CN102591204A - Realization method of Gauss system - Google Patents

Abstract

The invention discloses a realization method of a Gauss system, belonging to the measurement technology. Finite terms of the expansion of a Gauss prototype function are intercepted as an approximant of a Gauss function; a finite number of first-order systems with the same characteristics are cascaded; the system characteristics approximate the Gauss characteristics; when the cascade series of the first-order systems is gradually increased, the error of first-order system characteristics is gradually decreased compared with that of Gauss system characteristics; along with the increase of the cascade series, the approximation precision to the Gauss system is improved; the method is also suitable for a second-order system; with the instruction of the method, a Gauss system can be realized through the cascade approximation of the first-order systems and second-order systems with the same characteristics; and compared with the traditional system, the Gibbs phenomenon generated by the cut-off frequency is eliminated.

Description

The implementation method of a kind of Gauss system

Technical field

The invention belongs to the measuring system technical field, relate generally to the implementation method of a kind of Gauss system.

Background technology

In order to realize the undistorted transmission of signal, require system to have infinitely-great bandwidth and linear phase characteristic in theory, promptly desirable instrument should satisfy two conditions: 1) amplitude versus frequency characte is the constant with frequency-independent, 2) phase-frequency characteristic and frequency are linear.In fact, theoretic desirable instrument is impossible realize physically.Generally speaking; The bandwidth of actual signal all is limited; Even the spectrum distribution of signal can be extended to infinitely great frequency place, but owing to the contained energy of each frequency component along with increasing of frequency reduces, can ignore by the distortion that higher frequency components causes.System is transmitted undistortedly for assurance, if system can satisfy 1) amplitude versus frequency characte is a constant in certain frequency range, 2) phase-frequency characteristic and frequency are linear in this frequency range, and then this system also can be used as the instrument with undistorted characteristic.

For Design of Measurement System, people have carried out sufficient research to traditional desirable lowpass system model with rectangle amplitude-frequency response etc., and its time domain specification, frequency domain characteristic etc. had deep understanding.These ideal system models are non-causal systems; Physically can not realize; But available method of approaching is similar to realization; The system that is realized has produced Gibbs' effect because the decay at cut-off frequency place sharply changes, and promptly has overshoot and oscillatory occurences at the cutoff frequency place, and this is disadvantageous for the stable of measuring system.Desirable Gauss's system model has level and smooth step response characteristic, and the time frequency range long-pending minimum, therefore, can be used as the target of optimization system design.

Summary of the invention

The objective of the invention is in order to solve above-mentioned existing problem; On the basis of further investigation measuring system theory; With system with Gaussian response characteristic target as the optimization system design; Propose the method that a kind of Gauss system realizes, the system that is realized does not exist because the decay at cut-off frequency place sharply changes and has produced Gibbs' effect.

The objective of the invention is to realize like this:

The implementation method of a kind of Gauss system, exponential function e ^xExpansion

$e^x=\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+...,x\∈(-\∞,+\∞),$

Have in view of the above

$e^{x^2}=\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{{\left(x^2\right)}^n}{n!}=1+x^2+\frac{{\left(x^2\right)}^2}{2!}+...+\frac{{\left(x^2\right)}^n}{n!}+...,x\∈(-\∞,+\∞),$

The different power items of the above-mentioned function of intercepting

expansion are as Gaussian function approximating function formula; There is following relational expression to set up for object function

(1) for first-order system

$\underset{n\→\∞}{\lim }{\left[\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2}\right]}^n=e^{-{(\mathrm{\α\Ω}/{\mathrm{\Ω}}_c)}^2},$

Employing has cascade such as sensor, hardware filtering device, software filter and amplifying circuit, the detecting circuit of first-order system characteristic and approaches the Gauss system that realizes;

(2) for second-order system

$\underset{n\→\∞}{\lim }{\left[\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2+\frac{{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^4}{2}}\right]}^n=e^{-{(\mathrm{\α\Ω}/{\mathrm{\Ω}}_c)}^2},$

Employing has cascade such as sensor, hardware filtering device, software filter and amplifying circuit, the detecting circuit of second-order system characteristic and approaches the Gauss system that realizes.

