CN102631198A - Dynamic spectrum data processing method based on difference value extraction - Google Patents

Abstract

The invention discloses a dynamic spectrum data processing method based on difference extraction, which relates to the technical field of spectrum analysis, and synchronously collects photoplethysmography pulse waves under N wavelengths in the full band of the part to be measured, and sets the range of interval points; adopts difference extraction method to extract the difference dynamic spectrum group corresponding to each value SA within the range of the interval point number; compare the difference value dynamic spectrum group corresponding to each value SA, and select the group with the smallest degree of dispersion for superposition and average as the final dynamic spectrum result . The present invention can obtain a large number of difference dynamic spectra through difference calculation, realize more full utilization of experimental data, improve calculation efficiency, and reduce test complexity; The average effect eliminates the dynamic spectrum with gross errors, which greatly improves the signal-to-noise ratio of the dynamic spectrum and improves the accuracy of the non-invasive blood component detection of the dynamic spectrum.

Description

A Dynamic Spectral Data Processing Method Based on Difference Extraction

technical field

The invention relates to the technical field of spectrum analysis, in particular to a dynamic spectrum data processing method based on difference extraction that can improve the precision and efficiency of dynamic spectrum analysis.

Background technique

Among the many non-invasive optical detection methods of blood components, transmission spectroscopy has obvious advantages over other spectral measurement methods, among which dynamic spectroscopy can theoretically eliminate the interference of optical backgrounds such as skin and fat on the measured arterial blood spectrum of the pulsating part . The basic principle of dynamic spectroscopy is to irradiate the finger with light in the visible and near-infrared bands to obtain photoplethysmography waves containing blood component information at each wavelength. A dynamic spectrum can be composed. Since the light absorption of pulsating arterial blood is much weaker than that of the tissue background, coupled with the influence of factors such as spectral overlap, abnormal waveform interference, and limited sampling rate of the data acquisition system, how to make full use of the collected photoplethysmography data of each wavelength , it is particularly important to obtain high-quality dynamic spectra more quickly and effectively.

In order to obtain the difference in absorbance corresponding to the same blood volume change more simply and effectively, the peak-to-peak value of the photoplethysmogram (the difference between the maximum value and the minimum value in a single photoplethysmography cycle) is usually used to correspond to the pulsating artery The maximum amount of blood changes, and then constitute a dynamic spectrum. Existing dynamic spectrum extraction methods mainly include frequency domain extraction method (invention patent "method for non-invasive measurement of blood spectrum and components" publication number: CN101507607, publication date: August 19, 2009) and time domain single-shot extraction method (invention Patent "A Dynamic Spectral Data Processing Method Based on Single-edge Extraction Method" Publication No.: CN101912256A, Publication Date: December 15, 2010), both of them compose the dynamic spectrum by extracting the peak-to-peak value of the photoplethysmographic wave.

Through the analysis of the above two methods, it is found that both of them have the following deficiencies and defects:

1. The frequency domain extraction method uses the Fourier transform method to transform the logarithmic photoplethysmogram at each wavelength from the time domain to the frequency domain, and extracts the harmonic amplitude with the largest amplitude in the frequency domain to replace The peak-to-peak value of the logarithmic photoplethysmogram, this method is an indirect extraction method proposed to solve the problem of relatively difficult and large error in extracting the peak-to-peak value of the logarithmic photoplethysmogram in the time domain, although the photoplethysmography at each wavelength All the data are processed, but only the maximum harmonic component information is used, resulting in redundant calculations, reducing the efficiency of calculations, and it is difficult to suppress the influence of factors such as abnormal waveforms and baseline drift in the time-domain signals during the calculation process. Effective real-time assessment of data quality during calculations;

2. The time-domain single-shot extraction method preliminarily solves the difficulty of dynamic spectrum time-domain extraction, realizes the direct extraction of logarithmic pulse peak-to-peak values and can better suppress the influence of abnormal waveforms in photoplethysmography on the accuracy of dynamic spectra. Data processing The speed has been improved, but this method fails to make full use of the experimental data, there is still a large error in the location of the pulse wave peak, the experimental data processing procedure is cumbersome and complex, and the real-time monitoring ability is poor.

