A weak signal detection method based on graph fourier transform

Abstract

The application discloses a weak signal detection method based on graph Fourier transform, which comprises the following steps: firstly, calculating the power spectrum of a signal; then, respectively performing two kinds of graph transform on the power spectrum, one is undirected multiple loop graph transform, obtaining the self-loop weight value of each vertex, and defining the self-loop weight value as a graph signal; the other is simple undirected graph transform, obtaining the Laplace matrix of the graph; further, inverting the Laplace matrix and multiplying the inverted Laplace matrix with the graph signal to obtain the graph Fourier transform; finally, calculating the energy of the graph Fourier transform as the basis of signal existence detection, and completing the signal detection by comparing with a specific threshold. The self-loop information of the graph is defined as the graph signal, and the Fourier transform is performed on the graph signal, so that the self-loop information of the graph and the topological information represented by the graph Laplace matrix are effectively utilized, and the detection performance of the traditional graph domain spectrum sensing algorithm under low signal-to-noise ratio and fading channel is improved.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of signal processing, and particularly relates to a weak signal detection method based on graph Fourier transform. BACKGROUND

[0002] Noise signal detection is a classical topic in statistical signal processing, and is widely applied in cognitive radio, radar, communication electronic reconnaissance, deep space communication and other occasions. The signal detection problem can be generally modeled as a binary hypothesis testing model, and its general framework is as follows: by analyzing the difference of the observed signal under different hypotheses, a test statistic for distinguishing the presence or absence of the signal is obtained, and then a suitable threshold is set, and the presence or absence of the signal is determined by comparing the test statistic with the threshold. Therefore, the key link is the feature extraction link. For the defined feature, it is hoped to have certain separability and robustness. Traditional signal detection algorithms mainly include energy detection method, matched filter method, covariance matrix eigenvalue method, and cyclostationary test method. The common shortcoming of these methods is that a certain sample size is required. In recent years, the signal processing method based on graph domain transformation provides a new processing framework for signal detection and achieves good performance. However, the existing graph domain signal detection algorithm is mostly based on simple graph and does not consider the use of self-loop information, and the self-loop information reflects the correlation between signals of the same quantization level and also reflects the information of the signal itself in a certain sense. In addition, the existing graph domain signal detection algorithm only considers the topological features of the graph and does not define the graph signal, so it cannot effectively integrate more features and information of the graph and cannot use the processing tools such as graph Fourier transform in graph signal processing. SUMMARY

[0003] The present application provides a weak signal detection method based on graph Fourier transform to solve the problems in the prior art. Firstly, the power spectrum of the signal is calculated, and the self-loop weight value of each vertex and the Laplacian matrix of the graph are obtained by respectively performing undirected multiple ring graph transformation and simple undirected graph transformation on the power spectrum, and the self-loop weight value is defined as the graph signal. Further, the inverse of the Laplacian matrix is calculated and multiplied by the graph signal to obtain the Fourier transform of the graph signal. Finally, the energy of the graph Fourier transform is calculated as the basis for signal existence detection, and the signal detection is completed by comparing with a specific threshold.

[0004] To achieve the above object, the present application adopts the following technical scheme:

[0005] A weak signal detection method based on graph Fourier transform, characterized in that it comprises the following steps:

[0006] calculating the power spectrum of the observed signal, converting the power spectrum into an undirected multiple ring graph, and extracting the self-loop information of the undirected multiple ring graph as a graph signal;

[0007] The power spectrum is transformed into a simple undirected graph, and a Laplacian matrix and an eigenvector of the simple undirected graph are obtained;

[0008] The eigenvector of the Laplacian matrix is constructed into an eigenmatrix, and the eigenmatrix is multiplied by the graph signal to obtain a graph Fourier transform;

[0009] The energy of the graph Fourier transform is calculated as a detection statistic, and a corresponding decision threshold is set. By comparing the detection statistic and the decision threshold, the presence or absence of the signal is determined.

