Monitoring High-Dynamic Wideband Orthogonal Frequency Division Multiplexing Signal Under Weak Prior Knowledge

Abstract

In the context of the escalating requirement for high-throughput multimedia services, Orthogonal Frequency Division Multiplexing (OFDM) signal systems have become increasingly prevalent in Low Earth Orbit (LEO) satellite communications. In order to promote the judicious, efficient, and cost-effective deployment of satellite spectrum resources, there is a critical need to augment the capabilities of spectrum monitoring technology for uncollaborative LEO satellite OFDM signals. Addressing the complexities inherent in the broad bandwidth, substantial dynamic range, and weak prior knowledge associated with LEO satellite OFDM signal monitoring, this study introduces an innovative methodology. This approach harnesses a broadband parallel flexible filtering variable sampling technique to facilitate the real-time observation of satellite OFDM signals across a wide bandwidth spectrum. Moreover, the research presents a dynamic compensation technique, which utilizes ephemeris information, to mitigate frequency offset and amplitude fading issues within the monitored signals. Post compensation, the study conducts signal identification and parameter analysis, utilizing the intrinsic features of OFDM signals. This technique empowers real-time monitoring and the accurate analysis of satellite broadband OFDM signals, ensuring robust performance in scenarios characterized by weak prior knowledge and significant dynamic variations.

1. Introduction

Satellite communication network systems are poised to become a vital component of future global networks, as well as integral to the development of 5G and 6G technologies [1,2,3]. Contemporary Low Earth Orbit (LEO) satellite internet primarily employs broadband OFDM signals for communication within dynamic channels [4,5,6]. Ensuring the proper functioning of LEO satellite communication systems necessitates addressing the critical challenge of monitoring the OFDM signals of the dynamic channel environment of LEO satellites.

Presently, LEO communicate signals mainly use the Ka band (20–30 GHz) and Ku band (10–18 GHz). These millimeter-wave signals exhibit line-of-sight propagation traits and are acutely susceptible to atmospheric attenuation. Characterized by their dynamic nature and pronounced Doppler frequency shifts, LEO satellite channels, due to their lower orbits and higher velocities, necessitate dynamic compensation mechanisms [7,8,9]. The signal bandwidth of LEO constellations is rapidly expanding, with an average bandwidth extending up to 800 MHz (with plans for 2 GHz bandwidth signals), necessitating real-time monitoring and signal analysis capabilities that can accommodate ultra-wide bandwidths.

Existing monitoring systems are constrained by hardware limitations, with real-time analysis bandwidths typically spanning from 40 MHz to 160 MHz, sufficient for the analysis of signals from a single transponder. For the comprehensive analysis of multi-transponder, multi-signal, or wide bandwidth spectra, cyclic scanning and time-sharing analysis methods are employed, albeit with the inherent risk of missed detections due to the probabilistic nature of cyclic scanning [10,11,12]. To circumvent these limitations, ultra-wideband monitoring receivers and antenna systems are imperative, possessing the following critical performance attributes: an ultra-wideband reception spectrum, encompassing at least the Ka and Ku bands and extending to at least 30 GHz; ultra-large real-time bandwidth, necessitating devices capable of achieving at least 500 MHz in real-time; rapid frequency scanning, demanding swift and uninterrupted scanning across the entire frequency range to capture satellite signals; and high dynamic range, accommodating strong signals in close proximity and weak signals at greater distances [13].

Currently, in the domain of satellite signal detection and identification, research has predominantly been directed towards the modulation recognition of single-carrier signals, with a paucity of studies addressing the identification of LEO satellite OFDM signals. Feature-based modulation recognition methods and likelihood ratio-based modulation recognition methods are the predominant techniques employed. Likelihood ratio-based methods, while achieving optimal identification performance [14], are encumbered by high computational complexity and poor robustness. Consequently, feature-based methods have gained widespread adoption due to their low computational complexity, high identification efficiency, and robust performance, spurring extensive research efforts. Feature-based modulation identification methods encompass a myriad of approaches for extracting modulation mode recognition feature parameters, including the utilization of digital signal time-frequency characteristics [15], higher-order cyclostationary statistics [16], algorithms predicated on cyclic spectrum [17], and algorithms based on autocorrelation [18]. Furthermore, should the signal inadvertently forfeit its circularity as a consequence of the dynamic nature of the Low Earth Orbit satellite channel, the application of the Widely Linear Estimation technique would be appropriate for the subsequent analysis of the signal characteristics [19]. However, these studies have not adequately accounted for the actual channel conditions of LEO satellite communication, rendering the extracted features inapplicable to actual channel variations [20,21,22].

With the burgeoning development of deep learning algorithms, the application of neural networks in OFDM signal modulation identification has emerged as a cutting-edge trend [23,24,25]. Notable contributions include the proposal of an improved deep neural network (DNN) identification method by Xie et al. [26] and the utilization of long short-term memory (LSTM) neural networks and deep residual networks (ResNet) by Wang et al. to expedite the training process and enhance the accuracy of signal modulation mode identification [27]. Furthermore, Wang et al. [28] have employed a novel approach that abstracts the time-frequency characteristics of modulated signals using the short-time Fourier transform, followed by a conjunction with the instantaneous autocorrelation function to facilitate modulation classification. Li et al. [29] have leveraged the wavelet transform to abstract signal feature maps, which are subsequently classified using convolutional neural networks (CNNs) [30]. Although deep learning-based modulation identification methods have the capacity to extract a wealth of nonlinear features from signals [31], their practical application is hindered by the requirement for a substantial number of training samples and prolonged training times.

Existing research has the following limitations. Firstly, the majority of LEO satellite monitoring systems rely on spectrum scanning for spectrum analysis, characterized by limited real-time analysis bandwidth. Consequently, burst signals may be overlooked due to scanning durations and probabilistic occurrences. Additionally, real-time spectrum monitoring architectures are constrained by device computational resources and other factors. Secondly, the low orbits; high velocities; and intricate, dynamic channel conditions of LEO satellites directly impact satellite signals, inducing power attenuation and Doppler frequency offset. Thirdly, despite the prevalent use of OFDM signals in LEO satellites, research on the identification of LEO satellite OFDM signals is limited. Current research primarily concentrates on the modulation identification of single-carrier signals. To mitigate these issues, this paper introduces a monitoring methodology tailored for weak prior high-dynamic satellite wideband OFDM signals. Firstly, an architecture for the large-bandwidth real-time acquisition and monitoring of LEO satellite signals is employed. Within this framework, a variable-rate filtering algorithm based on digital resampling is designed, which enables flexible variable sampling rates in an all-digital parallel manner. This approach mitigates computational resource requirements and processing delays. Secondly, a dynamic compensation method leverages ephemeris information to transform the satellite’s coordinate system into the geodetic coordinate system, facilitating the extraction of ephemeris information and enabling dynamic compensation for LEO satellite signals. Finally, for LEO satellite OFDM signals, a multidimensional intrinsic feature extraction technique is implemented for joint identification.

