \frac{1}{2}}}$$

(30)

$$\mathrm{ES}={\displaystyle \frac{2\left|S\cap \tilde{S}\right|}{\left|S\right|+\left|\tilde{S}\right|}}$$

(31)

where $\tilde{S}$ is the estimated support set, and $S$ is the true support set. $\left|S\cap \tilde{S}\right|$ denotes the number of elements in the true support set $S$ that are included in the estimated support set $\tilde{S}$.

Generally speaking, the estimation of frequency and phase for time-limited sinusoidal signals in Gaussian white noise constitutes an estimation problem whose accuracy is constrained by the CRB. The lower bound for the estimation of sinusoidal signal frequency and phase is given by the following equation [37]:

$$\mathrm{var}[{\widehat{f}}_{CRB}]\ge {\displaystyle \frac{12f_s}{{(2\mathrm{\pi })}^{2}N(N^2-1)·\mathrm{SNR}}}$$

(32)

4.2. Comparison of the Frequency Estimation Performance Across Different Channels

The proposed algorithm estimates the frequency by selecting one channel from the sampled sequences. Thus, the selection of a channel is crucial for achieving maximum accuracy. To investigate the impact of the refinement accuracy of each channel on the overall frequency estimation accuracy, we conduct the following experiments.

The frequency estimation results are presented in Figure 6 for N ranging from 4000 to 12,000, with a SNR of 30 dB. It can be observed that the maximum inter-channel RMSE difference is 0.15 dB, indicating that the frequency estimation accuracy across different channels remains nearly identical for the same number of sampling points. For $N=12,000$, as the SNR varies from −20 dB to 20 dB, the frequency estimation results are shown in Figure 7. The RMSE curves of all channels nearly coincide, with a maximum difference of 0.2 dB, indicating that the choice of sampling channels does not affect the final frequency estimation when the sampling points and SNR are identical. Therefore, in subsequent experiments, a sampling channel is arbitrarily selected for frequency estimation.

4.3. Minimum Number of Channels Required to Obtain a Support Set

The number of sampling channels q, not only affects the accuracy of the support set but also influences the overall sampling rate and total data volume in the system.

The support set directly determines the frequency position of the original signal and plays a critical role in resolving ambiguity. It is therefore essential to ensure that the support set is estimated with perfect accuracy, that is $ES=1$. To reduce hardware costs and minimize unnecessary operations, it is essential to determine the minimum number of sampling channels required for accurate frequency estimation.

At an SNR of 30 dB, as the number of channels q, increases from two to twelve, the accuracy of the support sets obtained by different algorithms is presented in Figure 8. The accuracy of all algorithms improves significantly as the number of channels increases. Original MCS algorithms generally determine the support set for complete reconstruction through the CTF module. Specifically, the BPDN method requires five channels, while both the OMP and MUSIC methods require six channels. The proposed algorithm requires only three channels to accurately determine the support set, significantly reducing hardware costs.

Considering the effect of noise, the original MCS algorithm operates with $q=6$, while the proposed algorithm runs with q ranging from three to six. The support set accuracy is presented in Figure 9. When the proposed algorithm uses $q=3$, the minimum SNR required for complete reconstruction is 15 dB. In contrast, with a minimum of five channels, the proposed algorithm achieves complete reconstruction at an SNR as low as −20 dB. Conversely, the other three algorithms are more sensitive to noise, requiring a minimum SNR of 15 dB. This sensitivity is due to the CTF module constructing (6) with $\mathit{Y}\left(f\right)$, which contains noise and introduces additional noise components into V. If the noise level is too high, V deviates significantly, resulting in errors in the derived support set. However, the proposed algorithm determines the support set by directly identifying the amplitude maximum of $\mathit{Y}\left(f\right)$. As long as this amplitude maximum in the frequency domain is not completely masked by noise, the support set can be accurately derived.

To further evaluate the performance of the support set algorithm under different types of noise, a Poisson noise model with parameter $\lambda =10$ was introduced, as shown in Figure 10 and Figure 11. The results indicate that the proposed method exhibits performance comparable to that under Gaussian noise, achieving perfect results with only five ADCs.

To investigate the minimum SNR achievable by the proposed algorithm, the number of sampling channels was fixed at five, and the ES performance under different SNR conditions is illustrated in Figure 12. The results show that when N < 8000, the proposed algorithm can operate at a minimum SNR of −18 dB. When N > 8000, the minimum achievable SNR is further reduced to −20 dB.

