Solid particle component content online detection system and signal correction method

Abstract

The invention discloses a solid particle component content on-line detection system and a signal correction method, and relates to the technical field of measurement, in the system, a near-infrared light source emitting device is used for emitting a near-infrared light beam; the optical fiber transmission device is used for coupling the near-infrared light beam into a transmission optical fiber and transmitting the near-infrared light which is collected by the probe collection device and carries sample information to the near-infrared light detection device; the probe collecting device is used for arranging near-infrared light according to a set spatial position, the near-infrared light is collimated into parallel light by the optical fiber collimator, the parallel light is radiated to a sample in the sample box, and the reflected near-infrared light is collected to the optical fiber transmission device, and the signal processing device is used for processing the near-infrared light according to diffuse reflection absorbance of solid particles. Based on signal correction and a quantitative regression algorithm, the solid particle component content is detected, and the detection accuracy of the solid particle component content can be effectively improved.

Description

Technical Field

[0001] The present application relates to the field of measurement technology, and in particular to an online detection system for solid particle component content and a signal correction method. Background Art

[0002] The detection of solid particle component content is crucial to ensure product quality, optimize the production process, and improve equipment reliability and safety. Traditional detection methods have disadvantages such as complex operation, long time consumption, offline, and large errors, and cannot meet the relevant application requirements of fast, non-destructive, real-time, and accurate in modern industrial production processes. Near-infrared spectroscopy technology is currently the most widely used online spectral detection technology in the industrial field. It has high technical maturity, simple operation, and economical devices. It is used in many fields to analyze component content.

[0003] However, in the industrial production process, the spectral signal is easily affected by multiple interference factors in the complex on-site environment, resulting in increased errors in component content detection. Summary of the invention

[0004] The purpose of this application is to provide an online detection system for solid particle component content and a signal correction method, which can effectively improve the detection accuracy of solid particle component content.

[0005] To achieve the above objectives, this application provides the following solutions:

[0006] In a first aspect, the present application provides an online detection system for solid particle component content, comprising:

[0007] Near-infrared light source emitting device, optical fiber transmission device, probe collection device, near-infrared light detection device and signal processing device;

[0008] The near-infrared light source emitting device comprises a broadband light source and a focusing lens, and is used to emit a near-infrared light beam;

[0009] The optical fiber transmission device comprises an optical fiber coupler and a transmission optical fiber bundle; the optical fiber coupler is used to couple the near-infrared light beam into the transmission optical fiber; the transmission optical fiber bundle is composed of a plurality of transmission optical fibers, and is used to transmit the near-infrared light source to the probe collection device, and transmit the near-infrared light carrying sample information collected by the probe collection device to the near-infrared light detection device;

[0010] The probe collection device includes a fiber collimator, a sapphire light window and a passive mechanical component, which is used to arrange the near-infrared light transmitted by the fiber transmission device according to the set spatial position, and radiate the near-infrared light to the sample at the sample box after being collimated into parallel light by the fiber collimator, and collect the reflected near-infrared light at the fiber transmission device;

[0011] The near-infrared light detection device includes an optical fiber coupler and a photodetector, which are used to collect near-infrared spectrum signals of solid particles;

[0012] The signal processing device is used to calculate the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device to obtain the diffuse reflection absorbance of the solid particles, and detect the component content of the solid particles based on the diffuse reflection absorbance of the solid particles based on signal correction and quantitative regression algorithm.

[0013] Optionally, in the signal processing device, calculating the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device to obtain the diffuse reflection absorbance of the solid particles specifically includes:

[0014] Calculate the diffuse reflectance absorbance of solid particles;

[0015] Where k is the absorption coefficient, s is the scattering coefficient, and R ∞ is the diffuse reflectivity of solid particles.

[0016] Optionally, the focusing lens in the near-infrared light source emitting device is used to converge the near-infrared light into a small-spot near-infrared light beam; the lens diameter D1 of the focusing lens is 6.0 mm, the lens focal length d1 is 10.0 mm, and the lens material is quartz material.

