数字通信信号的调制方式识别研究
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摘要
通信信号的调制方式识别是截获信号处理领域的重要组成部分,它需要在复杂环境和有噪声干扰的条件下,不依赖于其它的先验知识,确定接收信号的调制方式,并提取相应的调制参数,为信号的进一步分析和处理提供依据。随着通信体制和信号调制方式的复杂化和多样化,通信环境日益复杂,使通信信号的调制方式识别成为军、民领域内十分重要的研究课题。
近几十年来,人们在通信信号调制方式识别方面进行了大量的研究,并提出了许多新的方法和思路。本文在已有方法的基础上,重点对常用数字通信信号调制方式的自动识别进行研究。本文的主要工作和创新之处在于：
首先,本文分析了常用数字调制信号的调制原理,并对基于希尔伯特变换的瞬时信息提取、常用的载波频率估计和码元速率估计方法进行了研究。
其次,针对数字调制方式识别在低信噪比下的应用,本文对基于瞬时特征参数的数字调制方式识别算法进行了两点改进：首先设计一种小波滤波器,在信号的预处理阶段对瞬时信息进行小波消噪处理,减弱噪声的影响；另外从中频信号中提取五个相对简单的瞬时特征参数,改善算法的复杂度,简化识别过程。从而提高了算法在低信噪比下的识别性能。
再次,本文在理论分析的基础上,提出了一种基于高阶累积量和星座图的联合数字调制方式识别算法。该算法针对较难识别的数字调制信号2FSK、4FSK和8PSK,将傅里叶变换与高阶累积量相结合,提取高性能的特征参数进行识别,同时采用改进的星座图聚类分析法提高MQAM信号的识别率。文中从理论上分析了该算法的有效性,并用计算机仿真验证了有效性。
最后,为了改善数字调制方式识别在小样本下的识别效果,给出了一种基于高阶累积量和支持向量机的数字调制方式识别算法。该算法采用一对一的多类构造法,使用交叉验证网格搜索的方法进行参数选取,构建了稳健的多类支持向量机分类器,并利用高阶累积量域内构造的特征向量,进行数字调制信号的识别。仿真结果表明,与BP神经网络法相比,该算法具有更好的小样本识别性能和推广泛化能力。
Modulation type recognition of communication signals is an important part in the field of intercepted signal processing, which needs to identify the modulation type and extract the modulation parameters of the received signals without any priori knowledge in a complex environment and noise interference conditions, and provide the basis for further analysis and signal processing. With the complication and the diversification of communication system and signal modulation type, the communication environment is more and more complicated, it makes that the research of the communication signals modulation type recognition becomes a very important topic in civil and military field.
In recent decades the people have done much reseach on modulation type recognition of communication signals, at the same time many new methods and ideas are also proposed in this area as well. Based on the existing method, modulation type recognition of common digital communication signals is mainly studied in the paper. Main work and innovation of this paper include:
Firstly, this paper analyses the modulation principle of common digital modulation signals, and studies instantaneous information extraction based on Hilbert transform, the common method of carrier frequency estimation and symbol rate estimation.
Secondly, for the application under the low signal-to-noise(SNR), the algorithm based on instantaneous characteristic parameters is improved in two aspects:on the one hand, a wavelet filter is designed to de-noise the instantaneous information in pre-processing of received signals; on the other hand, five relatively simple instantaneous characteristic parameters extracted from IF signal are adopted to improve algorithm complexity and simplify recognition procedure. As a result, recognition performance of algorithm is improved under low SNR.
Thirdly, given theoretical analysis, a recognition algorithm of digital modulation type based on high order cumulants(HOC) and constellation is presented in the paper. The algorithm combines Fourier transform with high order cumulants and extracts feature parameter of excellent performance to classify 2FSK,4FSK and 8PSK which are difficult to classify, and adopts improved constellation clustering analysis to improve the correct rate of MQAM modulation recognition. The paper analyses the effectiveness of algorithm in theory, and also verifies it by simulation experiment.
Finally, the paper gives a digital modulation type recognition algorithm based on high order cumulants and support vector machines to improve recognition effect under small sample. In order to build robust SVM classifier, one against one of multi-classifer is designed, and the method of parameter selection using cross-validating grid is adopted. Then, characteristic vector extracted in high order cumulants domain is adopted to identify digital modulation types.Simulation results show that the algorithm possesses better small sample recognition performance and generalization ability comparing with BP neural network.