In summary, we have made the following three main contributions:

• In response to the challenges posed by the wide bandwidth and high dynamics of signals from Low Earth Orbit satellites, we propose a flexible and adaptive filtering method that enables the parallel real-time acquisition and analysis of broadband signals. The method primarily relies on a dynamically adaptive interpolation filter, which adjusts its parameters in real-time according to the frontend sampling requirements. This architecture ensures that the received signal, after passing through the structure, conforms to a fixed oversampling rate, thereby reducing the complexity of subsequent signal analysis. This approach effectively addresses the complexity and large latency associated with the acquisition and processing of wideband dynamic signals.
• In response to the complex frequency offset variations in Low Earth Orbit satellite links, we propose a Doppler frequency offset compensation method based on the coordinate system transformation. The method predominantly leverages the ephemeris information of Low Earth Orbit satellites for frequency offset estimation. By investigating the rapid parsing of ephemeris data and frequency offset calculation and compensation methods across different coordinate systems, it achieves a reduction in computational complexity. This method effectively addresses the high complexity associated with Doppler frequency offset estimation in dynamic channels.
• In response to the challenges posed by the complex environment and lack of prior knowledge of non-cooperative signals in LEO satellite scenarios, we propose a modulation identification method based on the intrinsic features of OFDM. This method performs a multi-domain joint analysis and extraction of endogenous features of OFDM signals lacking prior knowledge, including time-frequency domain, higher-order domain, and entropy domain characteristics, and effectively enhances the accuracy of OFDM signal identification under weak prior knowledge.

The subsequent sections provide a brief introduction to the contents of this paper. Section 2 delineates the generation process of OFDM signals and the blind detection and reception model in detail and conducts an in-depth analysis of the cyclic correlation properties of CP-OFDM signals. In Section 3, we propose a flexible monitoring algorithm based on resampling filtering and elucidate the design principles of the resampling filter comprehensively. Section 4 introduces a Doppler frequency offset estimation algorithm predicated on satellite ephemeris and coordinate system transformation and elaborates on the implementation of the algorithm. In Section 5, we present a identification methodology for OFDM signals that leverages intrinsic features, including detection, structure identification, and subcarrier modulation identification, with thorough descriptions of these approaches. Simulation outcomes are presented in Section 6, and Section 7 provides a summary and conclusion of the paper.

2. System Model

2.1. System Model of OFDM Signal Monitoring

Figure 1 presents a model of a LEO satellite broadband OFDM signal monitoring system. At the OFDM communication link’s transmission end, the process begins with channel convolutional encoding of the input information sequence. The encoded data are then rearranged via row-column interleaving to improve error resistance, followed by modulation using PSK or QAM to meet various communication environment needs [32]. Pilot signals are inserted for receiver-end channel estimation, and after serial-to-parallel conversion and IFFT, the signal’s anti-interference capability is enhanced. A cyclic prefix and windowing functions are applied to reduce temporal interference and smooth signal edges. Finally, after PAPR suppression, the signal is transmitted with the synchronization header incorporated [33].

The blind monitoring system for OFDM signals consists of several key modules, including OFDM signal detection, parameter estimation, signal identification, and blind demodulation. The detection module performs real-time signal collection and OFDM signal existence checks, along with preprocessing. The parameter estimation module estimates critical parameters such as subcarrier spacing, sampling rate, number of subcarriers, and cyclic prefix length using joint time-frequency analysis on preprocessed data. The signal identification module extracts multidimensional features from the OFDM signal post-cyclic prefix removal and identifies the modulation method for each subcarrier.

This paper focuses on contributions to OFDM signal parameter estimation and identification, proposing a resampling filter-based wideband flexible monitoring method for the detection module, a dynamic compensation algorithm for satellite frequency offset based on ephemeris and coordinate system transformation for the estimation module, and an intrinsic feature-based OFDM signal identification method for the identification module.

These contributions are detailed in Section 3, Section 4, and Section 6 of the paper, respectively, providing a comprehensive approach to enhancing the monitoring and analysis of LEO satellite broadband OFDM signals.

2.2. Model of OFDM Multi-Carrier Signal

Upon the serial-to-parallel conversion of the symbol stream representing the original information, N parallel symbol streams are generated. These streams are modulated by different subcarriers and then transmitted using OFDM through inverse Fourier transform to the designated radio frequency signal. The signal is then transmitted through the transmitter. Let $X_k\left[l\right]$ denote the k-th OFDM transmission symbol on the l-th subcarrier. The total length of an OFDM symbol is denoted by G, where $G=N+D$. N represents the length of the useful symbols (sample points) and D denotes the length of the cyclic prefix (sample points). Due to serial-to-parallel conversion, the transmission time of G symbols is expanded to $G{T}_s$, where $T_s$ represents the period of symbol $X\left[l\right]$, and $G{T}_s$ is the duration of a single OFDM symbol, i.e., $T_{sym}=G{T}_s$. Let ${\psi }_{k,l}\left(t\right)$ represent the k-th OFDM signal on the l-th subcarrier.

$${\psi }_{k,l}\left(t\right)=\left\{\begin{array}{cc}{e}^{j2\pi f_l(t-k{T}_{sym})},\hfill & 0<t\le T_{sym},\hfill \\ 0,\hfill & other.\hfill \end{array}\right.$$

(1)

The expression for the baseband time-continuous OFDM signal is presented as

$$x_k\left(t\right)=\sum _{i=0}^{k-1}\sum _{l=0}^{G-1}{X}_l\left(l\right){\psi }_{k_{,}l}\left(t\right).$$

(2)

The baseband signal of OFDM, as represented by Equation (2), is sampled with the sampling time $t=k{T}_{sym}+n{T}_s$, where $T_s={\displaystyle \frac{T_{sym}}{G}}$ and $f_l={\displaystyle \frac{l}{T_{sym}}}$. This results in the discrete-time expression for the OFDM symbol as

$$x_k\left(n\right)=\sum _{i=0}^{k-1}\sum _{l=0}^{G-1}{X}_k\left(l\right)e^{j2\pi ln/G},n=0,1,\cdots ,N-1.$$

(3)

The baseband OFDM receive symbol expression can be readily obtained as

$$s_k\left(t\right)=\sum _{k=0}^{k-1}\sum _{l=0}^{G-1}{X}_k\left(l\right)e^{2\pi f_l(t-k{T}_{sym})},k{T}_{sym}<t<k{T}_{sym}+n{T}_s.$$

(4)

Considering the bandlimited nature of the channel, the baseband OFDM receive symbol can be expressed as $X_k\left(l\right)=a_{k,l}{g}_a(t-k{T}_s)$, where $a_{k,l}$ represents the corresponding level value and $g_a(t-k{T}_s)$ represents the pulse shape of the signal.

The time-domain expression for the l-th sample of the k-th OFDM symbol with a cyclic prefix equals:

$$c_{k,l}={\displaystyle \frac{1}{\sqrt{N}}}\sum _{n=0}^{G-1}{a}_{k,n}{e}^{\left(j2\pi \right(1-D)n/N)},l=0,1,\cdots ,G-1.$$

(5)

Hence, under the condition of additive Gaussian white noise, the multicarrier OFDM signal receive model expression is

$$r\left(t\right)=s_k\left(t\right)+n\left(t\right).$$

(6)

In the expression, $n\left(t\right)$ denotes a zero-mean Gaussian white noise, $n\left(t\right)\sim CN(0,2{\sigma }^2)$, where $2{\sigma }^2$ signifies the power of the noise.

2.3. OFDM Signal Cyclic Correlation Analysis

The cyclic prefix in an OFDM signal is intricately designed to reduce the impact of multipath fading. This prefix is a repetition of a segment of the signal within the OFDM symbol, resulting in a correlation between the cyclic prefix and the informative part of the signal. Furthermore, in the context of single-carrier signals, the transmission symbols are independently and identically distributed, such that correlation exists solely when there is no delay. Utilizing this property, the modulation type of an OFDM signal can be identified.

