Compton scattering background deduction and net peak area estimation method
By constructing an all-around peak signal Gaussian distribution function model in nuclear energy spectrum measurement technology and using Gaussian function to process the signal, the problem of difficulty in analyzing tail stacking pulse amplitude and baseline drift in the prior art is solved, and higher measurement accuracy and more accurate net peak area estimation are achieved.
Patent Information
- Application Number
- CN202510061844.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-15
- Publication Date
- 2025-04-25
AI Technical Summary
In the existing nuclear energy spectrum measurement technology, ladder filter forming algorithm and Gaussian filter forming algorithm are difficult to effectively solve the problem of tail stacking pulse amplitude analysis and the baseline deduction problem after baseline drift, and cannot meet the needs of real-time data processing.
By constructing an all-around peak signal Gaussian distribution function model with superimposed Compton scattered background, and using Gaussian function convolution, smoothing the signal, removing noise, and obtaining smooth spectral lines, thereby accurately estimating the net peak area.
The influence of Compton scattering platform is eliminated, the accuracy and accuracy of measurement is improved, and a more accurate net peak area is obtained, thus providing a better foundation for subsequent particle generation rate and energy spectrum characteristic analysis.
Smart Images
Figure CN119882010A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the technical field of nuclear energy spectrum measurement, and in particular to a Compton scattering background subtraction and net peak area estimation method. Background Art
[0002] Nuclear energy spectrum measurement technology integrates multiple disciplines such as electronic technology, nuclear detection technology, and computer technology. At present, it has become one of the important means of analyzing material composition, and plays an increasingly important role in medicine, geology, biology, environmental science, chemistry, archaeology and other disciplines. In nuclear radiation measurement, the energy of the incident particles is proportional to the amplitude of the pulse signal output by the nuclear detector. By measuring the amplitude of the pulse signal, the radiation energy can be analyzed. The acquisition and analysis of the energy spectrum is also one of the most important means in nuclear analysis methods. By acquiring and analyzing the energy spectrum of the radiation source, important information such as the structure of the radiation material, the type of constituent elements and the content can be obtained directly or indirectly. The traditional nuclear spectrometer that obtains the energy spectrum is mainly an analog nuclear spectrometer characterized by electronic devices to amplify and shape the nuclear signal, restore the baseline, discard the pile-up and hold the pulse signal peak.
[0003] Fully digital energy spectrometers based on digital filter shaping have gradually become the development trend of nuclear instruments. Currently, digital energy spectrometers mainly use trapezoidal filter shaping algorithms, which cannot solve the problem of tail pile-up pulse amplitude analysis and baseline subtraction after baseline drift. Gaussian filter shaping algorithms and baseline subtraction algorithms based on Gaussian pulses need to be further optimized to meet the needs of real-time data processing.
[0004] In view of this, the present invention provides a Compton scattering background subtraction and net peak area estimation method, which can smooth the signal and remove noise through Gaussian function convolution to obtain a smooth spectrum line and a more accurate net peak area. Summary of the invention
[0005] The object of the present invention is to provide a method for subtracting Compton scattering background and estimating net peak area, comprising: S1 constructing a Gaussian distribution function model of a full-energy peak signal superimposed on a Compton scattering background; S2 discretely taking values of a full-energy peak signal superimposed on a Compton scattering background and a Gaussian convolution function of a second derivative of a Gaussian function to obtain a full-energy peak discrete signal superimposed on a Compton scattering background and a convolution function discrete signal of a second derivative of a Gaussian function; S3 performing a convolution operation based on the full-energy peak discrete signal and the convolution function discrete signal to obtain a discrete processed signal; S4 determining the variance of the full-energy peak signal based on the discrete processed signal. ; S5 is based on the variance of the full energy peak signal , determine the variance of the Gaussian convolution function of the second derivative of the Gaussian function ; S6 is based on the variance of the Gaussian convolution function Update the convolution function discrete signal, and update the discrete processed signal based on the updated convolution function discrete signal; S7 update the variance of the full energy peak signal based on the updated discrete processed signal , and based on the variance of the updated full energy peak signal , update the variance of the Gaussian convolution function ; S8 repeatedly updates the variance of the full energy peak signal and the variance of the Gaussian convolution function , and obtain the final processed full energy peak signal variance And the final Gaussian convolution function variance , and obtain the final discrete peak value of the processed signal; S9 is based on the final processed full energy peak signal variance , the final Gaussian convolution function variance and the peak value, determine the net peak area of the full energy peak signal superimposed with the Compton scattering background.