近几十年来,人们在通信信号调制方式识别方面进行了大量的研究,并提出了许多新的方法和思路。本文在已有方法的基础上,重点对常用数字通信信号调制方式的自动识别进行研究。本文的主要工作和创新之处在于：
首先,本文分析了常用数字调制信号的调制原理,并对基于希尔伯特变换的瞬时信息提取、常用的载波频率估计和码元速率估计方法进行了研究。
其次,针对数字调制方式识别在低信噪比下的应用,本文对基于瞬时特征参数的数字调制方式识别算法进行了两点改进：首先设计一种小波滤波器,在信号的预处理阶段对瞬时信息进行小波消噪处理,减弱噪声的影响；另外从中频信号中提取五个相对简单的瞬时特征参数,改善算法的复杂度,简化识别过程。从而提高了算法在低信噪比下的识别性能。
再次,本文在理论分析的基础上,提出了一种基于高阶累积量和星座图的联合数字调制方式识别算法。该算法针对较难识别的数字调制信号2FSK、4FSK和8PSK,将傅里叶变换与高阶累积量相结合,提取高性能的特征参数进行识别,同时采用改进的星座图聚类分析法提高MQAM信号的识别率。文中从理论上分析了该算法的有效性,并用计算机仿真验证了有效性。
最后,为了改善数字调制方式识别在小样本下的识别效果,给出了一种基于高阶累积量和支持向量机的数字调制方式识别算法。该算法采用一对一的多类构造法,使用交叉验证网格搜索的方法进行参数选取,构建了稳健的多类支持向量机分类器,并利用高阶累积量域内构造的特征向量,进行数字调制信号的识别。仿真结果表明,与BP神经网络法相比,该算法具有更好的小样本识别性能和推广泛化能力。
Modulation type recognition of communication signals is an important part in the field of intercepted signal processing, which needs to identify the modulation type and extract the modulation parameters of the received signals without any priori knowledge in a complex environment and noise interference conditions, and provide the basis for further analysis and signal processing. With the complication and the diversification of communication system and signal modulation type, the communication environment is more and more complicated, it makes that the research of the communication signals modulation type recognition becomes a very important topic in civil and military field.
In recent decades the people have done much reseach on modulation type recognition of communication signals, at the same time many new methods and ideas are also proposed in this area as well. Based on the existing method, modulation type recognition of common digital communication signals is mainly studied in the paper. Main work and innovation of this paper include:
Firstly, this paper analyses the modulation principle of common digital modulation signals, and studies instantaneous information extraction based on Hilbert transform, the common method of carrier frequency estimation and symbol rate estimation.
Secondly, for the application under the low signal-to-noise(SNR), the algorithm based on instantaneous characteristic parameters is improved in two aspects:on the one hand, a wavelet filter is designed to de-noise the instantaneous information in pre-processing of received signals; on the other hand, five relatively simple instantaneous characteristic parameters extracted from IF signal are adopted to improve algorithm complexity and simplify recognition procedure. As a result, recognition performance of algorithm is improved under low SNR.
Thirdly, given theoretical analysis, a recognition algorithm of digital modulation type based on high order cumulants(HOC) and constellation is presented in the paper. The algorithm combines Fourier transform with high order cumulants and extracts feature parameter of excellent performance to classify 2FSK,4FSK and 8PSK which are difficult to classify, and adopts improved constellation clustering analysis to improve the correct rate of MQAM modulation recognition. The paper analyses the effectiveness of algorithm in theory, and also verifies it by simulation experiment.
Finally, the paper gives a digital modulation type recognition algorithm based on high order cumulants and support vector machines to improve recognition effect under small sample. In order to build robust SVM classifier, one against one of multi-classifer is designed, and the method of parameter selection using cross-validating grid is adopted. Then, characteristic vector extracted in high order cumulants domain is adopted to identify digital modulation types.Simulation results show that the algorithm possesses better small sample recognition performance and generalization ability comparing with BP neural network.
引文
[1]曹志刚,钱亚生.现代通信原理[M].北京：清华大学出版社,2006：8-302.
[2][美]Richard A.Poise著.陈鼎鼎等译.现代通信干扰原理与技术[M].北京：电子工业出版社,2005.
[3]何为,高峰,李跃波等.基于循环谱的MPSK信号识别[J].通信技术,2010,43(10)：17-18.