Compared to the original MCS algorithm, the proposed algorithm demonstrates significant improvements in both SNR and the required number of channels. In the subsequent experiments, the proposed algorithm uses $q=5$, while the original MCS algorithm uses $q=6$.

4.4. Experiments on the Bias in Preliminary Frequency Estimates

The preliminary frequency estimate defines the range for frequency refinement, establishing the foundation for spectrum refinement. To achieve more accurate frequency estimation, we conduct experiments on the deviation range of the preliminary frequency estimation. The results are compared with two approaches: reconstructing the signal using the original MCS algorithm followed by DFT, and performing DFT on the signal obtained at the Nyquist sampling rate. The SNR is set to 30 dB, with N varying from 4000 to 12,000, while all other parameters remain constant. The preliminary deviation in frequency estimation for all algorithms is presented in Figure 13, where the deviation curves exhibit a gradual decreasing trend. At $N=4000$, the deviation of the proposed algorithm from the original MCS method is 501 Hz, while that of the Nyquist sampling method is 1254 Hz. As N increases from 4000 to 12,000, the deviation of the proposed method from the original MCS method decreases by 336 Hz, and the deviation of the Nyquist sampling method decreases by 8406 Hz. This indicates that the proposed algorithm exhibits a smaller deviation range with the same number of sampling points, and its deviation is significantly smaller than the frequency resolution under these conditions. Therefore, spectrum refinement can be used to achieve more accurate frequency estimation.

In addition, we analyze the effect of noise on preliminary frequency estimation. With $N=12,000$, the SNR varies from −20 to 40 dB. As shown in Figure 14, as the SNR increases from −20 dB to −5 dB, the deviation of the preliminary frequency estimation in the proposed algorithm decreases by 3 Hz, while that of the Nyquist sampling method decreases by 43 Hz. However, when the SNR is lower than 15 dB, the deviation of the preliminary frequency estimation obtained by the original MCS method exceeds 8 MHz due to errors in acquiring the support set. This indicates that, compared to other methods, the proposed preliminary frequency estimation algorithm exhibits smaller deviations and better noise resistance.

4.5. Minimum Number of Samples to Achieve Frequency Estimation

The number of samples used in spectrum refinement is a key factor influencing its accuracy. Although increasing the number of sampling points enhances frequency resolution, it also directly increases computational complexity. To determine the minimum required number of sampling points, the SNR is set to 30 dB, and N is incremented from 4000 to 12,000 in steps of 200, while all other parameters remain unchanged.

The algorithm that reconstructs MCS and subsequently applies DFT and CZT is referred to as the original MCS algorithm, while the algorithm based on 200 MHz sampling followed by DFT and CZT is referred to as the Nyquist algorithm. Figure 15 presents a comparison of the proposed algorithm, the original MCS algorithm, MWC, CBMB and the Nyquist method. As the number of sampling points increases, the RMSE curve of the proposed method declines in stages, while the original MCS and Nyquist method exhibits a gradual change. This discrepancy arises because the number of circumferential convolution points used for spectral refinement must satisfy the condition $Ln=2^n$, and the number of refinement points LN is determined through Fourier interpolation of the sequence from a single sampling channel, where $LN<Ln$. For the original MCS algorithm, the number of refinement points after reconstruction is equivalent to normal sampling with LN points, whereas the Nyquist method utilizes only N points for spectrum refinement. In this paper, the proposed algorithm achieves its highest frequency estimation performance at $N=10,800$, with an RMSE of −2.1 dB. Although the original MCS algorithm achieves its optimal performance at $N=8400$, its computational complexity remains excessively high. The Nyquist method reaches a maximum RMSE of only 16 dB at $N=12,000$. Since the CBLB algorithm does not include a refinement stage and has low requirements on the number of sampling points, its frequency estimation RMSE is relatively large and remains nearly constant at around 37 dB. MWC is sensitive to sparse single-tone signals. When N is below 4200, a small proportion of the 5000 datasets exhibit significant frequency errors. For $N>4200$, the RMSE stabilizes at about 10 to 25 dB. These results indicate that under identical sampling point conditions, the proposed algorithm achieves higher accuracy than other methods.

