CN102867122A - Parabolic waveform fitting method of interference data - Google Patents

Abstract

The invention provides a parabolic waveform fitting method of interference data. In a parabolic waveform fitting process for received data by the method, complex inverse matrix calculation is not required. By the method, the efficiency is high and the signal processing efficiency can be improved. The method comprises the following steps of according to data with interference received at each time, calculating parameters A, B, C, D, E, F, G, H and I, and according to the calculated parameters, calculating parameters a, b and c in an infinite standard equation; and according to the parameters a, b and c, fitting a parabolic waveform. Inverse matrix calculation is not required, so that the calculation speed is high, and the requirement on processing the data in real time of a detector can be met.

Description

A kind of parabolic waveform approximating method of interfering data

Technical field

The present invention relates to a kind of Parabolic Fit method of interfering data, belong to the signal processing technology field.

Background technology

In photoelectricity, electromagnetism, laser and the detection process such as infrared, the signal of reception only has these random disturbance of rejecting often with random disturbance, just can extract correct receive data, thereby reach the purpose of detection.Usually these data always have some characteristic in noiseless situation, and for example signal of nuclear explosion is processed through some processes, and the waveform that arrives the data acquisition unit front end signal is similar to para-curve.According to these characteristics, utilize curve-fitting method just can reject random disturbance, obtain more accurate measurement data.Although the method for curve is a lot of at present, such as least square method, pseudo-inverse matrix method etc., but computation process more complicated, take least square method and pseudo-inverse matrix method as example, the two all relates to complicated inverse matrix computing, need to expend a large amount of operation time and resource, not only have a strong impact on real-time and the accuracy surveyed, also increased the development cost of detector.

Summary of the invention

The parabolic waveform approximating method that the purpose of this invention is to provide a kind of interfering data, utilize the method that the data that receive are being carried out in the process of Parabolic Fit, need not to carry out complicated inverse matrix and calculate, adopt the method efficient high, can improve the efficient that signal is processed.

Realize that technical scheme of the present invention is as follows:

A kind of parabolic waveform approximating method of interfering data is characterized in that:

From t ₀=0 constantly begins just to receive the situation with interfering data;

Step 101, calculating parameter A, B, C, D, E, F, G, H and I;

$\left\{\begin{array}{c}A=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ B=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ C=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right.$ $\left\{\begin{array}{c}D=\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}\\ E=\frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}\\ F=\frac{180}{N(N^2-1)(N^4-4)}\\ G=-\frac{18(2N+1)}{N(N-1)(N-2)}\\ H=-\frac{180}{N(N-1)(N^2-4)}\\ I=\frac{30}{N(N-1)(N-2)}\end{array}\right.$

Wherein, the band interfering data of y (k) for receiving k sampling instant, N is the sampling instant that receives last interfering data;

Step 102, calculating parameter a, b and c;

$\left\{\begin{array}{c}a=\mathrm{AI}+\mathrm{BH}+\mathrm{CF}\\ b=\mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ c=\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\end{array}\right.$

Step 103, calculate parabolic equation;

y＝at ²+bt+c。

Further, from t ₀≠ 0 constantly begins to receive the situation with interfering data:

Step 201, calculating parameter A, B, C, D, E, F, G, H and I;

$\left\{\begin{array}{c}A=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ B=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ C=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right.$ $\left\{\begin{array}{c}D=\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}\\ E=\frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}\\ F=\frac{180}{N(N^2-1)(N^4-4)}\\ G=-\frac{18(2N+1)}{N(N-1)(N-2)}\\ H=-\frac{180}{N(N-1)(N^2-4)}\\ I=\frac{30}{N(N-1)(N-2)}\end{array}\right.$

Wherein, the band interfering data of y (k) for receiving k sampling instant, N is the sampling instant that receives last interfering data;

Step 202, calculating parameter a, b and c;

$\left\{\begin{array}{c}a=\mathrm{AI}+\mathrm{BH}+\mathrm{CF}\\ b=\mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ c=\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\end{array}\right.$

Step 203, calculating parameter

And

$\left\{\begin{array}{c}a=\stackrel{\‾}{a},\\ b=(\stackrel{\‾}{b}-2\stackrel{\‾}{a}{t}_0)\\ c=\stackrel{\‾}{a}{t_0}^2-\stackrel{\‾}{b}{t}_0+\stackrel{\‾}{c}\end{array}\right.$

Step 204, calculate parabolic equation;

$y=\stackrel{\‾}{a}{t}^2+\stackrel{\‾}{b}t+\stackrel{\‾}{c}.$

Beneficial effect

The present invention is carrying out need not to carry out the technology of inverse matrix in the process of Parabolic Fit to the data that receive, and compared with prior art, the inventive method computing velocity is fast, can satisfy the real-time needs that detector is processed data.