In the k-th OFDM symbol, the auto-correlation of the time-domain data $c_{k,l}$ at the delayed sample point $n_{\tau }(0\le n_{\tau }\le N)$ equals

$$\begin{array}{c}E\left(c_{k,l}{c}_{k,l-n_{\tau }}^{*}\right)=1/NE[\sum _{n=0}^{N-1}{a}_{k,n}{e}^{\left(j2\pi \right(1-D)n/N)}·\sum _{n=0}^{N-1}{a}_{k,n}^{*}{e}^{(-j2\pi (1-n_{\tau }-D)n/N)}]=\\ {\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}E\left(a_{k,n}{a}_{k,n}^{*}\right)exp[j2\pi (l-D)n/N-\\ j2\pi \left(l-n_{\tau }-D\right)n/N]={\displaystyle \frac{{\sigma }_a^2}{N}}\sum _{n=0}^{N-1}exp\left(j2\pi n_{\tau }n/N\right).\end{array}$$

(7)

In practice, this function is approximated as

$$E\left(c_{k,l}{c}_{k,l-n}^{*}\right)=\left\{\begin{array}{cc}{\sigma }_a^2,\hfill & n_{\tau }=0,N,\hfill \\ 0,\hfill & others.\hfill \end{array}\right.$$

(8)

The correlation properties introduced by the cyclic prefix, disregarding the initial delay, frequency offset, and noise, result in the auto-correlation function of the received signal being

$$\begin{array}{c}{R}_r(t,\tau )=E[r\left(t\right)·r^{*}(t-\tau )]=\\ E[{\displaystyle \sum _k}{\displaystyle \sum _{l=0}^{G-1}}{c}_{k,l}g\left(t-l{T}_c-k{T}_s\right)·{\displaystyle \sum _k}{\displaystyle \sum _{l=0}^{G-1}}{c}_{k,l-n}^{*}{g}^{*}\left(t-{lT}_c-{kT}_s-\tau )\right]=\\ {\displaystyle \sum _k}E[{\displaystyle \sum _{l=0}^{G-1}}{c}_{k,l}g(t-l{T}_c-k{T}_s)·{\displaystyle \sum _{l=0}^{G-1}}{c}_{k,l-n}^{*}{\tau }^{g^{*}}(t-{lT}_c-{kT}_s-\tau )],\end{array}$$

(9)

wherein the delay $\tau =n_{\tau }{T}_c$.

As indicated by Equation (9), the cyclic period of the OFDM signal has two instances, which can be expressed as

$$\begin{array}{c}{R}_r(t,\tau )=\left\{\begin{array}{cc}{\sigma }_a^2\sum _k\sum _{l=0}^{G-1}g(t-l{T}_c-k{T}_s)·g^{*}(t-{lT}_c-{kT}_s-\tau ),\hfill & \left|\tau \right|<T_c,\hfill \\ {\sigma }_a^2\sum _k\sum _{l=N}^{G-1}g(t-l{T}_c-k{T}_s)·g^{*}(t-{lT}_c-{kT}_s-\tau ),\hfill & |{\tau }_N|<T_c,\hfill \\ 0,\hfill & \tau =others,\hfill \end{array}\right.\end{array}$$

(10)

wherein ${\tau }_N=|-N{T}_c|$.

From the above proof, it can be determined that the OFDM signal possesses second-order cyclostationary characteristics, with the OFDM signal having two cyclic periods, $T_c$ and $T_s$. The autocorrelation function $R_r(t,\tau )$ exhibits periodicity. Based on the properties of the Fourier transform, the Fourier transform with respect to t of the autocorrelation function $R_r(t,\tau )$ will result in discrete spectral lines. The positions of these spectral lines are at $\alpha ={\displaystyle \frac{m}{T_c}}$ or $\alpha ={\displaystyle \frac{m}{T_s}}$, where $m\in \mathbb{Z}$. Here, $\alpha$ is referred to as the cyclic frequency. Performing the Fourier transform of $R_r(t,\tau )$ with respect to t yields:

$$\begin{array}{cc}\hfill & |R_r^{\alpha }\left(\tau \right)|=\left\{\begin{array}{cc}{\displaystyle \frac{{\sigma }_a^2}{T_c}}|{\displaystyle \frac{sin\left[\pi \alpha \right(T_c-|\tau |\left)\right]}{\pi \alpha }}|,\hfill & \alpha =m/T_c,|\tau |\le T_c,m\in Z,\hfill \\ {\displaystyle \frac{{\sigma }_a^2}{T_s}}|{\displaystyle \frac{sin\left(\pi \alpha T_{c}D\right)}{sin\left(\pi \alpha T_c\right)}}| · |{\displaystyle \frac{sin\left[\pi \alpha \right(T_c-|{\tau }_N|)]}{\pi \alpha }}|,\hfill & \alpha =m/T_c,|{\tau }_N|\le T_c,m\in Z,\hfill \\ 0,\hfill & others.\hfill \end{array}\right.\hfill \end{array}$$

(11)

3. Resampling Filter-Based Wideband Flexible Monitoring

This section presents a broadband flexible monitoring approach based on resampling filters, which integrates resampling technology with low-pass filtering to enable the flexible adjustment of the sampling rate. The approach reduces the sampling rate to four times the symbol rate and subsequently applies low-pass filtering to achieve rate matching.

Due to the receiver’s lack of prior knowledge about the symbol rate during blind signal processing, a high sampling rate is often employed when sampling the signal. This leads to an excessively large observation bandwidth for the received signal, whereas the actual signal bandwidth is relatively narrow, which can adversely affect subsequent identification performance. In this paper, a digital resampling-based flexible low-pass filter is designed, which, in conjunction with the identified signal bandwidth, reduces the sampling rate to four times the symbol rate of the signal.

In this paper, we leverage digital resampling techniques to interpolate the target rate sampling points directly from the known sampling points using interpolation filters. In accordance with the Nyquist Sampling Theorem, as long as the sampling frequency is greater than twice the signal bandwidth, the original signal (y(t)) can be recovered from the sampled values $x\left(n{T}_s\right)$.

$$y\left(t\right)=\sum _{n=-\infty }^{+\infty }x\left(n{T}_s\right)h(t-n{T}_s).$$

(12)

In the expression, $T_s$ represents the sampling period, $h\left(t\right)={\displaystyle \frac{sin(\pi t/T_s)}{\pi t/T_s}}$

Subsequently, the recovered signal is subject to resampling to obtain

$$y\left(k{T}_i\right)=\sum _{n=-\infty }^{+\infty }x\left(n{T}_s\right)h(k{T}_i-n{T}_s)=\sum _{n=-\infty }^{+\infty }x\left(n{T}_s\right)h\left(T_s\left({\displaystyle \frac{k{T}_i}{T_s}}-n\right)\right).$$

(13)

Let $m_k=⌊{\displaystyle \frac{k{T}_i}{T_s}}⌋$ (where $\lfloor ·\rfloor$ denotes the floor function, which rounds down to the nearest integer), and ${\mu }_k={\displaystyle \frac{k{T}_i}{T_s}}-m_k$. Substituting these into Equation (13) yields.

$$y\left(k{T}_i\right)=\sum _{n=-\infty }^{+\infty }x\left(n{T}_s\right)h\left(\left(m_k+{\mu }_k-n\right)T_s\right).$$

(14)

