Wideband spectrum compression detection method with signal noise

Abstract

The present invention discloses a method for compression detection of broadband spectrum with signal noise, which specifically includes the following steps: first, considering broadband signals with Gaussian white noise, the sparsity estimation model based on binomial distribution is improved so that it can be applied to signals with noise, and the number of samples is reduced under the premise of ensuring accurate reconstruction. Secondly, the estimated upper bound of sparsity is used to improve the adaptive threshold denoising method, so as to reduce the influence of noise folding on signal reconstruction. Finally, simulation results show that the algorithm of the present invention can more accurately estimate the upper bound of sparsity for broadband signals with Gaussian white noise, and reduce the number of samples while ensuring the accuracy of signal reconstruction, thereby reducing the number of calculations of the adaptive threshold denoising algorithm.

Description

Technical Field

[0001] The present invention relates to a broadband spectrum compression detection solution and the field of mobile communications, and particularly to a sparsity estimation problem of a sparse signal with signal noise and a signal reconstruction problem. Background Art

[0002] With the rapid development of 5G networks and the advent of the Internet of Things (IoT), the demand for spectrum resources has increased dramatically. However, existing fixed spectrum allocation schemes result in low spectrum utilization. A survey by the Federal Communications Commission (FCC) in the United States shows that the utilization rate of licensed spectrum is only 15%-85%. Cognitive Radio (CR) is widely considered to be one of the effective solutions to address low spectrum utilization. When secondary users (SU) in CR discover idle spectrum in the licensed spectrum, CR allows SU to opportunistically access the licensed spectrum. To avoid interference with primary users (PU), the primary task of CR is to quickly and accurately find available idle spectrum through spectrum detection technology.

[0003] Using the traditional Nyquist sampling theorem to detect the spectrum of wideband signals requires a high-sampling-rate analog-to-digital converter (ADC). To address this high sampling rate issue, the paper [Z. Tian and G. B. Giannakis. Compressed Sensing for Wideband Cognitive Radios [C], 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP 07, 2007: IV-1357-IV-1360] proposed a wideband spectrum solution based on compressed sensing. Compressed sensing (CS) states that if a signal is sparse, it can be sampled at a rate much lower than the Nyquist rate. Using a specific reconstruction algorithm, the original signal can then be accurately reconstructed. However, CS requires that the signal itself be sparse, or that it be sparse under a sparse basis. Because the utilization of authorized spectrum in wideband is very low, this condition can be met in CR networks. Although CS technology can effectively reduce the sampling rate, it also brings a new problem, namely how to estimate the signal sparsity k. The accuracy of the sparsity determines the detection accuracy of broadband signals.

[0004] Due to the high dynamics of broadband signals, it is very difficult to obtain the accurate sparsity k. However, the number of samples and reconstruction accuracy are closely related to the sparsity k. Most greedy reconstruction algorithms require sparsity k as their input condition, such as the orthogonal matching pursuit (OMP) in the literature [Chen A, Krauthgamer R, Nadler B. Tight recovery guarantees for orthogonal matching pursuit under Gaussian noise [J]. 2020.], the compressed sampling matching pursuit algorithm (CoSaMP) and its improved μCoSaMP and other greedy reconstruction algorithms in the literature [Xiong Xiaoting, Xu Xuejie, Li Suwen. An improved compressed sampling matching pursuit algorithm [J]. Journal of Jiamusi University (Natural Science Edition), 2020, 38(05): 160-164.], in addition, the literature [Zhao Y B. Optimal $k$-thresholding algorithms for sparse optimization problems [J]. SIAM J.Optim., 2020, 30(1), 31-55.] A new optimal k-threshold algorithm is proposed, which also takes the sparsity k as known information. Therefore, how to accurately estimate the signal sparsity k becomes particularly important. Reference [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248.] Use the binomial distribution to accurately estimate the upper and lower bounds of sparsity. Reference [Bioglio V, Bianchi T, Magli E. On the fly estimation of the sparsity degreein Compressed Sensing using sparse sensing matrices [C] IEEE International Conference on Acoustics. IEEE, 2015.] Use the number of non-zero elements in the observation vector to estimate the upper bound of signal sparsity. Reference [Wang Jianming, Chen Jianhua. Adaptive rate compressed sensing of surveillance videos using energy matching [J]. Journal of Electronics and Information Technology, 2020, v.42(12):198-205.] By observing the second-order method of compressed sensing results, the energy of the signal is first estimated, and then the sparsity of the signal is estimated.Reference [Wang Y, Tian Z, Feng C. Sparsity Order Estimation and its Application in Compressive Spectrum Sensing for Cognitive Radios[J]. IEEE Transactions on Wireless Communications, 2012, 11(6): 2116-2125.] Monte Carlo simulations show that the estimated signal sparsity and the number of samples required to reconstruct the signal are both related to the sparsity k. Reference [Liu Fangzheng, Han Zhenzhong, Zeng Ruiqi. Radar signal reconstruction method under weak observation conditions based on variational mode decomposition and compressed sensing[J]. Journal of Electronics & Information Technology, 2021, 43(6): 9.] Assuming that the sparsity of the signal is known, the observation vector is first degraded and denoised, and then the orthogonal matching pursuit algorithm is used to reconstruct the original sparse signal.