Contents of the invention

In order to solve the problems of low operation efficiency in the current dynamic spectrum frequency domain extraction method and the inability to effectively evaluate and overcome the influence of abnormal waveforms in the operation, as well as the difficulties in pulse wave positioning and complex operation in the time domain single-shot extraction method, the present invention provides a A kind of dynamic spectral data processing method based on difference value extraction, described method comprises the following steps:

A method for processing dynamic spectral data based on difference extraction, said method comprising the following steps:

(1) Synchronously collect the photoplethysmography waves under the full band N wavelengths of the part to be measured, and set the range S of interval points;

(2) adopt difference extraction method to extract the difference value dynamic spectrum group corresponding to each numerical value SA in the range S of the interval point number;

(3) Compare the difference dynamic spectrum groups corresponding to the various numerical values SA, and select the group with the smallest degree of dispersion for superposition and averaging as the final dynamic spectrum result.

In step (2), the difference value dynamic spectrum group corresponding to each numerical value in the range S of the interval points extracted by the difference value extraction method specifically includes:

1) Take the logarithm of the full-band photoplethysmogram, obtain the full-band logarithmic pulse wave, select any of the numerical values SA within the range S of the interval points, and calculate the two values separated by the numerical value SA in chronological order. The absolute value of the sampling point difference is obtained to obtain the full-band difference sequence, wherein the length of the difference sequence at each wavelength is M-SA, and M is the number of sampling points;

2) superimposing and averaging the same position difference in the full-band difference sequence to obtain an average difference sequence;

根据所述平均差值

设置差值阈值范围，通过所述差值阈值范围对所述平均差值序列中差值进行筛选，获取筛选后L个差值，其中，i＝1，2，3...，M-SA，L的取值小于等于M-SA；3) Calculate the average difference for all the differences D _i in the average difference sequence

According to the mean difference

Set a difference threshold range, filter the difference in the average difference sequence through the difference threshold range, and obtain L difference values after filtering, wherein, i=1, 2, 3..., M-SA , the value of L is less than or equal to M-SA;

4) According to the positions of the L difference values after the screening, the difference values at the same position in the full-band difference value sequence are extracted in order of wavelength to form L initial difference value dynamic spectra;

5) Normalize the L initial differential dynamic spectra to obtain a normalized differential dynamic spectrum X _j , where j=1, 2, 3, ..., L;

6) Superimpose and average the normalized difference dynamic spectrum X _j to obtain the difference dynamic spectrum template

的相似程度；7) Euclidean distance is used to describe each normalized difference dynamic spectrum X _j and the difference dynamic spectrum template

degree of similarity;

8) According to the 3σ criterion and the degree of similarity, it is judged whether there is a gross error in each normalized difference dynamic spectrum X _j , and if there is, the corresponding normalized difference dynamic spectrum X _j is eliminated; if not, then After the screening is completed, a difference dynamic spectrum group corresponding to the value SA is finally obtained;

9) Steps 2) to 8) are executed cyclically, and the differential dynamic spectrum groups corresponding to other values within the range of the interval points are sequentially extracted.

The normalization of the L initial difference dynamic spectra, and obtaining the normalized difference dynamic spectra X _j specifically include:

Performing superposition and averaging on the L initial difference dynamic spectra to obtain an average optical path length difference dynamic spectrum;

Divide the spectral value of each wavelength in the initial difference dynamic spectrum by the spectral value corresponding to the average optical path length difference dynamic spectrum to obtain a set of proportional coefficients K _λ , where λ=1, 2, 3..., N ;

Superimpose and average all proportional coefficients K _λ to obtain an average optical path normalization coefficient

获取所述归一化差值动态光谱X_j。Multiply the spectral value of each wavelength in the initial difference dynamic spectrum by

The normalized difference dynamic spectrum X _j is obtained.

The degree of dispersion in step (3) is the smallest, that is, the value of the standard deviation σ is the smallest.

The beneficial effects of a dynamic spectral data processing method based on difference extraction provided by the present invention are:

Compared with the existing frequency domain extraction method and time domain difference value extraction method, the method provided by the present invention can obtain a large number of difference value dynamic spectra through difference operation, realize more full utilization of experimental data, and improve calculation efficiency , which reduces the complexity of the experiment; in the process of processing, the average effect of the difference sequence template is firstly used to effectively eliminate the low signal-to-noise ratio and singular difference dynamic spectrum, and secondly, the difference dynamic spectrum is used in the coarse error elimination process The average effect of the spectrum eliminates the dynamic spectrum with gross errors, which greatly improves the signal-to-noise ratio of the dynamic spectrum and improves the accuracy of the non-invasive blood component detection of the dynamic spectrum.

Description of drawings

Fig. 1 is a flow chart of a dynamic spectral data processing method based on difference extraction provided by the present invention;

Fig. 2 is a flow chart of extracting the difference dynamic spectrum group corresponding to each value in the interval point range S provided by the present invention.

