WO2020024320A1 - 一种基于细化傅里叶变换的信号分析方法及设备 - Google Patents

一种基于细化傅里叶变换的信号分析方法及设备 Download PDF

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WO2020024320A1
WO2020024320A1 PCT/CN2018/099521 CN2018099521W WO2020024320A1 WO 2020024320 A1 WO2020024320 A1 WO 2020024320A1 CN 2018099521 W CN2018099521 W CN 2018099521W WO 2020024320 A1 WO2020024320 A1 WO 2020024320A1
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fourier transform
signal
frequency
refined
formula
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轩建平
李锐
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华中科技大学
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R23/00—Arrangements for measuring frequencies; Arrangements for analysing frequency spectra
    • G01R23/16—Spectrum analysis; Fourier analysis

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  • the invention belongs to the field of signal processing, and more particularly, relates to a high-precision Fourier transform algorithm, which can improve the amplitude and frequency accuracy of the Fourier transform, and is therefore suitable for signal processing applications such as dense spectrum analysis, and is particularly applicable.
  • JTFA time-frequency analysis
  • Time-Frequency Analysis is an advantageous tool for non-steady-state signal analysis.
  • the time-frequency analysis method provides joint distribution information in the time and frequency domains, and clearly describes the relationship between signal frequency and time.
  • the basic idea of time-frequency analysis is that any changing signal is stable in a short time.
  • the main reason why the existing discrete Fourier transform algorithm (DFT) is not suitable for time-frequency analysis is that the frequency error of the short-time signal is large. The relationship between signal frequency and time cannot be clearly described.
  • DFT discrete Fourier transform algorithm
  • Both the steady state signal analysis and the non-steady state signal analysis are converted to the steady state signal for analysis.
  • the only difference is the sampling time t.
  • the stationary signal can have a longer sampling time t.
  • There is a short sampling time t so the signals described in this specification are assumed to be steady-state signals, and then the effect of sampling time on frequency and amplitude is discussed.
  • the frequency accuracy is 0.5 times the frequency resolution.
  • the higher the frequency resolution the higher the frequency accuracy.
  • the formula for calculating the frequency resolution is f s / N, where fs is the sampling frequency and N is the number of sampling points, where the number of sampling points N can be expressed as t ⁇ f s , so the actual frequency resolution is 1 / t. It can be seen from the above description that the shorter the sampling time t, the lower the frequency resolution, and the larger the frequency error, resulting in a failure to clearly reflect the relationship between frequency and time. This is why the discrete Fourier transform (DFT) algorithm is not suitable for time-frequency Analyze the cause.
  • the length of any signal is theoretically infinite in the time domain.
  • people usually intercept a limited length of signal in the time domain. This process is called windowing.
  • the actual signal spectrum is the convolution of the signal spectrum and the window function spectrum. With different window functions, the discrete Fourier transform (DFT) obtained is different.
  • DFT discrete Fourier transform
  • rectangular window functions are used more frequently. Taking the spectrum of rectangular window functions as an example, the frequency response of rectangular window functions is shown in Figure 1. .
  • the frequency and amplitude errors of the discrete Fourier transform are derived from the windowing of the non-integer period in the time domain.
  • a more general term is the truncation of the non-integer period. . If the signal happens to be truncated for the entire period, the discrete Fourier transform (DFT) spectrum is shown in Figure 3.
  • the accurate value is obtained at the characteristic frequency, and the non-essential frequency is 0.
  • the obtained frequency error and amplitude error are 0.
  • the small circle represents the frequency value point, and the distance between any two adjacent small circles represents the frequency resolution. Similar representations in the following figures will not be repeated. As shown in Fig.
  • the maximum frequency error is 0.5 times the frequency resolution, and the amplitude error can reach 36.4%.
  • the frequency resolution can be improved and the frequency error can be reduced, but the amplitude error cannot be reduced.
  • the sampling time is doubled, the frequency resolution is doubled, the frequency error is halved, and the amplitude error is maintained. 36.4% unchanged.
  • time-frequency analysis requires a short sampling time to meet the requirement that the signal is stable within the sampling time. Increasing the sampling time will cause the sampling time to be too long, and the stability of the sampling signal is difficult to guarantee. Therefore, increasing the sampling time is not suitable for the signal Time-frequency analysis.
