CN100499370C - Filter of accurate realizing signal differentiation, and method for implementing signal differentiation - Google Patents

Abstract

The disclosed filter includes multiple delay cells, multiple multiplication units and one addition unit. Number of multiplication units is one more than number of delay cells. Result of multiplying input signal and output signal, which is a delayed input signal by a delay cell, by a coefficient through a multiplication unit is output to the said addition unit. The addition unit outputs signal differential of input signal. The method uses FIR filter to carry out differential operation, providing higher computational accuracy, no accumulative error, equivalent precision between differential in high order and differential in first order, no problem of such that differential in higher order has larger error.

Description

Accurately realize the filter of signal differentiation and use it to realize the method for signal differentiation

Technical field

The present invention asks for the filter of input differential signal and uses this kind filter accurately to realize the method for signal differentiation relevant for a kind of.

Background technology

In signal processing, signal is carried out the processing method that differential is a kind of frequent use.Usually ask the differential of signal generally to be converted into the computing of asking difference at numeric field.This method has many shortcomings, and not only precision is not high, and has the accumulation of error when asking high subdifferential, can't use so that error is very big.As shown in Figure 1, difference mainly realizes that by a delay cell 101 and a subtrator signal x (n) input delay unit 101 obtains the signal x (n-1) that postpones a sampling period, signal x (n-1) and signal x (n) are input to subtrator 102, output differential signal Y (n).

Ask high subdifferential if desired, just above-mentioned differential scheme cascade is used.

More than use the scheme that difference method differentiates following shortcoming arranged:

1, difference method is differentiated, owing to be directly to use former and later two sample values to subtract each other, can exist than mistake, and error increases with the increase of sampling period.

2, use difference method and ask high subdifferential, can be owing to the accumulation of error cause resultant error very big, differentiating stage is inferior high more, and error is big more, makes the application difference method ask high subdifferential in fact not have the use meaning.

The result is that difference method is differentiated and used relatively difficulty, only is applied in occasions such as not high subdifferential of required precision or low subdifferential.

Summary of the invention

The invention provides and a kind ofly ask for the filter of input differential signal and use the method that this kind filter is realized signal differentiation, can effectively improve computational accuracy and can not produce the accumulation of error with the raising of differential exponent number.

The invention provides a kind of filter of accurate realization signal differentiation, comprise m delay cell, a m+1 multiplication unit and an adder unit, wherein m is the exponent number of filter, and m is more than or equal to the exponent number n that is differentiated; The sampled signal that will obtain after will sampling to input signal is imported the 1st multiplication unit and the 1st delay cell, and other each delay cell except that the 1st delay cell receives the signal of previous delay cell output respectively; After handling the signal that receives, described m delay units delay output signal to the 2nd respectively to m+1 multiplication unit, a described m+1 multiplication unit will output signal to described adder unit behind the coefficient of signal times with the filter of correspondence that receive respectively, and described adder unit carries out addition to the signal that receives and handles the differential signal that obtains described input signal; Wherein, the coefficient of the filter of described m+1 multiplication unit correspondence be respectively n! H ^-1(n) a m+1 factor, H ^-1(n) be matrix H ^-1The capable coefficient of n+1, described matrix H is by 0,1,2 of the m+1 that input signal is sampled pairing each sampling time of sample point, 3... form to m power, pairing each sampling time of described each sample point is the integral multiple of continuous sampling period, and in the described matrix H, and first is 1 in the capable capable vector of p, second is 1 of p the pairing sampling time of sample point to the m+1 item, 2,3... is to m power, wherein p=1,2......m+1; The 1st classifies the column vector that m+1 1 forms as, q classify as each sample point q-1 power of each self-corresponding sampling time, q=2,3......m+1.

The invention provides a kind of method of using described filter accurately to realize signal differentiation, comprise the following steps:

1) input signal is sampled;

2) sampled signal is input to the 1st multiplication unit and the 1st delay cell of described filter, other each delay cell except that the 1st delay cell receives the signal of previous delay cell output respectively, after handling the signal that receives, described m delay units delay output signal to the 2nd respectively to m+1 multiplication unit, a described m+1 multiplication unit will output signal to described adder unit behind the coefficient of signal times with the filter of correspondence that receive respectively, and described adder unit carries out addition to the signal that receives and handles the differential signal that obtains described input signal; Wherein, the coefficient of the filter of m+1 multiplication unit correspondence be respectively n! H ^-1(n) a m+1 factor, H ^-1(n) be matrix H ^-1The capable coefficient of n+1, described matrix H is 0,1,2 of m+1 pairing each sampling time of sample point that input signal is sampled, 3... the determinant of forming to m power, pairing each sampling time of described each sample point is the integral multiple of continuous sampling period, and in the described matrix H, and first is 1 in the capable capable vector of p, second is 1 of p the pairing sampling time of sample point to the m+1 item, 2,3... is to m power, wherein p=1,2......m+1; The 1st classifies the column vector that m+1 1 forms as, q classify as each sample point q-1 power of each self-corresponding sampling time, q=2,3......m+1;

3) read the signal that described filter is exported.