[0010] To optimize the above technical solutions, the following specific measures are taken:

[0011] Further, the power spectrum of the observation signal is:

[0012]

[0013] In the formula, R(k) is the power spectrum of the observation signal, r(n) is the discrete sample of the observation signal, and N is the number of observation signal samples.

[0014] Further, the power spectrum is converted into an undirected multiple loop graph, and the self-loop information of the undirected multiple loop graph is extracted as a graph signal, specifically:

[0015] After the power spectrum R(k) of the observation signal is normalized and quantized, the quantized sample Q(k) is obtained, and the quantized sample Q(k) is converted into an undirected multiple loop graph G(V, E, W), where V and E are the vertex set and edge set of the graph, respectively, and W is the weight matrix of the graph, which is numerically equal to the adjacency matrix of the graph, N0 is the number of vertices, and a ij represents the edge connection weight between the i-th vertex and the j-th vertex. If a ij ≠0 indicates that there is an edge connection between the two vertices, otherwise a ij =0 indicates that there is no connection between the two vertices.

[0016] The diagonal elements S(m)=a ij of the weight matrix w are extracted, m=1, 2,..., N0, i=j, as a graph signal S.

[0017] Further, the undirected multiple loop graph is determined by the amplitude change between adjacent samples of the quantized sample Q(k). For adjacent samples, the corresponding edge is considered to be connected, and the corresponding element in the adjacency matrix is incremented by 1, and finally the adjacency matrix is obtained.

[0018] Further, the power spectrum is transformed into a simple undirected graph, and a Laplacian matrix and an eigenvector of the simple undirected graph are obtained, specifically:

[0019] The power spectrum R(k) of the observed signal is normalized and quantized to obtain quantized samples Q(k), the quantized samples Q(k) are converted into a simple undirected graph G1(V, E) without repeated edges and self-loops, the adjacency matrix of which is B, the degree matrix is D, and the graph Laplacian is L=D-B, the graph Laplacian is eigen-decomposed to obtain:

[0020] L=UΛU T

[0021] In the formula, A is a diagonal matrix composed of eigenvalues of the graph Laplacian, and U is an eigenmatrix composed of eigenvectors of the graph Laplacian.

[0022] Further, the simple undirected graph is obtained as follows:

[0023] The adjacent relationship between each sample and its adjacent sample in the quantized sample Q(k) is investigated, and for adjacent samples, the corresponding edge is considered to be connected, and the corresponding element in the adjacency matrix is set to 1 as long as there is one connection, the adjacent relationship between the same samples is not counted, and the adjacent relationship between the same samples is not counted, and a simple undirected graph without repeated edges and self-loops is obtained.

[0024] Further, the eigenmatrix of the graph Laplacian is inverted, and the matrix is multiplied with the graph signal to obtain the graph Fourier transform, which is specifically:

[0025] The eigenmatrix U of the graph Laplacian is inverted to obtain U -1 =U T Then the graph Fourier transform of the graph signal S is:

[0026]

[0027] In the formula, is the graph Fourier transform, is an eigenmatrix with a dimension of N0xN0, is a graph signal with a dimension of N0x1, is a vector element of the graph Fourier transform.

[0028] Further, the energy of the graph Fourier transform is calculated as a detection statistic, and a corresponding decision threshold is set, and the presence or absence of the signal is determined by comparing the detection statistic and the decision threshold, which is specifically:

[0029] The energy of the graph Fourier transform is calculated as a detection statistic:

[0030]

[0031] In the formula, E x is the detection statistic;

[0032] The decision threshold is set by simulation training: a plurality of pure noise signals and noise signals are obtained, energy values of graph Fourier transform of the pure noise signals and the noise signals are obtained respectively, and a receiver operating curve is drawn, and a threshold corresponding to the set false alarm probability on the receiver operating curve is found as the decision threshold;

[0033] If the detection statistic E x is less than the decision threshold, it is determined that there is a signal, otherwise, it is determined that there is no signal.