Upon defining $m=m_k-n$, we obtain

$$y\left(k{T}_i\right)=\sum _{m=-\infty }^{+\infty }x\left(\left(m_k-m\right)T_s\right)h\left(\left({\mu }_k+m\right)T_s\right).$$

(15)

Equation (15) demonstrates that our approach merely uses the reconstruction of the original signal as an intermediate step, whereas in practice, new sampling values can be directly obtained through the weighted summation of known sampled values. However, it is impossible to achieve a completely ideal resampling process because we cannot perform an infinite series of weighted summations. Instead, we resort to using a finite-length FIR (Finite Impulse Response) filter with performance close to the ideal, as a substitute. The truncated form of Equation (15) after applying the FIR filter can be expressed as

$$y\left(k{T}_i\right)=\sum _{m=-I_1}^{I_2}x\left(\left(m_k-m\right)T_s\right)h\left(\left({\mu }_k+m\right)T_s\right)=\sum _{m=-I_1}^{I_2}{h}_I\left({\mu }_k+m\right)x\left(m_k-m\right).$$

(16)

In this paper, we employ a cubic interpolation polynomial to approximate the filter for calculating the weighted coefficients, aiming to achieve performance as close as possible to that of an ideal filter.

The impulse response expression for cubic interpolation is

$$h_c\left(t\right)=\left\{\begin{array}{cc}1-{\displaystyle \frac{1}{2}}\left|t\right|-{\left|t\right|}^2+{\displaystyle \frac{1}{2}}{\left|t\right|}^3,\hfill & \left|t\right|<1,\hfill \\ 1-{\displaystyle \frac{11}{6}}\left|t\right|+{\left|t\right|}^2-{\displaystyle \frac{1}{6}}{\left|t\right|}^3,\hfill & 1\le \left|t\right|<2,\hfill \\ 0,\hfill & \left|t\right|>2.\hfill \end{array}\right.$$

(17)

The frequency response is given by

$$H_c\left(f\right)={\displaystyle \frac{e^{-j4\pi f{t}_S}(2{\left(\pi f{T}_s\right)}^2+3){(e^{j2\pi f{t}_S}-1)}^4}{48{\left(\pi f{T}_s\right)}^4}}.$$

(18)

Substitute Equation (17) into Equation (16), and by setting $I_1=2$, $I_2=1$, we obtain the expression for the cubic interpolation coefficient with respect to $\mu$ as

$$\begin{array}{ccc}\hfill C_{-2}& ={\displaystyle \frac{1}{6}}{\mu }^3-{\displaystyle \frac{1}{6}}\mu ,\hfill & \hfill C_{-1}=-{\displaystyle \frac{1}{2}}{\mu }^3+{\displaystyle \frac{1}{2}}{\mu }^2+\mu ,\\ \hfill C_0& ={\displaystyle \frac{1}{2}}{\mu }^3-{\mu }^2-{\displaystyle \frac{1}{2}}\mu +1,\hfill & \hfill C_1=-{\displaystyle \frac{1}{6}}{\mu }^3+{\displaystyle \frac{1}{2}}{\mu }^2-{\displaystyle \frac{1}{3}}\mu .\end{array}$$

(19)

The architecture of the resampling filter employing cubic interpolation is represented in Figure 2.

For the baseband OFDM signal post-low-pass filtering, the aforementioned resampling filter based on cubic interpolation can be employed to achieve flexible and adjustable sampling rates, thereby facilitating the high-precision real-time monitoring of the broadband signal.

4. Satellite Frequency Offset Dynamic Compensation Algorithm Based on Ephemeris and Coordinate System Transformation

The LEO satellite exhibits characteristics of low altitude and high velocity, which inherently imbue the dynamic nature of the satellite’s trajectory. This dynamism consequently results in Doppler frequency shifts in the received signal [34]. This section delineates a Doppler frequency offset compensation algorithm tailored for LEO satellites. In the literature [35], the dynamics of LEO satellite signals are analyzed through the simulation of satellite orbits and propagation environments. The methodology employed in this paper necessitates a significantly reduced amount of a priori ephemeris information for dynamic compensation. Initially, the “two-line” satellite orbit element format is derived from the satellite ephemeris. Utilizing these elements, the satellite’s position and velocity in the Earth-Centered Inertial (ECI) coordinate system are computed. Subsequently, these coordinates are transformed into the Earth-Centered Fixed (ECF) coordinate system, facilitating the description of the satellite’s relative position and velocity in terms of latitude and longitude. Finally, the Doppler frequency offset for the LEO satellite link is calculated. In the following, a detailed introduction will be provided. The calculation process is illustrated in Figure 3.

As dictated by Kepler’s laws, the satellite’s trajectory is an ellipse that pierces through the Earth’s center plane. The satellite’s orbital path can be accurately defined by a suite of Keplerian orbital parameters. These parameters facilitate the estimation of the satellite’s relative position and velocity with respect to the Earth at any given moment within the ECI coordinate system. The Keplerian parameters that delineate the satellite’s orbital trajectory include the right ascension of the ascending node ($\mathsf{\Omega }$), the argument of perigee ($\omega$), the orbital inclination (i), the semi-major axis of the orbit (b), the eccentricity of the orbit (e), and the mean anomaly (M).

In Equation (20), n represents the mean motion, which can be calculated from the semi-major axis b of the orbit. $t_p$ denotes the epoch of perigee passage, and $\mu$ is the gravitational constant of the Earth. Utilizing the mean motion n in place of the semi-major axis b as an orbital parameter is a common practice currently adopted by the National Aeronautics and Space Administration (NASA) in the “two-line” satellite orbit element format. The corresponding six orbital parameters are $\mathsf{\Omega }$, i, e, $\omega$, M, and n.

$$\left\{\begin{array}{c}n=2\pi /T={(\mu /a^3)}^{1/2},\hfill \\ M=n(t-t_p).\hfill \end{array}\right.$$

(20)

The satellite ephemeris can be solved to determine the position and velocity of the satellite at any arbitrary time t within the ECI coordinate system as

$$\left\{\begin{array}{c}{\mathit{R}}_{\mathrm{s}}=a\left(\cos E-e\right)\mathit{P}+b\sqrt{1-e^2}\mathrm{sinE}\mathit{Q},\hfill \\ {\mathit{V}}_{\mathrm{s}}=\sqrt{\mathit{\mu }}\left[{\displaystyle \frac{-\sin E}{\sqrt{b}\left(1-e\cos E\right)}}\mathit{P}+{\displaystyle \frac{\sqrt{1-e^2}\cos E}{\sqrt{b}\left(1-e\cos E\right)}}\mathit{Q}\right].\hfill \end{array}\right.$$

(21)

Within the equation, E denotes the eccentric anomaly, which can be determined given the mean anomaly M and the eccentricity e as follows:

$$E-e\sin E=M.$$

(22)

The vectors P and Q can be determined, respectively, by Equations (15) and (16).

$$\mathit{P}=\left[\begin{array}{c}{P}_X\\ P_Y\\ P_Z\end{array}\right]=\left[\begin{array}{c}cos\omega \cos \mathsf{\Omega }-\sin \omega \sin \mathsf{\Omega }\cos i\\ \cos \omega \cos \mathsf{\Omega }+\sin \omega \sin \mathsf{\Omega }\cos i\\ \sin \omega \sin i\end{array}\right].$$