[0017] Optionally, the optical fiber in the transmission optical fiber bundle is a 6+1Y optical fiber structure, the core diameter of the optical fiber is 1000um, and the optical core material is low-hydroxy quartz.

[0018] Optionally, the lens diameter of the fiber collimator is D2=6.0 mm, and the lens focal length is d2=10.0 mm; the diameter of the sapphire window is D3=100 mm, and the transmittance is T≥98%; and the passive mechanical component is used to support the fiber collimator and the sapphire window.

[0019] Optionally, the quantitative regression algorithm is specifically:

[0020] c=β0+β1f′(λ)+ε;

[0021] Among them, β0 is the linear regression constant, β1 is the linear regression coefficient of the spectral absorbance SA′ of solid particles after correction in the near-infrared band, and ε is a random variable.

[0022] In a second aspect, the present application provides a signal correction method based on the above-mentioned solid particle component content online detection system, comprising:

[0023] Acquire near-infrared spectrum signals of solid particles collected by a near-infrared light detection device;

[0024] According to the near-infrared spectrum signal of solid particles, the wavelet transform decomposition method is used to obtain the components on different frequency scales;

[0025] Adjust the wavelet coefficients at different scales to remove interference signals from the near-infrared spectrum signal of solid particles, and obtain the processed near-infrared spectrum signal of solid particles;

[0026] The processed near-infrared spectrum signal of the solid particles is subjected to inverse wavelet transform to obtain the corrected near-infrared spectrum signal of the solid particles.

[0027] Optionally, according to the near-infrared spectrum signal of the solid particles, a wavelet transform decomposition method is used to obtain components on different frequency scales, specifically:

[0028]

[0029] Among them, c j,k is the wavelet coefficient, j is the decomposition scale, k is the position parameter on scale j; f(λ) is the spectral data, λ is the wavelength; ψ j,k (λ) is a wavelet function, which is obtained by scaling and shifting the mother wavelet function ψ(λ).

[0030] Optionally, adjusting the wavelet coefficients at different scales to remove interference signals from the near-infrared spectrum signal of solid particles to obtain a processed near-infrared spectrum signal of solid particles specifically includes:

[0031] Baseline removal of near infrared spectrum signals of solid particles; Remove noise from near infrared spectrum signals of solid particles;

[0032] Among them, J0 is the set layer number threshold, P is the set high-frequency wavelet coefficient threshold, c j,k is the wavelet coefficient, j is the decomposition scale, and k is the location parameter on scale j.

[0033] Optionally, the processed near-infrared spectrum signal of the solid particles is subjected to inverse wavelet transform to obtain a corrected near-infrared spectrum signal of the solid particles, specifically:

[0034]

[0035] In the formula, c 0,k is the approximation coefficient and φ(λ) is the scaling function.

[0036] According to the specific embodiments provided in this application, this application discloses the following technical effects:

[0037] The present application provides an online detection system for solid particle component content and a signal correction method. In the system, a near-infrared light source emitting device is composed of a broadband light source and a focusing lens, which is responsible for emitting a near-infrared light beam; the optical fiber transmission device includes an optical fiber coupler and a transmission optical fiber bundle, wherein the optical fiber coupler is used to couple the near-infrared light beam into the transmission optical fiber; the transmission optical fiber bundle is composed of a plurality of transmission optical fibers, and its function is to transmit the near-infrared light source to a probe collection device, and transmit the near-infrared light containing sample information collected by the probe collection device to a near-infrared light detection device; the probe collection device is composed of an optical fiber collimator, a sapphire optical window and passive mechanical components, and its function is to The near-infrared light transmitted by the optical fiber transmission device is arranged according to a predetermined spatial position, and converted into parallel light by an optical fiber collimator, and then radiated onto the sample in the sample box, and the reflected near-infrared light is collected and transmitted back to the optical fiber transmission device; the near-infrared light detection device is composed of an optical fiber coupler and a photodetector, and its function is to collect the near-infrared spectrum signal of the solid particles; the signal processing device is responsible for calculating the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device to obtain the diffuse reflection absorbance of the solid particles, and based on the diffuse reflection absorbance, the component content of the solid particles is detected through signal correction and quantitative regression algorithm. This application effectively solves the problem of inaccurate measurement of the component content of solid particles under the influence of multiple interference factors in industrial sites by designing a near-infrared spectrum solid particle component content detection device, and combining signal correction and quantitative regression algorithm, and significantly improves the accuracy of solid particle component content detection. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0039] Figure 1 A schematic diagram of the structure of an online detection system for solid particle component content provided in one embodiment of the present application;

[0040] Figure 2 A schematic diagram of optical fiber core distribution provided in one embodiment of the present application.