[0034] In addition, the present application also provides a computer readable storage medium, which stores a computer program, and the computer program makes the computer execute the weak signal detection method based on graph Fourier transform.

[0035] In addition, the present application also provides an electronic device, which comprises a memory, a processor and a computer program stored in the memory and executable on the processor, and when the processor executes the computer program, the weak signal detection method based on graph Fourier transform is realized.

[0036] The present application has the following beneficial effects: the present application provides a weak signal detection method based on graph Fourier transform, which extracts the energy characteristics of graph Fourier transform to determine the existence of signals in the observation signals. The method defines the self-loop information of the graph as a graph signal, performs Laplace transform on the graph signal, effectively utilizes the self-loop information of the graph and the topological information represented by the graph Laplace matrix, and uses the analysis tool of graph Fourier transform to effectively improve the detection performance of the traditional graph domain spectrum sensing algorithm under low signal-to-noise ratio and fading channel. BRIEF DESCRIPTION OF DRAWINGS

[0037] Figure 1 is a flowchart of the weak signal detection method based on graph Fourier transform provided by the present application.

[0038] Figure 2 is a comparison diagram of graph signals (i.e. self-loop values) of observation signal power spectrum under two kinds of assumptions.

[0039] Figure 3a and Figure 3b are schematic diagrams of graph topology and graph signals of observation signals under two kinds of assumptions of having and not having signals.

[0040] Figure 4 is a comparison diagram of energies of graph Fourier transform of observation signals under two kinds of assumptions.

[0041] Figure 5 is a comparison diagram of receiver operating curves of the method of the present application and the existing method. DETAILED DESCRIPTION

[0042] The application will be further described in detail in connection with the drawings.

[0043] In an embodiment, as shown in Figure 1 the application proposes a weak signal detection method based on graph Fourier transform, which specifically includes the following contents.

[0044] I. Definition of graph signal

[0045] Suppose the discrete samples of the observed signal are:

[0046] r(n) = s(n) + ω(n), 0 ≤ n ≤ N-1

[0047] where ω(n) is an additive Gaussian white noise process with mean 0 and variance σ 2 , s(n) is the discrete sample of the signal, and N is the number of observed signal samples. The power spectrum of the observed signal is:

[0048]

[0049] After normalization and quantization of the power spectrum R(k) of the observed signal, the quantized samples Q(k) are converted into an undirected multi-loop graph G(V, E, W), where V and E are the sets of vertices and edges of the graph, and w is the weight matrix of the graph, which is essentially the adjacency matrix of the graph where N0 is the number of vertices. Unlike a general simple undirected graph, the difference lies in that the amplitude change between adjacent samples of the quantized sequence Q(k) determines that for each quantized sample corresponding to different magnitude numbers, i.e. Q(k) = i, Q(k+1) = j, α, β are vertex numbers, the corresponding edge e αβ is considered to be connected, and the corresponding element a αβ of the adjacency matrix is added by 1 to obtain the adjacency matrix. The diagonal elements S(m) = a ij , m = 1, 2,..., N0, i = j of the weight matrix are extracted as the graph signal S.

[0050] II. Extraction of graph Laplacian matrix and its characteristic matrix

[0051] Continuing to use the quantized samples Q(k), the adjacency relationship between each sample and its adjacent sample is investigated. For each quantized sample corresponding to different magnitude numbers, i.e. Q(k) = i, Q(k+1) = j, α, β are vertex numbers, the corresponding edge e αβis connected, and the corresponding element of the adjacency matrix is set to 1 only once, but the adjacent relationship between the same quantization samples is not counted, i.e. a simple undirected graph without repeated edges and self-loops is obtained. The obtained graph is denoted as G1(V, E), the adjacency matrix of which is B, the degree matrix is D, and the graph Laplacian matrix is L = D - B. Since the graph G1(V, E) is a simple undirected graph, the graph Laplacian matrix is a semi-positive definite matrix. The graph Laplacian matrix is decomposed into eigenvalues and eigenvectors, and the following is obtained:

[0052] L = UΛU T

[0053] wherein A is a diagonal matrix composed of eigenvalues of the graph Laplacian matrix, U is a matrix composed of eigenvectors of the graph Laplacian matrix, and is called an eigenmatrix.