(23)

$$\mathit{Q}=\left[\begin{array}{c}{Q}_x\\ Q_y\\ Q_z\end{array}\right]=\left[\begin{array}{c}-sin\omega \cos \mathsf{\Omega }-\cos \omega \sin \mathsf{\Omega }\cos i\\ -\sin \omega \cos \mathsf{\Omega }+\cos \omega \sin \mathsf{\Omega }\cos i\\ \cos \omega \sin i\end{array}\right].$$

(24)

The transformation from the ECI coordinate system to the ECF coordinate system is necessary to facilitate the description of the satellite’s relative position and velocity in terms of latitude and longitude. This conversion requires the utilization of the relationships between the ECI coordinate system and the instantaneous mean equatorial coordinate system, the instantaneous mean equatorial coordinate system and the instantaneous true equatorial coordinate system, the instantaneous true equatorial coordinate system and the quasi-inertial coordinate system, and the quasi-inertial coordinate system and the ECF coordinate system. To determine the position and velocity of the terminal in the ECF coordinate system, it is first necessary to establish the transformation matrices for position and velocity as $\mathit{\alpha }$ and $\mathit{\beta }$ (which are constant matrices). Consequently, the position and velocity of the satellite in the ECF coordinate system can be expressed as:

$$\left\{\begin{array}{c}{\mathit{R}}_{\mathrm{s}}^{\prime }=\mathit{\alpha }·{\mathit{R}}_{\mathrm{s}},\hfill \\ {\mathit{V}}_{\mathrm{s}}^{\prime }=\mathit{\beta }·{\mathit{V}}_{\mathrm{s}}.\hfill \end{array}\right.$$

(25)

Typically, the geographical position of a receiver is represented using longitude (L), latitude (B), and elevation (H). This representation is converted to a vector ${\mathbf{R}}_0$ in the ECF coordinate system through Equation (26).

$${\mathit{R}}_0=\left[\begin{array}{c}{X}_0\\ Y_0\\ Z_0\end{array}\right]=\left[\begin{array}{c}(N+H)\cos B\cos L\\ (N+H)\cos B\sin L\\ \left[N\right(1-e_0^2)+H]\sin B\end{array}\right].$$

(26)

$$N^2={\displaystyle \frac{b_0}{\sqrt{1-e_0^2{sin}^{2}B}}}.$$

(27)

Within the context of the equation, $a_0$ represents the semi-major axis of the ellipsoid, with a value of $a_0=6378.137$ km; $e_0$ denotes the first eccentricity, where $e_0^2=0.0069438$.

Consequently, the relative position and relative velocity between the satellite and the ground station are given by:

$$\left\{\begin{array}{c}\mathit{R}={\mathit{R}}_{\mathit{s}}^{\prime }-{\mathit{R}}_{\mathbf{0}},\hfill \\ \mathit{V}={\mathit{V}}_{\mathit{s}}^{\prime }-{\mathit{V}}_{\mathbf{0}}.\hfill \end{array}\right.$$

(28)

Furthermore, based on the original definition of the Doppler shift, the following can be determined:

$$f_{\mathrm{D}}=f_0·{\displaystyle \frac{\left|V\right|cos\alpha }{c}}.$$

(29)

In the equation, $f_0$ denotes the nominal radio frequency of the received signal; c is the speed of light, with $c=3.0\times {10}^8$ m/s; and $\alpha$ represents the angle between the relative velocity V of the satellite with respect to the ground station and the line connecting the satellite and the ground station, denoted by $\mathbf{R}$. Within the OFDM parameter estimation module, the frequency offset compensation is applied to counteract the frequency offset generated by the actual motion, thereby ensuring that the processed signal is an OFDM signal from a Low Earth Orbit satellite with either no frequency offset or a very small frequency offset.

5. OFDM Signal Identification Based on Inherent Features

This sub-section presents an overview of OFDM signal detection, analysis, and modulation identification techniques. Initially, high-order cumulants are computed from the sampled data of LEO satellite signals, leveraging their statistical properties for OFDM signal detection. Subsequently, the correlation properties of the cyclic prefix inherent in OFDM signals are utilized to analyze and estimate signal parameters. Finally, the multidimensional endogenous features of each subcarrier within the OFDM signal are extracted to achieve modulation identification across the respective subcarriers.

5.1. High-Order Cumulant-Based OFDM Signal Detection

High-order cumulants provide an effective means for detecting non-Gaussian burst signals in a Gaussian noise environment. In an ideal Gaussian noise setting, the fourth-order cumulant is expected to be zero; hence, a non-zero measured value serves as a reliable indicator of the presence of a non-Gaussian signal. OFDM signals, which are composed of the superposition of multiple subcarriers, can be approximated to have a Gaussian amplitude distribution when the number of subcarriers is sufficiently large. This is in contrast to the amplitude distribution of single-carrier digital modulation signals, which do not exhibit Gaussian characteristics. Consequently, high-order cumulant-based OFDM signal detection algorithms leverage the asymptotic Gaussian distribution property of the OFDM signal’s time-domain envelope, as well as the suppressive effect of high-order cumulants on Gaussian random processes, thereby facilitating effective identification of the signal.

The characteristic function of a random variable $\phi \left(\upsilon \right)$ can be expressed as

$$\phi \left(v\right)=E\left\{e^{jx}\right\}={\int }_{-\infty }^{\infty }f\left(x\right)e^{jx}dx.$$

(30)

In the given formulation, $E\left\{·\right\}$ represents the expectation is being calculated, and $f\left(x\right)$ denotes the probability density function (PDF).

The k-th moment of a random variable can be obtained by taking the k-th derivative of the characteristic function $\phi \left(\upsilon \right)$ at the origin, as expressed in Equation (23).

$$M_k={\phi }^{\widehat{k}}\left(0\right)=E\left\{x^{\widehat{k}}\right\}={\displaystyle \frac{d^{\widehat{k}}\phi \left(v\right)}{d{v}^{\widehat{k}}}}{|}_{\nu =0}.$$

(31)

The k-th cumulant of a random variable can be derived by computing the k-th derivative of the logarithm of the characteristic function $\phi \left(\upsilon \right)$, evaluated at the origin, as expressed by the following equation:

$$C_k={\displaystyle \frac{d^{\widehat{k}}ln\left(\phi \left(\nu \right)\right)}{d{\nu }^k}}{|}_{v=0}.$$

(32)

The k-th moment and the k-th cumulant of a random variable satisfy the following relationship:

$$C_x\left(\mathbf{H}\right)=\sum _{{\bigcup }_{k=1}^q{H}_k=\mathbf{H}}{\left(-1\right)}^{q-1}\left(q-1\right)!\prod _{k=1}^q{M}_x\left(H_k\right).$$

(33)

In the expression provided, $\mathbf{H}=\left\{\begin{array}{c}1,2,...,l\end{array}\right\}$ denotes the set of symbols associated with the random variable $\mathbf{x}=\left\{\begin{array}{c}{x}_1,x_2,\dots ,x_l\end{array}\right\}$, and $(·)!$ represents the factorial operation.

It follows that the high-order moment and high-order cumulant of the complex random variable y can be expressed as

$$M_{p+q,p}=E\left[y^p{\left(y^{*}\right)}^q\right],$$

(34)

$$C_{p+q,p}=\mathrm{cum}\left[y_1,y_2,\dots ,y_p,y_1^{*},y_2^{*},\dots ,y_q^{*}\right].$$

(35)

In the context provided, $cum(·)$ refers to the operation of calculating cumulants.

