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CN109309640A - A blind signal format recognition method based on machine learning - Google Patents

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CN109309640A - A blind signal format recognition method based on machine learning - Google Patents

A blind signal format recognition method based on machine learning Download PDF

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CN109309640A
CN109309640A CN201811046509.4A CN201811046509A CN109309640A CN 109309640 A CN109309640 A CN 109309640A CN 201811046509 A CN201811046509 A CN 201811046509A CN 109309640 A CN109309640 A CN 109309640A Authority
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amplitude
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circle
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CN109309640B (en
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高明义
张俊峰
陈伟
沈纲祥
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Suzhou University
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    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04LTRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00Modulated-carrier systems
    • H04L27/0012Modulated-carrier systems arrangements for identifying the type of modulation
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    • G06COMPUTING; CALCULATING OR COUNTING
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    • G06F18/2321Non-hierarchical techniques using statistics or function optimisation, e.g. modelling of probability density functions
    • HELECTRICITY
    • H04ELECTRIC COMMUNICATION TECHNIQUE
    • H04BTRANSMISSION
    • H04B10/00Transmission systems employing electromagnetic waves other than radio-waves, e.g. infrared, visible or ultraviolet light, or employing corpuscular radiation, e.g. quantum communication
    • H04B10/60Receivers
    • H04B10/61Coherent receivers

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Abstract

本发明公开了一种基于机器学习的盲信号格式识别方法,包括以下步骤:(1)从相干光接收机后端获取信号,信号经过色散补偿和时钟恢复算法补偿之后,获得信号的振幅圆用作识别目标,振幅圆的圈数为识别特征参数;(2)将步骤(1)获得的振幅圆映射转化为振幅对角线,实现非线性到线性的映射;(3)利用聚类算法求出振幅对角线的聚类中心;(4)计算出决策图对数的差值,求出振幅圆的圈数,识别信号调制格式。本发明在对相干光通信数据的处理过程中,采用基于幅度特征的MFI方法,在没有任何其它先验知识的情况下,经过非线性转为线性的映射,在线性空间中无迭代地快速识别幅度对角线的聚类中心的个数,从而判定信号调制格式。该方法对噪声具有高鲁棒性,能适应目前通信系统中流行的M‑QAM的调制格式信号,为下一代认知光网络提供了技术基础。

The invention discloses a blind signal format recognition method based on machine learning, comprising the following steps: (1) Obtaining a signal from the back end of a coherent optical receiver, and after the signal is compensated by dispersion compensation and clock recovery algorithm, the amplitude circle of the signal is obtained for As the identification target, the number of circles of the amplitude circle is the identification characteristic parameter; (2) Convert the amplitude circle mapping obtained in step (1) into the amplitude diagonal line to realize the nonlinear to linear mapping; (3) Use the clustering algorithm to find (4) Calculate the difference of the logarithm of the decision diagram, obtain the circle number of the amplitude circle, and identify the signal modulation format. In the process of processing the coherent optical communication data, the present invention adopts the MFI method based on the amplitude feature, and without any other prior knowledge, through nonlinear to linear mapping, it can quickly identify in the linear space without iteration. The number of cluster centers on the amplitude diagonal to determine the signal modulation format. The method has high robustness to noise and can adapt to the M-QAM modulation format signals popular in current communication systems, providing a technical basis for the next generation of cognitive optical networks.

Description

Fanaticism format identification method based on machine learning

Technical field

The present invention relates to intelligent coherent fiber communication methods, and in particular to a kind of physical layer in elastic optical network Intelligent fanaticism format identification method.

Background technique

It, can be using according to business in order to increase the availability of frequency spectrum of wavelength-division multiplex (WDM) system with fixed band gap The elastic optical network of demand assignment difference band gap.In order to support the elastic optical network system of this distribution according to need resource, control is flat Face needs some information of physical layer, such as transmit signal modulation format, wavelength available and optical line by.In addition, modulation format Identify that (MFI) module is also most important for a coherent light receiving system, some of important Digital Signal Processing (DSP) Algorithm, such as adaptive equalization, carrier phase recovery and signal decision algorithm require to obtain signal modulation information in advance.Cause This, blind MFI algorithm is essential for following cognition optical-fiber network.