The invention has the advantages that: the implementation method of Ben Gaosi system is compared with traditional system, has eliminated because the Gibbs phenomenon that cut-off frequency produces.

Description of drawings

Fig. 1 is the amplitude versus frequency characte of n level first-order system cascade system and Gauss system

Fig. 2 is the amplitude versus frequency characte deviation curve of n level first-order system cascade system and Gauss system

Fig. 3 is the system responses characteristic of first-order system prototype

Fig. 4 is 8 grades of first-order system cascade system response characteristics

Fig. 5 is the amplitude versus frequency characte of n level second-order system cascade system and Gauss system

Fig. 6 is the amplitude versus frequency characte deviation curve of n level second-order system cascade system and Gauss system

Fig. 7 is the system responses characteristic of second-order system prototype

Fig. 8 is 4 grades of second-order system cascade system response characteristics

Fig. 9 is the sensor signal processing system schematic diagram

Figure 10 is a joint RC active filter circuit

Figure 11 is-the 2RC filtering circuit

Figure 12 realizes the schematic diagram of Gauss system for cascade

Embodiment:

For geometric measurement; When the inductance type micro-displacement sensor that uses is first-order system or second-order system; For such sensing system, when forming measuring system according to the basic physical model of geometric sense multimetering instrument, the system how to realize ideal with Gaussian response characteristic; Seem particularly important, carry out detailed analysis and study in the face of this problem down.

According to e ^xExpansion

$e^x=\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+...,x\∈(-\∞,+\∞),$

Have

$e^{x^2}=\underset{n=0}{\overset{\∞}{\mathrm{\Σ}}}\frac{{\left(x^2\right)}^n}{n!}=1+x^2+\frac{{\left(x^2\right)}^2}{2!}+...+\frac{{\left(x^2\right)}^n}{n!}+...,x\∈(-\∞,+\∞),$

The different power items of the above-mentioned function expansion of intercepting are as Gaussian function approximating function formula; According to central-limit theorem; For object function:

should have following relational expression to set up

For first-order system

$\underset{n\→\∞}{\lim }{\left[\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2}\right]}^n=e^{-{(\mathrm{\α\Ω}/{\mathrm{\Ω}}_c)}^2}---\left(1\right)$

The cascades such as sensor, amplifying circuit, detecting circuit, hardware filtering device and software filter that employing has the first-order system characteristic approach the Gauss system that realizes;

For second-order system

$\underset{n\→\∞}{\lim }{\left[\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2+\frac{{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^4}{2}}\right]}^n=e^{-{(\mathrm{\α\Ω}/{\mathrm{\Ω}}_c)}^2}---\left(2\right)$

The cascades such as sensor, amplifying circuit, detecting circuit, hardware filtering device and software filter that employing has the second-order system characteristic approach the Gauss system that realizes.

Instance 1 first-order system cascade approaches the Gauss system that realizes

1.1 the first-order system cascade approaches the central limit theorem of Gauss system

By aforementioned analysis, suppose that used sensor is a first-order system, and hope that the characteristic of other part all is a single order, when a lot of identical first-order system cascades, square can be described as of the amplitude versus frequency characte function of its system

${\left|H(\mathrm{\Ω}/{\mathrm{\Ω}}_c)\right|}_n={\left(\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2}\right)}^n$

So, can use the first-order system cascade to realize the Gauss system, will solve the represented limiting equation of formula (1)

$\underset{n\→\∞}{\lim }{\left[\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2}\right]}^n=e^{-{(\mathrm{\α\Ω}/{\mathrm{\Ω}}_c)}^2},$

Whether set up, this is the key that realizes the Gauss system.Proof procedure is following.