[0054] III. Calculation of the graph Fourier transform

[0055] The eigenmatrix U of the graph Laplacian matrix is inverted to obtain U -1 = U T , and the dimension of U is N0xN0. The graph Fourier transform of the graph signal s is as follows:

[0056]

[0057] IV. Calculation of the energy of the graph Fourier transform and signal detection

[0058] 1) The energy of the graph Fourier transform is calculated as a detection statistic according to the following formula, i.e.

[0059]

[0060] 2) A corresponding decision threshold η is set, and the presence or absence of a signal is determined by comparing the energy E x of the graph Fourier transform with the decision threshold η. If the energy E x of the graph Fourier transform is less than the decision threshold η, it is determined that there is a signal, otherwise, it is determined that there is no signal.

[0061] The decision threshold η can be obtained through appropriate simulation training, i.e. a plurality of groups of noise signals and noisy signals with a certain number of samples are obtained, the energy values of the graph Fourier transforms of the noise signals and the noisy signals are obtained by using the foregoing method, and then a receiver operating characteristic (ROC) curve is drawn to find a corresponding threshold with a false alarm probability of about 0.01 as the decision threshold. According to a large amount of simulation and experience, the threshold is generally between 0.3 and 0.4.

[0062] Figure 2 The graph signal of the power spectrum of the observed signal under the two hypotheses (i.e. the self-loop weight value). As can be seen from the graph, the self-loop weight value of the observed signal under the two hypotheses of the presence and absence of a signal is obviously different, and it is reasonable to define the self-loop weight value as a graph signal.

[0063] Figure 3a and Figure 3b The graph topology and graph signal converted from the observed signal power spectrum under two hypotheses are shown. It can be seen that the graph converted from the observed signal power spectrum under the H0 hypothesis is a complete graph, and the values of the first three vertices in the graph signal are large, while the graph converted from the observed signal power spectrum under the H1 hypothesis is a non-complete graph, and there is an isolated singular point, and only the values of the first and second vertices in the graph signal are large, and the others are approximately 0. In summary, the graph topology and the graph signal under the two hypotheses are obviously different, and the combination of the graph topology characteristics and the graph signal characteristics will further highlight the difference, and thus is naturally more conducive to signal detection.

[0064] The simulation conditions in the above Figure 2 , Figure 3a and Figure 3b are as follows: the observed signal under the H0 condition is Gaussian white noise, the observed signal under the H1 condition is a QPSK modulated signal contaminated by Gaussian white noise, the sample point number N = 300, the symbol rate R B = 2000 Baud, the carrier frequency f c = 800 KHz, the oversampling factor is 30, the sampling frequency is 6 MHz, the quantization level N0 = 8, and the signal-to-noise ratio is -6 dB. The simulation number is 1000, and the channel is a frequency selective channel.

[0065] Figure 4 The mean difference of the graph Fourier transform energy under the two hypotheses is given. The signal-to-noise ratio range in the graph is [-12:2] dB, and the step is 2 dB. The other simulation conditions are the same as Figure 2 and Figure 3a , Figure 3b . As shown in Figure 4 , under the condition of moderate signal-to-noise ratio, the energy of the graph Fourier transform generated by the observed signal power spectrum under the H0 and H1 hypotheses is obviously different, which is the experimental basis for the signal detection algorithm proposed in the present application.