Detailed ways

In order to make the object, technical solution and advantages of the present invention clearer, the implementation manner of the present invention will be further described in detail below in conjunction with the accompanying drawings.

In order to solve the problems of low operation efficiency in the current dynamic spectrum frequency domain extraction method and the inability to effectively evaluate and overcome the influence of abnormal waveforms in the operation, as well as the difficulties in pulse wave positioning and complex operation in the time domain single-shot extraction method, the embodiments of the present invention provide A dynamic spectral data processing method based on difference extraction is proposed, see Figure 1 and Figure 2, see the following description for details:

101: Synchronously collect the photoplethysmography waves of the part to be measured in the full band of N wavelengths, and set the range S of interval points;

Wherein, the number of sampling points of each photoplethysmogram is M, and the parts to be measured can be parts such as fingers and earlobes, which are not limited in the embodiments of the present invention during specific implementation;

According to the sampling rate and precision of the photoelectric plethysmography data acquisition device, the interval point range S is set in conjunction with the characteristics of the pulse wave of the human body. The embodiments of the present invention do not limit this during specific implementation;

102: Using the difference extraction method to extract the difference dynamic spectrum group corresponding to each value SA in the interval point range S;

Wherein, this step specifically includes steps 1021-1029, see the following description for details:

1021: Take the logarithm of the full-band photoplethysmography, obtain the full-band logarithmic pulse wave, select any value SA within the interval point range S, and calculate the absolute value of the difference between two sampling points separated by SA in chronological order , to obtain the full-band difference sequence;

Wherein, this step is specifically: according to the amended Lambert-Beer's law, logarithmically transform the photoplethysmography waves collected at all wavelengths to obtain full-band logarithmic pulse waves, and the length of the difference sequence under each wavelength is M- SA.

Wherein, the numerical value SA can select any numerical value within the interval point range S, for example: the interval point number range S is 1 to 5, and the numerical value SA can take a value of 1, 2, 3, 4 or 5. During specific implementation, the embodiment of the present invention There is no restriction on this.

1022: Obtain an average difference sequence by superimposing and averaging the differences at the same position in the full-band difference sequence;

Among them, since the photoplethysmography at each wavelength is collected synchronously at the same site, they have strict consistency in time and similarity in graph. The full-band difference sequence obtained by logarithmic and difference operations also has time consistency and graphic consistency, so the difference values at the same position in the difference sequence of each wavelength can be superimposed and averaged to obtain the average difference sequence.

根据平均差值

设置差值阈值范围，通过差值阈值范围对平均差值序列中差值进行筛选，获取筛选后L个差值；1023: Calculate the average difference for all differences D _i (i=1, 2, 3..., M-SA) in the average difference sequence

According to the average difference

Set the difference threshold range, filter the difference in the average difference sequence through the difference threshold range, and obtain L difference values after filtering;

根据平均差值

设置差值阈值范围，筛选得到差值范围内的L个差值，L的取值小于等于M-SA。其中，本发明实施例中的差值阈值范围选取为

具体实现时，还可以设置为其他的范围，本发明实施例对此不做限制。Wherein, this step is specifically: during the difference calculation process, there may be too small differences or abnormal difference sizes caused by abnormal waveforms, which seriously affect the signal-to-noise ratio of the difference dynamic spectrum and need to be eliminated. During processing, calculate the average difference for all differences D _i (i=1, 2, 3..., M-SA) in the average difference sequence

According to the average difference

Set the difference threshold range, filter to obtain L difference values within the difference range, and the value of L is less than or equal to M-SA. Wherein, the difference threshold range in the embodiment of the present invention is selected as

During specific implementation, other ranges may also be set, which is not limited in this embodiment of the present invention.

1024: According to the positions of the L difference values after screening, the difference values at the same position in the full-band difference value sequence are extracted in order of wavelength to form L initial difference value dynamic spectra;

Among them, according to the theory of dynamic spectrum, the difference of the same position in the full-band difference sequence can form a difference dynamic spectrum; since the average difference sequence is the "ideal sequence" of the difference sequence of each wavelength in the whole band, the average difference The selection of the difference in the sequence is essentially the selection of the difference at the same position in the full-band difference sequence, that is, the selection of the dynamic spectrum of the difference; according to the positions of the L differences obtained by screening, the corresponding L initial Difference Dynamic Spectrum.

1025: Normalize the L initial differential dynamic spectra to obtain normalized differential dynamic spectra X _j (j=1, 2, 3, ..., L);

Since there is a difference in optical path length between the initial difference dynamic spectra, it is necessary to normalize the initial difference dynamic spectra. Since the difference dynamic spectra at different times are similar but the optical path lengths are different, an average optical path length difference dynamic spectrum can be obtained by superimposing and averaging the L initial difference dynamic spectra.