Intelligent modulation format identification (MFI) method is roughly divided into two classes, decision-theoretic approach and feature extracting method.It is based on The MFI method of decision theory is a kind of more hypothesis verification methods, and such methods belong to bayesian theory, many priori is needed to know Know.The method of MFI based on feature extraction is modulated format identification by feature extraction, wherein relatively conventional is to utilize width Degree and identification feature of the constellation feature as MFI method.

In the prior art, the different location locating in coherent optical heterodyne communicatio according to MFI module, can will be based on feature extraction MFI method be divided into two classes.The first kind is that MFI module is placed on before carrier phase recovery module, extracts amplitude characteristic as knowledge It does not refer to.Amplitude recognition feature includes that papr (PAPR), amplitude histogram and the Stokes of signal amplitude are empty Between in amplitude distribution cumulative distribution function etc..Second class is that MFI module is placed on after carrier phase recovery module, is utilized To planisphere or eye figure directly distinguished by convolutional neural networks.To realize Classification and Identification, need to the classification in MFI module Device carries out pre-training, for example, by using artificial neural network (ANN), deep neural network (DNN) and convolutional neural networks network (CNN) algorithm, such methods realize that MFI needs the training of large sample to carry out Optimized model parameter.

Therefore, how to realize simple, strong noise robustness fanaticism format identification, have for the art important Meaning.

Summary of the invention

Goal of the invention of the invention is to provide a kind of fanaticism format identification method based on machine learning, is not needing it In the case where its priori knowledge, realization accurately identifies the modulation format of now widely used multi-signal, guarantees identification essence Degree and to the high robust of noise.

To achieve the above object of the invention, the technical solution adopted by the present invention is that: a kind of fanaticism lattice based on machine learning Formula recognition methods, for before carrier phase recovery, comprising the following steps:

(1) signal is obtained from coherent optical heterodyne communicatio rear end, signal obtains after dispersion compensation and clock recovery algorithm compensation The amplitude circle for obtaining signal is used as identification target, and the circle number of amplitude circle is identification feature parameter;

(2) amplitude diagonal line is converted by the amplitude circle mapping that step (1) obtains, realized non-linear to linear mapping;

(3) the cornerwise cluster centre of amplitude is found out using clustering algorithm;

(4) difference for calculating decision diagram logarithm finds out the circle number of amplitude circle, identification signal modulation format.

In above-mentioned technical proposal, in step (1), the compensated data of clock recovery algorithm are clock recovery data, docking The clock recovery data of receipts are pre-processed, it is normalized in the average energy based on clock recovery data, are obtained Data after normalizationp i :

In formula,x i It is clock recovery data,kIt is the total number of data point,iIt is data point.

In step (2), converting the cornerwise method of amplitude for amplitude circle mapping is, by the QAM in cartesian coordinate system Signal is transformed into polar coordinates, and in polar coordinates, for the variation range of every amplitude level all by noise dominant, noise is bigger, vibration Width level is wider, extracts the vertical axis ρ under polar coordinates as identification feature, as abscissa and is indulged with the amplitude information of complex signal Coordinate draws amplitude diagonal line.

In step (3), the clustering algorithm is used based on density-distance clustering algorithm.

It is described based on density-distance clustering algorithm specifically, two elements of algorithm are densityρiAnd distanceδi, densityρi Using gaussian kernel function, define:

WhereinI s ={ 1,2 ..., N }It is the tally set of data point,d ij It is dataiWith datajDistance,dc=ω* d F(N(N-1)/50) It is off value distance, ω is the weight parameter of dc, whereinF()It is round-off function,NIt is data point number;

δiIt is dataiThe minimum value of distance between the higher other data of density;

The cluster centre of cluster is these with relatively large local densityρiWith it is bigδiData point, using decision diagram, i.e.,ρ iIt is rightδiCurve can be with fast search cluster cluster centre;Pass through draftingγCurve is observed that the quantity of cluster centre, whereinγIt is defined as,

In above-mentioned technical proposal, for the data with the maximum local density,δiIt is the maximum of all data distances Value.

Preferably, weight parameter ω is 0.2.

In step (4), obtained γ value is arranged first in descending order, however uses logarithm γ value again,

According to formula, the X-coordinate of the smallest Y value is obtained, The circle number n as identified.