[0005] Most current studies on signal sparsity estimation do not consider signal noise, or only consider measurement noise. However, in practical applications, signal noise cannot be ignored. When considering signal noise, there are usually two ways to model it. The first is that the signal noise is deterministic and bounded; the second is that the signal noise is Gaussian white noise. In compressed sensing, signal noise will be multiplied after random measurement. This phenomenon is called noise folding (NF) [Arias-Castro E, Eldar Y C. Noise Folding in Compressed Sensing [J]. IEEE Signal Processing Letters, 2012, 18(8): 478-481.].

[0006] Due to the influence of NF, both signal sparsity estimation and signal reconstruction processes bring new challenges. In order to reduce the impact of noise NF on compressed sensing, there are two main processing methods: optimizing the sensing matrix and improving the reconstruction algorithm. In order to reduce the sampling of signal noise when sampling the signal, the literature [Pei Liye, Jiang Hua, Ma Yueliang. Compressed sensing denoising and reconstruction algorithm based on selective measurement [J]. Journal of Communications, 2017, 38(02): 106-114.] assumes that the noise information is known a priori, and designs the sensing matrix based on this a priori information to intelligently filter the noise component. The literature [PR Muduli, AK Mandal and A. Mukherjee. An Antinoise-Folding Algorithm for the Recovery of Biomedical Signals From Noisy Measurements [J]. IEEE Transactions on Instrumentation and Measurement, 2017: 2909-2916.] performs denoising on the estimated signal after signal reconstruction based on the data adaptive threshold denoising algorithm. The literature [X.Yang,Q.Cui,E.Dutkiewicz,X.Huang,X.Tao and G.Fang,Anti-noise-folding regularized subspace pursuit recovery algorithm for noisy sparse signals[C].2014IEEE WirelessCommunications and Networking Conference(WCNC),Istanbul,Turkey,2014,pp.275-280.] optimizes the reconstruction algorithm, introduces data preprocessing operations, and alleviates the impact of NF on signal reconstruction.

[0007] Currently, research on the NF problem in compressed sensing (CS) primarily focuses on the sensing matrix and reconstruction algorithms to minimize the impact of NF on CS. However, sparsity k is an essential parameter for minimizing the sampling rate while accurately reconstructing the original signal. Most current research on sparsity estimation only considers the impact of measurement error on the estimated samples. Existing research results are no longer applicable when the impact of signal noise on the estimated samples is considered. In order to solve the sparsity estimation problem of broadband sparse signals with signal noise, this paper improves the sparsity estimation model based on binomial distribution, and realizes the estimation of the upper bound of signal sparsity when the variance of the noise signal, the observation vector and the perception matrix are known. The literature [N.Mourad and JPReilly.Automaticthreshold estimation for Iterative Shrinkage Algorithms used with compressed sensing[C].2012IEEE International Conference on Acoustics,Speech and SignalProcessing(ICASSP),2012:2721-2724.] uses the sparsity upper bound to improve the adaptive threshold denoising algorithm and reduce the number of iterations. Summary of the Invention

[0008] When considering the influence of signal noise on the estimated samples, the present invention improves the sparsity estimation model based on the binomial distribution in order to solve the sparsity estimation problem of broadband sparse signals with signal noise. When the variance, observation vector and perception matrix of the noise signal are known, the upper bound of the signal sparsity can be estimated. The upper bound of the sparsity is used to improve the adaptive threshold denoising algorithm and reduce the number of iterations.

[0009] In order to achieve the above object, the technical solution adopted by the present invention is: a broadband spectrum compression detection method with signal noise, comprising the following steps:

[0010] 101. Pre-sampling to obtain estimated samples: The original signal is a sparse signal or becomes a sparse signal under a sparse basis transformation, and the estimated samples are obtained from the broadband sparse signal.

[0011] 102. Estimating the sparsity of a sparse signal from an estimated sample: First, analyze the probability and statistical characteristics of the noise components in the estimated sample. Then, based on the probability and statistical characteristics of the noise components, filter out most of the noise components in the estimated sample to obtain the upper bound of the signal's sparsity. Then, use the improved binomial distribution sparsity estimation model for sparsity estimation.

[0012] 103. Supplementary sampling: Use the estimated upper bound of signal sparsity to determine the number of samples and perform supplementary sampling.

[0013] Furthermore, the step 101 of pre-sampling to obtain estimated samples specifically includes:

[0014] Sparse signal: Let the broadband spectrum signal s = x + z with a length of N. If the signal is a sparse signal or is a sparse signal under the transformation of the sparse basis Ψ, that is,

[0015] X=Ψs

[0016] =Ψ(x+z) (1)

[0017] Where z represents the signal noise that obeys the Gaussian distribution, x represents the original signal, and Ψ is an N×N matrix. If there are only k non-zero values ​​in X after the transformation, then X is called the k-th order sparse signal of the signal s under the sparse basis Ψ transformation.

[0018] Obtain samples: In the compressed sensing method, the observation vector y is obtained through the M×N (M<<N) dimensional perception matrix A=ΘΨ, that is,

[0019]

[0020] Where M is the number of observations, Θ represents an M×N (M<<N) dimensional measurement matrix, and e=Az.

[0021] Furthermore, in step 102, the probability statistical characteristics of the noise component in the estimated sample are considered as the product of the signal noise and the measurement matrix, where the signal noise obeys the Gaussian distribution and the element a in the measurement matrix is ij Obey the binomial distribution, and let the element a in the measurement matrix ij Approximately a normal distribution.