  • Methods for correcting the amplitude spectrum or power spectrum which are respectively the ratio correction method, the energy center of gravity correction method, the FFT + FT spectrum continuous refinement analysis Fourier transform method, and the phase difference method.
  • Methods for correcting the amplitude spectrum or power spectrum which are respectively the ratio correction method, the energy center of gravity correction method, the FFT + FT spectrum continuous refinement analysis Fourier transform method, and the phase difference method.
  • the frequencies are too dense or continuous spectrum, because two or more adjacent frequency components are too dense, the side lobes will affect each other, so that each frequency component cannot be accurately corrected. Therefore, none of the above methods is suitable for analysis occasions with excessively dense frequencies or continuous spectrum, and thus is not suitable for time-frequency analysis.
  • the present invention provides a high-precision Fourier transform algorithm, which aims to improve the frequency resolution without increasing the sampling time, thereby solving the discrete Fourier transform ( DFT) technical problems of low amplitude accuracy and frequency accuracy, suitable for time-frequency analysis of signals.
  • DFT discrete Fourier transform
  • the present invention provides a signal analysis method based on a refined Fourier transform.
  • the signal to be analyzed x (n) is transformed as follows:
  • N represents the length of the signal x (n)
  • ⁇ is the pi
  • m 0,1,2, ..., (N / ⁇ ), ⁇ (0,1], and (N / ⁇ ) is rounded to N / ⁇ .
  • the present invention also provides a computer-readable storage medium.
  • the computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the method described above is implemented.
  • the present invention further provides a device for analyzing a signal based on a refined Fourier transform, which includes the computer-readable storage medium and the processor as described above, and the processor is configured to call and process the computer-readable storage.
  • a device for analyzing a signal based on a refined Fourier transform which includes the computer-readable storage medium and the processor as described above, and the processor is configured to call and process the computer-readable storage.
  • Computer programs stored on media which includes the computer-readable storage medium and the processor as described above, and the processor is configured to call and process the computer-readable storage.
  • the method of the present invention improves the frequency resolution without increasing the sampling time, thereby solving the problem that the frequency is too dense and the adjacent spectrum cannot be accurately corrected, making frequency dense or continuous spectrum correction possible;
  • the method of the present invention solves the problems that the discrete Fourier transform (DFT) method has low frequency and amplitude accuracy and is not suitable for time-frequency analysis;
  • DFT discrete Fourier transform
  • the method of the present invention has the advantage of accurately extracting each frequency component from a spectrum-dense signal
  • the method of the present invention is a complete Fourier transform method, which has an inverse transform method and satisfies the law of conservation of Parseval energy.
  • FIG. 1 is a schematic diagram of a frequency response of a rectangular window function
  • FIG. 2 is a schematic diagram of a non-integer period truncation of a periodic signal
  • FIG. 3 is a schematic diagram of a frequency response truncated over the entire period
  • Fig. 5 is a schematic diagram of the maximum error of truncation of the non-integer period after doubling the sampling time
  • RFT maximum Fourier transform
  • FIG. 7 is a schematic diagram of the maximum Fourier transform (RFT) maximum error when the parameter ⁇ is 0.25.
  • the present invention defines a Fourier transform method, called a Refined Fourier Transform (RFT, Refined Fourier Transform).
  • RFT Refined Fourier Transform
  • the definition formula for the RFT transform of a column of signals x (n) is shown in formula (1).
  • the method of the invention can improve the frequency resolution of the signal by 1 / ⁇ times.
  • the values of the parameter ⁇ are 0.5 and 0.25, respectively.
  • the value of parameter ⁇ is 0.5
  • the frequency resolution of the signal is twice that of the original ( Figure 4).
  • the value of parameter ⁇ is 0.25, and the frequency resolution of the signal is the original ( Figure 4) 4 times.
  • the frequency response of the window function remains unchanged, that is, the main lobe and The sidelobe width remains unchanged. Note that, unlike the method for increasing the sampling time shown in FIG. 5, the method of the present invention does not increase the sampling time, which is described in detail below.