Due to the above technical solutions, the present invention has the following advantages over the prior art:

The present invention is in the treatment process to coherent light communication data, using the MFI method based on amplitude characteristic, not any In the case where other priori knowledges, switch to linear mapping by non-linear, without iteratively quick identification width in linear space The number of cornerwise cluster centre is spent, therefore, it is determined that format modulation signal.This method has high robust to noise, adapts to The modulation format signal of M-QAM popular in communication system at present provides technical foundation for next generation's cognition optical-fiber network, can be with Save the hardware cost of future communication systems.

By the validity of experimental verification this method, for the format modulation signal recognition accuracy for reaching 95%, QPSK signal The optical signal to noise ratio (OSNR) of 13.2 dB of subsistence level, the OSNR of 16 QAM signal subsistence level, 13.3 dB, 64 QAM signals are most The low OSNR for needing 19.7 dB.Method of the invention simultaneously is to nonlinear fiber robustness with higher, in single mode optical fiber (SMF) in transmission experiment, have verified that accuracy of identification is unrelated with the incident power of 16/64 QAM signal.

Detailed description of the invention

Fig. 1 is 4/16/64-QAM signal amplitude figure, and optical signal to noise ratio (OSNR) is respectively 17 dB, 21 dB and 26 dB;

The explanatory diagram of the blind MFI algorithm of Fig. 2;

Fig. 3 is the identification situation in embodiment, average circle number (b) recognition accuracy for the identification amplitude circle that (a) is calculated and OSNR (dB) relationship;

Fig. 4 is the experimental system block diagram that blind MFI method is realized in embodiment;

Fig. 5 is in embodiment, (a) identify amplitude circle average circle number and (b) accuracy of identification of 4/16/64-QAM signal and The relationship of OSNR (dB), the amplitude image for the 64-QAM signal that insertion figure is 20.8-dB OSNR and the difference of logarithm γ value are bent Line;

Fig. 6 be the amplitude circle measured in embodiment average identification circle number and accuracy of identification and SMF in emit signal power (dBm) relational graph.

Specific embodiment

The invention will be further described with reference to the accompanying drawings and embodiments:

Embodiment one: compared with traditional K-means clustering algorithm that n observation is divided into known k cluster, quick clustering is calculated Method can quickly track cluster centre in the case where no priori knowledge, so that decision diagram is drawn out, final computing cluster Quantity.Therefore, present invention application quick clustering algorithm handles the amplitude data of signal, by identify the quantity of amplitude leyel come Judge the modulation format of signal.Fig. 1 shows the map of magnitudes of the 4/16/64-QAM with various OSNR.It is obvious that different tune The amplitude circle of signal processed has different circle numbers, that is, different amplitude leyel numbers.With the increasing of order of modulation and system noise Add, the map of magnitudes of signal becomes to obscure very much, it is difficult to distinguish.Therefore, similar and fuzzy map of magnitudes how is distinguished as ability The difficult point in domain.

A kind of fanaticism format identification method based on machine learning of the invention, comprising the following steps:

(1) signal is obtained from coherent optical heterodyne communicatio rear end, signal obtains after dispersion compensation and clock recovery algorithm compensation The amplitude circle for obtaining signal is used as identification target, and the circle number of amplitude circle is identification feature parameter;

Purpose is the circle number (that is, amplitude leyel number) by the amplitude circle of identification signal come the modulation format of decision signal;Wherein, The compensated data of clock recovery algorithm are clock recovery data, are pre-processed to received clock recovery data, when being based on It is normalized in the average energy that clock restores data, the data after being normalizedp i :

In formula,x i It is clock recovery data,kIt is the total number of data,iIt is data point.

(2) amplitude diagonal line is converted by the amplitude circle mapping that step (1) obtains, realization is non-linear to reflect to linear It penetrates;

Due to the amplitude circle of signal after clock recovery be it is nonlinear, be difficult to correctly find its cluster centre.The present invention It is converted.In Fig. 2 (a), the 64-QAM signal in cartesian coordinate system is transformed into polar coordinates.In polar coordinates In, system noise determines the amplitude variation of each amplitude, and noise is bigger, and amplitude longitudinal direction amplitude is wider.The present invention proposes to extract Vertical axis amplitude ρ under polar coordinates draws amplitude diagonal line (that is, with the amplitude information of complex signal as cross as identification feature The curve that coordinate and ordinate are drawn), as shown in Fig. 2 (b).