Embodiment

The below is elaborated to solution procedure of the present invention.

In the process of nuclear blast, usually the energy after according to certain frequency nuclear blast being occured is sampled, result according to sampling simulates over time curve of nuclear blast energy, the highest nuclear blast energy of curve acquisition from this match, and the nuclear blast energy reaches maximum time etc., so that follow-up research is used.Therefore the present invention proposes a kind of quick interfering data Parabolic Fit method that inverse matrix is calculated that do not need to carry out;

For initial time t ₀=0 situation is namely from t ₀=0 constantly begins just to receive the situation with interfering data:

Suppose that the band interfering data that receives i sampling instant is y (i), i=1,2 ..., N, and parabolical equation is

y＝at ²+bt+c

Therefore have

$\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)=\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 2^2\\ \·\·\·& & \\ 1& N& N^2\end{array}\right)\left(\begin{array}{c}c\\ b\\ a\end{array}\right)$

Note $P=\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 2^2\\ \·\·\·& & \\ 1& N& N^2\end{array}\right),$ And be referred to as the match matrix.So have

$P^T\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)=P^{T}P\left(\begin{array}{c}c\\ b\\ a\end{array}\right)$

$\left(\begin{array}{c}c\\ b\\ a\end{array}\right)={\left(P^{T}P\right)}^{-1}{P}^T\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)$

P wherein ^TThe transposition of representing matrix P, (P ^TP) ^-1Representing matrix (P ^TP) inverse matrix; Therefore

$\left(P^{T}P\right)=\left(\begin{array}{cccc}1& 1& \·\·\·& 1\\ 1& 2& \·\·\·& N\\ 1& 2^2& \·\·\·& N^2\end{array}\right)\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 2^2\\ \·\·\·& & \\ 1& N& N^2\end{array}\right)$

$=\left(\begin{array}{cccc}1& 1& \·\·\·& 1\\ 1& 2& \·\·\·& N\\ 1& 2^2& \·\·\·& N^2\end{array}\right)\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 2^2\\ \·\·\·& & \\ 1& N& N^2\end{array}\right)$

$=\left(\begin{array}{ccc}N& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}i& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^2\\ \underset{i=1}{\overset{N}{\mathrm{\Σ}}}i& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^2& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^3\\ \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^2& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^3& \underset{i=1}{\overset{N}{\mathrm{\Σ}}}{i}^4\end{array}\right)$

Utilize formula

$1+2+\·\·\·+N=\frac{N(N+1)}{2}$

$1^2+2^2+\·\·\·+N^2=\frac{N(N+1)(2N+1)}{6}$

$1^3+2^3+\·\·\·+N^3=\frac{N^2{(N+1)}^2}{4}$

$1^4+2^4+\·\·\·+N^4=\frac{N(N+1)(2N+1)(3N^3+3N-1)}{30}$

Obtain

$\left(P^{T}P\right)=\left(\begin{array}{ccc}N& \frac{N(N+1)}{2}& \frac{N(N+1)(2N+1)}{6}\\ \frac{N(N+1)}{2}& \frac{N(N+1)(2N+1)}{6}& \frac{N^2{(N+1)}^2}{4}\\ \frac{N(N+1)(2N+1)}{6}& \frac{N^2{(N+1)}^2}{4}& \frac{N(N+1)(2N+1)(3N^2+3N-1)}{30}\end{array}\right)$

To matrix

$\left(\begin{array}{cccccc}N& \frac{N(N+1)}{2}& \frac{N(N+1)(2N+1)}{6}& 1& 0& 0\\ \frac{N(N+1)}{2}& \frac{N(N+1)(2N+1)}{6}& \frac{N^2{(N+1)}^2}{4}& 0& 1& 0\\ \frac{N(N+1)(2N+1)}{6}& \frac{N^2{(N+1)}^2}{4}& \frac{N(N+1)(2N+1)(3N^2+3N-1)}{30}& 0& 0& 1\end{array}\right)$