[0066] Figure 5 is a comparison diagram of the receiver operating characteristic curve (ROC) of the present method under the condition of a signal-to-noise ratio of -6 dB and the commonly used signal detection algorithm. The simulation conditions are the same as Figure 2 and Figure 3a , Figure 3bThe selected comparison methods are: time domain graph transformation and quantile graph method (Time domain+quantile graph), power spectrum range graph method (Power Spectrum+block Range), power spectrum graph domain transformation method (Power Spectrum), autocorrelation function graph domain transformation method (Autocorrelation), and energy detection method (Energy). As shown in the figure, the area under the ROC of the method is larger, far from the symmetry line, and compared with the existing signal detection method, the method has the best detection performance.

[0067] In conclusion, the method of graph Fourier transform in graph signal processing is introduced into signal detection, the autocorrelation graph transformation of the observation signal power spectrum is carried out, the autocorrelation weight is obtained, the autocorrelation weight is defined as a graph signal, then the observation signal is transformed into a simple undirected graph, the graph Laplace matrix is extracted, the topological features of the graph are obtained, and finally the energy of the graph Fourier transform is calculated as a detection statistic. Experiments show that when there is no signal, the energy value of the graph Fourier transform is large, and when there is a signal, the value is small. According to this phenomenon, a suitable threshold value can be set, and the size of the Fourier transform energy value of the graph generated by the observation signal power spectrum is compared with the threshold value, so as to realize the detection of the signal. The autocorrelation information of the graph is defined as a graph signal, the Laplace transformation is carried out, the autocorrelation information of the graph and the topological information represented by the graph Laplace matrix are effectively utilized, and the analysis new tool of the graph Fourier transform is used, so that the detection performance of the traditional graph domain spectrum sensing algorithm under the low signal-to-noise ratio and fading channel is effectively improved.

[0068] In another embodiment, the application further provides a computer readable storage medium storing a computer program, and the computer program enables a computer to execute the weak signal detection method based on the graph Fourier transform as described in Embodiment One.

[0069] In another embodiment, the application further provides an electronic device, characterized in that it comprises a memory, a processor, and a computer program stored in the memory and executable on the processor, and when the processor executes the computer program, the method for detecting a weak signal based on the graph Fourier transform as described in Embodiment One is realized.

[0070] In the embodiments disclosed in the present application, the computer storage medium can be a tangible medium which can contain or store programs for use by or in connection with an instruction execution system, apparatus or device. The computer storage medium can include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus or device, or any suitable combination of the above. More specific examples of computer storage medium can include one or more wires, portable computer disks, hard drives, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), fiber optics, compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the above.

[0071] Those skilled in the art can realize that the units and algorithm steps of each example described in combination with the embodiments disclosed in the present application can be realized by electronic hardware or a combination of computer software and electronic hardware. Whether the functions are realized in hardware or software manner depends on the specific application and design constraints of the technical solutions. The skilled person can use different methods to realize the described functions for each specific application, but such implementation should not be considered beyond the scope of the present application.

[0072] The above is only the preferred embodiment of the present application, and the protection scope of the present application is not limited to the above-mentioned embodiments. Any technical solution falling within the concept of the present application shall fall within the protection scope of the present application. It should be noted that, for ordinary skilled persons in the art, some improvements and refinements without departing from the principles of the present application shall be considered within the protection scope of the present application.