[0041] Explanation of symbols:

[0042] Near-infrared broadband laser light source-1, focusing lens-2, first near-infrared fiber coupler-3, Y-type near-infrared transmission fiber bundle-4, second near-infrared fiber coupler-5, near-infrared fiber probe-6, fiber collimator-7, sapphire light window-8, sample box-9, near-infrared light detection device-10, signal processing device-11, Y-type fiber light source end core distribution-12, Y-type fiber probe end core distribution-13, Y-type fiber detection end core distribution-14. DETAILED DESCRIPTION

[0043] The following will be combined with the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0044] In order to make the above-mentioned objects, features and advantages of the present application more obvious and easy to understand, the present application is further described in detail below in conjunction with the accompanying drawings and specific implementation methods.

[0045] Embodiment 1

[0046] like Figure 1 As shown, this embodiment provides an online detection system for solid particle component content, comprising:

[0047] Near-infrared light source emitting device, optical fiber transmission device, probe collection device, near-infrared light detection device 10 and signal processing device 11;

[0048] The near-infrared light source emitting device comprises a broadband light source and a focusing lens 2, and is used to emit a near-infrared light beam;

[0049] The optical fiber transmission device comprises an optical fiber coupler and a transmission optical fiber bundle; the optical fiber coupler is used to couple the near infrared light beam into the transmission optical fiber; the transmission optical fiber bundle is composed of a plurality of transmission optical fibers, and is used to transmit the near infrared light source to the probe collection device, and transmit the near infrared light carrying the sample information collected by the probe collection device to the near infrared light detection device 10;

[0050] The probe collection device includes a fiber collimator 7, a sapphire light window 8 and a passive mechanical component, which is used to arrange the near-infrared light transmitted by the fiber transmission device according to the set spatial position, and collimate the near-infrared light into a parallel light beam by the fiber collimator 7 to radiate the sample at the sample box, and collect the reflected near-infrared light at the fiber transmission device;

[0051] The near-infrared light detection device 10 includes a fiber coupler and a photodetector, which is used to collect near-infrared spectrum signals of solid particles;

[0052] The signal processing device 11 is used to calculate the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device 10 to obtain the diffuse reflection absorbance of the solid particles, and detect the component content of the solid particles based on the diffuse reflection absorbance of the solid particles based on signal correction and quantitative regression algorithm.

[0053] Among them, the process of realizing the detection of solid particle component content using quantitative regression algorithm is as follows:

[0054] 1) Use a component analyzer to obtain the standard component content of solid particles.

[0055] 2) Use linear regression algorithm to invert the solid particle component content.

[0056] Among them, the linear regression model between the component content c and the corrected spectral absorbance f′(λ) is as follows:

[0057] c=β0+β1f′(λ)+ε.

[0058] Among them, β0 is the linear regression constant, β1 is the linear regression coefficient of the spectral signal f′(λ) after correction in the near-infrared band of solid particles, and ε is a random variable. The unknown coefficients β0 and β1 can be determined based on the component content data and the corresponding optical parameters.

[0059] Among them, in some embodiments, the near-infrared light source emitting device is mainly composed of a broadband light source and a focusing lens 2. The broadband light source can be a near-infrared broadband laser light source 1, which emits near-infrared light into free space, and after being converged by the focusing lens 2, forms a small spot near-infrared light beam with a diameter not exceeding 5 mm (spot diameter D0≤5mm). The focusing lens 2 has a lens diameter of 6.0 mm and a focal length of 10.0 mm, and is made of quartz material; and the main performance parameters of the near-infrared broadband laser light source 1 include: a wavelength range between 400 and 2400 nanometers, a maximum output power of 100 milliwatts, and power stability: ≤1%.