[4]杨小牛,楼才义,徐建良.软件无线电原理及应用[M].北京：电子工业出版社,2001：4-208.
[5]A.Polydor and K.Kim.On the Detection Classification of Quadrature Digital Modulations in Broad-band Noise[J].IEEE Trans.on Communications.August 1990: 1109-1211.
[6]S.S.Soliman,Z.S.Hsue.Signal classification using statistical moment[J].IEEE Trans. Commun,1992,40(5):908-916.
[7]C.Schreyoegg,J.Reichert. Modulation Classification of QAM Schemes Using the DFT of Phase Histogram Combined with Modulus Information.IEEE Military Communications Conference Proceedings,MILCOM97,USA,1997,3:1372-1376.
[8]Alltti-Veikko Rosti and Visa Koivunen.Classifieation Of MASK Modulated Signals Using The Mean Of Complex Envelope.Proeeedings of EUSIPCO-2000,2000:581-584.
[9]张炜,杨虎,张尔杨.基于近似对数似然比的数字调制信号自动识别算法[J].信号处理,2007,23(1)：58-62.
[10]Liedtke F F.Computer Simulation of an Automatic Classification Procedure for Digitially Modulated Communication Signals with Unknown Parameters [J]. Signal Processing,1984,6(4):311-323.
[11]Hsue.Z.S,Soliman.S.S.Automatic Modulation Classification Using Zero Crossing [J]. Radar and Signal Processing,1990,137(6).459-464.
[12]Nandi.A.K,Azzouz.E.E.Automatic identification of digital modulation[J].IEEE Signal Processing,1995,47(1):55-69.
[13]Nandi.A.K,Azzouz.E.E.Algorithms for automatic modulation recognition of communication signals[J].IEEE Trans.on communication,1988,46(4):431-436.
[14]Wong.M.L.D,Nandi.A.K. Automatic digital modulation recognition using artificial neuraland genetic Algorithm[J].IEEE. Signal Processing,2004,84:351-365.
[15]C.Louis,P.Sehier. Automatic Modulation Recognition with a Hierarchical Neural Network.IEEE Military Communications Conference,MILCOM9'94,USA,1994,3:713-717.
[16]Wei Dai,Wouzheng Wang,Jing Wang.Joint power estimation and modulation classification Using Second and higher Statistics.Wireless Communications and networking Conferences,2002,IEEE,Vol.1:155-158.
[17]Abdulnasir Hossen,Fakhri Al-Wadahi,Joseph A.Jervase.Classification of modulation signals using statistical signal characterization and artificial neural networks[J]. Engineering Applications of Arificial Intelligence,2006,(20):463-472.
[18]He Tao,Zhou Zheng'ou. Modulation Classification in Alpha-Stable Noise.ICCCAS, 2007,193-196.
[19]Liang Hong,Ho K C.Identification of digital modulation types using the wavelet tansform.MIL'99,VOL.1:427-731.
[20]高翔,李静辉,黄高明.改进的通信信号谱相关识别方法[J].电子信息对抗技术,2009,24(1)：20-23.
[21]包溪锐,吴瑛,周欣.基于高阶累积量的数字调制信号识别算法[J].信息工程大学学报,2007,8(4)：463-467.
[22]徐哲,胡世安,吴钦等.一种基于差分星座图的调制体制识别算法[J].计算机仿真,2009,26(11)：182-185.
[23]双冰.基于小波和分形理论的调制信号特征提取方法研究[J].信号处理,2005,21(3)：317-318.
[24]王艳聪,胡中豫,张云飞.基于谱相关特性和组合神经网络的数字调制信号识别[J].计算机应用研究,2009,26(11)：4234-4236.
[25]Shrime,Ataollah Ebrahimzadeh. Recognition of the Communication Signals Using Particle Swarm Optimization and Support Vector Machine Based on the Multi-Resolution Wavelet Analysis [J]. Wireless Personal Communications,2010:3-10.
[26]张贤达,保铮.通信信号处理[M].北京：国防工业处出版社,2002：17-25.
[27]朱华,黄辉宁,李永庆等.随机信号分析[M].北京：北京理工大学出版社,2005,288-304.
[28]李滔,李勇.基于正交变换的相位估计方法性能分析[J].电子与信息学报,2008,30(8)：1857-1860.
[29]汤素华,尹华锐等.一种多参数联合判决的调制识别方法[J].数据采集与处理,2003, 18(1)：27-31.
