CR signal modulation identification method based on VG transformation and hierarchical feature decision, medium and equipment

Abstract

The invention provides a CR signal modulation identification method based on VG transformation and hierarchical feature decision, a medium and equipment, and belongs to the technical field of signal processing. According to the basic thought of the method, three features are extracted, namely, the unsigned Laplace matrix quadratic form of an autocorrelation function, the adjacent matrix quadratic form of a power spectrum and the graph average degree of a square spectrum, and then the three features are used for constructing a hierarchical recognition network so as to complete recognition tasks of different observation signals. According to the method, a graph domain signal detection method in an irregular domain is adopted, observation signals are converted into VG, the unsigned Laplace matrix quadratic form, the adjacent matrix quadratic form and the average degree of the VG are obtained for hierarchical feature decision making, and CR signals can be effectively recognized under the low signal-to-noise ratio.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal processing, and in particular relates to a CR signal modulation recognition method, medium and device based on VG transformation and hierarchical feature decision. Background Art

[0002] In the field of cognitive radio spectrum sensing, from a narrow perspective, its core task is to detect the existence of the primary user signal; from a broad perspective, it involves further refinement of the modulation mode and parameters of the primary user signal. In this process, modulation recognition, as the key prerequisite for refined processing, occupies a vital position in cognitive radio scenarios. At present, modulation recognition methods can be mainly divided into two categories: one is a statistical method, which is simple and intuitive, but it has obvious shortcomings, that is, when the number of samples is limited, the features extracted by statistical methods are difficult to effectively improve in terms of separability and stability; the other is a method based on graph domain signal processing, which can effectively mine the potential topological structure characteristics of the signal without increasing the number of signal samples, thereby opening up a new dimension for feature extraction and has been widely used in communication signal detection and processing. However, most of the existing modulation recognition methods based on graph domain signal processing only use the topological structure characteristics of the signal, but ignore the statistical characteristics of the signal, that is, they fail to fully integrate the topological characteristics and statistical characteristics of the signal, which to a certain extent affects the overall performance of the algorithm. In addition, from the perspective of the mapping relationship between sample points and graph vertices, since most existing graph domain transformation methods are based on quantized graphs, the sample points of the signal are not mapped one-to-one with the vertices of the quantized graph, which leads to insufficient mining of the original topological features of the signal. In view of the above problems, how to choose a more appropriate graph transformation method that can more accurately reflect the signal topological features and organically combine it with statistical signal features is an important research direction for signal modulation recognition based on graph domain signal processing. Summary of the invention

[0003] In view of the deficiencies in the prior art, the present invention provides a cognitive radio (CR) signal modulation recognition method, medium and device based on visibility graph (VG) transformation and hierarchical feature decision.

[0004] To achieve the above object, the present invention adopts the following technical solutions:

[0005] In a first aspect, the present invention provides a CR signal modulation recognition method based on VG transformation and hierarchical feature decision, comprising:

[0006] Step 1: Obtain the observed signal and calculate its autocorrelation function. Sum the autocorrelation function in groups and perform the VG transform. Calculate the unsigned Laplacian matrix quadratic form of the VG transform. ; If , where η1 represents the decision threshold, then the observed signal is determined to be a 16QAM modulated signal; otherwise, the observed signal enters the decision in Step 2.

[0007] Step 2: Calculate the power spectrum of the observed signal. Sum the power spectrum in groups and perform the HVG transform. Calculate the adjacency matrix quadratic form χ of the HVG transform. A ; If χ A > η2, where η2 represents the decision threshold, then the observed signal is determined to be an LFM modulated signal; otherwise, the observed signal enters the decision in Step 3.

[0008] Step 3: Calculate the squared spectrum of the observed signal. Sum the squared spectrum in groups and perform uniform quantization. The obtained quantization sequence is used as the input of the HVG transform. Calculate the average degree E(k) of the adjacency matrix of the HVG transform. If E(k) < η3, where η3 represents the decision threshold, then the observed signal is determined to be a BPSK signal; otherwise, the observed signal is determined to be a QPSK signal.

[0009] Optionally, in Step 1, the process of calculating the unsigned Laplacian matrix quadratic form is as follows:

[0010] Step 1.1: Calculate the autocorrelation function C(m) of the observed signal as:

[0011]

[0012] where r(n) represents the observed signal, m represents the independent variable of the autocorrelation function, and N is the number of samples of the observed signal.