[0022] Specifically, the measurement matrix adopts a sparse random matrix, and each element a in the matrix ij All obey the binomial distribution, that is,

[0023] a ij ~B(M,β)(3)

[0024] According to the De Moivre-Laplace theorem, let the random variable η n (n=1,2,…) obeys the binomial distribution with parameters n,p(0<p<1), then for any have

[0025]

[0026] n represents the number of experiments, and p represents the probability of each experiment.

[0027] The De Moivre–Laplace theorem states that the normal distribution is the limiting distribution of the binomial distribution when the following two conditions are met:

[0028] ①np≥5;

[0029] ②np(1-p)≥5;

[0030] The binomial distribution can be approximated as a normal distribution;

[0031] The parameters in the measurement matrix are d = 6, np = M × β = d, which satisfies condition ①;

[0032] For condition ②, np(1-p)=d(1-d / M), when the number of samples M≥6·d, condition ② is satisfied.

[0033] Furthermore, the sparse random matrix is ​​constructed by randomly distributing d 1s in each column of the matrix, and the rest of the elements are 0, and each element a ij Independent and identically distributed, let each element a ij The probability of being 1 is β, then each element a ij The probability of being 0 is 1-β, and β=d / M.

[0034] Furthermore, the upper bound of the sparsity in step 102 is

[0035]

[0036] in

[0037]

[0038] In the formula Indicates rounding up, F v1,v2,χ represents the upper 100×(1-χ) quantile of the F distribution with degrees of freedom v1 and v2, χ is the significance level, ω=||y||0, v1=2(ω+1), v2=2(M-ω), α represents the error rate.

[0039] Furthermore, the construction of the improved binomial distribution sparsity estimation model in step 102 includes:

[0040] The signal noise and measurement matrix in the observation vector y are both normally distributed. After multiplying the two normal distributions, they are still compressed or amplified normal distributions, that is, the means are μ f 、μ g , and the standard deviations are σ f , σ g The two normal distributions are still normally distributed after multiplication, and the mean μ of the product is fg and standard deviation σ fg They are:

[0041]

[0042] According to the De Moivre-Laplace theorem, measure the element a in the matrix ij It can be approximated as a Gaussian distribution with mean d and variance d(1-d / M), then the mean is 0 and the variance is σ 2 After multiplying the Gaussian noise by , the mean and standard deviation of the product are obtained according to formulas (9) and (10):

[0043]

[0044] So far, the noise component in the observation vector y is a normal distribution with a mean of μ and a standard deviation of σ. Let

[0045]

[0046] At this time, ξ obeys the standard normal distribution, that is:

[0047]

[0048] The elements in the measurement matrix are approximated by the standard normal distribution from the binomial distribution. In order to filter out most of the noise components in the observation vector, according to the statistical characteristics of the normal distribution, given a probability p, the upper bound ξ1 can be obtained by looking up the standard normal distribution table. The obtained ξ1 value is substituted into formula (11) to obtain the interval upper bound γ of the noise component in the observation vector. The threshold γ of the noise component is used to first filter out most of the noise components in the observation vector. The number of observation vectors greater than γ is used to estimate the sparsity of the broadband signal with signal noise, that is:

[0049] h=sum(|y(:)|>γ) (13)

[0050] h represents the number of measurement vectors with a magnitude greater than γ.

[0051] Replace ω in formula (6) with h, and use formulas (5), (6) and (13) to obtain the upper bound of sparsity, as shown in formula (14):

[0052]

[0053] in

[0054]

[0055] Represents the sparsity upper bound of the improved binomial distribution sparsity estimation model.

[0056] Based on the above scheme, an adaptive threshold denoising algorithm is added after signal reconstruction to remove the noise part of the reconstructed signal, and the adaptive threshold denoising algorithm is improved using the sparsity upper bound to reduce the number of algorithm iterations.

[0057] The adaptive threshold denoising algorithm includes arranging the reconstructed signals in descending order of amplitude, selecting the largest amplitude w[1] and the smaller amplitude w[N1] of the reconstructed signals to construct a virtual line y=ax+b, where a=(w[N1]-w[1]) / (N1-1), and using the point-to-line distance formula to obtain:

[0058]

[0059] The optimal threshold is the value of the point farthest from the virtual straight line, that is:

[0060] τ=w[i*],i*=argmaxd(i) (17)

[0061] Due to the estimated signal sparsity Slightly larger than the actual sparsity of the sparse signal, take To ensure the accuracy of the estimated threshold τ.

[0062] The advantages and beneficial effects of the present invention are as follows:

[0063] 1. Currently, few studies on the sparsity estimation of broadband signals consider the presence of signal noise. Therefore, this paper improves the sparsity estimation model based on binomial distribution for broadband signals with Gaussian white noise, derives the probability distribution function of the noise in the observation vector, and filters out the noise component in the observation vector during sparsity estimation, thereby improving the sparsity estimation performance.