[30]阙隆树,冯全源.数字调制信号识别中的相位去卷叠门限研究[J].信号处理,2010,26(1)：56-58.
[31]刘贵江,冯小平.一种适用于数字调制信号的载波频率估计方法[J].系统工程与电子技术,2004,26(12)：1787-1788.
[32]Yulong Gao,Zhongzhao Zhang. Modulation Recognition Based on Combined Feature Parameter and Modified Probabilistic Neural Network. Proceedings of the 6th World Congress on Intelligent Control and Automation,2006:21-23.
[33]Wu Juan ping,Han Ying zheng,Zhang Jin mei,etc.Automatic modulation recognition of digital communication signals using statistical parameters. International Conference on communications, Circuits and Systems,2007:697-700.
[34]Ferreira A R,Da Silva.Wavelet denoising with evolutionary algorithms[J].Digital Signal Processing.2005,15(4):382-392.
[35]孙延奎.小波分析及应用[M].北京：机械工业出版社,2005.
[36]蒋云霄,杨俊安,钟子发.基于Donoho模型和高阶统计理论的小波消噪算法研究及应用[J].电路与系统学报,2007,12(1)：11-14.
[37]张达敏,王旭.对基于决策论的数字信号调制识别方法的改进[J].计算机应用,2009,29(12)：3228-3229.
[38]张贤达.时间序列分析-高阶统计量方法[M].北京：清华大学出版社,1996：22-58.
[39]Ataollah Ebrahimzadeh Shermeh, Reza Ghazalian.Recongntion of communication signals types using genetic algorithm and support vector machines based on higher order statistics [J].IEEE Digital signal processing,2010.
[40]Ananthram Swami.Hierarchical Digital Modulation Classification Using Cumulants[J]. Transactions on communications,2000,48(3):416-418.
[41]郭梯云,杨家炜,李建东.数字移动通信(第二版)[M].北京：人民邮电大学出版社,2001.
[42]Negar Ahmadi. Using fuzzy clustering and TTSAS algorithm for modulation classification based on constellation diagram [J]. Engineering Applications of Artificial Intelligence,2010,32(3):360-363.
[43]富勤学.多径信道下MPSK和MQAM信号的调制识别[D].西安：西安电子科技大学,2006.
[44]田玉双,王曰海,李式臣等.存在定时误差的MQAM信号调制方式识别[J].江南大学 学报,2008,7(2)：179-181.
[45]黄英,雷菁.卫星通信中调制识别算法研究[J].系统工程与电子,2009,31(16)：1303-1305.
[46]顾学迈,丁楠.卫星通信常用数字调制方式自动识别算法研究[J].哈尔滨工业大学学报,2007,30(1)：93-96.
[47]韩刚,张文红,李建东.基于高阶累积量和支持向量机的调制识别研究[J].系统工程与电子,2003,25(8)：1007-1009.
[48]Li Jian dong,Feng Xiang.Modulation classification algorithm based on support vector machines and cyclic cumulants[J].Systems Engineering and Electronics,2007,29(4):520-523.
[49]Cortes C,Vapnik V.N.Support vector networks[J].Machine Learning,1995,20:144-152.
[50]赵晖.支持向量机分类方法及其在文本分类中的应用研究[D].大连：大连理工大学,2005.
[51]温熙森.模式识别与状态监控[M].北京：科学出版社,2007.10：315-340.
[52]吴青.基于优化理论的支持向量机学习算法研究[D].西安：西安电子科技大学.2009.
[53]Ataollah Ebrahim Zadeh.Automatic recognition of radio signals using a hybrid intelligent technique [J]. Expert Systems with Applications,2010,37(8):5806-5807.
[54]苟博,黄贤武.支持向量机多类多分类方法[J].数据采集与处理,2006,21(3)：334-339.
[55]Hsu C W,Lin C J.A comparison of methods for multiclass support vector machines[J]. IEEE Transactions on Neural Networks,2002,13(2):415-425.
[56]Madzarov G,Gjorqji D,Chorbev L. A multi-class SVM classifier utilizing binary decision tree[J]. Informatica,2009,33(2):233-236.
[57]Keerthi S S,Lin C J.Asymptotic behaviors of support vector machines with Gaussian kernel[J].Neural Computation,2003,15(7):1667-1689.
[2][美]Richard A.Poise著.陈鼎鼎等译.现代通信干扰原理与技术[M].北京：电子工业出版社,2005.