Proof: order

$G(\mathrm{\Ω}/{\mathrm{\Ω}}_c)={\left(\frac{1}{1+{({\mathrm{\β}}_n\mathrm{\Ω}/{\mathrm{\Ω}}_c)}^2}\right)}^n$

At cutoff frequency Ω=Ω _cThe damping capacity at place is G (Ω/Ω _cDuring)=p, have

${\left(\frac{1}{1+{\left({\mathrm{\β}}_n\right)}^2}\right)}^n=p$

$1+{\left({\mathrm{\β}}_n\right)}^2=p^{-\frac{1}{n}}$

${\mathrm{\β}}_n=\sqrt{p^{-\frac{1}{n}}-1}---\left(3\right)$

If

q＝(β _nΩ/Ω _c) ²

Then

$G(\mathrm{\Ω}/{\mathrm{\Ω}}_c)={\left(\frac{1}{1+q}\right)}^n$

If

${\mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">n</figure-callout>}}^2=p^{-\frac{1}{n}}-1=\frac{1}{t}$

Then

$q=\frac{1}{t}\&CenterDot;{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2$

$n=\frac{\ln \frac{1}{p}}{\ln (1+\frac{1}{t})}---\left(4\right)$

When n →+∞, promptly t →+during ∞,

$\underset{t\&RightArrow;+\&infin;}{\lim }G(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)=\underset{t\&RightArrow;\&infin;}{\lim }{\left(\frac{1}{1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}\right)}^{\frac{\ln \frac{1}{p}}{\ln (1+\frac{1}{t})}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{-\frac{t}{t}\&CenterDot;\frac{\ln \frac{1}{p}}{\ln (1+\frac{1}{t})}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{-t\&CenterDot;\frac{\ln \frac{1}{p}}{t\&CenterDot;\ln (1+\frac{1}{t})}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{-t\frac{\ln \frac{1}{p}}{\ln {(1+\frac{1}{t})}^t}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{\frac{1}{\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}\&CenterDot;\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2\&CenterDot;(-t)\&CenterDot;\frac{\ln \frac{1}{p}}{\ln {(1+\frac{1}{t})}^t}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{\frac{1}{\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}\&CenterDot;{(-1)\&CenterDot;(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2\&CenterDot;\frac{\ln \frac{1}{p}}{\ln {(1+\frac{1}{t})}^t}}---\left(5\right)$

Turn to according to the limit

formula (5):

$\underset{t\&RightArrow;+\&infin;}{\lim }G(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)=e^{-\ln \frac{1}{p}\&CenterDot;{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}=e^{{-(\mathrm{\&alpha;\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}$

In the formula--- $\mathrm{\&alpha;}=\sqrt{\mathrm{Ln}\frac{1}{p}}$

The first-order system cascade approaches the central limit theorem of Gauss system and must demonstrate,prove.Formula (1) shows the cascade of infinite a plurality of first-order systems, and its system performance is just ad infinitum approached gaussian characteristics.Also promptly, limited cascade with first-order system of identical characteristics, its system performance is that of gaussian characteristics is approximate.

According to With

When making p=0.707, get α=0.5887, the cascade progression n=1 for first-order system, has β at 2,4,8,16 o'clock ₁=0.6436, β ₂=0.435, β ₄=0.3008, β ₈=0.2104, β ₁₆=0.148, make the cascaded functions of first-order system

Gauss's system function Amplitude-frequency response, as shown in Figure 1.

The amplitude error of first-order system cascade systems at different levels and Gauss system amplitude-frequency response is as shown in Figure 2, and its amplitude peak deviation is listed in table 1.

The amplitude peak deviation that the cascade of table 1n level first-order system approaches the Gauss system

Can know that by Fig. 2 and table 1 when the cascade progression of first-order system increased gradually, its system performance and Gauss's system performance ratio error were mutually reducing gradually, this means the increase along with cascade progression, with the approximation accuracy that improves the Gauss system.