[0060] Among them, in some embodiments, the optical fiber transmission device is mainly composed of an optical fiber coupler and a transmission optical fiber bundle. The function of the optical fiber coupler is to efficiently couple the near-infrared light beam converged by the focusing lens 2 into the transmission optical fiber, and its coupling efficiency can reach ≥90%. The device includes a first near-infrared optical fiber coupler 3 and a second near-infrared optical fiber coupler 5. The first near-infrared optical fiber coupler 3 and the second near-infrared optical fiber coupler 5 are respectively located at the two ends of a Y-type near-infrared transmission optical fiber bundle 4; the transmission optical fiber bundle can use a Y-type near-infrared transmission optical fiber bundle 4, which is composed of a plurality of optical fibers. It is responsible for transmitting the near-infrared light source to the probe end, and transmitting the near-infrared light carrying sample information collected by the probe end to the detector end. Each optical fiber adopts a 6+1Y-type structure with a core diameter of 1000um. The optical core material is low-hydroxyl quartz, and the cladding material is a polymer. Specifically, if Figure 2 What is shown involves the core distribution 12 of the Y-type optical fiber light source end, the core distribution 13 of the Y-type optical fiber probe end, and the core distribution 14 of the Y-type optical fiber detection end.

[0061] Among them, in some embodiments, the probe collection device is mainly composed of a fiber collimator 7, a sapphire light window 8 and passive mechanical components. The fiber collimator 7 is responsible for collimating the near-infrared light transmitted by the transmission fiber bundle into a parallel light beam, and vertically injecting it into the sapphire light window 8. After the light beam passes through the sapphire light window 8, it is parallelly irradiated on the surface of the solid particle sample, resulting in diffuse reflection. The near-infrared light carrying the sample information is transmitted in the opposite direction along the original path, and finally reaches the input end of the collection fiber bundle. Specific parameters include: the lens diameter of the fiber collimator 7 is D2=6.0mm, and the focal length of the lens is d2=10.0mm; the diameter of the sapphire light window 8 is D3=100mm, and the transmittance is T≥98%; the passive mechanical components mainly undertake the functions of support and fixation.

[0062] The near-infrared light detection device 10 is mainly composed of a fiber coupler and a photodetector. The fiber coupler couples the near-infrared light transmitted by the fiber transmission device into the photodetector end, and the photodetector collects the diffuse reflection spectrum information of solid particles in the near-infrared band. The bandwidth range of the near-infrared photodetector is: 1000-2200nm.

[0063] In some embodiments, the signal processing device 11 calculates the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device 10 to obtain the diffuse reflection absorbance of the solid particles, specifically including:

[0064] Calculate the diffuse reflectance absorbance of solid particles;

[0065] Where k is the absorption coefficient, s is the scattering coefficient, and R ∞ is the diffuse reflectivity of solid particles.

[0066] Since the Kubelka-Munk equation is linearly related to the component concentration, the following formula is often used to express diffuse reflectance absorbance and is applied to the drawing of near-infrared spectra.

[0067]

[0068] Embodiment 2

[0069] This embodiment provides a signal correction method based on the above-mentioned solid particle component content online detection system, including:

[0070] The near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device 10 is obtained.

[0071] According to the near-infrared spectrum signal of solid particles, the wavelet transform decomposition method is used to obtain components on different frequency scales.

[0072] The wavelet coefficients at different scales are adjusted to remove interference signals from the near-infrared spectrum signal of solid particles, and the processed near-infrared spectrum signal of solid particles is obtained.

[0073] The processed near-infrared spectrum signal of the solid particles is subjected to inverse wavelet transform to obtain the corrected near-infrared spectrum signal of the solid particles.

[0074] Use a component analyzer to obtain the standard component content of solid particles.

[0075] The linear regression algorithm is used to invert the solid particle component content.