[0006] Furthermore, the full energy peak signal function is a Gaussian distribution function, and its expression is: ;in, is the full energy peak signal function, is the full energy peak signal amplitude, is the energy spectrum signal channel address, is the variance of the full energy peak signal, is a natural exponential function; the Compton scattering background function superimposed on the full energy peak signal is set to a linear curve function, and the Gaussian distribution function model of the full energy peak signal is set to the form of the addition of the Compton scattering background function and the full energy peak signal function, and its expression is: ;in, is the full energy peak signal superimposed with the Compton scattering background. is the full energy peak signal amplitude, is the energy spectrum signal channel address, is the variance of the full energy peak signal, is the natural exponential function, is the slope of the first order curve, is the first order curve intercept.
[0007] Furthermore, the Gaussian convolution function is obtained by applying the Gaussian function Perform a second derivative to obtain: the Gaussian function The expression is set to: ;in, is the Gaussian function set, is the variance of the Gaussian function, is the natural exponential function, is the energy spectrum signal channel address; the expression of the Gaussian convolution function is: ;in, is the signal of the Gaussian convolution function, is the variance of the Gaussian function, is the natural exponential function, is the energy spectrum signal channel address; the signal of the Gaussian convolution function Discrete values are taken to obtain the discrete signal of the convolution function, which is expressed as follows: ;in, is the discrete signal of the convolution function, is the first integer variable, is the maximum integer value, is the variance of the Gaussian function, is a natural exponential function.
[0008] Furthermore, the expression of the discrete processed signal is: ;in, is the discrete Compton scattering full energy peak signal, is a discrete Gaussian convolution signal, is the first integer variable, whose value is an integer. is the maximum value of an integer, which is a positive integer. is the second integer variable.
[0009] Further, step S4 includes: S4-1 determining the number of samples of the discrete processed signal from the peak position to the amplitude position of 0; S4-2 obtaining the previous amplitude value of the previous sampling point at the discrete processed signal amplitude position of 0; S4-3 obtaining the subsequent amplitude value of the subsequent sampling point at the discrete processed signal amplitude position of 0; S4-4 determining the variance of the full-energy peak signal based on the number of samples, the previous amplitude value, the subsequent amplitude value and the variance of the Gaussian convolution function.
[0010] Furthermore, the variance of the full energy peak signal is expressed as: ;in, is the variance of the processed full energy peak signal, is the variance of the Gaussian convolution function, is the number of samples, is the previous value, The latter value.
[0011] Furthermore, the variance of the Gaussian convolution function is taken as the integer value of the variance of the processed full energy peak signal.
[0012] Furthermore, the peak value of the final discrete processed signal is the amplitude at the peak position of the full energy peak of the processed signal after Gaussian convolution.
[0013] Furthermore, step S9 includes: S9-1 obtains the optimal processed variance through dichotomy; S9-2 obtains the ideal amplitude value of the final processed full-energy peak signal and the actual amplitude value of the discrete final processed full-energy peak signal; S9-3 determines the height correction factor based on the ideal amplitude value and the actual amplitude value; S9-4 determines the net peak area based on the height correction factor and the optimal processed variance.
[0014] Furthermore, the optimal processed variance is calculated as follows: ;in, is the optimal processed variance, is the final processed full energy peak signal variance, j is the third integer, taking a positive value, is the error range.
[0015] Furthermore, the actual amplitude value is the amplitude at the peak position of the full energy peak of the processed signal after Gaussian convolution, and the ideal amplitude value is obtained by convolving the Compton scattering full energy peak signal and the convolution signal, and its expression is: ;in, is the ideal amplitude value of the processed signal, is the final processed full energy peak signal variance, is the final Gaussian convolution function variance, is the amplitude of the initial processed signal before Gaussian convolution.