[0064] 2. Calculate the number of samples based on the upper bound of the estimated signal sparsity, and reduce the number of samples while ensuring that the signal can be accurately reconstructed. At the same time, use the upper bound of the sparsity to improve the adaptive threshold denoising and reduce the number of algorithm iterations, so that the threshold can be determined more accurately and quickly. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is a schematic diagram of the process of the present invention;

[0066] Figure 2 The sparsity estimation under different signal-to-noise ratios of the present invention;

[0067] Figure 3 Comparison of the success rate of sparsity upper bound estimation for different signal-to-noise ratios and different methods;

[0068] Figure 4 Comparison of sampling rates among different methods;

[0069] Figure 5 Comparison of the number of iterations between the improved adaptive threshold denoising and the unimproved one;

[0070] Figure 6 Comparison of reconstruction accuracy between the case where adaptive threshold denoising is added after reconstruction and the case where adaptive threshold denoising is not added;

[0071] Figure 7 Comparison of detection probabilities of different methods;

[0072] Figure 8 Schematic diagram of the threshold estimation method. DETAILED DESCRIPTION

[0073] The present invention designs Figure 1 The algorithm flow chart shown is based on Figure 1 The present invention provides a wideband spectrum compression detection method with signal noise, which aims at the sparsity estimation problem with signal noise and designs the following steps:

[0074] 101. Pre-sampling to obtain estimated samples: Assume that the original signal is a sparse signal or becomes a sparse signal under a sparse basis transformation. Then obtain estimated samples from the broadband sparse signal.

[0075] The pre-sampling to obtain estimated samples specifically includes:

[0076] 201. Sparse signal: Assume that the broadband spectrum signal s=x+z with a length of N, if the signal is a sparse signal or is a sparse signal under the transformation of the sparse basis Ψ, that is,

[0077] X=Ψs

[0078] =Ψ(x+z) (1)

[0079] Where z represents the signal noise that follows a Gaussian distribution, x represents the original signal, and Ψ is an N×N matrix. If there are only k non-zero values ​​in X after the transformation, that is, ||X||0=k, k<<N. Then X is said to be the k-th order sparse signal of the signal s under the sparse basis Ψ transformation.

[0080] 202. Obtaining samples: In CS, the observation vector y is obtained through the M×N (M<<N)-dimensional perception matrix A=ΘΨ. Since the noise folding phenomenon is caused by signal noise, the present invention mainly considers the effect of signal noise on compressed sensing, that is,

[0081]

[0082] Where M is the number of observations, M=O(k×log(N / k)), Θ represents the measurement matrix of M×N (M<<N) dimensions, and e=Az.

[0083] 102. Sparsity estimation: Sparsity estimation is performed using estimated samples. However, since the estimated samples are affected by noise, there are noise components with a large degree of dispersion in the estimated samples. If the confidence interval is obtained by using the relationship between the number of non-zero elements ω in the sample and the estimated samples, there will be errors in the sparsity estimation results. Based on the binomial distribution sparsity estimation method, the present invention first derives and analyzes the probability and statistical characteristics of the noise components in the estimated samples, and then filters out most of the noise components in the estimated samples based on the probability and statistical characteristics of the noise components to obtain the upper bound of the sparsity of the signal. Specifically including:

[0084] (1) Based on the binomial distribution sparsity estimation model;

[0085] (2) Normal approximation of the binomial distribution;

[0086] (3) Improved sparsity estimation model.

[0087] The step (1) is based on the binomial distribution sparsity estimation model, specifically including: in order to ensure that the sparse signal does not lose information during the compressed sensing process, that is, when the perception matrix satisfies the restricted isometry property (RIP), the sparse signal can be accurately reconstructed using the reconstruction algorithm. When the number of observations M is large enough, the sparse random matrix satisfies the RIP property. The construction method of the sparse random matrix is ​​to randomly distribute d 1s in each column of the matrix, and the rest of the elements are 0, and each element a ij Independent and identically distributed, let each element a ij The probability of being 1 is β, then each element a ij The probability of being 0 is 1-β, and β=d / M, that is:

[0088] Using the sparse random matrix as the measurement matrix, we first calculate the number of non-zero elements ω in the estimated sample Me, and then estimate the upper bound of the signal sparsity based on the exact confidence interval of the binomial distribution. Right now

[0089]

[0090] in

[0091]

[0092] In the formula Indicates rounding up, F v1,v2,χ represents the upper 100×(1-χ) quantile of the F distribution with degrees of freedom v1 and v2, χ is the significance level, ω=||y||0, v1=2(ω+1), v2=2(M-ω), α represents the error rate.

[0093] The normal approximation of the binomial distribution in step (2) specifically includes: due to the influence of noise folding, the estimated sample no longer contains only signal information, but also contains noise information. Therefore, before using the estimated sample to estimate the signal sparsity, it is necessary to filter out most of the noise components to ensure the accuracy of the sparsity estimation. Since y can be regarded as the product of the signal noise and the measurement matrix, where the signal noise obeys a Gaussian distribution, and the element a in the measurement matrix ij Obey binomial distribution. In order to obtain the probability statistical characteristics of the noise component in the observation vector y, the present invention transforms the element a in the measurement matrix into ij Approximately a normal distribution.

