CN111490752A - Method for calculating spectral derivative by digital filter - Google Patents
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Abstract
The invention provides a method for solving a spectral derivative by a digital filter, which comprises the following steps of 1, selecting a spectrum Sp to be solved; step 2, constructing a convolution transfer function H; step 3, solving the derivative of the spectrum Sp; the convolution transfer function H is obtained by: designing a transfer function PF of a smoothing filter, carrying out inverse Fourier transform on the PF to obtain an array H0, and obtaining a convolution transfer function H from an array H0; the step of calculating the derivative of the spectrum Sp comprises directly calculating the convolution of H and the spectrum data Sp, and obtaining a 0-order result after selective excision; and substituting the H difference for derivation to obtain difference results of each step, calculating the convolution of the difference results and Sp, and obtaining derivatives of Sp of n order after selective excision. The invention can calculate each order derivative spectrum which is easy to clearly identify; the derivative spectrogram obtained by the method can obtain clearer and more definite information, and has important significance for the identification, identification and deep analysis of the spectrogram.
Description
Technical Field
The invention relates to the field, in particular to a method for solving a spectral derivative by a digital filter.
Background
The derivative spectroscopy is a mature and effective method in ultraviolet-visible spectrum analysis, and has an important role in solving the direct measurement problem of a complex multi-component system because the resolution capability of the spectrum and the selectivity can be effectively improved. Meanwhile, for the near infrared spectrum, the method is a conventional data preprocessing method for eliminating scattering drift by solving a first derivative. However, other spectra, especially raman and infrared spectra with sharp and narrow peak types, are not applied much, and the reason is not that the method is not applicable, but an effective derivative spectrum acquisition means is lacked; for ultraviolet-visible spectrum and near infrared spectrum, derivative spectroscopy basically adopts 1-order treatment due to the limitation of derivation capability, and the realization difficulty of derivative spectroscopy stopping at 2-order and more than 2-order is large.
The difficulty of spectral derivation is noise interference, where the amplitude of signals at different frequencies decays exponentially with frequency. Direct derivation, except for very high signal-to-noise ratio, the high frequency part of the noise in the signal will completely cover the real signal after derivative processing. Therefore, an effective derivative noise reduction method must be used to filter out the high frequency part of the signal and retain the true signal of relatively low frequency. The reason that derivative spectroscopy can be implemented over a broad peak of the ultraviolet or near infrared spectrum is that the lower the true signal frequency to be denoised, the greater the difference from the noise high frequency, the less difficult the derivative calculation. On narrow peaks such as Raman and mid-infrared, the frequency is close to the high frequency of noise, and the problem difficulty is obviously improved.
The spectrum derivation method has two solutions of software and hardware, and the hardware method of the electronic differential element has limited effect except that the equipment and design cost is increased, and is realized by depending on software of a signal processing algorithm more often. In the algorithm approach, the currently available S-G digital differential filter is limited to the signal-to-noise ratio of the signal to achieve better results. The S-G algorithm has limited precision, large jump degree of parameter setting and cannot be finely adjusted, so that the calculation error of the second derivative is obviously increased, and the peak form of the second derivative has obvious error along with the increase of the filtering strength.
The existing S-G filter has better direct smooth performance on spectral data, but as the derivative order number rises, the precision of the S-G filter is limited, so that the error cannot be finely adjusted, and the derivative calculation error of more than 2 orders is obvious. Since the filtering noise reduction-derivation method has a reliable mathematical principle, it is critical to reduce the calculation error to improve the adjustable accuracy of the method.
Polynomial fitting, represented by S-G filtering, least squares optimized filtering, is reduced to a simplified FIR filter in principle. The direct problem encountered in the polynomial fitting approach is the dragon lattice phenomenon caused by adopting high-order polynomial fitting; the higher the precision, the higher the required polynomial order, and the greater the probability of the dragon lattice phenomenon. Therefore, the selectable orders are limited, the adjustable range is usually an integer of 2 to 8 orders, and the range of the selectable parameters is small; in addition, the width of the S-G filter frame is limited, and can only be an odd number, the step amplitude is minimum 2, and the fine-tuning subdivision cannot be further performed.
High-precision high-order derivative measurement or calculation is a key technology in the design and manufacture of international high-end instruments, and due to the lack of effective algorithms, instrument hardware with very high signal-to-noise ratio has to be relied on, so that the manufacturing cost of products is promoted. The high-precision digital filtering derivation algorithm has very high technical and industrial values in the aspects of instrument research and development and product performance.
The inventor finds that the calculation principle of the digital filter is convolution calculation of the original signal and the transfer function, and the accuracy of the transfer function can be improved to realize the purpose of improving the adjustability of the filter. Further, the derivative calculation of the original signal may be performed by convolution of the original signal with the derivative of the transfer function, depending on the nature of the convolution.
