基于非先验函数系的信号识别
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摘要
信号分析在众多的科研领域都有运用,它涉及科学研究、生产技术领域,而且涉及医疗诊断,与人们的日常生活密切相关。对信号处理的研究和运用在不同的学科之间相互交叉,共同推进了信号处理研究的发展。
信号分析方法多,本文尝试从运算方法的角度去分析、比较各种方法,不仅仅以基函数的不同去区分各种方法。
实际工程信号处理中,DFT是最常用的信号处理方法,DFT分析的核心是建立正交基,但是要建立一个正交基能够准确识别任意一个频率的正余弦信号是做不到的,这样一个正交基是超现实的,因此需要“频谱校正”。DFT先验地建立正交基,同时要满足逆变换,保证能够重构信号,存在频率间隔。基函数的频率与信号频率不吻合将产生误差,先验地选取基函数是DFT分析误差的原因。基于上述分析,提出非先验策略。一个工程信号,只有有限个不同频率的信号。找到这有限个频率对应的基函数,由这些基函数组成一个函数系,就能够对这一信号进行准确分析。这个问题转化为非先验函数的寻找和函数系的建立问题。在这一想法指导下,提出了非先验的计算分析方法,对单频、非密集频谱、一般密集谱和超密集谱由简到繁的各种情况进行了研究,结果表明非先验的分析方法都能够找到一个函数系与实际存在的信号吻合,这样可消除泄露误差。目前的频谱校正方法,以解方程为主要手段,解方程适用于单频和非密集谱,对于密集谱解方程的方法复杂,如果同时考虑负频谱项,多个密集谱的情况,解方程实际难以实现。非先验函数的找寻在数学上基于优化计算。在解决负频谱影响(超低频信号识别)和密集谱的识别问题上尤其显示其优越性,而这两方面是目前频谱校正研究的关注点。非先验方法基函数的选取,依据实际信号的情况而定,没有频率间隔的限制；从非先验分析的角度看,理论上信号截断对正余弦信号的准确识别没有影响,信号截断是实际工程信号处理中的必然,在信号截断的情况下仍然能够进行准确识别具有实际工程意义。
在完成识别的基础上,研究了用非先验方法逼近信号。在识别正余弦信号的研究中分析了减法具有消除交叉干扰的作用,由于减法非常简单,以至于它的这一作用没有得到重视。基本非先验方法有两个核心运算一个是内积运算,一个是减法运算,减法运算保证了这一方法的收敛性。一旦非先验的函数系被确定,则能够进行最佳逼近运算。与DFT方法对比,非先验方法的分析机理不是插值逼近。非先验方法的分析机理是一种逐步逼近,采用DFT分析工程信号是一种构造型的插值分析方法。本文分析了逼近展开与识别的不同,这两者有着多方面的区别,对它们的分析应该采用不同的路线,识别和逼近展开的核心区别在于信息熵不同,识别的信息量大,识别的结果在时间上可延拓,而一般逼近展开则不能够。通过算例对比了非先验分析与DFT和DCT的逼近速度,结果表明非先验方法的逼近效率高,非先验方法具有更广的基函数使用空间,能够更加灵活地适用于不同的信号。
卷积也是信号处理中常见的运算,卷积公式Y(ω)=H(ω)X(ω)有其适用范围,只能运用于能量有限的信号,对于Y(ω)和X(ω)都受到噪声干扰时,卷积公式Y(ω)=H(ω)X(ω)和反向滤波公式X(ω)=Y(ω)/H(ω)都会带来大的误差。提出基函数卷积运算,这种卷积运算对功率信号和能量信号都适用；与噪声限值相结合,基函数卷积也能够适用于信号受到噪声污染的情况。并且将其运用于有较强噪声干扰情况下的反卷积运算,诊断结果有较高的精度。
将非先验函数系分析方法运用于阻尼识别。实际上阻尼识别和正余弦信号识别是同一类问题,可以采用相同的技术路线,不同点仅在于基函数的选取不同。提出了非先验函数系的阻尼识别方法,理论上这一方法对信号的长度没有特定的要求,是一种准确识别方法。同时将非先验分析用于双传声器声强测量,进行了模拟计算和实验验证。结果证明这种方法可以避免泄漏误差,准确地测量出声强值。
DFT等先验基方法在思想方面强调了理想性,但是要达到这种理想条件有很大的难度。以正、余弦信号的识别为例,不能够现实地建立一个完备的正交函数基保证对任意频率的正、余弦信号的准确识别。先验基分析中,基的建立占主导地位；非先验分析中,实际信号占主导地位,基函数的选取和确定要视信号而定。
Signal analysis is applied to various scientific research fields. It not only involves in research and production technology, but also involves in medical diagnosis. It is closely related to daily life of people. The research and application of signal analysis as well as its intercross of the difference subjects promote the development of the research in signal processing.