[0016] Furthermore, the expression of the height correction factor is: ;in, is the height correction factor, is the ideal amplitude value, is the actual value of the amplitude.
[0017] Further, the expression of the net peak area is: ;in, is the net peak area, is the optimal Gaussian convolution function variance, is the optimal processed full energy peak signal variance, is the peak value of the final discrete processed signal, is the height correction factor.
[0018] The technical solution of the embodiment of the present invention has at least the following advantages and beneficial effects:
[0019] By convolving the Compton scattering full energy peak signal with the discrete convolution signal, the influence of the Compton scattering platform can be eliminated, the accuracy and precision of the measurement can be improved, and the properties of matter can be better studied as well as the basic processes in nuclear physics and particle physics can be explored.
[0020] A more accurate net peak area can be obtained through the optimal processed variance and height correction factor, paving the way for obtaining more accurate particle generation rate and energy spectrum characteristics in the future. BRIEF DESCRIPTION OF THE DRAWINGS
[0021] Figure 1 An exemplary flow chart of a Compton scattering background subtraction and net peak area estimation method provided by the present invention;
[0022] Figure 2 An exemplary schematic diagram of a discrete processed signal provided by the present invention;
[0023] Figure 3 An exemplary schematic diagram of calculating the altitude correction factor provided by the present invention. DETAILED DESCRIPTION
[0024] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Generally, the components of the embodiments of the present invention described and shown in the drawings here can be arranged and designed in various different configurations.
[0025] Figure 1 An exemplary flow chart of a Compton scattering background subtraction and net peak area estimation method provided by the present invention.
[0026] Step 1: construct a Gaussian distribution function model of the full energy peak signal superimposed with the Compton scattering background.
[0027] Among them, the mathematical model of the full energy peak signal is a Gaussian distribution function, and its expression is:
[0028] ;
[0029] in, is the mathematical model of the full energy peak signal, H is the amplitude of the processed signal, c is the channel address of the energy spectrum signal, is the processed variance of the processed signal, is a natural exponential function.
[0030] The Compton scattering background function superimposed on the processed signal is set as a linear curve function, and the expression of the full energy peak signal after superimposing the Compton scattering background is:
[0031] ;
[0032] in, is the full energy peak signal after superimposing the Compton scattering background. is the processed signal amplitude, is the energy spectrum signal channel address, is the processed variance of the processed signal, is the natural exponential function, is the slope of the first order curve, is the first order curve intercept.
[0033] Step 2, discretely taking values of the full energy peak signal of the superimposed Compton scattering background and the Gaussian convolution function of the second derivative of the Gaussian function, to obtain the full energy peak discrete signal of the superimposed Compton scattering background and the convolution function discrete signal of the second derivative of the Gaussian function.
[0034] Among them, the Gaussian convolution function signal is obtained by Taking the second derivative, we get: Gaussian function The expression is set to:
[0035] ;
[0036] in, is the Gaussian function set, is the variance of the Gaussian convolution function signal, is the natural exponential function, It is the energy spectrum signal channel address.
[0037] The expression of the convolution signal of the convolution function of the second derivative of the Gaussian function is:
[0038] ;
[0039] in, is the convolution function of the second derivative of the Gaussian function, is the variance of the Gaussian convolution function signal, is the natural exponential function, It is the energy spectrum signal channel address.
[0040] Furthermore, the expression of the convolution function signal of the second derivative of the discrete Gaussian function is:
[0041] ;
[0042] in, is the convolution function signal of the second derivative of the discrete Gaussian function, n is the first integer variable, is the maximum integer value, is the variance of the Gaussian convolution function signal, is a natural exponential function.
[0043] Step 3, performing a convolution operation on the full energy peak signal of the discrete superimposed Compton scattering background and the convolution function signal of the second derivative of the discrete Gaussian function to obtain a discrete processed signal.