[0013] Step 1.2: Divide the autocorrelation function C(m) into groups of every λ samples, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain the grouped summation sequence y(l), 1 ≤ l ≤ L. The process is as follows:

[0014]

[0015] Step 1.3: Perform the VG transform on the grouped summation sequence y(l) to obtain the adjacency matrix A.

[0016] Step 1.4: Calculate the unsigned Laplacian matrix L w = Α + D, where D is the degree matrix of the adjacency matrix A. Then calculate the quadratic form w of L , y = [y(1), y(2),..., y(L)] T .

[0017] Optionally, in step 1.3, the visibility rule of the VG transform is as follows:

[0018]

[0019] In the formula, A(x a ,y a ), B(x b ,y b ), C(x c ,y c ) are three arbitrary points in the grouped summation sequence y(l), and satisfy x a < x c < x b .

[0020] Optionally, in step 2, the process of calculating the quadratic form χ A of the adjacency matrix of the HVG transform is as follows:

[0021] Step 2.1: Calculate the power spectrum R(k) of the observed signal as:

[0022]

[0023] In the formula, r(n) represents the observed signal, k represents the independent variable of the power spectrum, and N is the number of samples of the observed signal;

[0024] Step 2.2: Divide R(k) into groups of every λ samples, and a total of L = [N / λ] groups are formed; perform a summation process on each group to obtain the grouped summation sequence r(l), 1 ≤ l ≤ L, and the process is as follows:

[0025]

[0026] Step 2.3: Perform the HVG transform on the grouped summation sequence r(l) to obtain the adjacency matrix A;

[0027] Step 2.4: Calculate the quadratic form χ A = r T · A · r, r = (r(1), r(2),... r(l)) T .

[0028] Optionally, in step 2.3, the visibility rule of the HVG transform is as follows:

[0029]

[0030] In the formula, A(x a ,y a ), B(x b ,y b ), C(x c ,yc ) are three arbitrary points in the grouped summation sequence r(l), and satisfy x a < x c < x b .

[0031] Optionally, in step 3, the process of calculating the average degree E(k) of the adjacency matrix of the HVG transform is as follows:

[0032] Step 3.1: Calculate the squared spectrum Z(k) of the observed signal as:

[0033]

[0034] where r(n) represents the observed signal, k represents the independent variable of the squared spectrum, and N is the number of samples of the observed signal;

[0035] Step 3.2: Divide the squared spectrum Z(k) into groups of every λ samples, and a total of L = [N / λ] groups are formed; perform a summation process on each group to obtain the grouped summation sequence z(l), 1 ≤ l ≤ L, and the process is as follows:

[0036]

[0037] Step 3.3: First, convert the grouped summation sequence z(l) to the [0,1] interval in the maximum-minimum normalization manner to obtain the normalized sequence U z (k):

[0038]

[0039] where θ min and θ max represent the minimum and maximum values of z(l) respectively;

[0040] Then, perform uniform quantization on the normalized sequence U z (k) to obtain the quantization sequence Q(k):

[0041]

[0042] where M is the number of quantization levels;

[0043] Step 3.4: Perform the HVG transform on the quantization sequence Q(k) to obtain the adjacency matrix A, and calculate its average degree E(k):

[0044]

[0045] where k i is the sum of the number of all edges connected to vertex i, and P(k i ) is the degree distribution of the graph.

[0046] In a second aspect, the present invention provides a computer-readable storage medium storing a computer program, which causes a computer to execute the CR signal modulation recognition method based on VG transformation and hierarchical feature decision as described in the first aspect.

[0047] In a third aspect, the present invention provides an electronic device, including: a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, the CR signal modulation recognition method based on VG transformation and hierarchical feature decision as described in the first aspect is implemented.

[0048] The beneficial effects of the present invention are as follows: The visual graph method adopted by the present invention can mine more subtle structural features of signals, convert signals from the observation domain to the graph domain to obtain the topological features of signals, and fully integrate the original statistical characteristics of signals. At the same time, based on Figure 2 the quadratic form, different graph representation matrices are selected to construct recognition feature quantities. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flowchart of the CR signal modulation recognition method based on VG transformation and hierarchical feature decision.