[0094] The present invention uses a sparse random matrix as the measurement matrix, and each element a in the matrix ij All obey the binomial distribution, that is,

[0095] a ij ~B(M,β) (5)

[0096] De Moivre-Laplace Theorem: Suppose that the random variable ηn (n=1,2,…) obeys a binomial distribution with parameters n,p (0<p<1). have

[0097]

[0098] The De Moivre-Laplace theorem states that the normal distribution is the limiting distribution of the binomial distribution when the following two conditions are met:

[0099] ①np≥5;

[0100] ②np(1-p)≥5;

[0101] The binomial distribution can be approximated as a normal distribution, that is:

[0102] B(n,p)≈N(np,np(1-p)) (7)

[0103] The parameters d=6, np=M×β=d in the measurement matrix of the present invention satisfy the condition ①;

[0104] For condition ②, np(1-p)=d(1-d / M), when the number of samples M≥6·d, condition ② can be satisfied because the number of samples M=O(klog(N / k)), which is easy to satisfy for broadband signals. The proof is complete.

[0105] In summary, the element a in the measurement matrix in the present invention is ij It can be approximated as a normal distribution.

[0106] The improved sparsity estimation model of step (3) specifically includes: since each element in the random matrix. According to formula (7), this binomial distribution can be approximated as a normal distribution, namely:

[0107] a ij ~B(M,dM)≈N(d,d(1-d / M)) (8)

[0108] At the same time, the signal noise is also a normal distribution N0~N(0,σ 2 ).

[0109] After multiplying two normal distributions, they are still compressed or amplified normal distributions, that is, the means are μ f 、μ g , and the standard deviations are σ f , σ g The two normal distributions are still normally distributed after multiplication, and the mean μ of the product is fg and standard deviation σ fg They are:

[0110]

[0111] According to the De Moivre-Laplace theorem, measure the element a in the matrix ij It can be approximated as a Gaussian distribution with mean d and variance d(1-d / M), then the mean is 0 and the variance is σ 2 After multiplying the Gaussian noise by , the mean and standard deviation of the product are obtained according to formulas (11) and (12):

[0112]

[0113] So far, the noise component in the observation vector y is a normal distribution with a mean of μ and a standard deviation of σ. Let

[0114]

[0115] At this time, ξ obeys the standard normal distribution, that is:

[0116]

[0117] The present invention approximates the elements in the measurement matrix from a binomial distribution to a standard normal distribution. In order to filter out most of the noise components in the observation vector, based on the statistical characteristics of the normal distribution, given a probability p, the upper bound ξ1 can be obtained by consulting the standard normal distribution table. The obtained ξ1 value is substituted into formula (13) to obtain the interval upper bound γ of the noise component in the observation vector. The threshold γ of the noise component is first used to filter out most of the noise components in the observation vector. The number of observation vectors greater than γ is used to estimate the sparsity of the broadband signal with signal noise, that is:

[0118] h=sum(|y(:)|>γ) (15)

[0119] h represents the number of measurement vectors with a magnitude greater than γ.

[0120] Substituting ω in formula (4) with h, we can obtain the upper bound of sparsity from formulas (3), (4), and (15), as shown in formula (116):

[0121]

[0122] in

[0123]

[0124] Represents the sparsity upper bound of the improved binomial distribution sparsity estimation model.

[0125] 103. Supplementary sampling: Then use the estimated upper bound of the signal sparsity to determine the number of samples, and perform supplementary sampling to ensure that the original signal can be accurately reconstructed while reducing the number of samples as much as possible to improve the reliability of the detection scheme.

[0126] 104. Adaptive threshold denoising algorithm: In order to reduce the probability of false alarm, an adaptive threshold denoising algorithm is added after signal reconstruction, and the sparsity upper bound is used to improve the adaptive threshold denoising algorithm to reduce the number of algorithm iterations.

[0127] The adaptive threshold denoising algorithm in step 104 has the following specific steps:

[0128] Due to the unknown signal sparsity and the influence of noise folding on signal reconstruction, the signal reconstructed by compressed sensing may have a non-zero value at the zero frequency of the original signal, which will increase the probability of false alarm. Since the variance of the Gaussian white noise considered in the present invention is small, an adaptive threshold denoising algorithm is added after signal reconstruction to remove the noise part of the reconstructed signal.

[0129] The signal reconstruction algorithm mainly discusses Basis Pursuit De-Noising (BPDN). The main goal of the BPDN algorithm is to find the optimal solution to the underdetermined equation, namely:

[0130] min||s||1s.t||As-y||2≤ε (18)

[0131] Where ε>||c||2, c represents the measurement noise, and ε represents the reconstruction error.

[0132] The BPDN algorithm minimizes sparse signal errors while ensuring the sparsest possible signal representation. While BPDN can achieve some denoising effects on traditional observation vectors containing Gaussian measurement noise, the present invention primarily considers signal noise, resulting in poor signal reconstruction performance for BPDN. Because the variance of the signal noise considered in the present invention is relatively small, the amplitude of the reconstructed signal noise is smaller than the signal amplitude. Therefore, the present invention employs threshold denoising to remove noise from the reconstructed sparse signal.

[0133] The threshold denoising algorithm of the present invention adopts the adaptive threshold denoising algorithm and improves the adaptive threshold denoising algorithm based on the estimated sparsity. The algorithm idea is as follows: Figure 8 As shown. The reconstructed signals are arranged in descending order of amplitude. The solid line represents the correct solution vector, and the dotted line represents the noise component in the reconstructed signal. Select the largest amplitude w[1] and the smaller amplitude w[N1] of the reconstructed signal to construct a virtual line y=ax+b, where a=(w[N1]-w[1]) / (N1-1), and the point where N1 is the smallest and not 0. Using the distance formula from the point to the line, we get:

[0134]

[0135] d(i) represents the distance from the i-th point to the straight line, b represents the intercept of the straight line, w[i] represents the amplitude of the i-th point, and a represents the slope of the straight line.