  • Equation (3) Equation (3)
  • formula (3) can be expressed as formula (4):
  • formula (5) By replacing the first 1 / T on the right side of formula (5) with ⁇ , formula (5) becomes the following formula (6):
  • formula (8) According to the time shift of the Fourier transform, formula (7) can be written as formula (8):
  • the discrete Fourier transform (DFT) of a long signal can be replaced by a refined Fourier transform (RFT) of a short signal.
  • RFT discrete Fourier transform
  • the frequency is the same when the frequency resolution is improved.
  • the resolution is increased by 1 / ⁇ , and the amplitude accuracy remains unchanged.
  • the lengths of the window functions of the long signal and the short signal are different, the frequency response of the window function is different, and the widths of the main lobe and the side lobe of the window function are also different.
  • the RFT [x (n), ⁇ ] transform of the present invention is different from the traditional DFT (x (n)) processing method for increasing the sampling time, as shown in Figs. 5 and 6.
  • the frequency error is the same, but the amplitude error is different.
  • the refined Fourier transform (RFT) is suitable for the analysis of short signals, which makes it possible to use the refined Fourier transform (RFT) for time-frequency analysis.
  • the method of the present invention improves the frequency resolution without increasing the sampling time, thereby solving the problem that the frequencies are too dense and the adjacent spectrum cannot be accurately corrected, so that frequency dense or continuous spectrum correction becomes possible.
  • the inverse transform algorithm of the algorithm of the present invention is similar to the inverse transform algorithm of DFT.
  • the RFT response of the signal R (m, ⁇ ) is convolved with e 2 ⁇ j ⁇ mn / N , and then divided by N / ⁇ .
  • the Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT) are essentially the same.
  • the results obtained by the FFT calculation are no different from the DFT.
  • the Fast Fourier Transform algorithm only improves the Fourier Transform algorithm. Speed of calculations and reduced computer memory usage.