[0050] Figure 2 is the difference in the mean value of the unsigned Laplacian matrix quadratic form calculated after performing VG transformation on the grouped sum autocorrelation functions of four observed signals at different signal-to-noise ratios.

[0051] Figure 3 is the difference in the mean value of the adjacency matrix quadratic form calculated after performing HVG transformation on the grouped sum power spectra of three observed signals at different signal-to-noise ratios.

[0052] Figure 4 is the difference in the mean value of the average degree of the adjacency matrix calculated after quantifying and performing HVG transformation on the grouped sum squared spectra of two observed signals at different signal-to-noise ratios.

[0053] Figure 5 is the recognition accuracy rate of four observed signals in the method of the present invention at different SNRs.

[0054] Figure 6 is a schematic diagram comparing the recognition accuracy rates of the method of the present invention with other comparison methods. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0055] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application.

[0056] In one embodiment, the present invention proposes a CR signal modulation recognition method based on VG transformation and hierarchical feature decision, as Figure 1As shown, it specifically includes the following steps:

[0057] Step 1: Obtain the observation signal, calculate the autocorrelation function (take half) of the observation signal samples and sum them in groups, then perform the VG transform and calculate the unsigned quadratic form χ of its Laplacian matrix L , if the obtained χ L > η1, then determine the observation signal as a 16QAM modulation signal; otherwise, proceed to the next decision on the quadratic form of the adjacency matrix of the power spectrum.

[0058] In this embodiment, the observation signal is Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK), 16-Quadrature Amplitude Modulation (16QAM), and Linear Frequency Modulation (LFM) superimposed with Gaussian white noise.

[0059] The observation signal r(t) contaminated by Additive White Gaussian Noise (AWGN) can be expressed as:

[0060] r(t) = s(t) + w(t)

[0061] where s(t) is the modulated signal and w(t) is a real Gaussian white noise with zero mean and variance σ 2 .

[0062] BPSK carries information by adjusting the phase of the carrier, and its amplitude and frequency remain constant. BPSK usually uses the initial phases 0 and π to correspond to binary "1" and "0" respectively, and its signal model is:

[0063]

[0064] where N C is the number of symbols, T0 is the symbol duration, ξ(n) represents the level value of the nth symbol, g(t) is the baseband pulse waveform with a duration of T0, which is assumed to be a rectangular pulse with a height of 1 and a pulse width of T0 here; ω c = 2πf c is the carrier angular frequency, where f c is the carrier frequency, is the absolute phase shift of the nth symbol, and generally takes:

[0065]

[0066] Wherein, ξ(n) is expressed as:

[0067]

[0068] When sending the binary symbol "0", the phase is 0; when sending the binary symbol "1", the phase is π.

[0069] QPSK has 4 initial carrier phases that can be taken, usually set to 0, π / 2, π, 3π / 2 respectively, and its signal model is:

[0070]

[0071] Wherein, the absolute phase offset of the nth symbol is:

[0072]

[0073] Quadrature amplitude modulation is a signal method that combines amplitude and phase modulation. Compared with the aforementioned modulation methods, it has a higher bandwidth utilization rate and lower requirements for bit signal-to-noise ratio. The 16QAM in this embodiment can be regarded as the superposition of two orthogonal four-level amplitude shift keying signals, and its mathematical expression can be represented as:

[0074] e 16QAM (t) = m I (t)cosω c t + m Q (t)sinω c t

[0075] Wherein, cosω c t is the in-phase carrier, sinω c t is the quadrature carrier, and m I (t) and m Q (t) are the modulation amplitudes.

[0076] For LFM waveform modulation, the most significant feature is that the frequency of the signal shows a linear upward or downward change trend over time. In a frequency modulation signal, if the frequency changes linearly with time, this frequency modulation is called linear frequency modulation, that is, an LFM signal, and its signal model is:

[0077]

[0078] Wherein, f c is the carrier frequency, is a rectangular signal, is the frequency modulation slope.

[0079] In this embodiment, step 1 specifically includes:

[0080] Step 1.1: Calculate the autocorrelation function (take half) of the four observation signals, that is:

[0081]

[0082] Wherein, r(n) represents the observed signal, m represents the independent variable of the autocorrelation function, and N is the number of samples of the observed signal.