It is difficult to master numerous signal analysis methods. This paper attempts to grasp the different analysis methods from the perspective of calculation operation method. It is without using the different basis functions to distinguish the different analysis method.
For practical engineering signal, DFT is the most commonly used signal processing method. The idea of orthogonal basis for DFT was widely accepted. However, we can't accurately identify any frequency of the sine or cosine signal by construction an orthogonal basis function. The reason is that such an orthogonal basis is surreal. The orthogonal prior basis of DFT is constructed; at the same time the inverse transform of DFT must be satisfied in order to reconstruct the signal. The orthogonal prior basis of DFT exist frequency separation. The origin error of the DFT analysis is produced when the frequency of the orthogonal prior basis discordance with the signal frequency. A non-prior basis strategy is proposed based on the above analysis. For an engineering signal, there are only a finite number of different frequency components. After searched the basis function of its corresponding frequencies, then the mathematic function system is formed by these basis functions. We can accurately analyze the signals by the basis function system. The problem is transformed into the issue of establishing and searching the non-prior basis function. Under the guidance of this idea, non-priori basis function strategy proposed. Single-frequency spectrum, non-dense spectrum, common dense spectrum and ultra-dense spectrum are calculated and analyzed from the simple to complex various situations. The method of non-prior basis function can find a function system coincide with the real signal, so as to eliminate leakage errors. Constructed the equation is the necessary mean for the current spectrum correction method. Solving the equations is suitable for single-frequency and non-dense spectrum. For dense spectrum, solving equations is too complicated. If taking into account the negative spectrum, solving equation is difficult to implement for multi-frequencies signal. Searching for non-prior basis function is based on optimization calculation. The method of non-prior basis function has its advantage of solving negative frequency impact (ultra-low frequency signal recognition) and dense spectrum identification. The selection of non-prior basis function of non-prior method is depended on the practical signal without limitation of frequency intervals. The truncation of the signal has no influence on precisely diagnosing the sine cosine signal in theory. However, the truncation of the signal is an inevitable process in practical engineering signal.
After identification of a practical engineering signal, the series expansion approximation is studied. The subtraction is researched for identification of sine and cosine signal to eliminate cross-interference. Because the subtraction is too simple, so it did not get too much attention for the approximation. Non-prior basis function approach has two core operations:one is the inner product, one is subtraction. The subtraction is to ensure convergence of this method. Once the Non-prior basis function system has been identified, then we can make the best approximation operator. Compared with the DFT method, the approximation mechanism of non-prior basis method is not same as the interpolated approximation mechanism. The approach mechanism of non-prior method is a gradual approach, and DFT is a tectonic interpolation approach. This article analyzes the difference of approach and identification method. They are essentially different; the analysis should be taken on different routes. The core difference of identification and approximation is that the signal entropy is different. Identification has large semaphore and its result can be extended, while the approach method can not be extended. Approximation rate is compared between the non-prior approximation and DFT approximation by an example. The results show that the non-prior approximation has much higher approximated efficiency and extensive application fields of basis function. Therefore, it can be applied to different signals flexibly.
Convolution is also a common signal processing operation. The convolution formula Y(ω)=H(ω)X(ω) has its application range. It can be applied for energy limited signal. When Y(ω) and X(ω) are subjected to noise interference, the convolution formula Y(ω)=H(ω)X(ω) and the inverse filter formula X(ω)=Y(ω)/H(ω) will bring back a large error. Basis function convolution operation is proposed for power signals and energy signals analysis. Combined with noise limits, basis functions convolution can be applied to the situation of noise interference. The diagnosed results demonstrated that when the deconvolution operation is applied to the strong noise pollution, high accuracy is also achieved.
Under the guidance idea of non-priori basis function system, damping identification is solved as an application. The approximation and identification are discussed. Then we can get a conclusion that damping identification and cosine signal identification is the same kind of problem. They can use the same technology route. The only difference is the basis function selected. The damping identification method using non-prior basis function system is proposed. There is no specific need of the length of the signal for this approach in theory. So it is an accurate identification. The success of damping identification proves that the method of non-prior basis function system is reasonable. The non-prior basis analysis method is also applied in the two-microphone sound intensity measurement. Both simulation example and the experiment are carried out. The result verifies that the method can avoid leakage errors and measure the sound intensity accurately.
The non-prior method of DFT emphasizes on its ideal situation. However, it is difficult to meet the ideal condition, even it is possibly a surreal condition. So we can't built a complete orthogonal basis function to ensure precisely diagnose any frequency sine cosine signal. In prior basis analysis, the construction of basis function plays dominant role, while in non-prior basis analysis, the practical engineering signal plays dominant role. The selection and determination of basis function is depended upon the practical signal.