Described each sample point is chosen for the center symmetry with t=0.

The sampling frequency of described each sample point is at least the twice of input signal highest frequency.

The present invention uses the FIR filter and differentiates, and computational accuracy is higher, and precision is relevant with the exponent number of FIR filter, and the exponent number of FIR filter is big more, and the precision of being differentiated is high more.

The error free accumulation of method of the present invention asks the precision of a high subdifferential and a subdifferential suitable, can not exist the differential order high more, the problem that error is big more.

Description of drawings

Fig. 1 is for asking the realization block diagram of signal differentiation with difference method in the prior art.

Fig. 2 asks the process schematic diagram of signal differentiation with the FIR filter for the present invention.

Fig. 3 is the present invention's FIR filter construction block diagram.

Embodiment

The method schematic diagram that the present invention asks signal differentiation with the FIR filter as shown in Figure 2, the sampled signal x (kT) of x (t) is input to the FIR filter, output signal is exactly the n subdifferential signal X of x (t) ⁽ⁿ⁾(t) sampling X ⁽ⁿ⁾(kT).

The detailed block diagram of the FIR filter that realization is differentiated to input signal as shown in Figure 3.M is the exponent number of FIR filter among Fig. 3, and n is the exponent number of being differentiated, and m is more than or equal to n.The FIR filter comprises m delay cell and m+1 multiplication unit and an adder unit.The output signal process multiplication unit of input signal and delay cell multiply by the coefficient h (k) of corresponding filter respectively, k=0, and 1 ..., m.Each signal with multiplication unit output carries out sum operation through adder unit then, and the output signal of this adder unit is the n rank differential signal of input signal.

The exponent number m of filter is big more, and the differential precision of being asked is high more, and amount of calculation is big more simultaneously, and filter is complicated more.

The coefficient h of filter (k) changes with the change of the exponent number m of exponent number n that is differentiated and filter, and the acquiring method of the coefficient h of filter (k) is as follows:

Ask for the differential of a certain moment signal, constantly as initial point, signal can be expressed as a multinomial with this:

x(t)＝a ₀+a ₁t+a ₂t ²+a ₃t ³+a ₄t ⁴+...+a _nt ⁿ+a _n+1t ⁿ⁺¹+...

Then

$\frac{\mathrm{dx}\left(t\right)}{\mathrm{dt}}=x\&prime;\left(t\right)=a_1+2a_{2}t+3a_3{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="C200310118266E00103.png"\; state="\{\{state\}\}">t</figure-callout>}}^2+...+n{a}_n{t}^{n+1}+...$

$\frac{d^{2}x\left(t\right)}{{\mathrm{<figure-callout\; id="2"\; label="dt"\; filenames="C200310118266E00103.png"\; state="\{\{state\}\}">dt</figure-callout>}}^2}=x\&prime;\&prime;\left(t\right)=2a_2+3\&times;2a_{3}t+...+n\&times;(n-1)a_n{t}^{n-2}+...$

$\frac{d^{n}x\left(t\right)}{{\mathrm{dt}}^n}=x^{\left(n\right)}\left(t\right)=n!a_n+\frac{(n+1)!}{1}{a}_{n+1}t+......$

As seen t=0 n subdifferential constantly is:

$\frac{d^{n}x\left(0\right)}{{\mathrm{dt}}^n}=n!a_n$

That is to say, ask t=0 n subdifferential constantly only need obtain polynomial coefficient a _nGet final product.

The polynomial expression formula of simplifying signal is the m+1 item, and m is more than or equal to n,

x(t)＝a ₀+a ₁t+a ₂t ²+...+a _mt ^m

With near the value of constantly m+1 the sample point of t=0, the substitution following formula is formed m+1 item equation group, separates this equation group, just can obtain m+1 item coefficient.

With choose t=-kT ,-(k-1) T ..., 0 ... (k-1) T, kT m+1=2k+1 the sample value of putting constantly is example.Order

$X=\left[\begin{array}{c}x(-\mathrm{kT})\\ x(-(k-1)T)\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ x(-T)\\ x\left(0\right)\\ x\left(T\right)\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ x\left((k-1)T\right)\\ x\left(\mathrm{kT}\right)\end{array}\right]$

$A=\left[\begin{array}{c}{a}_0\\ {\mathrm{<figure-callout\; id="1"\; label="a"\; filenames="C200310118266E00101.png,C200310118266E00103.png"\; state="\{\{state\}\}">a</figure-callout>}}_1\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ a_k\\ \&CenterDot;\&CenterDot;\&CenterDot;\\ a_{2k}\end{array}\right]$