Disclosure of Invention
To address the deficiencies of the related art as discussed above, the present invention provides a method for a digital filter to derive spectral derivatives.
The invention discloses a method for solving a spectral derivative by a digital filter, which is realized by the following technical scheme:
a method for deriving a spectral derivative for a digital filter, comprising:
and 3, solving the derivative of the spectrum Sp through the convolution transfer function H.
Further, the convolution transfer function H in step 2 is obtained by:
step a, designing a transfer function PF of a smoothing filter;
b, performing inverse Fourier transform on the transfer function PF, and outputting a real part array of a transform array to obtain an array H0;
and c, obtaining a convolution transfer function H according to the array H0 obtained in the step b.
Further, the step 3 of calculating the derivative of the spectrum Sp includes calculating a 0-order result of Sp and calculating an n-order result of Sp; wherein n is a positive integer.
Further, the 0-order result of Sp is obtained by the following method:
directly calculating convolution of H and the filtered spectral data Sp;
if the H contains m elements, cutting off each m/2 elements from the front and back of the convolution result, wherein the rest array elements correspond to the result of Sp smoothing, namely the 0-order result;
wherein m is a non-negative integer.
Further, the nth order result of Sp is obtained by the following method:
using H difference to replace derivation, obtaining the difference results of H (1), H (2), … …, H (n) and other orders in sequence, calculating the convolution of the difference H (n) of H and Sp;
if the number m of elements in H (n) is even, cutting off m/2 elements before and after the convolution result respectively, and leaving the elements as the nth derivative result of Sp;
if the number m of the elements of H (n) is odd, cutting off (m +1)/2 elements respectively from the front and the back of the convolution result, and cutting off (m-1)/2 elements from the back end, wherein the remaining elements are the nth derivative result of Sp.
Further, the transfer function PF in step a is represented as an array, and the transfer function PF includes a band-stop section Wr and a band-pass section Wp;
the band-stop segment Wr selects the right half of the distribution function describing the bandpass behavior of the signal for setting the percentage of passable frequencies, the passable frequency range including low-frequency signals;
the band-pass section Wp is used for allowing the frequency to pass by 100%;
wherein, Wr and Wp are nonnegative integers, and the smaller the number, the stronger the filtering strength.
Further, the array H0 in step b refers to an array from the head of the output real part array to a half position of the whole array.
Further, the convolution transfer function H in the step c is obtained by combining H0 with H1 obtained by turning H0 back and forth, and dividing the combined value by the sum of all elements;
the formula of the convolution transfer function H is:
H＝[H1，H0]/sum([H1，H0])。
further, the formula of the transfer function PF is as follows:
compared with the prior art, the derivation method has the following advantages:
1) the method comprises the steps of sequentially obtaining each order of derivative (difference) of a transfer function, and then obtaining the convolution of an original signal and each order of derivative (difference) of the transfer function, so that the spectrum of the multiple orders of derivative is obtained;
2) the invention can calculate each order derivative spectrum which is easy to clearly identify;
3) the derivative spectrogram obtained by the method can obtain clearer and more definite information;
4) the method has important significance for the identification, the identification and the deep analysis of the spectrogram.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
FIG. 1 is a graph showing a simulated frequency pass-blocking amplitude function PF in example 1 of the present invention;
fig. 2 is a schematic diagram of the real part output of the frequency blocking and passing amplitude function PF after inverse fourier transform in embodiment 1 of the present invention;
FIG. 3 is a schematic diagram showing the structure of a transfer function H in example 1 of the present invention;
FIG. 4 is a schematic diagram of the derivative (difference) transfer functions of the respective orders in embodiment 1 of the present invention;
FIG. 5 is a convolution result of the simulated spectrum Sp and the 0 th to 4 th order transfer function in example 1 of the present invention;
FIG. 6 shows the results of the derivatives of the first and second orders outputted in example 1 of the present invention;
FIG. 7 shows the direct difference result of the spectral data Sp1 in example 2 of the present invention;
fig. 8 shows the derivative result of applying the derivative filter of the present invention to the spectral data Sp1 in embodiment 2 of the present invention;
FIG. 9 is a comparison of the results of different treatments in example 2 of the present invention;
FIG. 10 shows the results of the spectral treatment of aspirin according to the method of the present invention in example 3 of the present invention.
Detailed Description
The following description of the embodiments of the present invention is provided for illustrative purposes, and other advantages of the present invention will be readily apparent to those skilled in the art from the disclosure herein.
A method for a digital filter to derive spectral derivatives, comprising: step 1, selecting a spectrum Sp to be subjected to derivation; step 2, constructing a convolution transfer function H for derivation calculation; and 3, solving the derivative of the spectrum Sp through the convolution transfer function H.