Claims

1. A method for detecting a weak signal based on graph Fourier transform, characterized in that, It comprises the following steps: The power spectrum of the observation signal is calculated, the power spectrum is converted into an undirected multiple loop graph, and loop information of the undirected multiple loop graph is extracted as a graph signal, specifically as follows: The power spectrum R(k) of the observation signal is normalized and quantized to obtain quantized samples Q(k), and the quantized samples Q(k) are converted into an undirected multi-loop graph G(V, E, W), where V and E are the vertex set and edge set of the graph respectively, and W is the weight matrix of the graph, which is numerically equal to the adjacency matrix of the graph The same, N0 is the number of vertices, a ij represents the edge connection weight between the ith vertex and the jth vertex, if a ij ≠0 represents the existence of edge connection between two vertices, otherwise a ij =0 represents no connection between two vertices; extracting the diagonal elements S(m) = a of the weight matrix W ij , m = 1, 2,..., N0, i = j, as a graph signal S; The undirected multiple loop graph is determined by amplitude variation between adjacent samples of the quantized sample Q(k), and for adjacent samples, the corresponding edge is considered to be connected, and the corresponding element in the adjacency matrix is added by 1, and finally the adjacency matrix is obtained; The power spectrum is subjected to simple undirected graph transformation to obtain a Laplacian matrix of the simple undirected graph and an eigenvector thereof; the simple undirected graph is obtained in the following manner: The adjacent relationship between each sample of the quantized sample Q(k) and its adjacent sample is investigated, and for adjacent samples, the corresponding edge is considered to be connected, and the corresponding element in the adjacency matrix is set to 1 as long as there is one connection, without repeated counting and without counting the adjacent relationship between the same samples, and then a simple undirected graph without repeated edges and loops is obtained; The characteristic matrix composed of the eigenvector of the Laplacian matrix is inverted, and matrix multiplication operation is performed on the inverted characteristic matrix and the graph signal to obtain a graph Fourier transform; The energy of the graph Fourier transform is calculated as a detection statistic, and a corresponding decision threshold is set, and the presence or absence of the signal is determined by comparing the detection statistic and the decision threshold, specifically as follows: The energy of the graph Fourier transform is calculated as a detection statistic: wherein E X is the detection statistic; χ m is a vector element of the graph Fourier transform; The decision threshold η is set in a simulation training manner: a plurality of pure noise signals and noisy signals are obtained, the energy values of the graph Fourier transform of the pure noise signals and the noisy signals are obtained respectively, and a receiver operating curve is drawn, and the threshold corresponding to the set false alarm probability on the receiver operating curve is taken as the decision threshold; If the detection statistic E X is less than a decision threshold, then a signal is determined to be present, otherwise a signal is determined to be absent.

2. The method of claim 1, wherein the method is based on graph Fourier transform. The power spectrum of the observation signal is as follows: In the formula, R(k) is the power spectrum of the observation signal, r(n) is the discrete sample of the observation signal, and N is the sample number of the observation signal.

3. The method of claim 1, wherein the method is based on graph Fourier transform. The power spectrum is subjected to simple undirected graph transformation to obtain a Laplacian matrix of the simple undirected graph and an eigenvector thereof, specifically as follows: After the power spectrum R(k) of the observation signal is normalized and quantized, the quantized sample Q(k) is obtained, the quantized sample Q(k) is converted into a simple undirected graph G1(V, E) without repeated edges and loops, the adjacency matrix thereof is B, the degree matrix is D, the graph Laplacian matrix is L=D-B, the graph Laplacian matrix is characteristic decomposed to obtain: L = UΛU T In the formula, Λ is a diagonal matrix composed of the eigenvalues of the graph Laplacian matrix, and U is a characteristic matrix composed of the eigenvectors of the graph Laplacian matrix.

4. The method of claim 3, wherein the method is based on graph Fourier transform. The characteristic matrix composed of the eigenvector of the Laplacian matrix is inverted, and matrix multiplication operation is performed on the inverted characteristic matrix and the graph signal to obtain a graph Fourier transform, specifically as follows: The characteristic matrix U of the graph Laplacian is inverted to obtain U -1 = U T The graph Fourier transform of the signal S is then given by wherein is the graph Fourier transform, denotes a feature matrix of dimension N0xN0, denotes a graph signal of dimension N0x1, is a vector element of the graph Fourier transform.

5. A computer readable storage medium storing a computer program, characterized in that, The computer program enables the computer to execute the weak signal detection method based on the graph Fourier transform according to any one of claims 1-4.

6. An electronic device, comprising: It comprises: A memory, a processor, and a computer program stored on the memory and executable on the processor, and when the processor executes the computer program, the weak signal detection method based on the graph Fourier transform according to any one of claims 1-4 is implemented.

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