The problem of obtaining the cluster centre of amplitude circle as a result, it is cornerwise to be converted into the amplitude how obtained in Fig. 2 (b) Cluster centre.Therefore, non-linear clustering problem can be converted to linear clustering problem by the mapping proposed by the invention.

(3) the cornerwise cluster centre of amplitude is found out using clustering algorithm;

That is densityρiAnd distanceδi.Make density p i used here as gaussian kernel function, define:

I s ={ 1,2 ..., N }It is the tally set of data point,d ij It is dataiWith datajDistance.dc=ω* d F(N(N-1)/50) It is off value distance, whereinF()It is round-off function,NIt is data point number,ωIt is weight parameter.δiIt is several According toiThe minimum value of distance between the higher other data of density.But for the data with the maximum local density,δi It is the maximum value of all data distances.Cluster centers are exactly these with relatively large local densityρiWith it is bigδiData Point.Therefore, it can use decision diagram (i.e.ρiIt is rightδiCurve graph), the cluster centre of fast search cluster.As shown in Figure 2 (c), side What block indicated is cluster centre, and what other stains indicated is that other cluster points or cluster are dizzy.Generally, pass through draftingγCurve can be seen The quantity of cluster centre is observed, as shown in Figure 2 (d) shows, whereinγIt is defined as,

In Fig. 2 (d), depict 20 it is biggishγValue, and it is arranged in descending order.As can be seen that the 9th point in Fig. 2 (d) And there is variation between the 10th point.Since cluster centre is with higherγValue, and other clusters point is with lowerγValue.Cause This, occursγIt is worth the place of variation, that is, cluster centre excessively arrives the place of cluster point.Therefore, pass through searchingγValue mutation Position may thereby determine that the number of cluster centre.And it can be determined that the modulation format of signal using the number of cluster centre.

However, in Fig. 2 (d)γValue mutation is not it is obvious that being difficult to differentiate between.It is proposed pair in the present inventionγValue takes logarithm, from And make log (γ)Value mutation increases, as shown in Fig. 2 (e).In figure, the 9th log (γ)Value and the 10th log (γ)Value is obvious It is different, so as to be clearly observed cluster centre (have higher log (γ)Value), such as the relatively high hurdle institute in Fig. 2 (e) Show.

In order to the quantity of cluster centre in automatic computing cluster, the present invention utilizes following formula,

By calculate adjacent log (γ)The difference of value, as shown in Fig. 2 (f).The position of difference maximum point is exactly the number of cluster centre Amount.

(4) difference for calculating decision diagram logarithm finds out the circle number of amplitude circle, identification signal modulation format.

According to the numerical value of the X-coordinate point under the minimum value of the Y of upper step acquisition, the width of the amplitude circle of QAM signal can be determined Level number n value is spent, so as to judge the modulation format of signal.

To further understand the present invention, the feasibility of the invention using Simulation, and further have evaluated the hair The performance of the MFI method of bright proposition.In analogue system, 12.5Gbaud 4/8/16/32/64/256-QAM signal passes through additivity Gaussian white noise channel.The value of OSNR decides the distribution of amplitudes of these signals, therefore has significant impact to MFI performance.

Fig. 3 depicts the function of the circle number (that is, amplitude leyel number) and recognition accuracy of amplitude circle as OSNR value.Scheming The amplitude distribution figure of the 256-QAM signal of 30-dB OSNR and the circle number n of identification are inserted in 3(a).To each of each OSNR A modulated signal has used 100 test datas respectively.Since the map of magnitudes of poor signal is relatively fuzzyyer, it is difficult to it differentiates, because The circle number of the signal amplitude circle identified in this each test may not be identical.In emulation and experiment, the circle number of amplitude circle is equal For the average of the circle number of the amplitude circle of all tests.The MFI scheme of the invention can be used to identify within the scope of different OSNR QAM signal.In addition, QPSK and 8-QAM signal of the OSNR less than 10 dB has the amplitude distribution of diverging, therefore they will be wrong It accidentally identifies, and obtains biggish amplitude leyel number, such as Fig. 3 (a).For the lower OSNR signal of about 12-dB, it is difficult correct Identify 32/64/ 56-QAM modulation format.Such as Fig. 3 (b), reach 95% accuracy of identification, QPSK signal subsistence level 7-dB OSNR, 8-QAM signal subsistence level 10-dB OSNR, 16-QAM signal subsistence level 16-dB OSNR, 32-QAM signal are minimum Need 21-dB OSNR, 64-QAM signal subsistence level 22-dB OSNR, 256-QAM signal subsistence level 33-dB OSNR.