[3]何为,高峰,李跃波等.基于循环谱的MPSK信号识别[J].通信技术,2010,43(10)：17-18.
[4]杨小牛,楼才义,徐建良.软件无线电原理及应用[M].北京：电子工业出版社,2001：4-208.
[5]A.Polydor and K.Kim.On the Detection Classification of Quadrature Digital Modulations in Broad-band Noise[J].IEEE Trans.on Communications.August 1990: 1109-1211.
[6]S.S.Soliman,Z.S.Hsue.Signal classification using statistical moment[J].IEEE Trans. Commun,1992,40(5):908-916.
[7]C.Schreyoegg,J.Reichert. Modulation Classification of QAM Schemes Using the DFT of Phase Histogram Combined with Modulus Information.IEEE Military Communications Conference Proceedings,MILCOM97,USA,1997,3:1372-1376.
[8]Alltti-Veikko Rosti and Visa Koivunen.Classifieation Of MASK Modulated Signals Using The Mean Of Complex Envelope.Proeeedings of EUSIPCO-2000,2000:581-584.
[9]张炜,杨虎,张尔杨.基于近似对数似然比的数字调制信号自动识别算法[J].信号处理,2007,23(1)：58-62.
[10]Liedtke F F.Computer Simulation of an Automatic Classification Procedure for Digitially Modulated Communication Signals with Unknown Parameters [J]. Signal Processing,1984,6(4):311-323.
[11]Hsue.Z.S,Soliman.S.S.Automatic Modulation Classification Using Zero Crossing [J]. Radar and Signal Processing,1990,137(6).459-464.
[12]Nandi.A.K,Azzouz.E.E.Automatic identification of digital modulation[J].IEEE Signal Processing,1995,47(1):55-69.
[13]Nandi.A.K,Azzouz.E.E.Algorithms for automatic modulation recognition of communication signals[J].IEEE Trans.on communication,1988,46(4):431-436.
[14]Wong.M.L.D,Nandi.A.K. Automatic digital modulation recognition using artificial neuraland genetic Algorithm[J].IEEE. Signal Processing,2004,84:351-365.
[15]C.Louis,P.Sehier. Automatic Modulation Recognition with a Hierarchical Neural Network.IEEE Military Communications Conference,MILCOM9'94,USA,1994,3:713-717.
[16]Wei Dai,Wouzheng Wang,Jing Wang.Joint power estimation and modulation classification Using Second and higher Statistics.Wireless Communications and networking Conferences,2002,IEEE,Vol.1:155-158.
[17]Abdulnasir Hossen,Fakhri Al-Wadahi,Joseph A.Jervase.Classification of modulation signals using statistical signal characterization and artificial neural networks[J]. Engineering Applications of Arificial Intelligence,2006,(20):463-472.
[18]He Tao,Zhou Zheng'ou. Modulation Classification in Alpha-Stable Noise.ICCCAS, 2007,193-196.
[19]Liang Hong,Ho K C.Identification of digital modulation types using the wavelet tansform.MIL'99,VOL.1:427-731.
[20]高翔,李静辉,黄高明.改进的通信信号谱相关识别方法[J].电子信息对抗技术,2009,24(1)：20-23.
[21]包溪锐,吴瑛,周欣.基于高阶累积量的数字调制信号识别算法[J].信息工程大学学报,2007,8(4)：463-467.
[22]徐哲,胡世安,吴钦等.一种基于差分星座图的调制体制识别算法[J].计算机仿真,2009,26(11)：182-185.
[23]双冰.基于小波和分形理论的调制信号特征提取方法研究[J].信号处理,2005,21(3)：317-318.
[24]王艳聪,胡中豫,张云飞.基于谱相关特性和组合神经网络的数字调制信号识别[J].计算机应用研究,2009,26(11)：4234-4236.
[25]Shrime,Ataollah Ebrahimzadeh. Recognition of the Communication Signals Using Particle Swarm Optimization and Support Vector Machine Based on the Multi-Resolution Wavelet Analysis [J]. Wireless Personal Communications,2010:3-10.
[26]张贤达,保铮.通信信号处理[M].北京：国防工业处出版社,2002：17-25.
[27]朱华,黄辉宁,李永庆等.随机信号分析[M].北京：北京理工大学出版社,2005,288-304.
[28]李滔,李勇.基于正交变换的相位估计方法性能分析[J].电子与信息学报,2008,30(8)：1857-1860.