[0076] In some embodiments, the wavelet transform decomposition method is used according to the near-infrared spectrum signal of the solid particles to obtain the frequency components at different scales, which can be specifically as follows:

[0077] According to the formula And the formula The spectral signal is decomposed into components at different scales through wavelet transform.

[0078] Among them, c j,k is the wavelet coefficient, j is the decomposition scale, k is the position parameter on scale j; f(λ) is the spectral data, λ is the wavelength; ψ j,k (λ) is a wavelet function, which is obtained by scaling and shifting the mother wavelet function ψ(λ).

[0079] Among them, the wavelet coefficients at different scales are adjusted to remove interference signals from the near-infrared spectrum signal of solid particles, and the processed near-infrared spectrum signal of solid particles is obtained, which can be specifically as follows:

[0080] Baseline removal is:

[0081]

[0082] Where J0 is the set layer number threshold, and the wavelet coefficients below this layer number are set to zero.

[0083] The noise removal is:

[0084]

[0085] Where P is the set high-frequency wavelet coefficient threshold.

[0086] The processed wavelet coefficient c′ j ' ,k Reconstructed by inverse wavelet transform, the corrected spectral signal is obtained

[0087] f′(λ)=∑ j,k c″ j,k ψ j,k (λ)+∑ k c 0,k φ(λ-k).

[0088] In the formula, c 0,k is the approximation coefficient and φ(λ) is the scaling function.

[0089] In some embodiments, the process of detecting the content of solid particles using a quantitative regression algorithm is as follows:

[0090] 1) Use a component analyzer to obtain the standard component content of solid particles.

[0091] 2) Use linear regression algorithm to invert the solid particle component content.

[0092] Among them, the linear regression model between the component content c and the corrected spectral absorbance f′(λ) is as follows:

[0093] c=β0+β1f′(λ)+ε.

[0094] Among them, β0 is the linear regression constant, β1 is the linear regression coefficient of the spectral signal f′(λ) after correction in the near-infrared band of solid particles, and ε is a random variable. The unknown coefficients β0 and β1 can be determined based on the component content data and the corresponding optical parameters.

[0095] In summary, this application has the following technical effects:

[0096] 1) Through signal correction, the present application can effectively reduce the measurement errors caused by multiple interference factors in industrial sites, thereby improving the accuracy and stability of component content detection.

[0097] 2) The introduction of the signal correction method can effectively reduce the interference of the external environment (such as vibration or sample movement) on the optical path and spectral signals, thereby improving the reliability of the detection system and making it suitable for use in more complex environmental conditions.

[0098] 3) After the spectral signal is corrected, the spectral data more truly reflects the internal characteristics of the sample. Combined with the quantitative regression algorithm, the component content can be predicted more accurately.

[0099] The technical features of the above embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0100] This article uses specific examples to illustrate the principles and implementation methods of this application. The description of the above embodiments is only used to help understand the method and core ideas of this application. At the same time, for those skilled in the art, according to the ideas of this application, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting this application.

Claims

1. An online detection system for solid particle component content, characterized in that: The solid particle component content online detection system comprises: a near-infrared light source emitting device, an optical fiber transmission device, a probe collection device, a near-infrared light detection device and a signal processing device; The near-infrared light source emitting device comprises a broadband light source and a focusing lens, and is used to emit a near-infrared light beam; The optical fiber transmission device comprises an optical fiber coupler and a transmission optical fiber bundle; the optical fiber coupler is used to couple the near-infrared light beam into the transmission optical fiber; the transmission optical fiber bundle is composed of a plurality of transmission optical fibers, and is used to transmit the near-infrared light source to the probe collection device, and transmit the near-infrared light carrying sample information collected by the probe collection device to the near-infrared light detection device; The probe collection device includes a fiber collimator, a sapphire light window and a passive mechanical component, which is used to arrange the near-infrared light transmitted by the fiber transmission device according to the set spatial position, and radiate the near-infrared light to the sample at the sample box after being collimated into parallel light by the fiber collimator, and collect the reflected near-infrared light at the fiber transmission device; The near-infrared light detection device includes an optical fiber coupler and a photodetector, which are used to collect near-infrared spectrum signals of solid particles; The signal processing device is used to calculate the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device to obtain the diffuse reflection absorbance of the solid particles, and detect the component content of the solid particles based on the diffuse reflection absorbance of the solid particles based on signal correction and quantitative regression algorithm.