信号分析方法多,本文尝试从运算方法的角度去分析、比较各种方法,不仅仅以基函数的不同去区分各种方法。
实际工程信号处理中,DFT是最常用的信号处理方法,DFT分析的核心是建立正交基,但是要建立一个正交基能够准确识别任意一个频率的正余弦信号是做不到的,这样一个正交基是超现实的,因此需要“频谱校正”。DFT先验地建立正交基,同时要满足逆变换,保证能够重构信号,存在频率间隔。基函数的频率与信号频率不吻合将产生误差,先验地选取基函数是DFT分析误差的原因。基于上述分析,提出非先验策略。一个工程信号,只有有限个不同频率的信号。找到这有限个频率对应的基函数,由这些基函数组成一个函数系,就能够对这一信号进行准确分析。这个问题转化为非先验函数的寻找和函数系的建立问题。在这一想法指导下,提出了非先验的计算分析方法,对单频、非密集频谱、一般密集谱和超密集谱由简到繁的各种情况进行了研究,结果表明非先验的分析方法都能够找到一个函数系与实际存在的信号吻合,这样可消除泄露误差。目前的频谱校正方法,以解方程为主要手段,解方程适用于单频和非密集谱,对于密集谱解方程的方法复杂,如果同时考虑负频谱项,多个密集谱的情况,解方程实际难以实现。非先验函数的找寻在数学上基于优化计算。在解决负频谱影响(超低频信号识别)和密集谱的识别问题上尤其显示其优越性,而这两方面是目前频谱校正研究的关注点。非先验方法基函数的选取,依据实际信号的情况而定,没有频率间隔的限制；从非先验分析的角度看,理论上信号截断对正余弦信号的准确识别没有影响,信号截断是实际工程信号处理中的必然,在信号截断的情况下仍然能够进行准确识别具有实际工程意义。
在完成识别的基础上,研究了用非先验方法逼近信号。在识别正余弦信号的研究中分析了减法具有消除交叉干扰的作用,由于减法非常简单,以至于它的这一作用没有得到重视。基本非先验方法有两个核心运算一个是内积运算,一个是减法运算,减法运算保证了这一方法的收敛性。一旦非先验的函数系被确定,则能够进行最佳逼近运算。与DFT方法对比,非先验方法的分析机理不是插值逼近。非先验方法的分析机理是一种逐步逼近,采用DFT分析工程信号是一种构造型的插值分析方法。本文分析了逼近展开与识别的不同,这两者有着多方面的区别,对它们的分析应该采用不同的路线,识别和逼近展开的核心区别在于信息熵不同,识别的信息量大,识别的结果在时间上可延拓,而一般逼近展开则不能够。通过算例对比了非先验分析与DFT和DCT的逼近速度,结果表明非先验方法的逼近效率高,非先验方法具有更广的基函数使用空间,能够更加灵活地适用于不同的信号。
卷积也是信号处理中常见的运算,卷积公式Y(ω)=H(ω)X(ω)有其适用范围,只能运用于能量有限的信号,对于Y(ω)和X(ω)都受到噪声干扰时,卷积公式Y(ω)=H(ω)X(ω)和反向滤波公式X(ω)=Y(ω)/H(ω)都会带来大的误差。提出基函数卷积运算,这种卷积运算对功率信号和能量信号都适用；与噪声限值相结合,基函数卷积也能够适用于信号受到噪声污染的情况。并且将其运用于有较强噪声干扰情况下的反卷积运算,诊断结果有较高的精度。
将非先验函数系分析方法运用于阻尼识别。实际上阻尼识别和正余弦信号识别是同一类问题,可以采用相同的技术路线,不同点仅在于基函数的选取不同。提出了非先验函数系的阻尼识别方法,理论上这一方法对信号的长度没有特定的要求,是一种准确识别方法。同时将非先验分析用于双传声器声强测量,进行了模拟计算和实验验证。结果证明这种方法可以避免泄漏误差,准确地测量出声强值。
DFT等先验基方法在思想方面强调了理想性,但是要达到这种理想条件有很大的难度。以正、余弦信号的识别为例,不能够现实地建立一个完备的正交函数基保证对任意频率的正、余弦信号的准确识别。先验基分析中,基的建立占主导地位；非先验分析中,实际信号占主导地位,基函数的选取和确定要视信号而定。
Signal analysis is applied to various scientific research fields. It not only involves in research and production technology, but also involves in medical diagnosis. It is closely related to daily life of people. The research and application of signal analysis as well as its intercross of the difference subjects promote the development of the research in signal processing.