[0044] By convolving the convolution function signal of the second derivative of the Gaussian function with the Compton scattering full energy peak signal, the influence of the Compton scattering platform can be eliminated. The processed signal after the convolution process can be expressed as:
[0045] ;
[0046] in, is the Compton scattering full energy peak signal, is the convolution function signal of the second derivative of the Gaussian function, is the variance of the convolution function signal of the second derivative of the Gaussian function, is the processed variance of the processed signal, is the processed signal amplitude, It is the energy spectrum signal channel address.
[0047] The discrete processed signal is obtained by convolving the convolution function signal of the second derivative of the discrete Gaussian function and the full energy peak signal of the superimposed Compton scattering background; the expression of the discrete processed signal is:
[0048] ;
[0049] in, is the discrete Compton scattering full energy peak signal, is the convolution function signal of the second derivative of the discrete Gaussian function, is the first integer variable, whose value is an integer, is the maximum value of an integer, which is a positive integer. is the second integer variable. To ensure A positive integer with a certain precision. The higher the precision, The larger the value, the higher the If the value is too large, when two peaks are close to each other, the convolution calculation will affect each other, resulting in inaccurate calculation results. The absolute value of is greater than 0.001 to meet the calculation accuracy requirements. See the schematic diagram of Figure 2 shown.
[0050] Step 4: Determine the variance of the full energy peak signal based on the discrete processed signal . Determine the full energy peak signal variance ,include:
[0051] Determine the number of samples of the discrete processed signal from the peak position to the position where the amplitude is 0.
[0052] Get the previous amplitude value of the previous sampling point where the discrete processed signal amplitude is 0.
[0053] Get the next amplitude of the next sampling point where the discrete processed signal amplitude is 0.
[0054] The processed variance is determined based on the number of samples, the previous value, the next value, and the variance of the Gaussian convolution function.
[0055] Step 5: Variance of the full energy peak signal , determine the variance of the convolution function of the second derivative of the Gaussian function .
[0056] Among them, the full energy peak signal variance Variance of the convolution function with the second derivative of the Gaussian function The expression of the relationship is:
[0057]
[0058] in, is the full energy peak signal variance, is the variance of the convolution function of the second derivative of the Gaussian function, is the number of samples, is the previous value, The latter value.
[0059] Among them, the convolution function variance The selection principle is: The value is greater than or equal to the full energy peak signal variance That is, the width of the convolution Gaussian function can completely cover the width of the full-energy peak, and the processed signal is fully characterized during the convolution processing. The larger the value, the better. When the distance between the all-round peak 2 and the all-round peak 3 is close, If the value is too large, the convolution Gaussian function will be too wide and cover two full energy peaks at the same time. The convolution process will be affected by the other full energy peak, resulting in an inaccurate characterization of the processed signal. Round up, Just take this value.
[0060] Step 6: Variance of the convolution function based on the second derivative of the Gaussian function The convolution function signal of the second derivative of the discrete Gaussian function is updated, and the discrete processed signal is updated based on the updated convolution function signal of the second derivative of the discrete Gaussian function.
[0061] Step 7: Update the variance of the full energy peak signal based on the updated discrete processed signal , and based on the variance of the updated full energy peak signal , update the variance of the convolution function of the second derivative of the Gaussian function .
[0062] Step 8: Repeat the update of the full energy peak signal variance The variance of the convolution function of the second derivative of the Gaussian function , and obtain the final processed full energy peak signal variance And the final Gaussian convolution function variance , and obtain the final discrete peak value of the processed signal.
[0063] Step 9, based on the final full energy peak signal variance , the final convolution function variance And the final discretized processed signal peak is determined to determine the net peak area of the full energy peak signal superimposed with the Compton scattering background.
[0064] For example, the processing flow of this method is to first estimate the variance of the processed full energy peak signal based on experience The variance w value of the Gaussian convolution function is selected, and then the variance of the processed full energy peak signal is calculated after the Gaussian function convolution processing of the second derivative is performed. Since the value of w is determined based on empirical estimation, the variance of the processed full energy peak signal is calculated It is also a rough value, and then the variance of the processed full energy peak signal is Based on the rough value, the exact value of the variance w of the Gaussian convolution function is determined by rounding up, the convolution function of the second derivative of the Gaussian function is updated, and the variance of the processed full energy peak signal is calculated after the processed full energy peak signal is convolved with the Gaussian function again. The precise value of the peak area of the processed signal is finally calculated.