[0136] The optimal threshold is the value of the point farthest from the virtual straight line, that is:

[0137] τ=w[i*],i*=argmaxd(i) (20)

[0138] τ represents the estimation threshold.

[0139] This method can obtain an optimal estimate without requiring known noise statistics, while also adaptively adjusting the threshold as the signal changes. To ensure accurate determination of the optimal threshold when signal sparsity is unknown, a larger N1 is typically chosen. If N1 is smaller than the signal sparsity k, not only noise but also some signal will be filtered out, increasing the probability of false alarms. While a larger N1 ensures accurate threshold estimation, it comes at the expense of increased time overhead for the adaptive denoising algorithm.

[0140] In order to accurately obtain the threshold and reduce the time cost of the adaptive threshold denoising algorithm, the present invention estimates the upper bound of the signal sparsity. This method is improved to reduce the time cost of the threshold denoising algorithm. Slightly larger than the actual sparsity of the sparse signal, let Only more calculations are made times, but the accuracy of the estimated threshold τ is guaranteed.

[0141] The process of the improved adaptive threshold denoising algorithm is as follows:

[0142]

[0143]

[0144] In order to verify the present invention, we conducted simulation experiments on the MATLAB platform. In order to verify the effectiveness of the broadband spectrum detection scheme proposed in the present invention, the improved sparsity estimation model and the adaptive threshold denoising algorithm were simulated respectively, mainly from five aspects: sparsity estimation, sampling rate, time overhead of the adaptive threshold denoising algorithm, reconstruction accuracy, and detection probability. The first two groups of experiments show that the improved sparsity estimation model of the present invention can estimate the sparsity of sparse signals with signal noise and the sampling rate of the present invention. The sampling rate is compared with the improved performance of the binomial distribution sparsity estimation model without considering signal noise in the literature [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48 (02): 243-248]. The third group of experiments is to compare and analyze the number of iterations of the improved algorithm with the adaptive threshold algorithm. The fourth group of experiments is to compare and analyze the reconstruction of the improved adaptive threshold denoising algorithm using only the BPDN algorithm. The last group of experiments is to compare and analyze the detection probability with the algorithm proposed in the literature [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248] and the literature [X. Yang, Q. Cui, E. Dutkiewicz, X. Huang, X. Tao and G. Fang, Anti-noise-folding regularized subspace pursuit recovery algorithm for noisy sparse signals [C]. 2014 IEEE Wireless Communications and Networking Conference (WCNC), Istanbul, Turkey, 2014, pp. 275-280]. That is, the detection probability is used to illustrate the size of the false alarm and false alarm probability. In the simulation process, assuming that the signal length N = 512 and the number of samples d = 6, confidence 1-α = 0.99, sparsity k = 10, noise variance σ 2 =0.01.

[0145] Figure 2 It is the simulation result of the sparsity upper bound estimation. As can be seen from the figure, the present invention can successfully estimate the upper bound of the sparsity of the signal when SNR=10, SNR=5, SNR=0, and SNR=-5. This shows that the improved sparsity estimation model of the present invention is suitable for sparse signals with noise, and can accurately estimate the upper bound of the sparsity for different SNRs, and is robust to noise. In addition, it can be seen from the figure that under different SNR conditions, the upper bound of the sparsity estimated by the present invention continues to increase with the increase of sparsity, and the increase amplitude is basically consistent with the increase amplitude of the true sparsity. This shows that the improved sparsity estimation model of the present invention can adapt to the high dynamics of cognitive radio networks, and can adaptively adjust the number of samples according to the change of sparsity to improve the accuracy of signal reconstruction.

[0146] Figure 3 This is a simulation result diagram showing the success rate of the sparsity upper bound estimation as the estimated sample Me changes. The main purpose is to compare the estimation success rate of the improved sparsity estimation model of the present invention at different signal-to-noise ratios with the sparsity estimation success rate of Reference 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved wideband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248] without considering the signal noise. Where the sparsity k = 10, the estimated sample Me is equally spaced at intervals of 10 in the interval [10, 80].

[0147] Figure 4The sampling rates of different methods were compared. When the signal sparsity is constant, the sampling rate gradually decreases as the signal length increases. When the signal length is the same, the sampling rates of the two methods are greater than the sampling rate of the actual sparsity. This is because both methods use the upper bound of the estimated sparsity to calculate the number of samples M. Although it increases the sampling cost to a certain extent, it ensures that the original signal can be accurately reconstructed. When the signal length is small, the sampling rate of the present invention is basically the same as that of Document 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248]. When the signal length is large, the sampling rate of the present invention is slightly higher than the sampling rate of Document 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248]. It shows that the improved sparsity estimation model of the present invention is suitable for the sparsity estimation of broadband sparse signals with noise, and can be compared with the case in reference 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248] which does not consider signal noise, thereby greatly reducing the sampling rate of broadband sparse signals with noise.