Make Applying Elementary Row Operations and can get matrix

$\left(\begin{array}{cccccc}1& 0& 0& \frac{3(3N^2+3N+2)}{N(N-1)(N-2)}& \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{30}{N(N-1)(N-2)}\\ 0& 1& 0& \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}& \frac{-180}{N(N-1)(N^2-4)}\\ 0& 0& 1& \frac{30}{N(N-1)(N-2)}& \frac{-180}{N(N-1)(N^2-4)}& \frac{180}{N(N^2-1)(N^2-4)}\end{array}\right)$

Thus

${\left(P^{T}P\right)}^{-1}=\left(\begin{array}{ccc}\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}& \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{30}{N(N-1)(N-2)}\\ \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}& \frac{-180}{N(N-1)(N^2-4)}\\ \frac{30}{N(N-1)(N-2)}& \frac{-180}{N(N-1)(N^2-4)}& \frac{180}{N(N^2-1)(N^2-4)}\end{array}\right)$

$\left(\begin{array}{c}c\\ b\\ a\end{array}\right)={\left(P^{T}P\right)}^{-1}{P}^T\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)$

$=\left(\begin{array}{ccc}\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}& \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{30}{N(N-1)(N-2)}\\ \frac{-18(2N+1)}{N(N-1)(N-2)}& \frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}& \frac{-180}{N(N-1)(N^2-4)}\\ \frac{30}{N(N-1)(N-2)}& \frac{-180}{N(N-1)(N^2-4)}& \frac{180}{N(N^2-1)(N^2-4)}\end{array}\right)\left(\begin{array}{cccc}1& 1& \·\·\·& 1\\ 1& 2& \·\·\·& N\\ 1& 2^2& \·\·\·& N^2\end{array}\right)\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)$

$=\frac{1}{N(N-1)(N-2)}\left(\begin{array}{ccc}\text{3}(3N^2+3N+2)& -18(2N+1)& 30\\ -18(2N+1)& \frac{12(2N+1)(8N+11)}{(N+1)(N+2)}& \frac{-180}{N+2}\\ 30& \frac{-180}{N+2}& \frac{180}{(N+1)(N+2)}\end{array}\right)\left(\begin{array}{cccc}1& 1& \·\·\·& 1\\ 1& 2& \·\·\·& N\\ 1& 2^2& \·\·\·& N^2\end{array}\right)\left(\begin{array}{c}y\left(1\right)\\ y\left(2\right)\\ \·\\ \·\\ \·\\ y\left(N\right)\end{array}\right)$

$\frac{1}{N(N-1)(N-2)}\left(\begin{array}{ccc}3(3N^2+3N+2)& -18(2N+1)& 30\\ -18(2N+1)& \frac{12(2N+1)(8N+11)}{(N+1)(N+2)}& \frac{-180}{N+2}\\ 30& \frac{-180}{N+2}& \frac{180}{(N+1)(N+2)}\end{array}\right)\left(\begin{array}{c}\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ \underset{k}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ \underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right)$

Note

$\left\{\begin{array}{c}A=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ B=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ C=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right.$ $\left\{\begin{array}{c}D=\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}\\ E=\frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}\\ F=\frac{180}{N(N^2-1)(N^2-4)}\\ G=-\frac{18(2N+1)}{N(N-1)(N-2)}\\ H=-\frac{180}{N(N-1)(N^2-4)}\\ I=\frac{30}{N(N-1)(N-2)}\end{array}\right.$

Then

$\left(\begin{array}{c}c\\ b\\ a\end{array}\right)=\left(\begin{array}{ccc}D& G& I\\ G& E& H\\ I& H& F\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\end{array}\right)=\left(\begin{array}{ccc}D& G& I\\ G& E& H\\ I& H& F\end{array}\right)=\left(\begin{array}{c}\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\\ \mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ \mathrm{AI}+\mathrm{BH}+\mathrm{CF}\end{array}\right)$

Namely

$\left\{\begin{array}{c}a=\mathrm{AI}+\mathrm{BH}+\mathrm{CF}\\ b=\mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ c=\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\end{array}\right.$

2) for initial time t ₀≠ 0, from t ₀≠ 0 constantly begins to receive the situation with interfering data:

Suppose that the band interfering data that receives i sampling instant is y (i), i=t ₀+ 1, t ₀+ 2 ..., t ₀+ N, the para-curve for the treatment of match is y=at ²+ bt+c.The numerical value of element is larger in the above-mentioned match matrix at this moment, therefore easily causes many adverse consequencess.The first easily causes overflowing in calculating, and it two is that computing is more complicated.For this reason, can be with sampled data along the horizontal ordinate t that moves to left ₀, the data that obtain are