Further, the convolution transfer function H in step 2 is obtained by: step a, designing a transfer function PF of a smoothing filter; b, performing inverse Fourier transform on the transfer function PF, and outputting a real part array of a transform array to obtain an array H0; and c, obtaining a convolution transfer function H according to the array H0 obtained in the step b.
Further, the step 3 of calculating the derivative of the spectrum Sp includes calculating a 0-order result of Sp and calculating an n-order result of Sp; wherein n is a positive integer.
Further, the 0-order result of Sp is obtained by the following method: directly calculating convolution of H and the filtered spectral data Sp; if the H contains m elements, cutting off each m/2 elements from the front and back of the convolution result, wherein the rest array elements correspond to the result of Sp smoothing, namely the 0-order result; wherein m is a non-negative integer.
Further, the nth order result of Sp is obtained by the following method: using H difference to replace derivation, obtaining the difference results of H (1), H (2), … …, H (n) and other orders in sequence, calculating the convolution of the difference H (n) of H and Sp; if the number m of elements in H (n) is even, cutting off m/2 elements before and after the convolution result respectively, and leaving the elements as the nth derivative result of Sp; if the number m of the elements of H (n) is odd, cutting off (m +1)/2 elements respectively from the front and the back of the convolution result, cutting off (m-1)/2 elements from the back end, and leaving the elements as the nth derivative result of Sp;
where H-differencing is common knowledge of numerical calculations, discrete calculations are to calculate the differential, instead of the differential calculation.
Further, the transfer function PF in step a is represented as an array, and the transfer function PF includes a band-stop section Wr and a band-pass section Wp; the band-stop segment Wr selects the right half of the distribution function describing the bandpass behavior of the signal for setting the percentage of passable frequencies, the passable frequency range including low-frequency signals; the band-pass section Wp is used for allowing the frequency to pass by 100%; wherein, Wr and Wp are nonnegative integers, and the smaller the number, the stronger the filtering strength.
Further, the array H0 in step b refers to an array from the head of the output real part array to a half position of the whole array.
Further, the convolution transfer function H in the step c is obtained by combining H0 with H1 obtained by turning H0 back and forth, and dividing the combined value by the sum of all elements; the formula of the convolution transfer function H is:
H＝[H1，H0]/sum([H1，H0])。
further, the formula of the transfer function PF is as follows:
example 1
The present embodiment provides a method for calculating a spectral derivative by a digital filter, including:
firstly, selecting a spectrum Sp to be subjected to derivation; then, with Wr and Wp both being 500, the transfer function H, and its 1, 2, 3, 4-order transfer functions H (1), H (2), H (3), H (4) are constructed.
The distribution function for describing the bandpass behavior of the signal in this embodiment is gaussian.
As shown in fig. 1. Setting an array with Wp elements of 1 to form a band-pass part; then, constructing the right half part of a Gaussian peak by taking Wr as the peak width, and normalizing to obtain a band-stop part; and combining the two parts to obtain a complete frequency blocking amplitude function PF.
And performing inverse Fourier transform on the frequency-blocking amplitude function PF, and taking a real part of an inverse Fourier transform result to output, as shown in FIG. 2.
The first half of the inverse Fourier transform real part of the frequency-blocking amplitude function PF is taken as H0, H0 is turned into H1, and H1 and H0 are combined to form a transfer function H, as shown in FIG. 3.
H (1) to H (4) shown in fig. 4 are each order derivative (difference) transfer functions obtained by differentiating H.
Fig. 5 shows the convolution of a simulated gaussian peak Sp with a transfer function of order 0 to 4. And (5) correspondingly cutting off each m/2 data points before and after the convolution result to obtain each derivative result, as shown in figure 6.
Example 2
The present embodiment provides a method for a digital filter to derive a spectral derivative:
the distribution function for describing the bandpass behavior of the signal in this embodiment is gaussian.
First, fitted spectral data Sp1 was obtained by constructing 5 gaussian peaks of different widths, superimposing white noise of 1% intensity at the maximum peak, see fig. 7 (a). Fig. 7(b) - (e) are direct 1-4 order difference results, respectively, and it can be seen that after more than 1 order, the signal is covered by noise and is hardly recognizable.
By adopting the derivation method provided by the invention, smoothing and 1-4 order derivative calculation are completed in sequence. As a result, as shown in fig. 8, when the filter parameters Wp and Wr are 500 and 800, the signal in the 4 th derivative can still be effectively identified.
Fig. 9 is a comparison of the results of the noise-free direct 2 nd order difference, the S-G filtered 2 nd order derivative and the 2 nd order derivative of the signal for the three cases of the method of the present invention, where the narrowest peak width of the signal, here the most significant part of the error in the derivation process, is shown. It can be seen that the present invention has better performance and adjustability than S-G filtering, which is currently considered to be the best filter derivation method.