[0148] Figure 5 The following figure compares the number of iterations. When the sparsity k = 10, the estimated samples increase at two equal intervals in the interval [40, 60]. The simulation results show that the adaptive threshold denoising algorithm in Reference 13 [N. Mourad and JP Reilly. Automatic threshold estimation for iterative shrinkage algorithms used with compressed sensing [C]. 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012: 2721-2724] only requires around 60 iterations. However, the adaptive algorithm improved by using the upper bound of sparsity significantly reduces the number of iterations to only slightly more than the actual sparsity of the signal. This significantly reduces the time overhead of the adaptive threshold denoising algorithm, meeting the high dynamic requirements of cognitive networks.

[0149] Figure 6 This is a comparison of Basis Pursuit De-Noising (BPDN) followed by adaptive threshold denoising and BPDN algorithm reconstruction. e=60, the sparsity k is 10 and 20 respectively, the parameters of adaptive threshold denoising are: For sparse signals with signal noise, the reconstruction effect using only the BPDN algorithm is unsatisfactory when the sparsity is k=10 and k=20. The position of non-zero elements in the reconstructed signal is severely distorted compared to the original signal. The present invention adds adaptive threshold denoising after the BPDN algorithm. Although the amplitude of the reconstructed non-zero elements is different from that of the original signal, the position and number of the reconstructed non-zero elements are the same as those in the original signal. The purpose of spectrum detection is to detect the presence of a signal, not to restore the specific value of the signal. Therefore, the algorithm proposed in this invention is suitable for signals with noise. At the same time, the detection scheme proposed in this invention can also adaptively adjust the number of samples according to changes in sparsity, and can more accurately reconstruct the original signal.

[0150] Figure 7 This is the simulation result of the detection accuracy changing with the signal sparsity k. The estimated sample Me=60, and the sparsity k takes values ​​at equal intervals of 2 in the interval [10,30]. Both the present invention and document 13 [N.Mourad and JPReilly.Automaticthreshold estimation for Iterative Shrinkage Algorithms used with compressedsensing[C].2012IEEE International Conference on Acoustics,Speech and SignalProcessing(ICASSP),2012:2721-2724] add Gaussian white noise with SNR=10 to the original sparse signal, while the original signal of document 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved wideband compressed spectrum detection scheme based on binomial distribution[J]. Journal of Electronics, 2020, 48(02):243-248] does not contain signal noise. Detection accuracy = 1-false alarm probability-false alarm probability. From Figure 7It can be seen that with the increase of sparsity k, the detection accuracy of the three detection algorithms shows a downward trend. This is because with the increase of signal sparsity k, the number of estimated samples does not change, so the success rate of estimating signal sparsity k decreases, which ultimately affects the detection accuracy of broadband signals. In the simulation result figure, the detection accuracy of document 3 [Ma Bin, Wang Hongming, Xie Xianzhong. Improved broadband compressed spectrum detection scheme based on binomial distribution [J]. Journal of Electronics, 2020, 48(02): 243-248] is the highest, and the detection accuracy of the other two methods is lower than this detection probability. The reason is that both the present invention and document 13 [N. Mourad and JP Reilly. Automatic threshold estimation for Iterative Shrinkage Algorithms used with compressed sensing [C]. 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012: 2721-2724] add Gaussian white noise to the original sparse signal. After random measurement, the signal noise is amplified exponentially, which affects the signal reconstruction effect. The detection accuracy of the present invention is superior to that of Reference 13 [N. Mourad and JP Reilly. Automatic threshold estimation for iterative shrinkage algorithms used with compressed sensing [C]. 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012: 2721-2724] because the present invention improves the threshold denoising algorithm and can adaptively adjust the N1 parameter in the threshold denoising algorithm according to the estimated signal sparsity, thereby increasing the detection accuracy. This indicates that the improved adaptive threshold denoising algorithm of the present invention has a better denoising effect on the reconstructed signal.

[0151] It should also be noted that the terms "comprises," "includes," or any other variations thereof are intended to encompass non-exclusive inclusion, such that a process, method, commodity, or apparatus that includes a series of elements includes not only those elements but also other elements not explicitly listed, or includes elements inherent to such process, method, commodity, or apparatus. In the absence of further limitations, an element defined by the phrase "comprises a ..." does not exclude the presence of other identical elements in the process, method, commodity, or apparatus that includes the element.

[0152] The above embodiments should be understood as merely illustrating the present invention and not as limiting the scope of protection of the present invention. After reading the contents of the present invention, technicians may make various changes or modifications to the present invention, and these equivalent changes and modifications also fall within the scope defined by the claims of the present invention.

Claims

1. A broadband spectrum compression detection method with signal noise, characterized in that: It includes the following steps:

101. Pre-sampling to obtain estimated samples: The original signal is a sparse signal or is sparse under the transformation of a certain sparse basis. Estimated samples are obtained from the wideband sparse signal.