[0083] Step 1.2: Divide the autocorrelation function C(m) of the observed signal into groups of every λ samples, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain a grouped summation sequence y(l), where 1 ≤ l ≤ L, and the process is as follows:

[0084]

[0085] Step 1.3: Perform a VG transform on the above sequence y(l) to obtain an adjacency matrix A, and the visibility rule of VG is:

[0086]

[0087] Wherein, A(x a ,y a ), B(x b ,y b ), C(x c ,y c ) are three arbitrary points in the sequence, and satisfy x a < x c < x b .

[0088] Step 1.4: Let the unsigned Laplacian matrix of the VG transform be L w , that is:

[0089] L w = Α + D

[0090] Wherein, A is the adjacency matrix obtained by the above VG transform, and D is the degree matrix.

[0091] Then calculate the quadratic form of L w

[0092]

[0093] Wherein, y = (y(1), y(2),..., y(L)) T .

[0094] Set a threshold η1. If the obtained then judge the observed signal as a type I2 signal, otherwise enter the next judgment of the quadratic form of the adjacency matrix of the power spectrum.

[0095] Figure 2After a series of processes on four observed signals and then performing a VG transformation, the mean value χ of the quadratic form of the unsigned Laplacian matrix is calculated. L The differences at different signal-to-noise ratios. As can be seen from the figure, the of the 16QAM signal is much larger than the other three signals. Therefore, a decision can be made by setting the threshold η1.

[0096] Then, the signals of type I1 are further subdivided as follows:

[0097]

[0098] Step 2: Calculate the power spectrum of the observed signal samples, group and sum them, then perform an HVG transformation and calculate the quadratic form χ of its adjacency matrix. A If the obtained χ A > η2, the observed signal is determined to be an LFM modulation signal; otherwise, proceed to the next step of judging the average degree of the squared spectrum.

[0099] In this embodiment, Step 2 specifically includes:

[0100] Step 2.1: Calculate the power spectrum of the signals of type I1, that is:

[0101]

[0102] Step 2.2: Divide every λ samples of R(k) into a group, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain the grouped summation sequence r(l), where 1 ≤ l ≤ L. The process is as follows:

[0103]

[0104] Step 2.3: Perform an HVG transformation on the above sequence r(l) to obtain the adjacency matrix A. The visibility rule of HVG is:

[0105]

[0106] In the formula, A(x a , y a ), B(x b , y b ), C(x c , y c ) are three arbitrary points in the sequence, and satisfy x a < x c < x b .

[0107] Step 2.4: Then calculate the quadratic form χ A of the adjacency matrix, that is:

[0108] χ A = rT ·A·r

[0109] where \(r=(r(1),r(2),\cdots,r(l))\) T 。

[0110] Set a threshold \(\eta_2\). If the obtained \(\chi\) A \(>\eta_2\), then the observed signal is determined to be an LFM modulated signal; otherwise, proceed to the next step of judging the average degree of the squared spectrum.

[0111] Figure 3 is the mean value \(\chi\) of the quadratic form of the adjacency matrix calculated after performing the HVG transform on the grouped summation power spectra of three observed signals A The difference at different signal-to-noise ratios. As can be seen from the figure, the \(\chi\) of the LFM signal A is greater than the other two signals. Therefore, the judgment can be made by setting the threshold \(\eta_2\).

[0112] Step 3: Calculate the squared spectrum of the observed signal, extract the grouped summation sequence and quantize it as the input of the HVG transform. If the average degree \(E(k)\) of the adjacency matrix of the obtained HVG \(<\eta_3\), then the observed signal is determined to be a BPSK signal; otherwise, the observed signal is determined to be a QPSK signal.

[0113] In this embodiment, Step 3 specifically includes:

[0114] Step 3.1: Calculate the squared spectrum of the I3 type signal, that is:

[0115]

[0116] Step 3.2: Divide the squared spectrum \(Z(k)\) of the observed signal into groups of every \(\lambda\) samples, and a total of \(L = [N / \lambda]\) groups are formed. Perform a summation process on each group to obtain the grouped summation sequence \(z(l)\), \(1\leq l\leq L\), and the process is as follows:

[0117]

[0118] Step 3.3: Perform uniform quantization on the above sequence \(z(l)\), that is:

[0119] Convert the squared spectrum grouped summation sequence \(z(l)\) to the interval \([0,1]\) in the maximum-minimum normalization manner, that is

[0120]

[0121] where \(\theta\) min and \(\theta\) max represent the minimum value and the maximum value of \(z(l)\) respectively.