As can be seen in fig. 9, the two parameters selected by the present invention have a large margin of adjustability, so that a smoother curve can be finely adjusted while maintaining agreement with the true signal at the peak, while S-G filtering requires a trade-off between curve smoothing and reality.
Example 3
Raman spectra of analytically pure aspirin were collected under 785nm laser excitation with an integration time of 1 ms. The derivative calculation method of the invention is adopted to carry out 4-order derivative calculation on the actually measured aspirin Raman spectrum to obtain a 4-order derivative spectrum, as shown in figure 10.
By applying the method, the fourth-order derivative spectrum which is easy to clearly identify can be calculated, and the original spectrogram and the derivative spectrogram are compared, so that the original overlapped Raman peaks on the derivative spectrum are completely separated, and clearer and more definite information can be obtained from the derivative spectrum, and the method has important significance for the identification, identification and deep analysis of the spectrogram.
Claims (9)
1. A method for deriving a spectral derivative for a digital filter, comprising:
step 1, selecting a spectrum Sp to be subjected to derivation;
step 2, constructing a convolution transfer function H for derivation calculation;
and 3, solving the derivative of the spectrum Sp through the convolution transfer function H.
2. A method for deriving a spectral derivative by a digital filter according to claim 1, wherein the convolution transfer function H of step 2 is obtained by:
step a, designing a transfer function PF of a smoothing filter;
b, performing inverse Fourier transform on the transfer function PF, and outputting a real part array of a transform array to obtain an array H0;
and c, obtaining a convolution transfer function H according to the array H0 obtained in the step b.
3. A method for deriving a spectral derivative for a digital filter according to claim 1 or 2, wherein said deriving a spectral derivative Sp in step 3 comprises deriving a 0 th order of Sp and deriving an nth order of Sp; wherein n is a positive integer.
4. A method for deriving spectral derivatives for a digital filter as defined in claim 3 wherein the 0 th order result of Sp is obtained by:
directly calculating convolution of H and the filtered spectral data Sp;
if the H contains m elements, cutting off each m/2 elements from the front and back of the convolution result, wherein the rest array elements correspond to the result of Sp smoothing, namely the 0-order result;
wherein m is a non-negative integer.
5. A method for use in a digital filter for deriving spectral derivatives as claimed in claim 3 or 4, wherein said nth order result for Sp is obtained by:
using H difference to replace derivation, obtaining the difference results of H (1), H (2), … …, H (n) and other orders in sequence, calculating the convolution of the difference H (n) of H and Sp;
if the number m of elements in H (n) is even, cutting off m/2 elements before and after the convolution result respectively, and leaving the elements as the nth derivative result of Sp;
if the number m of the elements of H (n) is odd, cutting off (m +1)/2 elements respectively from the front and the back of the convolution result, and cutting off (m-1)/2 elements from the back end, wherein the remaining elements are the nth derivative result of Sp.
6. A method for deriving spectral derivatives for a digital filter as claimed in claim 3 wherein step a said transfer function PF is represented by an array, said transfer function PF comprising a band stop section Wr and a band pass section Wp;
the band-stop segment Wr selects the right half of the distribution function describing the bandpass behavior of the signal for setting the percentage of frequencies that can pass;
the band-pass section Wp is used for allowing the frequency to pass by 100%;
wherein, Wr and Wp are non-negative integers.
7. The method of claim 3, wherein the array H0 in step b is an array from the head of the output real part array to half the position of the whole array.
8. The method for deriving the spectral derivative of a digital filter according to claim 2 or 7, wherein the convolution transfer function H of step c is obtained by combining H0 with H1 obtained by flipping H0 back and forth, and dividing by the sum of all elements;
the formula of the convolution transfer function H is:
H＝[H1，H0]/sum([H1，H0])。
9. a method for deriving a spectral derivative for a digital filter according to claim 3, wherein said transfer function PF is formulated as follows:
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| US4553248A (en) * | 1983-06-10 | 1985-11-12 | International Business Machines Corporation | Analog adaptive magnitude equalizer |
| CN101534933A (en) * | 2006-05-26 | 2009-09-16 | 沃特世科技公司 | Ion detection and parameter estimation for n-dimensional data |
| CN105814603A (en) * | 2013-05-30 | 2016-07-27 | 帝国创新有限公司 | Method and apparatus |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4553248A (en) * | 1983-06-10 | 1985-11-12 | International Business Machines Corporation | Analog adaptive magnitude equalizer |
| CN101534933A (en) * | 2006-05-26 | 2009-09-16 | 沃特世科技公司 | Ion detection and parameter estimation for n-dimensional data |
| CN105814603A (en) * | 2013-05-30 | 2016-07-27 | 帝国创新有限公司 | Method and apparatus |
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| 文玉梅等: ""用多项式近似的图象逆滤波及空间移变系统图象的恢复"" * |
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