  • the refined Fourier transform (RFT) and the discrete Fourier transform (DFT) of the present invention have similar fast transform algorithms.
  • the results obtained by calculating the fast Fourier transform form of Formula 1 of the present invention are the same as those of the present invention.
  • the calculation results of the method are no different, so the method for performing the above signal processing by using the fast Fourier transform form of the method of the present invention should also be included in the protection scope of the present invention.

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Abstract

一种基于细化傅里叶变换的信号分析方法，属于信号处理领域。该方法是将待分析的信号x(n)与e -2 πj αmn/N求卷积，获得x(n)的细化傅里叶变换形式；其中，N表示信号x(n)的长度，j为虚数单位，π为圆周率，m的取值范围为m＝0,1,2,…,(N/α)，α∈(0,1]，(N/α)为N/α取整。该方法解决了离散傅里叶变换(DFT)方法频率和幅值精度低和不适用于时频分析等问题；与众多幅值谱或者功率谱校正方法相比，该方法具有从频谱密集信号中准确提取各个频率成分的优势；此外，该方法是一种完备的傅里叶变换方法，其存在逆变换方法且满足帕斯瓦尔(Parseval)能量守恒定律。

Description

一种基于细化傅里叶变换的信号分析方法及设备 【技术领域】
本发明属于信号处理领域，更具体地，涉及一种高精度的傅里叶变换算法，能够提高傅里叶变换的幅值及频率精度，从而适用于密集频谱分析等信号处理应用场合，尤其适用于时频分析(JTFA)。
【背景技术】
时频分析(JTFA)是非稳态信号分析的有利工具，时频分析方法提供了时间域和频率域的联合分布信息，清晰地描述了信号频率随着时间变化的关系。时频分析的基本思想是任意一个变化的信号在短时间内是平稳的，现有的离散傅里叶变换算法(DFT)不适用于时频分析的主要原因是短时间信号的频率误差大，不能清晰描述信号频率随着时间变化的关系。
不管是稳态信号的分析还是非稳态信号的分析，都被转换为稳态信号进行分析，所不同的是采样时间t而已，平稳信号可以有较长的采样时间t，非平稳信号只能有短采样时间t，因而本说明书中所述的信号都假定为稳态信号，然后来讨论采样时间对频率和幅值的影响。
众所周知，对离散傅里叶变换(DFT)算法来说，频率的精度为频率分辨率的0.5倍，频率分辨率越高，频率的精度也越高。频率分辨率的计算公式为f s/N，上式中fs为采样频率，N为采样点数，其中采样点数N可以表示为t×f s，因而实际频率分辨率为1/t。从上述说明可以看出采样时间t越短，频率分辨率越低，频率误差越大，导致不能清晰反映频率随着时间变化的关系，这就是离散傅里叶变换(DFT)算法不适合时频分析的原因。
对任意一个信号，理论上其长度在时间域内是无限长。为了分析该信号，人们通常在时间域内截取一段有限长度的信号，这个过程为称为加窗。实际的信号频谱是信号的频谱与窗函数频谱的卷积。采用不同的窗函数， 得到的离散傅里叶变换(DFT)也就不同，工程实践中矩形窗函数使用较多，以矩形窗函数的频谱为例，矩形窗函数的频率响应如图1所示。
离散傅里叶变换(DFT)的频率误差和幅值误差来源于时域非整周期加窗，更为通用的说法是非整周期截断，如图2，为一个周期信号的时域非整周期截断。如果信号恰好是整周期截断，那么其离散傅里叶变换(DFT)频谱如图3所示，特征频率处取得准确值，非特征频率处为0，得到的频率误差和幅值误差都为0，小圆圈表示频率取值点，任意相邻两个小圆圈之间的距离表示频率分辨率，后面图中类似的表示不再赘述。如图4所示，如果信号是非整周期截断，那么其频率最大误差为频率分辨率的0.5倍，幅值误差可以达到36.4％。如图5所示，通过增加采样时间，可以提高频率分辨率，减少频率误差，但是不能减少幅值误差，采样时间增加一倍，频率分辨率提高一倍，频率误差减半，幅值误差保持36.4％不变。但是，时频分析要求采样时间短，才能满足信号在采样时间内为稳定信号，而增加采样时间会导致采样时间过长，采样信号的稳定性难以保证，因此，增加采样时间不适合用于信号的时频分析。