[29]汤素华,尹华锐等.一种多参数联合判决的调制识别方法[J].数据采集与处理,2003, 18(1)：27-31.
[30]阙隆树,冯全源.数字调制信号识别中的相位去卷叠门限研究[J].信号处理,2010,26(1)：56-58.
[31]刘贵江,冯小平.一种适用于数字调制信号的载波频率估计方法[J].系统工程与电子技术,2004,26(12)：1787-1788.
[32]Yulong Gao,Zhongzhao Zhang. Modulation Recognition Based on Combined Feature Parameter and Modified Probabilistic Neural Network. Proceedings of the 6th World Congress on Intelligent Control and Automation,2006:21-23.
[33]Wu Juan ping,Han Ying zheng,Zhang Jin mei,etc.Automatic modulation recognition of digital communication signals using statistical parameters. International Conference on communications, Circuits and Systems,2007:697-700.
[34]Ferreira A R,Da Silva.Wavelet denoising with evolutionary algorithms[J].Digital Signal Processing.2005,15(4):382-392.
[35]孙延奎.小波分析及应用[M].北京：机械工业出版社,2005.
[36]蒋云霄,杨俊安,钟子发.基于Donoho模型和高阶统计理论的小波消噪算法研究及应用[J].电路与系统学报,2007,12(1)：11-14.
[37]张达敏,王旭.对基于决策论的数字信号调制识别方法的改进[J].计算机应用,2009,29(12)：3228-3229.
[38]张贤达.时间序列分析-高阶统计量方法[M].北京：清华大学出版社,1996：22-58.
[39]Ataollah Ebrahimzadeh Shermeh, Reza Ghazalian.Recongntion of communication signals types using genetic algorithm and support vector machines based on higher order statistics [J].IEEE Digital signal processing,2010.
[40]Ananthram Swami.Hierarchical Digital Modulation Classification Using Cumulants[J]. Transactions on communications,2000,48(3):416-418.
[41]郭梯云,杨家炜,李建东.数字移动通信(第二版)[M].北京：人民邮电大学出版社,2001.
[42]Negar Ahmadi. Using fuzzy clustering and TTSAS algorithm for modulation classification based on constellation diagram [J]. Engineering Applications of Artificial Intelligence,2010,32(3):360-363.
[43]富勤学.多径信道下MPSK和MQAM信号的调制识别[D].西安：西安电子科技大学,2006.
[44]田玉双,王曰海,李式臣等.存在定时误差的MQAM信号调制方式识别[J].江南大学 学报,2008,7(2)：179-181.
[45]黄英,雷菁.卫星通信中调制识别算法研究[J].系统工程与电子,2009,31(16)：1303-1305.
[46]顾学迈,丁楠.卫星通信常用数字调制方式自动识别算法研究[J].哈尔滨工业大学学报,2007,30(1)：93-96.
[47]韩刚,张文红,李建东.基于高阶累积量和支持向量机的调制识别研究[J].系统工程与电子,2003,25(8)：1007-1009.
[48]Li Jian dong,Feng Xiang.Modulation classification algorithm based on support vector machines and cyclic cumulants[J].Systems Engineering and Electronics,2007,29(4):520-523.
[49]Cortes C,Vapnik V.N.Support vector networks[J].Machine Learning,1995,20:144-152.
[50]赵晖.支持向量机分类方法及其在文本分类中的应用研究[D].大连：大连理工大学,2005.
[51]温熙森.模式识别与状态监控[M].北京：科学出版社,2007.10：315-340.
[52]吴青.基于优化理论的支持向量机学习算法研究[D].西安：西安电子科技大学.2009.
[53]Ataollah Ebrahim Zadeh.Automatic recognition of radio signals using a hybrid intelligent technique [J]. Expert Systems with Applications,2010,37(8):5806-5807.
[54]苟博,黄贤武.支持向量机多类多分类方法[J].数据采集与处理,2006,21(3)：334-339.
[55]Hsu C W,Lin C J.A comparison of methods for multiclass support vector machines[J]. IEEE Transactions on Neural Networks,2002,13(2):415-425.
[56]Madzarov G,Gjorqji D,Chorbev L. A multi-class SVM classifier utilizing binary decision tree[J]. Informatica,2009,33(2):233-236.
[57]Keerthi S S,Lin C J.Asymptotic behaviors of support vector machines with Gaussian kernel[J].Neural Computation,2003,15(7):1667-1689.