$H=\left[\begin{array}{c}1,{(-\mathrm{kT})}^1,{(-\mathrm{kT})}^2,...,{(-\mathrm{kT})}^{2\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="C200310118266E00101.png,C200310118266E00103.png"\; state="\{\{state\}\}">k</figure-callout>}}\\ 1,{(-(k-1)T)}^1,{(-(k-1)T)}^2,...,{(-(k-1)T)}^{2k}\\ \&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\\ 1,{(-T)}^1,{(-T)}^2,...,{(-T)}^{2\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="C200310118266E00101.png,C200310118266E00103.png"\; state="\{\{state\}\}">k</figure-callout>}}\\ \mathrm{1,0,0},...,0\\ 1,T,{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="C200310118266E00103.png"\; state="\{\{state\}\}">T</figure-callout>}}^2,...,T^{2k}\\ \&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\\ 1,\left(\mathrm{kT}\right),{\left(\mathrm{kT}\right)}^2,....{\left(\mathrm{kT}\right)}^{2k}\end{array}\right]$

Wherein T is sampling period, then

X＝HA

Solve an equation and obtain:

A＝H ^-1X

Make H ^-1The n+1 row coefficient be H ^-1(n), n is smaller or equal to 2k, and promptly n is smaller or equal to m, then

a _n＝H ^-1(n)X

Be that t=0 n subdifferential constantly is

$\frac{d^{n}y\left(0\right)}{{\mathrm{dt}}^n}=n!H^{-1}\left(n\right)X$

As seen with n! H ^-1(n) a 2k+1 factor is as the coefficient of FIR filter, constitutes filter, and signal x (k) is carried out filtering, and the output of filter is exactly the signal after the n subdifferential of signal x (k).

The above; only for the preferable embodiment of the present invention, but protection scope of the present invention is not limited thereto, and anyly is familiar with those skilled in the art in the technical scope that the present invention discloses; the variation that can expect easily or replacement all should be encompassed within protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection range of claims.

Claims (4)

1, a kind of filter of accurate realization signal differentiation comprises m delay cell, a m+1 multiplication unit and an adder unit, and wherein m is the exponent number of filter, and m is more than or equal to the exponent number n that is differentiated:

The sampled signal that will obtain after will sampling to input signal is imported the 1st multiplication unit and the 1st delay cell, and other each delay cell except that the 1st delay cell receives the signal of previous delay cell output respectively;

After handling the signal that receives, described m delay units delay output signal to the 2nd respectively to m+1 multiplication unit;

A described m+1 multiplication unit will output signal to described adder unit behind the coefficient of signal times with the filter of correspondence that receive respectively;

Described adder unit carries out addition to the signal that receives and handles the differential signal that obtains described input signal;

Wherein, the coefficient of the filter of described m+1 multiplication unit correspondence be respectively n! H ^-1(n) a m+1 factor, H ^-1(n) be matrix H ^-1The capable coefficient of n+1, described matrix H is by 0,1,2 of the m+1 that input signal is sampled pairing each sampling time of sample point, 3... form to m power, pairing each sampling time of described each sample point is the integral multiple of continuous sampling period, and in the described matrix H, and first is 1 in the capable capable vector of p, second is 1 of p the pairing sampling time of sample point to the m+1 item, 2,3... is to m power, wherein p=1,2......m+1; The 1st classifies the column vector that m+1 1 forms as, q classify as each sample point q-1 power of each self-corresponding sampling time, q=2,3......m+1.

2, a kind of method of using the described filter of claim 1 accurately to realize signal differentiation is characterized in that, comprises the following steps:

1) input signal is sampled;

2) sampled signal is input to the 1st multiplication unit and the 1st delay cell of described filter, other each delay cell except that the 1st delay cell receives the signal of previous delay cell output respectively, after handling the signal that receives, described m delay units delay output signal to the 2nd respectively to m+1 multiplication unit, a described m+1 multiplication unit will output signal to described adder unit behind the coefficient of signal times with the filter of correspondence that receive respectively, and described adder unit carries out addition to the signal that receives and handles the differential signal that obtains described input signal; Wherein, the coefficient of the filter of m+1 multiplication unit correspondence be respectively n! H ^-1(n) a m+1 factor, H ^-1(n) be matrix H ^-1The capable coefficient of n+1, described matrix H is 0,1,2 of m+1 pairing each sampling time of sample point that input signal is sampled, 3... the determinant of forming to m power, pairing each sampling time of described each sample point is the integral multiple of continuous sampling period, and in the described matrix H, and first is 1 in the capable capable vector of p, second is 1 of p the pairing sampling time of sample point to the m+1 item, 2,3... is to m power, wherein p=1,2......m+1; The 1st classifies the column vector that m+1 1 forms as, q classify as each sample point q-1 power of each self-corresponding sampling time, q=2,3......m+1.

3) read the signal that described filter is exported.

3, the method for accurate realization signal differentiation as claimed in claim 2 is characterized in that, described each sample point is chosen for the center symmetry with t=0.

4, the method for accurate realization signal differentiation as claimed in claim 3 is characterized in that, the sampling frequency of described each sample point is at least the twice of input signal highest frequency.

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