102. Estimating the sparsity of the sparse signal from the estimated samples: First, analyze the probability statistical characteristics of the noise components in the estimated samples. Then, according to the probability statistical characteristics of the noise components, filter out most of the noise components in the estimated samples to obtain the upper bound of the sparsity of the signal. An improved binomial distribution sparsity estimation model is used for sparsity estimation. The upper bound of the sparsity is where In the formula Indicates rounding up, F v1,v2,χ represents the upper 100×(1-χ) quantile of the F distribution with degrees of freedom v1 and v2, χ is the significance level, ω=‖‖y‖‖0, v1=2(ω+1), v2=2(M-ω), α represents the error rate; The construction of the improved binomial distribution sparsity estimation model includes: The signal noise and measurement matrix in the observation vector y are both normally distributed. After multiplying the two normal distributions, they are still compressed or amplified normal distributions, that is, the means are μ f 、μ g , and the standard deviations are σ f , σ g The two normal distributions are still normally distributed after multiplication, and the mean μ of the product is fg and standard deviation σ fg They are: According to the De Moivre-Laplace theorem, the element a in the matrix is ​​measured ij It can be approximated as a Gaussian distribution with mean d and variance d(1-d / M), then the mean is 0 and the variance is σ 2 After multiplying the Gaussian noise by , the mean and standard deviation of the product are obtained according to formulas (5) and (6): At this point, it is obtained that the noise component in the observation vector y follows a normal distribution with a mean of μ and a standard deviation of σ. Let At this time, ξ follows a standard normal distribution, that is: The elements in the measurement matrix are approximated from a binomial distribution to a standard normal distribution. In order to filter out most of the noise components in the observation vector, according to the statistical characteristics of the normal distribution, a probability p is given, and the upper bound ξ1 can be obtained by looking up the standard normal distribution table. Substitute the obtained value of ξ1 into formula (7) to get the interval upper bound γ of the noise component in the observation vector. First, use the threshold γ of the noise component to filter out most of the noise components in the observation vector, and use the number of elements greater than γ in the observation vector to estimate the sparsity of the wideband signal with signal noise, that is: h = sum(|y(:)| > γ) (9) h represents the number of elements in the measurement vector with an amplitude greater than γ. Replace ω in formula (2) with h. That is, from formulas (1), (2) and (9), the upper bound of the sparsity is obtained, as shown in formula (10): where represents the upper bound of the sparsity of the improved binomial distribution sparsity estimation model, M represents the number of observations, β = d / M; 103. Supplementary sampling: Determine the number of samplings using the estimated upper bound of the signal sparsity and perform supplementary sampling.

2. The broadband spectrum compression detection method with signal noise according to claim 1, characterized in that: The step 101 of pre-sampling to obtain estimated samples specifically includes: Sparse signal: Let the wideband spectrum signal s with length N be s = x + z. If the signal is a sparse signal or is sparse under the transformation of the sparse basis Ψ, that is X = Ψs = Ψ(x + z) (12) where z represents the signal noise following a Gaussian distribution, x represents the original signal, and Ψ is an N×N matrix. If there are only k non-zero values in X after transformation, then X is called a k-order sparse signal of the signal s under the transformation of the sparse basis Ψ. Obtaining samples: In the compressive sensing method, the observation vector y is obtained through the M×N (M << N) dimensional sensing matrix A = ΘΨ, that is where M represents the number of observation values, Θ represents an M×N (M << N) dimensional measurement matrix, and e = Az.

3. The broadband spectrum compression detection method with signal noise according to claim 2, characterized in that: In step 102, the probability statistical characteristics of the noise component in the estimated sample are considered as the product of the signal noise and the measurement matrix, where the signal noise obeys the Gaussian distribution and the element a in the measurement matrix is ij Obey the binomial distribution, and let the element a in the measurement matrix ij Approximately a normal distribution.

4. The broadband spectrum compression detection method with signal noise according to claim 3, characterized in that: The measurement matrix adopts a sparse random matrix, and each element a in the matrix ij All obey the binomial distribution, that is, a ij ~B(M,β)(14) According to the De Moivre-Laplace theorem, let the random variable η n (n=1,2,…) obeys the binomial distribution with parameters n,p(0<p<1), then for any have n represents the number of experiments, and p represents the probability of each experiment. The De Moivre - Laplace theorem shows that the normal distribution is the limiting distribution of the binomial distribution. When the following two conditions are met: ① np ≥ 5; ② np(1 - p) ≥ 5; The binomial distribution can be approximated as a normal distribution. The parameter d = 6 in the measurement matrix, and np = M×β = d, which meets condition ①. For condition ②, np(1 - p) = d(1 - d / M). When the number of samplings M ≥ 6·d, condition ② is met.

5. The broadband spectrum compression detection method with signal noise according to claim 4, characterized in that: The sparse random matrix is ​​constructed by randomly distributing d 1s in each column of the matrix, and the rest of the elements are 0, and each element a ij Independent and identically distributed, let each element a ij The probability of being 1 is β, then each element a ij The probability of being 0 is 1-β, and β=d / M.

6. The broadband spectrum compression detection method with signal noise according to claim 1, characterized in that: It also includes an adaptive threshold denoising step: after signal reconstruction, an adaptive threshold denoising algorithm is added to remove the noise part of the reconstructed signal, and the sparsity upper bound is used to improve the adaptive threshold denoising algorithm to reduce the number of algorithm iterations.

7. The method for detecting wideband spectrum compression with signal noise according to claim 6, characterized in that: The adaptive threshold denoising algorithm includes arranging the reconstructed signals in descending order of amplitude, selecting the largest amplitude w[1] and the smaller amplitude w[N1] of the reconstructed signals to construct a virtual line y=ax+b, where a=(w[N1]-w[1]) / (N1-1), and using the point-to-line distance formula to obtain: The optimal threshold is the value of the point farthest from the virtual straight line, that is: τ=w[i*],i*=argmaxd(i) (17) Due to the estimated signal sparsity Slightly larger than the actual sparsity of the sparse signal, take To ensure the accuracy of the estimated threshold τ.

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