To investigate the impact of noise on algorithm performance, $N=10,800$ is set, and the SNR is varied from −20 dB to 30 dB, while all other parameters remain constant, as presented in Figure 16. The proposed algorithm achieves a minimum operational SNR of −20 dB with an RMSE of 16 dB and is capable of reaching the Cramér–Rao lower bound at low signal-to-noise ratios. The lowest SNR attained by the Nyquist method is −20 dB, but it exhibits a higher RMSE of 29 dB, indicating significant errors. MWC is capable of reaching an SNR of 0 dB, with the RMSE remaining around 16 dB. The CBMB algorithm has an SNR threshold of 0 dB. Compared with the original MCS method, it exhibits slightly improved noise immunity. The original MCS algorithm requires a minimum SNR of 15 dB, while the proposed algorithm operates at −20 dB, corresponding to a 35 dB reduction in the SNR threshold. At N = 10,800, the proposed algorithm achieves a 35 dB reduction in the SNR threshold compared with the original MCS method under Gaussian white noise. Among the four algorithms, the proposed algorithm demonstrates the highest noise immunity.

To investigate the relationship between the performance of the proposed algorithm and the CRLB, the number of sampling channels was fixed at five, and the RMSE under different SNR conditions is shown in Figure 17. It can be observed that the number of sampling points has a relatively minor influence on whether the algorithm can approach the CRLB. The proposed algorithm approaches the CRLB very closely over an SNR range from −20 dB to 5 dB.

Table 1. Maximum relative frequency estimation error of each algorithm for different SNR.

Relative Error (ppm)	SNR
	−20 dB	−10 dB	0 dB	10 dB	20 dB	30 dB
Nyquist	100	8	2	0.8	0.5	0.4
Original MCS	-	-	-	-	0.018	0.015
Proposed	1.5	0.3	0.1	0.04	0.02	0.015

presents the maximum relative frequency error of each algorithm at N = 10,800. It can be observed that, under different SNR conditions, the proposed algorithm achieves approximately an order-of-magnitude improvement in accuracy compared with the Nyquist method. When the SNR exceeds 20 dB, both the original MCS method and the proposed method exhibit very small estimation errors. When the SNR is below 20 dB, the estimation error becomes very large for the original MCS method due to failures in support set acquisition.

We further examined the effect of the number of channels on the accuracy of frequency estimation. The first sampling channel was selected for spectral refinement under the conditions of SNR = −10 dB and N = 8000, and the corresponding frequency estimation results for different numbers of channels are summarized in

Table 2. Frequency estimation error for different number of sampling channels.

Frequency (Hz)	Errors (Hz)
	Three Channels	Four Channels	Five Channels	Six Channels
61,235,567.890	−17.109	−17.109	−17.109	−17.109
61,236,567.890	13,526,885.000	−21.015	−21.015	−21.015
61,237,567.890	13,524,901.000	−36.640	−36.640	−36.640
61,238,567.890	13,522,897.000	−32.734	−32.734	−32.734
61,239,567.890	−21.015	−21.015	−21.015	−21.015
61,240,567.890	−9.296	−9.296	−9.296	−9.296
61,241,567.890	10.235	10.235	10.235	10.235
61,242,567.890	10.235	10.235	10.235	10.235
61,243,567.890	−21.015	−21.015	−21.015	−21.015
61,244,567.890	13,510,909.000	13,510,909.000	−44.452	−44.452
61,245,567.890	−17.109	−17.109	−17.109	−17.109
61,245,667.890	−23.359	−23.359	−23.359	−23.359
61,245,767.890	−29.609	−29.609	−29.609	−29.609
61,245,867.890	−28.046	−28.046	−28.046	−28.046
61,245,967.890	−18.671	−18.671	−18.671	−18.671

. The results show that when the number of channels is small, frequency estimation errors are significant due to inaccuracies in determining the support set. A comparison between five and six channels demonstrates that, once the support set is accurately obtained, the number of channels has no noticeable impact on the accuracy of frequency estimation.

The runtime and memory consumption of the algorithm were evaluated through MATLAB R2023b simulations. As shown in

Table 3. Execution time and memory consumption of different algorithms.

Algorithm	Execution Time (s)		Memory Consumption (kb)
	Mean	Standard Deviation
Nyquist	0.1702	0.0063	11,704.00
MCS	0.4042	0.0158	25,036.00
MWC	0.4613	0.0097	39,400.00
Proposed	0.2588	0.0082	14,104.00

, the results indicate that the proposed algorithm requires less memory and shorter runtime compared with other MCS-based methods, thereby demonstrating superior real-time performance.