[0122] Then, for the normalized sequence \(U\) z(k) Perform uniform quantization (the number of quantization levels is M), and the quantization sequence can be obtained:

[0123]

[0124] Step 3.4: Perform HVG transformation on the above sequence Q(k) to obtain the adjacency matrix A, and calculate its average degree E(k), that is:

[0125]

[0126] In the formula, k i is defined as the sum of the number of all edges connected to the vertex i, and P(k i ) is the degree distribution of the graph.

[0127] Then set the threshold η3. If E(k) < η3, the observed signal is judged as a BPSK signal; otherwise, the observed signal is judged as a QPSK signal.

[0128] Figure 4 is the difference in the average degree E(k) calculated under different signal-to-noise ratios after quantization and HVG transformation of the grouped sum square spectra of BPSK and QPSK signals. It can be seen from the figure that the E(k) of the QPSK signal is greater than that of the BPSK signal. Therefore, the decision can be made by setting the threshold η3.

[0129] Figure 5 is the recognition correct rate of the four modulation signals under different SNRs. It can be seen from the figure that the recognition correct rate of this method for type I2 signals has always been relatively high, and the recognition correct rate for type I1 signals increases with the increase of SNR.

[0130] Figure 6 is a comparison schematic diagram of the recognition correct rate curves of the method of the present invention, the sine wave extraction method, and the joint characteristic parameter method. In order to make full use of the data and avoid experimental contingency, the simulation conditions are as follows: 1000 simulations are performed, and the signal-to-noise ratio is set to -12 to -4 dB. It can be seen that the recognition correct rate of the method of the present invention is better than other methods.

[0131] In another embodiment, the present invention proposes a computer-readable storage medium storing a computer program, and the computer program causes a computer to execute the CR signal modulation recognition method based on VG transformation and hierarchical feature decision in the foregoing embodiment.

[0132] In another embodiment, the present invention proposes an electronic device, including: a memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, the CR signal modulation recognition method based on VG transformation and hierarchical feature decision in the foregoing embodiment is implemented.

[0133] In the embodiments disclosed in the present application, a computer storage medium may be a tangible medium that can contain or store a program for use by or in connection with an instruction execution system, apparatus, or device. The computer storage medium may include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of the computer storage medium would include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CDROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0134] Those of ordinary skill in the art will recognize that the units and algorithm steps of the examples described in connection with the embodiments disclosed in the present application can be implemented in electronic hardware or a combination of computer software and electronic hardware. Whether these functions are executed in hardware or software depends on the specific application and design constraints of the technical solution. Skilled artisans may use different methods for each specific application to implement the described functions, but such implementation should not be considered to exceed the scope of the present application.

[0135] The above are only the preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions falling within the idea of the present invention belong to the protection scope of the present invention. It should be noted that for those of ordinary skill in the art of this technology, several improvements and refinements made without departing from the principle of the present invention should be regarded as within the protection scope of the present invention.

Claims

1. A CR signal modulation recognition method based on VG transformation and hierarchical feature decision, characterized in that, Including: Step 1: Obtain the observed signal and calculate its autocorrelation function, sum the grouped autocorrelation functions and perform a VG transform, and calculate the unsigned quadratic form of the Laplacian matrix of the VG transform If η1 represents the decision threshold, then the observed signal is determined to be a 16QAM modulated signal; otherwise, the observed signal enters the decision in Step 2 Step 2: Calculate the power spectrum of the observed signal, sum the grouped power spectra and perform the HVG transform, and calculate the quadratic form χ of the adjacency matrix of the HVG transform A ; If χ A > η2, where η2 represents the decision threshold, then the observed signal is determined to be an LFM modulated signal; otherwise, the observed signal proceeds to the decision in Step 3 Step 3: Calculate the squared spectrum of the observed signal, sum the grouped squared spectra and perform uniform quantization. The obtained quantization sequence is used as the input of the HVG transform, and calculate the average degree E(k) of the adjacency matrix of the HVG transform. If E(k) < η3, where η3 represents the decision threshold, then the observed signal is judged as a BPSK signal; otherwise, the observed signal is judged as a QPSK signal.