The second-order cumulant of a zero-mean stationary signal $x\left(n\right)$ can be expressed as

$$C_{20}\left({\tau }_1\right)=E\left\{x\left(n\right)x(n+{\tau }_1)\right\},$$

(36)

$$C_{21}\left({\tau }_1\right)=E\left\{x^{*}\left(n\right)x\left(n+{\tau }_1\right)\right\}.$$

(37)

The fourth-order cumulant of the signal $x\left(n\right)$ can be represented as

$$\begin{array}{c}{\mathrm{C}}_{42}\left({\tau }_1,{\tau }_2,{\tau }_3\right)=E\left\{x^{*}\left(n\right)x\left(n+{\tau }_1\right)x\left(n+{\tau }_2\right)x^{*}\left(n+{\tau }_3\right)\right\}\\ -E\left\{x^{*}\left(n\right)x\left(n+{\tau }_1\right)\right\}E\left\{x\left(n+{\tau }_2\right)x^{*}\left(n+{\tau }_3\right)\right\}\\ -E\left\{x^{*}\left(n\right)x(n+{\tau }_2)\right\}E\left\{x\left(n+{\tau }_1\right)x^{*}\left(n+{\tau }_3\right)\right\}\\ -E\left\{x^{*}\left(n\right)x\left(n+{\tau }_3\right)\right\}E\left\{x\left(n+{\tau }_1\right)x\left(n+{\tau }_2\right)\right\}.\end{array}$$

(38)

In the expression provided, ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$ represent time delays, which typically can be set to 0.

The OFDM received signal $r\left(t\right)$ can be represented as

$$\left\{\begin{array}{c}r\left(t\right)={\displaystyle \frac{1}{\sqrt{N}}}\sum _{n=0}^{N-1}s\left(t\right),\hfill \\ s\left(t\right)=s_{n}exp\left[j2\pi \left(f_c+n\Delta f\right)\left(t-T_{\mathrm{sym}}\right)+{\phi }_0\right]*\mathrm{g}\left(t-T_{\mathrm{sym}}\right).\hfill \end{array}\right.$$

(39)

In the expression provided, $s_n$ is the transmitted signal, $f_c$ is the carrier frequency, and ${\varphi }_0$ is the initial phase.

Consequently, the fourth-order cumulant of the said signal is expressed as

$$C_{4x}\left[\mathrm{r}\left(t\right)\right]={\displaystyle \frac{1}{\sqrt{N_{\mathrm{SC}}^4}}}\sum _{n=-N_{\mathrm{SC}}/2}^{N_{\mathrm{SC}}/2}{C}_{4x}\left[\sum _{k=1}^{\infty }s\left(t\right)\right].$$

(40)

From the aforementioned expression, it can be observed that due to the independence of the subcarriers in OFDM, the fourth-order cumulant of an OFDM signal is reduced by a factor of ${\displaystyle \frac{1}{N_{\mathrm{SC}}}})$ compared to that of a single-carrier system.

In this section, we differentiate between OFDM and single-carrier signals through the use of eigenvalue $d_1$ for detection and discrimination purposes.

$$d_1={\displaystyle \frac{C_{42}}{C_{21}^2}}=2\left(k_{20}-1\right)=2\left({\displaystyle \frac{M_{42}}{M_{21}^2}}-1\right).$$

(41)

As illustrated in

Table 1. Theoretical $d_1$ values for single-carrier and OFDM signals.

Signal	${\mathit{d}}_1$
MPSK	0.00
MFSK	0.00
MQAM	0.62–0.76
OFDM	2.00

, the theoretical eigenvalue $d_1$ for single-carrier and OFDM signals is presented. It is evident that the $d_1$ feature of OFDM signals is distinctly different from that of single-carrier signals, allowing for their identification through the establishment of decision thresholds.

The procedural flow of the OFDM signal detection algorithm based on higher-order cumulants is depicted in Algorithm 1.

Algorithm 1 Procedure Flow of the OFDM Signal Detection Algorithm Based on Higher-Order Cumulants

Input: receive signal $r\left(n\right)$;

Output: Detection Outcome: Differentiation Between OFDM Signal and Single-Carrier Signal;

1: Set the identification threshold ${\lambda }_{mul}$ as indicated in Table 1; 2: Calculate the higher-order moments $M_{42}$ and $M_{21}$ of the received signal in accordance with Equation (34); 3: Compute the feature value $d_1$ based on Equation (41). 4: if  $d_1<{\lambda }_{mul}$  then 5:     Identify the signal as an OFDM signal and output the identification result 6: else 7:     Identify the signal as a single-carrier signal and output the identification outcome. 8: end if

5.2. Analysis of OFDM Signal Parameters Utilizing Cyclic Correlation Characteristics

In Section 2.3, an analysis of the correlation properties of CP-OFDM has been conducted. Leveraging the correlation characteristics introduced by the CP to the OFDM signal, it is feasible to estimate the effective time length $T_{\mathrm{use}}$ of the OFDM symbol, the total symbol length $T_s$, and the length of the cyclic prefix $T_{\mathrm{cp}}$ with minimal prior knowledge. Additionally, corresponding estimations can be made for the number of subcarriers K and the oversampling rate $\rho$. The steps for estimating OFDM parameters are illustrated in Figure 4.

Given that the sampling frequency $F_s$ employed by the receiver is known, the accurate estimation of $N_s$, $N_{\mathrm{use}}$, and $N_{\mathrm{CP}}$ allows for the straightforward determination of $T_s$, $T_{\mathrm{use}}$, and $T_{\mathrm{CP}}$ by multiplying $N_s$, $N_{\mathrm{use}}$, and $N_{\mathrm{CP}}$ by ${\displaystyle \frac{1}{F_s}}$, respectively. Consequently, the most critical parameters to estimate initially are $N_s$, $N_{\mathrm{use}}$, and $N_{\mathrm{CP}}$.

According to Section 2.3, the second-order cyclic cumulant $C[\alpha ;\tau ]$ is expressed as

$$C[\alpha ;\tau ]={\displaystyle \frac{1}{N}}\sum _{n=0}^{N}r\left[n\right]r^{*}[n+\tau ]e^{-j2\pi \alpha n}\tau =0,1,...,M.$$

(42)

It follows that the cyclic correlation function possesses the following properties:

$$C[n;\tau ]=C[n+N_s;\tau ],C[n;\tau ]={\displaystyle \frac{1}{N}}\sum _{\alpha =0}^{N}C[\alpha ;\tau ]e^{j2\pi \alpha n}.$$

(43)

The expression above denotes that $N_s$ is the cyclic factor and also represents the length of a complete symbol period for the OFDM signal, which corresponds to the total symbol time duration. Therefore, within $N_s$ samples, the correlation function satisfies the following property: $C[n;\tau ]=C[n+N_{\mathrm{use}};\tau ]$ for $n\in N_{\mathrm{CP}}$. This indicates that the cyclic cumulant exhibits a bipartite peak characteristic: one due to the cyclic nature at every $N\mathrm{use}$ points, and the other because of the cyclic nature at every $N_s$ points. However, when $\tau =N_{\mathrm{use}}$, the peak caused by $N_{\mathrm{use}}$ vanishes, and at this point, the peaks of the cyclic cumulant $C[\alpha ;\tau ]$ represent $\pm k/T_s$, where $k=0,1,2,\dots$. The spacing between these peaks allows us to determine the number of samples corresponding to the total symbol time duration $N_s$. Hence, it is necessary to first estimate the number of samples corresponding to the effective symbol time duration $N_{\mathrm{use}}$ using the maximum likelihood method.