Since the average optical path length difference dynamic spectrum has a high signal-to-noise ratio, use this as a standard to normalize each initial difference dynamic spectrum, and finally make each difference dynamic spectrum and the average optical path length difference dynamic spectrum have the same of light path length.

③用初始差值动态光谱中各波长的光谱值乘以获取归一化差值动态光谱X_j。Taking a certain differential dynamic spectrum as an example, the normalization steps are as follows: ①Divide the spectral value of each wavelength in the initial differential dynamic spectrum by the average optical path length corresponding to the spectral value of the differential dynamic spectrum to obtain a set of proportional coefficients K _λ (λ=1, 2, 3..., N); ② superimpose and average all proportional coefficients K _λ to obtain an average optical path normalization coefficient

③ Multiply the spectral value of each wavelength in the initial difference dynamic spectrum by Obtain the normalized difference dynamic spectrum X _j .

1026: Superimpose and average the normalized difference dynamic spectrum X _j to obtain the difference dynamic spectrum template

的相似程度；1027: Use Euclidean distance to describe each normalized difference dynamic spectrum X _j and difference dynamic spectrum template

degree of similarity;

之间的距离为

以

来描述二者的相似性，

越小，则表明二者的相似性越高。Wherein, this step is specifically: according to the definition of Euclidean distance, each normalized difference dynamic spectrum X _j and the difference dynamic spectrum template

The distance between

by

To describe the similarity between the two,

The smaller the value, the higher the similarity between the two.

X_j，λ，

分别为X_j，

在波长λ对应的光谱值，λ＝1，2，3，...，N。in,

Xj _,λ ,

Respectively X _j ,

The spectral value corresponding to the wavelength λ, λ=1, 2, 3, . . . , N.

1028: According to the 3σ criterion and the degree of similarity, judge whether there is a gross error in each normalized difference dynamic spectrum X _j , if yes, delete the corresponding normalized difference dynamic spectrum X _j ; if not, the screening ends, and finally Obtain a differential dynamic spectrum group corresponding to the value SA;

During the measurement process, due to interference such as external noise or baseline drift, these factors will produce gross errors and affect the accuracy of the dynamic spectrum. Therefore, it is necessary to eliminate the normalized difference dynamic spectrum containing gross errors to improve the signal-to-noise ratio of the dynamic spectrum.

之间的平均欧氏距离残差v_j、标准差σ；若某一归一化差值动态光谱的残差大于3σ，即|v_j|＞3σ，则认为该归一化差值动态光谱含有粗大误差并予以剔除，否则予以保留；对所有归一化差值动态光谱完成所述动态光谱模板下的一轮粗大误差剔除后，对筛选剩余归一化差值动态光谱重新执行步骤1026～步骤1028，重新获得差值动态光谱模板来对含粗大误差动态光谱进行剔除；每经过一轮筛选后归一化差值光谱数量L会相应减少，直至所有含有粗大误差的归一化差值动态光谱被剔除；最终得到一组与数值SA相对应的一个差值动态光谱组。Among them, the gross error elimination step is specifically: calculating each normalized difference dynamic spectrum X _j and the difference dynamic spectrum template

The average Euclidean distance between Residual v _j , standard deviation σ; if the residual of a certain normalized difference dynamic spectrum is greater than 3σ, that is, |v _j |>3σ, it is considered that the normalized difference dynamic spectrum contains gross errors and is eliminated. Otherwise, keep it; after completing a round of coarse error elimination under the dynamic spectrum template for all normalized difference dynamic spectra, re-execute steps 1026 to 1028 for the screened remaining normalized difference dynamic spectra to regain the difference Dynamic spectrum templates are used to remove the dynamic spectra with gross errors; after each round of screening, the number L of normalized difference spectra will be reduced accordingly until all normalized difference dynamic spectra containing gross errors are eliminated; finally a Group A difference dynamic spectrum group corresponding to the value SA.

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; L</span>}}\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; ,</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; d</span>}}$

$\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; =</span>\sqrt{\frac{\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}$

1029: cyclically execute steps 1022-1028, and sequentially extract the difference dynamic spectrum groups corresponding to other values within the interval point range S.

103: Compare the difference dynamic spectrum groups corresponding to each value, and select the group with the smallest degree of dispersion for superposition and averaging as the final dynamic spectrum result.

Among them, the smallest degree of dispersion is the smallest value of σ.

The 3σ judgment criterion applied in the method of the embodiment of the present invention is a well-known technology in data processing methods, and is well known to engineers and technicians in the field.