It is difficult to master numerous signal analysis methods. This paper attempts to grasp the different analysis methods from the perspective of calculation operation method. It is without using the different basis functions to distinguish the different analysis method.
For practical engineering signal, DFT is the most commonly used signal processing method. The idea of orthogonal basis for DFT was widely accepted. However, we can't accurately identify any frequency of the sine or cosine signal by construction an orthogonal basis function. The reason is that such an orthogonal basis is surreal. The orthogonal prior basis of DFT is constructed; at the same time the inverse transform of DFT must be satisfied in order to reconstruct the signal. The orthogonal prior basis of DFT exist frequency separation. The origin error of the DFT analysis is produced when the frequency of the orthogonal prior basis discordance with the signal frequency. A non-prior basis strategy is proposed based on the above analysis. For an engineering signal, there are only a finite number of different frequency components. After searched the basis function of its corresponding frequencies, then the mathematic function system is formed by these basis functions. We can accurately analyze the signals by the basis function system. The problem is transformed into the issue of establishing and searching the non-prior basis function. Under the guidance of this idea, non-priori basis function strategy proposed. Single-frequency spectrum, non-dense spectrum, common dense spectrum and ultra-dense spectrum are calculated and analyzed from the simple to complex various situations. The method of non-prior basis function can find a function system coincide with the real signal, so as to eliminate leakage errors. Constructed the equation is the necessary mean for the current spectrum correction method. Solving the equations is suitable for single-frequency and non-dense spectrum. For dense spectrum, solving equations is too complicated. If taking into account the negative spectrum, solving equation is difficult to implement for multi-frequencies signal. Searching for non-prior basis function is based on optimization calculation. The method of non-prior basis function has its advantage of solving negative frequency impact (ultra-low frequency signal recognition) and dense spectrum identification. The selection of non-prior basis function of non-prior method is depended on the practical signal without limitation of frequency intervals. The truncation of the signal has no influence on precisely diagnosing the sine cosine signal in theory. However, the truncation of the signal is an inevitable process in practical engineering signal.
After identification of a practical engineering signal, the series expansion approximation is studied. The subtraction is researched for identification of sine and cosine signal to eliminate cross-interference. Because the subtraction is too simple, so it did not get too much attention for the approximation. Non-prior basis function approach has two core operations:one is the inner product, one is subtraction. The subtraction is to ensure convergence of this method. Once the Non-prior basis function system has been identified, then we can make the best approximation operator. Compared with the DFT method, the approximation mechanism of non-prior basis method is not same as the interpolated approximation mechanism. The approach mechanism of non-prior method is a gradual approach, and DFT is a tectonic interpolation approach. This article analyzes the difference of approach and identification method. They are essentially different; the analysis should be taken on different routes. The core difference of identification and approximation is that the signal entropy is different. Identification has large semaphore and its result can be extended, while the approach method can not be extended. Approximation rate is compared between the non-prior approximation and DFT approximation by an example. The results show that the non-prior approximation has much higher approximated efficiency and extensive application fields of basis function. Therefore, it can be applied to different signals flexibly.
Convolution is also a common signal processing operation. The convolution formula Y(ω)=H(ω)X(ω) has its application range. It can be applied for energy limited signal. When Y(ω) and X(ω) are subjected to noise interference, the convolution formula Y(ω)=H(ω)X(ω) and the inverse filter formula X(ω)=Y(ω)/H(ω) will bring back a large error. Basis function convolution operation is proposed for power signals and energy signals analysis. Combined with noise limits, basis functions convolution can be applied to the situation of noise interference. The diagnosed results demonstrated that when the deconvolution operation is applied to the strong noise pollution, high accuracy is also achieved.
Under the guidance idea of non-priori basis function system, damping identification is solved as an application. The approximation and identification are discussed. Then we can get a conclusion that damping identification and cosine signal identification is the same kind of problem. They can use the same technology route. The only difference is the basis function selected. The damping identification method using non-prior basis function system is proposed. There is no specific need of the length of the signal for this approach in theory. So it is an accurate identification. The success of damping identification proves that the method of non-prior basis function system is reasonable. The non-prior basis analysis method is also applied in the two-microphone sound intensity measurement. Both simulation example and the experiment are carried out. The result verifies that the method can avoid leakage errors and measure the sound intensity accurately.
The non-prior method of DFT emphasizes on its ideal situation. However, it is difficult to meet the ideal condition, even it is possibly a surreal condition. So we can't built a complete orthogonal basis function to ensure precisely diagnose any frequency sine cosine signal. In prior basis analysis, the construction of basis function plays dominant role, while in non-prior basis analysis, the practical engineering signal plays dominant role. The selection and determination of basis function is depended upon the practical signal.
引文
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