$${\widehat{N}}_{use}=arg\underset{\tau }{max}C[0;\tau ].$$

(44)

Consequently, the effective duration of the associated OFDM symbol can be expressed as

$${\widehat{T}}_{use}={\widehat{N}}_{use}\times 1/F_s.$$

(45)

It can be derived that the total duration of the OFDM symbol can be estimated as

$${\widehat{T}}_{sl}={\displaystyle \frac{1}{arg{max}_{\frac{1}{{\widehat{T}}_{use}+0.5^{*}{\widehat{T}}_{use}}\le \alpha \le \frac{1}{{\widehat{T}}_{use}}}C[\alpha ;N_{use}]}}.$$

(46)

Through the aforementioned estimation, the estimated value of the number of samples corresponding to the length of the cyclic prefix (CP) can be determined as

$${\widehat{N}}_{CP}={\widehat{N}}_s-{\widehat{N}}_{use}.$$

(47)

Thus, the estimated value for the length of the cyclic prefix $T_{\mathrm{CP}}$ is determined as $T_{CP}=N_{CP}\times 1/F_s$.

In LEO satellite systems, the number of OFDM subcarriers is $2^k$. Therefore, based on the signal bandwidth $\widehat{B}w$ and the effective symbol time duration, the number of subcarriers can be estimated.

$$\widehat{K}=2^{⌊{log}_2\left(\widehat{B}{w}^{*}{\widehat{T}}_{use}\right)⌋}.$$

(48)

Due to the discrepancy in sampling rates between the receiver and the transmitter, the estimation formula for the oversampling factor is determined as

$${\widehat{\rho }}_1={\displaystyle \frac{N_{use}}{\widehat{K}}},{\widehat{\rho }}_2=⌊{\displaystyle \frac{N_{use}}{\widehat{K}}}⌋.$$

(49)

5.3. Multidimensional Feature-Based OFDM Signal Identification

Based on the parameter analysis results described earlier, the cyclic prefix is removed from the received low-orbit satellite OFDM baseband signal. Subsequently, the Fast Fourier Transform (FFT) is applied to isolate the subcarrier signals across individual subchannels. By leveraging the multidimensional feature vectors of these subcarrier signals in conjunction, the modulation scheme employed for each subcarrier can be identified.

This paper introduces a multidimensional domain feature extraction technique, which yields a total of 13 types of signal features. The specific categories of these signal features are depicted in Figure 5.

Higher-order cumulants possess the property of reflecting the higher-order statistical characteristics of modulated signals, offering robustness against fading and effective suppression of Gaussian white noise. Moreover, different digital modulation signals exhibit distinct higher-order cumulants. Consequently, these cumulants with their unique features can serve as a basis for classifying modulated signals.

Based on Equations (34) and (35), it is feasible to compute the second-order, fourth-order, sixth-order, and eighth-order cumulants:

$$\begin{array}{cc}\hfill & C_{20}=M_{20},\hfill \\ \hfill & C_{21}=M_{21},\hfill \\ \hfill & C_{40}=M_{40}-3M_{20}^2,\hfill \\ \hfill & C_{41}=M_{41}-3M_{21}{M}_{20},\hfill \\ \hfill & C_{42}=M_{42}-{\left|M_{20}\right|}^2-2M_{21}^2,\hfill \\ \hfill & C_{60}=M_{60}-15M_{40}{M}_{20}+30M_{21}^2,\hfill \\ \hfill & C_{61}=M_{61}-5{\mathrm{M}}_{40}{\mathrm{M}}_{21}-10M_{20}{M}_{41}\hfill \\ \hfill & +30M_{21}{M}_{20}^3,\hfill \\ \hfill & C_{63}=M_{63}-6{\mathrm{M}}_{41}{\mathrm{M}}_{20}-9M_{21}{M}_{42}\hfill \\ \hfill & +18M_{21}{M}_{20}^2+12M_{21}^3,\hfill \\ \hfill & C_{80}=M_{80}-28M_{20}{M}_{60}-35M_{40}^2\hfill \\ \hfill & +420M_{40}{M}_{20}^2-630M_{20}^4.\hfill \end{array}$$

(50)

Wavelet transform is a time-frequency analysis method for signals, characterized by its multi-resolution analysis capabilities, as well as its localization properties in both time and frequency domains. By employing wavelet transform to decompose the signal at different scales, it is possible to delineate the details of various modulation type signals. The use of different wavelet bases to analyze the same problem can yield distinct results. Common wavelet functions include the Haar, Daubechies, Coflet, and Symlet families, and the Morlet wavelet, among others, owing to its simplicity and ease of computation; the Haar wavelet is suitable for engineering applications. Therefore, this paper adopts the Haar wavelet. The Haar wavelet is defined as

$$\mathsf{\Psi }\left(t\right)=\left\{\begin{array}{cc}1,\hfill & -0.5<t\le 0,\hfill \\ -1,\hfill & 0<t\le 0.5,\hfill \\ 0,\hfill & others.\hfill \end{array}\right.$$

(51)

The wavelet basis function, denoted as ${\mathsf{\Psi }}_{(a,b)}\left(t\right)$, is defined as

$${\mathsf{\Psi }}_{(a,b)}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{a}}},\hfill & -0.5a<t\le 0,\hfill \\ -{\displaystyle \frac{1}{\sqrt{a}}},\hfill & 0<t\le 0.5a,\hfill \\ 0,\hfill & others.\hfill \end{array}\right.$$

(52)

In the entropy domain, features such as the Renyi entropy, power spectrum Shannon entropy, and power spectrum exponential entropy are utilized as characteristic parameters for modulation identification. The definition of the $\alpha$-th order Renyi entropy is

$$H_{\alpha }\left(X\right)=-{\displaystyle \frac{1}{\alpha -1}}log{E}_X\left[P{\left(X\right)}^{\alpha -1}\right],$$

(53)

where X and $P\left(X\right)$ represent random variables, and E denotes the expectation operator.

When $\alpha -1=s$ approaches 0, the Renyi entropy reduces to the Shannon entropy, whose formula is given by

$$H=E[-{log}_2{p}_i]=\sum _{i=1}^n{p}_i{log}_2{p}_i.$$

(54)

Here, E represents the expectation operator, and $p_i$ is the probability.

The power spectrum exponential entropy signifies the uncertainty of signal energy under the division of the power spectrum, providing a quantitative description of the complexity of energy distribution in the frequency domain. The power spectrum exponential entropy is defined as

$$H=\mathrm{E}\left[{\mathrm{e}}^{1-p_i}\right]=\sum _{i=1}^n{p}_i{\mathrm{e}}^{1-p_i},$$

(55)

where H represents the entropy value, E denotes the expectation operator, and $p_i$ is the probability.

6. Simulation Experiment and Analysis

6.1. Dataset Description

To validate the monitoring capability of the proposed algorithm for low-orbit satellite wideband OFDM signals, a simulation dataset was constructed for testing. The dataset configuration is as shown in

Table 2. OFDM parameter configuration.