[0065] The expression for the net peak area of the processed signal is:
[0066] ;
[0067] in, is the net peak area, is the final processed full energy peak variance, is the final convolution function variance, is the final discrete processed signal peak value.
[0068] In some embodiments, determining the net peak area of the Compton scattering total energy peak signal comprises:
[0069] Through the dichotomy method, the best processed variance is obtained. For example, The corresponding address when the amplitude is 0 It has nothing to do with H, a and b. Let's assume H=1000, a=0, b=0. Taking different values, the final processed variance estimated is ,and The final processed variance is known to be , the best can be estimated by the dichotomy method The specific implementation method is to select the appropriate ,calculate Make sure the value is less than ,right Use the halving test to find a positive integer n such that , then the expression of the optimal processed variance is:
[0070] ;
[0071] in, is the optimal processed variance, is the final processed variance, j is the third integer, taking a positive value, is the error range.
[0072] The ideal amplitude value of the processed signal is obtained; the processed signal is obtained by convolving the Compton scattering full energy peak signal and the convolution function signal.
[0073] Get the actual value of the amplitude of the discrete processed signal.
[0074] like Figure 3 As shown, when and After the value is determined, H takes the value unit 1, and the height after convolution processing can be obtained The ideal value , but due to the influence of calculation accuracy, the height after convolution processing The actual value of , based on the ideal amplitude value and the actual amplitude value, determine the altitude correction factor; the expression of the altitude correction factor is:
[0075] ;
[0076] in, is the height correction factor, is the ideal amplitude value, is the actual value of the amplitude.
[0077] Based on the height correction factor, the net peak area is determined; the expression for the net peak area is:
[0078] ;
[0079] in, is the net peak area, is the optimal convolution Gaussian function signal variance, is the optimal processed variance, is the peak value of the final discrete processed signal, is the height correction factor.
[0080] Figure 2 An exemplary schematic diagram of a discrete processed signal provided by the present invention.
[0081] like Figure 2 As shown, the channel address range is 0-120, the standard count rate is -0.6-1, and the coordinates of point D are , the coordinates of point A are , the coordinates of point B are , the coordinates of point C are .
[0082] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.
Claims
1. A Compton scattering background subtraction and net peak area estimation method, characterized in that: include: S1 constructs a Gaussian distribution function model of the full energy peak signal superimposed with the Compton scattering background; S2 discretizes the full energy peak signal of the superimposed Compton scattering background and the Gaussian convolution function of the second derivative of the Gaussian function to obtain the full energy peak discrete signal of the superimposed Compton scattering background and the convolution function discrete signal of the second derivative of the Gaussian function; S3 performs a convolution operation based on the full energy peak discrete signal and the convolution function discrete signal to obtain a discrete processed signal; S4 determines the variance of the full energy peak signal based on the discrete processed signal ; S5 is based on the variance of the full energy peak signal , determine the variance of the Gaussian convolution function of the second derivative of the Gaussian function ; S6 is based on the variance of the Gaussian convolution function Updating the convolution function discrete signal, and updating the discrete processed signal based on the updated convolution function discrete signal; S7 updates the variance of the full energy peak signal based on the updated discrete processed signal , and based on the variance of the updated full energy peak signal , update the variance of the Gaussian convolution function ; S8 repeatedly updates the variance of the full energy peak signal and the variance of the Gaussian convolution function , and obtain the final processed full energy peak signal variance And the final Gaussian convolution function variance , and obtain the final discrete peak value of the processed signal; S9 is based on the final processed full energy peak signal variance , the final Gaussian convolution function variance And the peak value, determine the net peak area of the full energy peak signal superimposed with the Compton scattering background.
2. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: The full energy peak signal function is a Gaussian distribution function, and its expression is: ； in, is the full energy peak signal function, is the full energy peak signal amplitude, is the energy spectrum signal channel address, is the variance of the full energy peak signal, is the natural exponential function; The Compton scattering background function superimposed on the full energy peak signal is set to a linear curve function, and the Gaussian distribution function model of the full energy peak signal is set to the form of the addition of the Compton scattering background function and the full energy peak signal function, and its expression is: ； in, is the full energy peak signal superimposed with the Compton scattering background. is the full energy peak signal amplitude, is the energy spectrum signal channel address, is the variance of the full energy peak signal, is the natural exponential function, is the slope of the first order curve, is the first order curve intercept.
3. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: The Gaussian convolution function is obtained by Perform a second derivative to obtain: the Gaussian function The expression is set to: ； in, is the Gaussian function set, is the variance of the Gaussian function, is the natural exponential function, is the energy spectrum signal channel address; The expression of the Gaussian convolution function is: ； in, is the signal of the Gaussian convolution function, is the variance of the Gaussian function, is the natural exponential function, is the energy spectrum signal channel address; The Gaussian convolution function of the signal Discrete values are taken to obtain the discrete signal of the convolution function, which is expressed as follows: ； in, is the discrete signal of the convolution function, is the first integer variable, is the maximum integer value, is the variance of the Gaussian function, is a natural exponential function.
4. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: The expression of the discrete processed signal is: ； in, is the discrete Compton scattering full energy peak signal, is a discrete Gaussian convolution signal, is the first integer variable, whose value is an integer. is the maximum value of an integer, which is a positive integer. is the second integer variable.
5. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: Step S4 includes: S4-1 determines the number of samples of the discrete processed signal from the peak position to the position where the amplitude is 0; S4-2 obtains the previous amplitude value of the previous sampling point at which the amplitude of the discrete processed signal is 0; S4-3 obtains the next amplitude value of the next sampling point where the amplitude of the discrete processed signal is 0; S4-4 determines the variance of the full energy peak signal based on the number of samples, the previous amplitude value, the next amplitude value and the variance of the Gaussian convolution function.
6. The Compton scattering background subtraction and net peak area estimation method according to claim 5, characterized in that: The expression of the variance of the full energy peak signal is: ； in, is the variance of the processed full energy peak signal, is the variance of the Gaussian convolution function, is the number of samples, is the previous value, The latter value.
7. The Compton scattering background subtraction and net peak area estimation method according to claim 6, characterized in that: The variance of the Gaussian convolution function is taken as the integer value of the variance of the processed full energy peak signal.
8. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: The peak value of the final discrete processed signal is the amplitude at the peak position of the full energy peak of the processed signal after Gaussian convolution.
9. The Compton scattering background subtraction and net peak area estimation method according to claim 1, characterized in that: Step S9 includes: S9-1 obtains the best treated variance through dichotomy; S9-2 obtains the ideal amplitude value of the final processed full-energy peak signal and the actual amplitude value of the discrete final processed full-energy peak signal; S9-3 determining an altitude correction factor based on the ideal amplitude value and the actual amplitude value; S9-4 determines the net peak area based on the height correction factor and the optimal processed variance.
10. The Compton scattering background subtraction and net peak area estimation method according to claim 9, characterized in that: The optimal processed variance is calculated as follows: ； in, is the optimal processed variance, is the final processed full energy peak signal variance, j is the third integer, taking a positive value, is the error range.
11. The Compton scattering background subtraction and net peak area estimation method according to claim 9, characterized in that: The actual amplitude value is the amplitude at the peak position of the full energy peak of the processed signal after Gaussian convolution, and the ideal amplitude value is obtained by convolving the Compton scattering full energy peak signal and the convolution signal, and its expression is: ； in, is the ideal amplitude value of the processed signal, is the final processed full energy peak signal variance, is the final Gaussian convolution function variance, is the amplitude of the initial processed signal before Gaussian convolution.
12. The Compton scattering background subtraction and net peak area estimation method according to claim 9, characterized in that: The expression of the altitude correction factor is: ； in, is the height correction factor, is the ideal amplitude value, is the actual value of the amplitude.
13. The Compton scattering background subtraction and net peak area estimation method according to claim 9, characterized in that: The expression of the net peak area is: ； in, is the net peak area, is the optimal Gaussian convolution function variance, is the optimal processed full-energy peak signal variance, is the peak value of the final discrete processed signal, is the height correction factor.