2. The CR signal modulation recognition method based on VG transform and hierarchical feature decision according to claim 1, characterized in that: In step 1, the process of calculating the unsigned quadratic form of the Laplacian matrix of the VG transform is as follows: Step 1.1: Calculate the autocorrelation function C(m) of the observed signal as: where r(n) represents the observed signal, m represents the independent variable of the autocorrelation function, and N is the number of samples of the observed signal; Step 1.2: Divide the autocorrelation function C(m) into groups of every λ samples, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain the grouped summation sequence y(l), where 1 ≤ l ≤ L. The process is as follows: Step 1.3: Perform a VG transform on the grouped summation sequence y(l) to obtain the adjacency matrix A; Step 1.4: Calculate the unsigned Laplacian matrix L of the VG transformation w = Α + D, where D is the degree matrix of the adjacency matrix A; then calculate the quadratic form of L w y = (y(1), y(2),..., y(L)) T .​ 3. The CR signal modulation recognition method based on VG transform and hierarchical feature decision according to claim 2, wherein: In Step 1.3, the visibility rule of the VG transform is: wherein, A(x a , y a ), B(x b , y b ), C(x c , y c ) are three arbitrary points in the grouped summation sequence y(l), and satisfy x a < x c < x b .

4. The CR signal modulation recognition method based on VG transformation and hierarchical feature decision as described in claim 1, wherein: In step 2, the process of calculating the quadratic form χ of the adjacency matrix of the HVG transform A is as follows: Step 2.1: Calculate the power spectrum R(k) of the observed signal as: where r(n) represents the observed signal, k represents the independent variable of the power spectrum, and N is the number of samples of the observed signal; Step 2.2: Divide R(k) into groups of every λ samples, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain the grouped summation sequence r(l), where 1 ≤ l ≤ L. The process is as follows: Step 2.3: Perform an HVG transform on the grouped summation sequence r(l) to obtain the adjacency matrix A; Step 2.4: Calculate the quadratic form χ of the adjacency matrix of the HVG transformation A = r T · A · r, where r = [r(1), r(2),... r(l)] T .

5. The CR signal modulation recognition method based on VG transformation and hierarchical feature decision according to claim 4, characterized in that: In Step 2.3, the visibility rule of the HVG transform is: wherein, A(x a , y a ), B(x b , y b ), C(x c , y c ) are three arbitrary points in the grouped summation sequence r(l), and satisfy x a < x c < x b .

6. The CR signal modulation recognition method based on VG transform and hierarchical feature decision as claimed in claim 1, wherein: In Step 3, the process of calculating the average degree E(k) of the adjacency matrix of the HVG transform is: Step 3.1: Calculate the squared spectrum Z(k) of the observed signal as: where r(n) represents the observed signal, k represents the independent variable of the squared spectrum, and N is the number of samples of the observed signal; Step 3.2: Divide the squared spectrum Z(k) into groups of every λ samples, and a total of L = [N / λ] groups are formed. Perform a summation process on each group to obtain the grouped summation sequence z(l), where 1 ≤ l ≤ L. The process is as follows: Step 3.3: First, convert the grouped summation sequence z(l) to the interval [0, 1] in the maximum-minimum normalization manner to obtain the normalized sequence U z (k): where θ min and θ max represent the minimum and maximum values of z(l), respectively; Then, perform uniform quantization on the normalized sequence U z (k) to obtain the quantization sequence Q(k): where M is the number of quantization levels; Step 3.4: Perform an HVG transform on the quantization sequence Q(k) to obtain the adjacency matrix A, and calculate its average degree E(k): where k i is the sum of all the edge numbers connected to vertex i, and P(k i ) is the degree distribution of the graph.

7. A computer-readable storage medium storing a computer program, characterized in that, The computer program causes the computer to execute the CR signal modulation recognition method based on the VG transform and hierarchical feature decision as described in any one of claims 1 - 6.

8. An electronic device, characterized in that, Including: A memory, a processor, and a computer program stored on the memory and executable on the processor. When the processor executes the computer program, it implements the CR signal modulation recognition method based on the VG transform and hierarchical feature decision as described in any one of claims 1 - 6.

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