中国的丁康等人在《振动工程学报》2003年3月，第16卷第1期发表的《平稳和非平稳振动信号的若干处理方法及发展》记载了如下内容：目前国内外有四种对幅值谱或者功率谱进行校正的方法，它们分别是比值校正法、能量重心校正法、FFT+FT谱连续细化分析傅里叶变换法和相位差法。但是，以上方法由于在频率过于密集或者连续谱场合中，邻近的两个或多个频率成分由于过于密集，旁瓣会互相影响，导致无法准确修正出各个频率成分。因此，以上方法都不适用于频率过于密集的分析场合或者连续谱，从而也不适用于时频分析。
【发明内容】
针对现有技术的以上缺陷或改进需求，本发明提供了一种高精度傅里 叶变换算法，其目的在于在不增加采样时间的前提下，提高频率分辨率，从而解决离散傅里叶变换(DFT)幅值精度和频率精度低的技术问题，适用于信号的时频分析。
为了实现上述目的，本发明提供了一种基于细化傅里叶变换的信号分析方法，对待分析的信号x(n)进行如下变换：
Figure PCTCN2018099521-appb-000001
上式中，N表示信号x(n)的长度，
j为虚数单位，
π为圆周率，
m的取值范围为m＝0,1,2,…,(N/α)，α∈(0,1]，(N/α)为N/α取整。
为了实现上述目的，本发明还提供了一种计算机可读存储介质，该计算机可读存储介质上存储有计算机程序，该计算机程序被处理器执行时实现如前所述的方法。
为了实现上述目的，本发明还提供了一种基于细化傅里叶变换的信号分析的设备，包括如前所述的计算机可读存储介质以及处理器，处理器用于调用和处理计算机可读存储介质中存储的计算机程序。
总体而言，本发明所构思的以上技术方案与现有技术相比，具有如下
有益效果：
1、本发明的方法在无需增加采样时间的情况下提高了频率分辨率，从而解决了频率过于密集而导致无法准确修正出相邻频谱的问题，使得频率密集或者连续谱校正成为可能；
2、本发明的方法解决了离散傅里叶变换(DFT)方法频率和幅值精度低和不适用于时频分析等问题；
3、与众多幅值谱或者功率谱校正方法相比，本发明的方法具有从频谱密集信号中准确提取各个频率成分的优势；
4、本发明的方法是一种完备的傅里叶变换方法，其存在逆变换方法且满足帕斯瓦尔(Parseval)能量守恒定律。
【附图说明】
图1是矩形窗函数的频率响应示意图；
图2是一个周期信号的非整周期截断示意图；
图3是整周期截断的频率响应示意图；
图4是非整周期截断导致的最大误差示意图；
图5是倍增采样时间后，非整周期截断最大误差示意图；
图6是参数α的取值为0.5时，细化的傅里叶变换(RFT)最大误差示意图；
图7是参数α的取值为0.25时，细化的傅里叶变换(RFT)最大误差示意图。
【具体实施方式】
为了使本发明的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本发明进行进一步详细说明。应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。此外，下面所描述的本发明各个实施方式中所涉及到的技术特征只要彼此之间未构成冲突就可以相互组合。
本发明定义了一种傅里叶变换方法，称为细化的傅里叶变换(RFT，Refined Fourier Transform)，对一列信号x(n)的RFT变换的定义公式如公式(1)所示。
Figure PCTCN2018099521-appb-000002
上式中N表示信号x(n)的长度，亦即信号x(n)中包含的采样点数量，j为虚数单位，π为圆周率，其中m的取值范围为m＝0,1,2,…,(N/α)，α∈(0,1]， (N/α)表示对N/α取整，通常可以直接使用运算程序默认的取整模式即可。一般来说，N/α较大时，向上或向下取整的差别可以忽略不计；此外，由于取整本身就只有向上或者向下两种可能，如果有特殊需求，则自行检验不同取整方式的差别后，根据实际需求进行选择即可。
本发明的方法能够将信号的频率分辨率提高1/α倍。如图6和图7所示，参数α的取值分别为0.5和0.25。图6中，参数α的取值为0.5，信号的频率分辨率为原来(图4)的2倍，图7中，参数α的取值为0.25，信号的频率分辨率为原来(图4)的4倍。
参考图6和图7所示，由于本发明的处理过程中信号的窗函数长度N和采样频率fs保持不变，因而窗函数的频率响应保持不变，也就是说矩形窗函数的主瓣和旁瓣宽度保持不变。注意与图5所示的增加采样时间的方法相区别，本发明的方法并没有增加采样时间，下文有详细说明。
由于窗函数的频率响应保持不变，引入α后，m值由0～N变为0～N/α，导致频谱的宽度发生了变化，从而提高了频率分辨率，频率精度和幅值精度相比于图4都提高了，参考图6和图7所示。图6与图4相比，当参数α的取值为0.5时，频率精度提高了一倍，幅值最大相对误差变为10％。
当参数α的取值为0.25时，频率精度提高了三倍，幅值最大相对误差变为2.6％。依次类推，当α的取值越接近0，幅值精度和频率精度就越高。
下面通过另一个例子对本发明采用细化的傅里叶变换(RFT，Refined Fourier Transform)提高频率分辨率，进而提高频率精度和幅值精度的原理进行说明。
对任意一个稳态信号，随着时间的增加，其在时间域内不断的重复出现，这使得采用一个较短的信号准确表示一个较长的信号成为可能。这里我们假设一个长信号x(n)的长度为T×N(当然，在其他实施例中也可以将短信号当做长信号进行下述变换)，其离散傅里叶变换可以表示为：