1.2 single order simulation system prototype and system responses characteristic thereof

When getting 8 grades of first-order system cascades,

$G_8\left(\mathrm{\&Omega;}\right)=\frac{1}{1+{({\mathrm{\&beta;}}_8\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}$

By the amplitude square principle of design:

G ₈(Ω)＝|H _a8(jΩ)| ²＝H _a8(jΩ)H _a8(-jΩ)

With s=j Ω substitution following formula, then have:

$G_8\left(s\right)=H_{a8}\left(s\right)H_{a8}(-s)=\frac{1}{1-({\mathrm{\&beta;}}_8^2/{\mathrm{\&Omega;}}_c^2){\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}$

Its limit is:

Choose the limit s of the left half-plane on s plane ₂As the limit of simulation system function H (s), establishing gain constant is K ₀, obtain

$H_{a8}\left(s\right)=\frac{K_0}{s-s_2}---\left(6\right)$

By H _A8(s) | _S=0=H (j Ω) | Ω=0 solves

H _A8(s) be design Gauss and approach the H of system _{β 8}(s) single order simulation system prototype.

Single order simulation system prototype H _A8(s) unit impulse response

$h_8\left(t\right)=L^{-1}\left[\frac{K_0}{s-s_2}\right]$

$=K_0{e}^{s_{2}t}---\left(7\right)$

Unit-step response

$y\left(t\right)=L^{-1}\left[\frac{1}{s}\frac{K_0}{s-s_2}\right]$ (8)

$=-\frac{s_2}{K_0}(1-e^{s_{2}t})$

Make the unit impact response and the unit-step response curve of single order simulation system according to formula (7) and formula (8), as shown in Figure 3.

1.3 single order simulation system prototype cascade and system responses characteristic thereof

8 grades of single orders are simulated when system-level, the unit impulse response of system

$h_8\left(t\right)=L^{-1}\left[{\left(\frac{K_0}{s-s_2}\right)}^8\right]$

$=\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^8}{7!}{t}^7{e}^{s_2{s}_{2}t}---\left(9\right)$

Unit-step response

$y\left(t\right)={\&Integral;}_0^t{h}_8\left(x\right)\mathrm{dx}$

$=\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^8}{7!}{\&Integral;}_0^t{x}^7{e}^{s_{2}x}\mathrm{dx}$

Then

$y\left(t\right)=\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^8}{7!}({{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^7{t}^7{e}^{s_{2}t}-7{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^6{t}^6{e}^{s_{2}t}+42{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^5{t}^5{e}^{s_{2}t}-210{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^4{t}^4{e}^{s_{2}t}+$ (10)

$840{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^3{t}^3{e}^{s_{2}t}-2520{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^2{t}^2{e}^{s_{2}t}+5040{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2{\mathrm{te}}^{s_{2}t}-5040e^{s_{2}t}+5040)/{{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2}^8$

According to formula (9) and formula (10), make 8 grades of first-order system cascaded system unit impact response curves and unit-step response curve respectively, as shown in Figure 4.

Can know that by Fig. 4 the unit impulse response of the cascade system of 8 grades of first-order systems and unit-step response family curve very approach the resonse characteristic of Gauss system.

Instance 2 second-order system cascades approach the Gauss system that realizes

2.1 the second-order system cascade approaches the central limit theorem of Gauss system

When supposing that used sensor is second-order system, if will use a plurality of second-order systems to realize the Gauss system, then demand is demonstrate,proved the limit

$\underset{n\&RightArrow;\&infin;}{\lim }{\left[\frac{1}{1+{(\mathrm{\&beta;\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{{(\mathrm{\&beta;\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}{2}}\right]}^n=e^{-{(\mathrm{\&alpha;\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}$

Set up.Issued a certificate below.