2. The on-line detection system for solid particle component content according to claim 1, characterized in that: In the signal processing device, the near-infrared spectrum signal of the solid particles collected by the near-infrared light detection device is calculated to obtain the diffuse reflection absorbance of the solid particles, specifically including: According to the formula Calculate the diffuse reflectance absorbance of solid particles; Where k is the absorption coefficient, s is the scattering coefficient, and R ∞ is the diffuse reflectivity of solid particles.

3. The on-line detection system for solid particle component content according to claim 2 is characterized in that: The focusing lens in the near-infrared light source emitting device is used to converge the near-infrared light into a small-spot near-infrared light beam; the lens diameter D1 of the focusing lens is 6.0 mm, the lens focal length d1 is 10.0 mm, and the lens material is quartz material.

4. The on-line detection system for solid particle component content according to claim 3 is characterized in that: The optical fiber in the transmission optical fiber bundle is a 6+1Y optical fiber structure, the core diameter of the optical fiber is 1000um, and the optical core material is low-hydroxy quartz.

5. The on-line detection system for solid particle component content according to claim 4 is characterized in that: The lens diameter of the fiber collimator is D2=6.0mm, and the lens focal length is d2=10.0mm; the diameter of the sapphire light window is D3=100mm, and the transmittance is T≥98%; the passive mechanical component is used to support the fiber collimator and the sapphire light window.

6. The on-line detection system for solid particle component content according to claim 5, characterized in that: The quantitative regression algorithm is specifically: c=β0+β1f′(λ)+ε; Among them, β0 is the linear regression constant, β1 is the linear regression coefficient of the spectral absorbance SA′ of solid particles after correction in the near-infrared band, and ε is a random variable.

7. A signal correction method for an online detection system for solid particle component content according to any one of claims 1 to 6, characterized in that: include: Acquire near-infrared spectrum signals of solid particles collected by a near-infrared light detection device; According to the near-infrared spectrum signal of solid particles, the wavelet transform decomposition method is used to obtain the components on different frequency scales; Adjust the wavelet coefficients at different scales to remove interference signals from the near-infrared spectrum signal of solid particles, and obtain the processed near-infrared spectrum signal of solid particles; The processed near-infrared spectrum signal of the solid particles is subjected to inverse wavelet transform to obtain the corrected near-infrared spectrum signal of the solid particles.

8. The signal correction method of the solid particle component content online detection system according to claim 7, characterized in that: According to the near-infrared spectrum signal of solid particles, the wavelet transform decomposition method is used to obtain the components on different frequency scales, specifically: Among them, c j,k is the wavelet coefficient, j is the decomposition scale, k is the position parameter on scale j; f(λ) is the spectral data, λ is the wavelength; ψ j,k (λ) is a wavelet function, which is obtained by scaling and shifting the mother wavelet function ψ(λ).

9. The signal correction method according to claim 7, characterized in that: Adjust the wavelet coefficients at different scales, remove interference signals from the near-infrared spectrum signal of solid particles, and obtain the processed near-infrared spectrum signal of solid particles, specifically including: According to the formula Baseline removal of near infrared spectrum signals of solid particles; According to the formula Remove noise from near infrared spectrum signals of solid particles; Among them, J0 is the set layer number threshold, P is the set high-frequency wavelet coefficient threshold, c j,k is the wavelet coefficient, j is the decomposition scale, and k is the location parameter on scale j.

10. The signal correction method according to claim 7, characterized in that: The inverse wavelet transform is performed on the processed near-infrared spectrum signal of the solid particles to obtain the corrected near-infrared spectrum signal of the solid particles, specifically: f′(λ)=∑ j,k c″ j,k ψ j,k (λ)+∑ k c 0,k φ(λ-k) In the formula, c 0,k is the approximation coefficient and φ(λ) is the scaling function.

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