$\stackrel{\_}{y}\left(i\right)=y(t_0+i),i=\mathrm{1,2}\·\·\·,N$

Utilize the above-mentioned contrary method of closing to obtain corresponding para-curve In coefficient

Get by coordinate translation is special

$a{t}^2+\mathrm{bt}+c=\stackrel{\_}{a}{(t-t_0)}^2+\stackrel{\_}{b}(t-t_0)+\stackrel{\_}{c}$

Namely

$a{t}^2+\mathrm{bt}+c=\stackrel{\_}{a}{t}^2+{(\stackrel{\_}{b}-2\stackrel{\_}{a}{t}_0)t+\stackrel{\_}{a}{t}_0}^2-\stackrel{\_}{b}{t}_0+\stackrel{\_}{c}$

Therefore

$\left\{\begin{array}{c}a=\stackrel{\_}{a},\\ b=(\stackrel{\_}{b}-2\stackrel{\_}{a}{t}_0)\\ c={\stackrel{\_}{a}{t}_0}^2-\stackrel{\_}{b}{t}_0+\stackrel{\_}{c}\end{array}\right.$

In sum, above is preferred embodiment of the present invention only, is not for limiting protection scope of the present invention.Within the spirit and principles in the present invention all, any modification of doing, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (2)

1. the parabolic waveform approximating method of an interfering data is characterized in that:

From t ₀=0 constantly begins just to receive the situation with interfering data;

Step 101, calculating parameter A, B, C, D, E, F, G, H and I;

$\left\{\begin{array}{c}A=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ B=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ C=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right.$ $\left\{\begin{array}{c}D=\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}\\ E=\frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}\\ F=\frac{180}{N(N^2-1)(N^4-4)}\\ G=-\frac{18(2N+1)}{N(N-1)(N-2)}\\ H=-\frac{180}{N(N-1)(N^2-4)}\\ I=\frac{30}{N(N-1)(N-2)}\end{array}\right.$

Wherein, the band interfering data of y (k) for receiving k sampling instant, N is the sampling instant that receives last interfering data;

Step 102, calculating parameter a, b and c;

$\left\{\begin{array}{c}a=\mathrm{AI}+\mathrm{BH}+\mathrm{CF}\\ b=\mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ c=\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\end{array}\right.$

Step 103, calculate parabolic equation;

y＝at ²+bt+c。

2. the parabolic waveform approximating method of an interfering data is characterized in that:

From t ₀≠ 0 constantly begins to receive the situation with interfering data:

Step 201, calculating parameter A, B, C, D, E, F, G, H and I;

$\left\{\begin{array}{c}A=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}y\left(k\right)\\ B=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}\mathrm{ky}\left(k\right)\\ C=\underset{k=1}{\overset{N}{\mathrm{\Σ}}}{k}^{2}y\left(k\right)\end{array}\right.$ $\left\{\begin{array}{c}D=\frac{3(3N^2+3N+2)}{N(N-1)(N-2)}\\ E=\frac{12(2N+1)(8N+11)}{N(N^2-1)(N^2-4)}\\ F=\frac{180}{N(N^2-1)(N^4-4)}\\ G=-\frac{18(2N+1)}{N(N-1)(N-2)}\\ H=\frac{180}{N(N-1)(N^2-4)}\\ I=\frac{30}{N(N-1)(N-2)}\end{array}\right.$

Wherein, the band interfering data of y (k) for receiving k sampling instant, N is the sampling instant that receives last interfering data;

Step 202, calculating parameter a, b and c;

$\left\{\begin{array}{c}a=\mathrm{AI}+\mathrm{BH}+\mathrm{CF}\\ b=\mathrm{AG}+\mathrm{BE}+\mathrm{CH}\\ c=\mathrm{AD}+\mathrm{BG}+\mathrm{CI}\end{array}\right.$

Step 203, calculating parameter

And

$\left\{\begin{array}{c}a=\stackrel{\‾}{a},\\ b=(\stackrel{\‾}{b}-2\stackrel{\‾}{a}{t}_0)\\ c=\stackrel{\‾}{a}{t_0}^2-\stackrel{\‾}{b}{t}_0+\stackrel{\‾}{c}\end{array}\right.$

Step 204, calculate parabolic equation;

$y=\stackrel{\‾}{a}{t}^2+\stackrel{\‾}{b}t+\stackrel{\‾}{c}.$