In summary, the embodiment of the present invention provides a dynamic spectral data processing method based on difference value extraction. Compared with the existing frequency domain extraction method and time domain difference value extraction method, this method can obtain a large number of The dynamic spectrum of difference value realizes the full utilization of experimental data, improves the calculation efficiency and reduces the complexity of experiment; The effective elimination of the singular difference dynamic spectrum, and secondly, the average effect of the difference dynamic spectrum is used to eliminate the dynamic spectrum with gross errors in the process of gross error elimination, which greatly improves the signal-to-noise ratio of the dynamic spectrum and improves the performance of the dynamic spectrum. Accuracy of Noninvasive Blood Component Testing.

Those skilled in the art can understand that the accompanying drawing is only a schematic diagram of a preferred embodiment, and the serial numbers of the above-mentioned embodiments of the present invention are for description only, and do not represent the advantages and disadvantages of the embodiments.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included in the protection of the present invention. within range.

Claims (4)

1. A dynamic spectral data processing method based on difference extraction is characterized by comprising the following steps:

(1) synchronously collecting the photoplethysmography of the part to be measured under the full-wave-band N wavelengths, and setting an interval point number range S;

(2) extracting a difference dynamic spectrum group corresponding to each numerical value SA in the interval point number range S by adopting a difference extraction method;

(3) and comparing the difference dynamic spectrum groups corresponding to the numerical values SA, and selecting one group with the minimum dispersion degree to perform superposition averaging to obtain a final dynamic spectrum result.

2. The method as claimed in claim 1, wherein the step (2) of extracting the difference dynamic spectrum group corresponding to each value in the interval point number range S by using the difference extraction method specifically includes:

1) taking logarithm of the full-waveband photoelectric volume pulse waves to obtain full-waveband logarithmic pulse waves, selecting any numerical value SA in the interval point number range S, calculating the absolute value of the difference value of two sampling points which are separated by the numerical value SA according to time sequence to obtain a full-waveband difference value sequence, wherein the length of the difference value sequence under each wavelength is M-SA, and M is the number of the sampling points;

2) carrying out superposition averaging on the difference values at the same position in the full-waveband difference value sequence to obtain an average difference value sequence;

3) for all difference values D in the average difference value sequence_iCalculating the average difference

According to the average difference valueSetting a difference threshold range, screening the differences in the average difference sequence through the difference threshold range, and acquiring L screened differences, wherein the value of L is less than or equal to that of M-SA (1, 2, 3.), and the value of M-SA;

4) according to the positions of the L screened difference values, extracting the difference values at the same position in the full-waveband difference value sequence according to the wavelength sequence to form L initial difference value dynamic spectrums;

5) normalizing the L initial difference dynamic spectrums to obtain a normalized difference dynamic spectrum X_jWherein j is 1, 2, 3, …, L;

6) dynamic spectrum X of the normalized difference_jCarrying out superposition averaging to obtain a dynamic spectrum template of difference values

7) Dynamic spectrum X for describing various normalized difference values by using Euclidean distance_jAnd the difference value dynamic spectrum template

The degree of similarity of (c);

8) judging the dynamic spectrum X of each normalized difference value according to the 3 sigma criterion and the similarity degree_jWhether a gross error exists or not, if so, rejecting the corresponding normalized difference dynamic spectrum X_j(ii) a If the difference value does not exist, the screening is finished, and finally a difference value dynamic spectrum group corresponding to the numerical value SA is obtained;

9) and circularly executing the steps 2) -8), and sequentially extracting the difference dynamic spectrum groups corresponding to other numerical values in the interval point number range.

3. The difference extraction-based dynamic spectrum data processing method as claimed in claim 2, wherein the L initial difference dynamic spectra are normalized to obtain a normalized difference dynamic spectrum X_jThe method specifically comprises the following steps:

carrying out superposition averaging on the L initial difference dynamic spectrums to obtain an average optical path length difference dynamic spectrum;

dividing the spectrum value of each wavelength in the initial difference dynamic spectrum by the spectrum value corresponding to the average optical path length difference dynamic spectrum to obtain a group of proportionality coefficients K_λWherein λ 1, 2, 3, N;

for all proportionality coefficients K_λCarrying out superposition averaging to obtain an average optical path normalization coefficient

Multiplying the spectrum value of each wavelength in the initial difference dynamic spectrum by

Obtaining the normalized difference dynamic spectrumX_j。

4. The method according to claim 1, wherein the minimum degree of dispersion in step (3) is the minimum value of standard deviation σ.

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