In addition, testing the performance of the MFI method of invention proposition using Fig. 4 experiment.By adjusting in Fig. 4 experimental provision First adjustable attenuator (VOA), take away single mode optical fiber (SMF), the signal with different OSNR values can be measured, and calculate MFI recognition accuracy.

Fig. 5 depicts the relationship of the circle number and accuracy of identification and OSNR value for the amplitude circle that experiment measures.To reach 95% Accuracy of identification, QPSK signal subsistence level 13.2-dB OSNR, 16-QAM signal subsistence level 13.3-dB OSNR, 64-QAM letter Number subsistence level 19.7-dB OSNR, wherein each modulation format signal uses 50 test datas.As shown in Figure 3 and Figure 5, should Experimental measurements are very similar with simulation result, existing OSNR deviation be characteristic due to optical-fibre channel and photoelectric subassembly and Caused by quantity difference using modulated signal.

To assess proposed MFI to the robustness of nonlinear fiber.The front position of preamplifier in an experiment, insertion SMF hop (including EDFA, VOA and SMF).Because 16-QAM and 64-QAM signal is more susceptible to the influence of nonlinear fiber, Therefore experiment measures the accuracy of identification of both signals and the relationship of injection SMF signal power.Fig. 6 depict 16-QAM and The accuracy of identification of 64-QAM signal and the relationship of injection SMF signal power.Insertion figure is minimum signal power and peak signal function The map of magnitudes and planisphere of corresponding signal when rate.From the planisphere of insertion, spontaneous radiation (ASE) can clearly be observed that The influence of noise and nonlinear fiber.In the case of low signal power, ASE noise is bigger, causes the fuzzy of planisphere.High RST Under power situation, nonlinear fiber causes signal constellation (in digital modulation) phase rotation.Because nonlinear fiber has distorted the planisphere of signal, but What the map of magnitudes of signal is influenced without, therefore MFI scheme of the invention has very strong robustness to nonlinear fiber.Note The signal power for entering SMF influences less, as shown in the triangle marked curve in Fig. 6 the accuracy of identification of measurement.

Claims (8)