Proof: order

$G(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)={\left(\frac{1}{1+{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{1}{2}{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}\right)}^n$

Make cutoff frequency Ω=Ω _cDamping capacity G (Ω/the Ω at place _c)=p when (0＜p＜1), then has

${(1+{\left({\mathrm{\&beta;}}_n\right)}^2+\frac{1}{2}{\left({\mathrm{\&beta;}}_n\right)}^4)}^{-n}=p$

$1+{\left({\mathrm{\&beta;}}_n\right)}^2+\frac{1}{2}{\left({\mathrm{\&beta;}}_n\right)}^4=p^{-\frac{1}{n}}$

${\mathrm{\&beta;}}_n=\sqrt{\sqrt{2\&times;p^{-\frac{1}{n}}-1}-1}---\left(11\right)$

If

q＝(β _nΩ/Ω _c) ²

Then

$G(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)={\left(\frac{1}{1+q+{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">q</figure-callout>}}^2/2}\right)}^n$

If ${\mathrm{\&beta;}}_{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">n</figure-callout>}}^2=\sqrt{2\&times;p^{-\frac{1}{n}}-1}-1=\sqrt{2\&times;p^{-\frac{1}{n}}-1}-1=\frac{1}{t}$

Then

$q=\frac{1}{t}\&CenterDot;{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2$

$n=\frac{\ln \frac{1}{p}}{\ln \left[\frac{{(1+\frac{1}{t})}^2+1}{2}\right]}---\left(12\right)$

When n →+∞, promptly t →+during ∞,

$\underset{t\&RightArrow;+\&infin;}{\lim }G\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)=\underset{t\&RightArrow;\&infin;}{\lim }{\left(\frac{1}{1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4}\right)}^{\frac{\ln \frac{1}{p}}{\ln \left[\frac{{(\frac{1}{t}+1)}^2+1}{2}\right]}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4)}^{-\frac{t}{t}\&CenterDot;\frac{\ln \frac{1}{p}}{\ln \left[\frac{{(\frac{1}{t}+1)}^2+1}{2}\right]}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4)}^{-t\&CenterDot;\frac{\ln \frac{1}{p}}{t\&CenterDot;\ln \left[\frac{{(\frac{1}{t}+1)}^2+1}{2}\right]}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4)}^{-t\&CenterDot;\frac{\ln \frac{1}{p}}{\ln {\left[\frac{{(\frac{1}{t}+1)}^2+1}{2}\right]}^t}}---\left(13\right)$

Can derive obtains by

:

$\underset{t\&RightArrow;+\&infin;}{\lim }\ln {\left[\frac{{(1+\frac{1}{t})}^2+1}{2}\right]}^t=\underset{t\&RightArrow;\&infin;}{\lim }\ln {(1+\frac{1+2t}{2t^2})}^{\frac{{2\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}{1+2t}\&CenterDot;\frac{1+2t}{2t}}$

$=\underset{t\&RightArrow;+\&infin;}{\lim }\ln \left({\mathrm{<figure-callout\; id="1"\; label="e"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">e</figure-callout>}}^{\frac{1+2t}{2t}}\right)$

$=\underset{t\&RightArrow;+\&infin;}{\lim }\frac{1+2t}{2t}$

$=\underset{t\&RightArrow;+\&infin;}{\lim }1+\frac{1}{2t}$

$=1$

Therefore, formula (13) turns to

$\underset{t\&RightArrow;+\&infin;}{\lim }G(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{1}{2t^2}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4)}^{-t\&CenterDot;\ln \frac{1}{p}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{\left(\right[1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2]\&CenterDot;[1+\frac{\frac{1}{2t^2}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}{1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}\left]\right)}^{-t\&CenterDot;\ln \frac{1}{p}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2)}^{-t\&CenterDot;\ln \frac{1}{p}}\&CenterDot;\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{\frac{1}{2t^2}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}{1+\frac{1}{t}{(\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2})}^{-t\&CenterDot;\ln \frac{1}{p}}---\left(14\right)$

For the subitem in the formula (14)

$\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4}{1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2})}^{-t\&CenterDot;\ln \frac{1}{p}}$

Have

$1+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4>1+\frac{\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4}{1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2}>1,t>0$