OFDM Parameter	Value
Number of effective sample points ($N_{use}$)	640
Number of sample points ($N_s$)	800
Number of samples in the CP ($N_{cp}$)	160
Number of subcarriers ($N_{sc}$)	64
Bandwidth ($BW$)	200 MHz
Oversampling factor ($\rho$)	25
Sampling rate ($Fs$)	5 Gsps
Modulation	BPSK QPSK 8QPSK 16QAM 64QAM
Pulse shaping	Rectangular pulse
Number of Monte Carlo runs	1000

. This setup aims to test the robustness of the model in the face of real low-orbit satellite dynamic channel environments. Our objective is to assess the performance of the model in analyzing and recognizing OFDM signals under dynamic channels, and therefore, frequency offsets and time delays were set to better simulate real-world application scenarios.

To realistically simulate low-orbit satellite OFDM signals, 64 subcarriers are divided into eight users, with each user comprising eight consecutive subcarriers. The modulation scheme for each subcarrier is randomly selected from [BPSK, QPSK, 8PSK, 16QAM, and 64QAM]. Additionally, the Doppler frequency offset of the OFDM signal is caused by the dynamics of the low-orbit satellite, and thus its frequency offset range is set to [−100 KHz, 100 KHz] in this paper. The length of the time delay sampling points is randomly determined within the range of [0, $N_{cp}$] points. In this paper, the channel is characterized by Additive White Gaussian Noise (AWGN) signals, with the channel parameters listed in

Table 3. Channel simulation parameter configuration.

Channel Parameter	Value
Time delay	[0, 160]
Doppler shift	[−100 KHz, 100 KHz]
SNR	[−3 dB, 14 dB]

.

6.2. Simulation Results

6.2.1. OFDM Signal Detection Simulation

In this section, two OFDM signal detection algorithms are investigated through simulation, including the burst signal detection algorithm based on autocorrelation and the proposed burst signal detection algorithm based on higher-order cumulants.

Both of these algorithms demonstrate detection success rates for continuous and burst OFDM signals under AWGN channels, as illustrated in Figure 6.

The analysis in Figure 6 reveals that under an AWGN channel, the burst signal detection performance of the higher-order cumulant algorithm is optimal, yet its performance for continuous signal detection is poor. At SNR = −4 dB, the correct detection rate of burst signals reaches 99%, and at SNR = 0 dB, the correct detection rate of continuous signals reaches 99%. This is because the higher-order cumulant can extract higher-order statistical information from the signal, which is affected by noise at low SNRs. This makes it prone to misidentifying continuous signals as burst signals. However, the overall performance is superior to that of the autocorrelation method and can well adapt to the low-orbit satellite channel.

The use of higher-order cumulant features can simultaneously distinguish between OFDM and single-carrier signals, as shown in Figure 7. These features are insensitive to noise, resulting in a good distinction effect.

6.2.2. OFDM Signal Parameter Analysis Simulation

Figure 8 presents the estimation performance graph with normalized MSE as the reference for various SNR conditions when the number of OFDM symbols used for simulation is 5. Figure 9 shows the estimation performance graph with normalized MSE as the reference for various SNR conditions when the number of OFDM symbols used for simulation is 10 (Note: due to the fact that $N_{use}$, the number of subcarriers, and the oversampling factor are fully estimated correctly when the symbol count is high, their curves exceed the display range after the normalized MSE reaches zero, indicating that their curves or parameter points have disappeared from the graph.). By comparing Figure 8 and Figure 9, it can be observed that the normalized MSE for parameter estimation changes due to the different number of OFDM symbols used in the simulation, i.e., the error in parameter estimation changes. Figure 9 exhibits a better normalized MSE index than Figure 8, and by observing the normalized MSE values in Figure 9, it can be seen that only $N_s$ and $N_{CP}$ have estimation errors under these conditions. Although the deviation values of $N_s$ and $N_{CP}$ are the same, their normalized representation in the normalized MSE index leads to $N_s$ displaying better performance than $N_{CP}$.

6.2.3. OFDM Signal Identification Simulation

With the increase in SNR, the accuracy of modulation identification for each subcarrier of the OFDM signal continues to improve, as shown in Figure 10. The accuracy change curve for SNRs ranging from 0 to 15 dB is plotted. It can be seen that the average identification rate exceeds 90% when the SNR is greater than 5 dB. Among them, the trend of identification rate growth for the five modulation schemes is similar. When the SNR is greater than 7 dB, the identification rate of all signals exceeds 90%, and for QAM signals, which have a variety of signal types with similar amplitude and phase characteristics, it is more difficult to distinguish. When the SNR is greater than 9 dB, the identification rate of all modulation signals exceeds 95%.

6.3. Method Limitations and Future Work

Although the method proposed in this study has demonstrated promising results, it is important to acknowledge certain limitations, and it is crucial to provide avenues for further exploration in future research. Firstly, due to challenges related to the lack of publicly accessible datasets and hardware limitations in acquiring real low-orbit satellite signals, we were restricted to generating signal datasets only in a simulated environment. While this allowed for controlled experiments, the performance may differ when applied to real-world low-orbit satellite signals, which are influenced by more complex noise characteristics, signal distortions, and environmental factors. Moreover, this study only considered low-orbit satellite OFDM signals in an AWGN environment. In more complex electromagnetic environments, a variety of noise types may arise, including interference from other signals, atmospheric noise, and jamming techniques. The robustness of the model to these more complex noise conditions remains unexplored and represents an ongoing research area. Future work will focus on evaluating and improving the detection and identification of low-orbit satellite broadband multi-type signals under dynamic channels.

7. Conclusions

In this paper, we first propose a wideband flexible monitoring algorithm based on resampling filtering, which adaptively adjusts the receiver sampling rate to four times the signal bandwidth for low-orbit satellite wideband OFDM signals, thereby improving analysis sensitivity. Secondly, we introduce an algorithm for Doppler frequency offset estimation based on satellite ephemeris and coordinate system transformation, which achieves the high-precision estimation of Doppler frequency offsets in the dynamic channel of low-orbit satellites, thereby mitigating the impact of channel dynamics on OFDM signal analysis and identification. Finally, we present a detection, analysis, and identification algorithm based on intrinsic features of OFDM signals. Initially, the OFDM signal is differentiated from single-carrier signals using the cumulative feature of OFDM signals. Subsequently, the algorithm estimates parameters such as symbol length, CP length, the number of subcarriers, and the oversampling rate using the cyclic transformation domain feature of OFDM signal CP. After removing the cyclic prefix, the algorithm extracts each subcarrier of the OFDM signal and jointly identifies the modulation schemes of these subcarriers based on their multi-domain intrinsic features. The simulation experiment conducted on OFDM signal detection under the presence of delay and Doppler in an AWGN channel shows that the low-orbit satellite broadband OFDM signal monitoring method proposed in this paper can accurately detect OFDM signals in the absence of prior information and differentiate them from single-carrier signals. The detection accuracy can reach over 90% at SNR $\ge 2$ dB. Additionally, the proposed method can reliably identify the structural parameters of OFDM signals. When the number of accumulated symbols exceeds 10, some parameters can achieve a 100% correct rate. Finally, the method can recognize the modulation schemes of each subcarrier. At SNR $\ge 7$ dB, the average identification rate can reach over 90%.

All analyses and assumptions presented in this paper are predicated on the intact structure of the CP-OFDM signals, which possess cyclostationary characteristics. However, due to the complex dynamic nature of LEO satellite channels, these cyclic properties may be compromised during transmission. Consequently, confining the analysis to cyclic symbols can result in a loss of degrees of freedom. Therefore, the application of conjugate cyclic statistics and widely-linear signal processing methods can be employed to analyze non-cyclic symbols, thereby enhancing system performance.