1.一种基于机器学习的盲信号格式识别方法,用于载波相位恢复之前,其特征在于,包括以下步骤:1. a blind signal format identification method based on machine learning, before being used for carrier phase recovery, is characterized in that, comprises the following steps: (1) 从相干光接收机后端获取信号,信号经过色散补偿和时钟恢复算法补偿之后,获得信号的振幅圆用作识别目标,振幅圆的圈数为识别特征参数;(1) Obtain the signal from the back end of the coherent optical receiver. After the signal is compensated by dispersion compensation and clock recovery algorithm, the amplitude circle of the obtained signal is used as the identification target, and the number of circles of the amplitude circle is the identification characteristic parameter; (2) 将步骤(1) 获得的振幅圆映射转化为振幅对角线,实现非线性到线性的映射;(2) Convert the amplitude circle map obtained in step (1) into an amplitude diagonal line to realize nonlinear to linear mapping; (3) 利用聚类算法求出振幅对角线的聚类中心;(3) Use the clustering algorithm to find the cluster center of the amplitude diagonal; (4) 计算出决策图对数的差值,求出振幅圆的圈数,识别信号调制格式。(4) Calculate the difference of the logarithm of the decision diagram, find the number of turns of the amplitude circle, and identify the signal modulation format. 2.根据权利要求1所述的基于机器学习的盲信号格式识别方法,其特征在于:步骤(1)中,时钟恢复算法补偿后的数据为时钟恢复数据,对接收的时钟恢复数据进行预处理,基于时钟恢复数据的平均能量对其进行归一化处理,获得归一化后的数据p i 2. the blind signal format identification method based on machine learning according to claim 1, is characterized in that: in step (1), the data after the clock recovery algorithm compensation is clock recovery data, and the clock recovery data received is preprocessed , and normalize it based on the average energy of the clock recovery data to obtain the normalized data p i : 式中,x i 是时钟恢复数据,k是数据的总个数,i是数据点。In the formula, x i is the clock recovery data, k is the total number of data, and i is the data point. 3.根据权利要求1所述的基于机器学习的盲信号格式识别方法,其特征在于:步骤(2)中,将振幅圆映射转化为振幅对角线的方法是,将笛卡尔坐标系中的QAM信号转换到极坐标中,在极坐标中,每条振幅水平的变化范围都由噪声主导,噪声越大,振幅水平越宽,提取极坐标下的垂直轴ρ作为识别特征,用复信号的幅度信息做为横坐标和纵坐标绘制幅度对角线。3. the blind signal format recognition method based on machine learning according to claim 1, is characterized in that: in step (2), the method that amplitude circle mapping is converted into amplitude diagonal is, in Cartesian coordinate system The QAM signal is converted into polar coordinates. In polar coordinates, the variation range of each amplitude level is dominated by noise. The greater the noise, the wider the amplitude level. The vertical axis ρ in polar coordinates is extracted as the identification feature, and the complex signal is used. The amplitude information is used as the abscissa and ordinate to draw the amplitude diagonal. 4.根据权利要求1所述的基于机器学习的盲信号格式识别方法,其特征在于:步骤(3)中,所述聚类算法采用基于密度-距离的聚类算法。4 . The blind signal format recognition method based on machine learning according to claim 1 , wherein: in step (3), the clustering algorithm adopts a density-distance-based clustering algorithm. 5 . 5.根据权利要求4所述的基于机器学习的盲信号格式识别方法,其特征在于:所述基于密度-距离的聚类算法具体为,算法的二要素是密度ρi和距离δi,密度ρi采用高斯核函数,定义:5. the blind signal format recognition method based on machine learning according to claim 4, is characterized in that: described density-distance-based clustering algorithm is specifically, the two elements of algorithm are density ρi and distance δi , density ρi adopts Gaussian kernel function, defined: 其中I s = {1,2,...,N}是数据点的标签集,d ij 是数据i与数据j的距离, dc=ω* d f(N(N-1)/ 50) 是截止值距离,ω是dc的权重参数,其中f(·)是舍入函数,N是数据点个数;where Is = {1, 2, ..., N} is the label set of data points, d ij is the distance of data i from data j , dc=ω* d f(N(N-1)/50) is cutoff distance, ω is the weight parameter of dc, where f( ) is the rounding function, and N is the number of data points; δi是数据i与密度较高的其它数据之间距离的最小值; δi is the minimum distance between data i and other data with higher density; 集群的聚类中心是这些具有相对大的局部密度ρi和大的δi的数据点,使用决策图,即ρ iδi曲线可以快速搜索集群聚类中心;通过绘制γ曲线可以观察到聚类中心的数量,其中γ定义为,The cluster centers of the clusters are these data points with relatively large local density ρi and large δi . Using a decision diagram, i.e. ρi vs. δi curve , the cluster center can be quickly searched; the cluster center can be observed by plotting the γ curve The number of , where γ is defined as, . 6.根据权利要求5所述的基于机器学习的盲信号格式识别方法,其特征在于:对于具有最大局部密度的数据而言,δi是所有数据距离的最大值。6 . The blind signal format recognition method based on machine learning according to claim 5 , wherein: for the data with the largest local density, δi is the maximum value of all data distances. 7 . 7.根据权利要求5所述的基于机器学习的盲信号格式识别方法,其特征在于:权重参数ω为0.2。7 . The blind signal format recognition method based on machine learning according to claim 5 , wherein the weight parameter ω is 0.2. 8 . 8.根据权利要求1所述的基于机器学习的盲信号格式识别方法,其特征在于:步骤(4)中,首先将得到的γ值按降序排列,然而再采用对数γ值,8. The blind signal format recognition method based on machine learning according to claim 1, is characterized in that: in step (4), at first the obtained γ values are arranged in descending order, but then logarithmic γ values are adopted, 根据公式,得到最小的Y值的X坐标,即为识别的圈数n。According to the formula , and the X coordinate of the smallest Y value is obtained, which is the number of recognized circles n.

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