And

$\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4)}^{-t\&CenterDot;\frac{\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2}}=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4)}^{2{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2\&CenterDot;{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^{-4}\&CenterDot;\frac{1}{2{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\&CenterDot;{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4\&CenterDot;(-t\&CenterDot;\frac{\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{2})}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{e}^{-\frac{\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{4t}\&CenterDot;{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4}$

$=1$

So $\underset{t\&RightArrow;\&infin;}{\mathrm{Lim}}{(1+\frac{\frac{1}{2t^2}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^4}{1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2})}^{-t\&CenterDot;\frac{\mathrm{<figure-callout\; id="2"\; label="Ln"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">Ln</figure-callout>}2}{2}}=1$

Therefore, formula (14) turns to

$\underset{t\&RightArrow;+\&infin;}{\lim }G\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2)}^{-t\&CenterDot;\ln \frac{1}{p}}$

$=\underset{t\&RightArrow;\&infin;}{\lim }{(1+\frac{1}{t}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2)}^{-t\&CenterDot;{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^{-2}\&CenterDot;{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2\&CenterDot;\ln \frac{1}{p}}$

$=e^{-\ln \frac{1}{p}{\left(\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2}$

$=e^{-{\left(\mathrm{\&alpha;}\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2}$

In the formula--- $\mathrm{\&alpha;}=\sqrt{\mathrm{Ln}\frac{1}{p}}$

To sum up $\underset{n\&RightArrow;\&infin;}{\mathrm{Lim}}{\left(\frac{1}{1+{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{1}{2}{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}\right)}^n=e^{-{\left(\mathrm{\&alpha;}\frac{\mathrm{\&Omega;}}{{\mathrm{\&Omega;}}_c}\right)}^2}$

The second-order system cascade approaches the central limit theorem of Gauss system and must demonstrate,prove.Formula (2) shows the cascade of infinite a plurality of second-order systems, and its system performance is just ad infinitum approached gaussian characteristics.Also promptly, limited cascade with second-order system of identical characteristics, its system performance is gaussian characteristics-individual being similar to.

According to

With

When making p=0.707, α=0.5887 is arranged then, work as n=1,2,4,8,16 o'clock, β ₁=0.5935, β ₂=0.4172, β ₄=0.2945, β ₈=0.2082, β ₁₆=0.1472, it is more as shown in Figure 5 to map.

The amplitude error of second-order system cascade systems at different levels and Gauss system amplitude-frequency response is as shown in Figure 6, and its amplitude peak deviation is listed in table 2.

The amplitude peak deviation that the cascade of table 2n level second-order system approaches the Gauss system

Can know that by Fig. 6 when the cascade progression of second-order system increased gradually, its characteristic and Gauss's system performance ratio error were mutually reducing gradually, this means the increase along with cascade progression, with the approximation accuracy that improves the Gauss system; And the cascade of second-order system is faster than the cascade velocity of approch of first-order system.

2.2 second order simulation system prototype and system responses characteristic thereof

When getting 4 grades of second-order system cascades,

$G_4\left(\mathrm{\&Omega;}\right)=\frac{1}{1+{({\mathrm{\&beta;}}_4\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{1}{2}{({\mathrm{\&beta;}}_4\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}$

By the amplitude square principle of design ^[89,90]:

G ₄(Ω)＝|H _a4(jΩ)| ²

＝H _a4(jΩ)H _a4(-jΩ)

With s=j Ω substitution following formula, then have:

$G_4\left(s\right)=H_{a4}\left(s\right)H_{a4}(-s)$

$=\frac{1}{1-({\mathrm{\&beta;}}_4^2/{\mathrm{\&Omega;}}_c^2){\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}^2+({\mathrm{\&beta;}}_4^4/2{\mathrm{\&Omega;}}_c^4){\mathrm{<figure-callout\; id="4"\; label="s"\; filenames="HSA00000675174500021.png,HSA00000675174500022.png"\; state="\{\{state\}\}">s</figure-callout>}}^4}$