In summary, the proposed algorithm enables accurate frequency estimation over the $[4,100]$ MHz range while offering higher accuracy, fewer sampling points, and better noise immunity compared to existing methods. In conclusion, the number of sampling points should be flexibly determined based on specific measurement requirements and experimental conditions. In practical measurements, when the measured distance is long, signal attenuation is significant, or the signal-to-noise ratio is low, a larger number of sampling points can be adopted to achieve higher frequency estimation accuracy. Increasing the number of sampling points not only enhances spectral resolution and reduces estimation error but also helps suppress the impact of noise on signal characteristics, thereby improving the overall stability and reliability of the measurement. Conversely, when the signal quality is high, the system requires a rapid response, or computational and storage resources are limited, fewer sampling points can be employed to reduce data processing load, minimize system latency, and improve real-time performance.

4.6. Visualization Example of Frequency Estimation Using the MCS Algorithm

To visualize the results, 500 test cases were generated. To meet the undersampling requirement, the test frequency was incremented from 61 MHz to 61.5 MHz in steps of 1000 Hz. With $N=10,800$, the frequency estimation error curve is shown in Figure 18. It can be observed that the proposed algorithm achieves a frequency estimation error within 1.5 Hz at SNR = 30 dB, with minimal fluctuation across frequencies. At SNR = −10 dB, the frequency estimation error remains within 30 Hz. This demonstrates that the proposed algorithm not only eliminates the need for complex reconstruction but also achieves high-accuracy frequency estimation, maintaining a certain level of accuracy even at low SNR values.

4.7. Impact of Channel Mismatch on the Algorithm

The MCS structure employs a delay mechanism to construct a sensing matrix, from which we derive the support set. In practical hardware circuits, ADC sampling is affected by non-ideal factors such as gain mismatch, timing (delay) errors, and inter-channel synchronization errors. To investigate the impact of delay errors on the proposed algorithm, we use five ADCs with N = 10,800 and set the sampling pattern as C = {3; 6; 11; 16; 22}. Each ADC delay corresponds to the Nyquist sampling interval associated with the sampling pattern, and the maximum delay error is varied from 0 to 1. As shown in Figure 19, when the delay error is smaller than 0.4, the support set can still be accurately identified. The MCS structure employs a delay mechanism to construct a sensing matrix, from which we obtain the support set. ADC delay errors have only a minor influence on the accuracy of spectral refinement.

Gain errors arise from differences in amplification among the sub-channels, resulting in amplitude discrepancies in the sampled signals. In the corresponding experiment, we keep the number of ADCs and the number of sampling points unchanged, and fix the delay error at 0.4. To study inter-channel mismatch, we assign different gains to each ADC, with the gain ranging from 1 to 2. Figure 20 illustrates the effect of the maximum gain error in different channels on the frequency estimation performance. The results indicate that frequency estimation can still be achieved when the gain error is within 1.5. Moreover, gain mismatch has a significant impact on support set acquisition, whereas its influence on spectral refinement accuracy is relatively small.

5. Conclusions

In this paper, we propose an MCS-based frequency estimation algorithm that eliminates the need for signal reconstruction, thereby reducing the sampling rate, improving computational speed, and saving storage space. Compared with the traditional MCS algorithm, the proposed method requires fewer ADCs and achieves a wider noise tolerance range, with the SNR threshold reduced by 35 dB. This method locates the segmented frequency interval of the signal through the support set and performs frequency estimation under spectral refinement. This enables frequency estimation using only a single sampling channel when the support set is obtained, thereby significantly simplifying the algorithmic structure. Moreover, the proposed support set acquisition method resolves the ambiguity in frequency estimation and demonstrates strong robustness. Simulation results verify its effectiveness in single-tone scenarios generated by a single target. In future work, MCS will be implemented on FPGA and further extended to other scenarios, such as multi-target positioning and multi-target measurement. This technique can be applied to autonomous vehicles and embedded lidar systems, effectively mitigating the engineering challenges arising from cost constraints and high sampling rate requirements. Furthermore, the proposed algorithm also exhibits extensibility to fields such as cognitive radio and medical measurement.