Figure PCTCN2018099521-appb-000003
将上述长信号分成T个长度为N的短信号，那么公式2的求和过程就可以写为公式(3)：
Figure PCTCN2018099521-appb-000004
为了便于表示，我们将公式(3)中的各个求和运算用F i(m)取代，那么公式(3)就可以表示为公式(4)：
X(m)＝F 1(m)+F 2(m)+…+F i(m)+…+F T(m)  (4)
从上述公式(4)中取出F i(m)进行分析，将F i(m)的求和运算域进行变换得到下述的公式(5)：
Figure PCTCN2018099521-appb-000005
将公式(5)右边的第一个1/T用α代替，公式(5)就变成下述公式(6)：
Figure PCTCN2018099521-appb-000006
参考公式(1)，将公式(1)代入公式(6)，可以得到公式(7)：
Figure PCTCN2018099521-appb-000007
根据傅里叶变换的时移性，公式(7)可以写成公式(8)：
F i(m)＝RFT[x 1(k)，α]  (8)
结合公式(8)和公式(4)可知，理论上一个稳态信号的离散傅里叶变换(DFT)在频域内不断的重复。
将公式(8)带入公式(4)，我们可以得到公式(9)：
DFT(x(n))＝X(m)＝T×RFT[x 1(k)，α]  (9)
从公式(9)可知：一个长信号的离散傅里叶变换(DFT)可以由一个短信号的细化傅里叶变换(RFT)代替。理论上，采用RFT[x(n),α]变换得到的结果与对DFT(x(n))增加T＝1/α倍采样时间达到的效果，在提升频率分辨率时是一样的，频率分辨率提高了1/α倍，幅值精度保持不变。但是，由于长信号和短信号的窗函数长度不同，窗函数的频率响应不同，窗函数的主瓣和旁瓣宽度也不同。因此，实际上本发明的RFT[x(n),α]变换与传统的DFT(x(n))增加采样时间的处理方法是有所区别的，如图5和图6所示，两者的频率误差相同，但是幅值误差不同。
通过观察图6发现，RFT[x(n),α]变换在提高频率分辨率的基础上，幅值精度也提高了，这使得本发明的细化傅里叶变换(RFT)的优势更为独特。根据上面的分析，细化傅里叶变换(RFT)适用于短信号的分析，这使得细化傅里叶变换(RFT)用于时频分析成为可能。
本发明的方法在无需增加采样时间的情况下提高了频率分辨率，从而解决了频率过于密集而导致无法准确修正出相邻频谱的问题，使得频率密集或者连续谱校正成为可能。
本发明算法的逆变换算法与DFT的逆变换算法类似，信号的RFT响应R(m,α)与e 2πjαmn/N求卷积，然后除以N/α，具体的逆变换计算公式如公式(10)所示，本发明算法的逆变换过程可以准确地还原得到原始信号x(n)，其中n的取值范围为n＝0,1,2,…,N。
Figure PCTCN2018099521-appb-000008
本发明算法满足帕斯瓦尔(Parseval)能量守恒定律，经过本发明算法变换，其能量守恒方程可以表示为公式(11)。
Figure PCTCN2018099521-appb-000009
此外，快速傅里叶变换(FFT)与离散傅里叶变换(DFT)本质上是相同的，FFT计算得到的结果与DFT无异，快速傅里叶变换算法仅仅是提高了傅里叶变换算法的计算速度和减少计算机内存的使用。本发明的细化的傅里叶变换(RFT)与离散傅里叶变换(DFT)具有类似的快速变换算法，本发明的公式1的快速傅里叶变换形式计算得到的结果与本发明的上述方法的计算结果无异，因而利用本发明的方法的快速傅里叶变换形式进行上述信号处理的方法，也应包含在本发明的保护范围之内。
本领域的技术人员容易理解，以上所述仅为本发明的较佳实施例而已，并不用以限制本发明，凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等，均应包含在本发明的保护范围之内。

Claims (4)

  1. 一种基于细化傅里叶变换的信号分析方法，其特征在于，对待分析的信号x(n)进行如下变换：
    Figure PCTCN2018099521-appb-100001
    上式中，N表示信号x(n)的长度，
    j为虚数单位，
    π为圆周率，
    m的取值范围为m＝0,1,2,…,(N/α)，α∈(0,1]，(N/α)为N/α取整。
  2. 如权利要求1所述的一种基于细化傅里叶变换的信号分析方法，其特征在于，包括公式(1)的快速傅里叶变换形式。
  3. 一种计算机可读存储介质，其特征在于，该计算机可读存储介质上存储有计算机程序，该计算机程序被处理器执行时实现如权利要求1或2所述的方法。
  4. 一种基于细化傅里叶变换的信号分析的设备，其特征在于，包括如权利要求3所述的计算机可读存储介质以及处理器，处理器用于调用和处理计算机可读存储介质中存储的计算机程序。
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