Its limit is: $s_1=\frac{{\mathrm{\&Omega;}}_c}{{\mathrm{\&beta;}}_4}(1.0987+j0.4551),$ ${\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_2=\frac{{\mathrm{\&Omega;}}_c}{{\mathrm{\&beta;}}_4}(-1.0987+j0.4551),$ ${\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_3=\frac{{\mathrm{\&Omega;}}_c}{{\mathrm{\&beta;}}_4}(-1.0987-j0.4551),$ ${\mathrm{<figure-callout\; id="4"\; label="s"\; filenames="HSA00000675174500021.png,HSA00000675174500022.png"\; state="\{\{state\}\}">s</figure-callout>}}_4=\frac{{\mathrm{\&Omega;}}_c}{{\mathrm{\&beta;}}_4}(1.0987-j0.4551).$ Choose the limit s of the left half-plane on S plane ₂, s ₃As the limit of simulation system function H (s), establishing gain constant is K ₀, obtain

$H_{a4}\left(s\right)=\frac{K_0}{(s-s_2)(s-s_3)}---\left(15\right)$

By H _A4(s) | _S=0=H (j Ω) | _Ω=0Condition obtain

H _A4Be design Gauss and approach the H of system _{β 4}Second order simulation system prototype.

The unit impulse response of second order simulation system prototype

X(s)＝1

$\frac{Y\left(s\right)}{X\left(s\right)}=\frac{K_0}{(s-s_2)(s-s_3)}$

$Y\left(s\right)=\frac{K_0}{s_2-s_3}[\frac{1}{s-s_2}-\frac{1}{s-s_3}]$

$h\left(t\right)=\frac{K_0}{s_2-s_3}(e^{s_{2}t}-e^{s_{3}t})---\left(16\right)$

Unit-step response

$X\left(s\right)=\frac{1}{s}$

$\frac{Y\left(s\right)}{X\left(s\right)}=\frac{K_0}{(s-s_2)(s-s_3)}$

$Y\left(s\right)=\frac{1}{s}\&CenterDot;\frac{K_0}{(s-s_2)(s-s_3)}$

$=\frac{K_0}{s_2{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_3}\frac{1}{s}+\frac{K_0}{s_2(s_2-s_3)}\frac{1}{s-s_2}-\frac{K_0}{s_3(s_2-s_3)}\frac{1}{s-{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_3}$

Then

$y\left(t\right)=\frac{K_0}{s_2{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_3}+\frac{K_0}{s_2(s_2-s_3)}{e}^{s_{2}t}-\frac{K_0}{s_3(s_2-s_3)}{e}^{s_{3}t}---\left(17\right)$

According to formula (16) and formula (17), it is as shown in Figure 7 to map.

2.3 second order simulation system prototype cascade and system responses characteristic thereof

The cascade of 4 grades of second order simulation system prototypes is

its system unit impulse response

$\frac{Y\left(s\right)}{X\left(s\right)}={\left[\frac{K_0}{(s-s_2)(s-s_3)}\right]}^4$

$Y\left(s\right)=\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^4}{{(s-s_2)}^4{(s-s_3)}^4}$

Then

$h\left(t\right)=L^{-1}\left[\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^4}{{(s-s_2)}^4{(s-s_3)}^4}\right]$

$={{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^4\frac{t^3{e}^{s_2\mathrm{<figure-callout\; id="3"\; label="t"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}}{3!}*\frac{t^3{e}^{s_3\mathrm{<figure-callout\; id="3"\; label="t"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}}{3!}$

$=\frac{{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">K</figure-callout>}}_0}^4}{81}{t}^3{e}^{s_{2}t}(6-6e^{s_{3}t}+e^{s_3\mathrm{<figure-callout\; id="3"\; label="t\; s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}{s_{}}^{}{{}_3}^3{t}^3-3e^{s_3\mathrm{<figure-callout\; id="3"\; label="t\; s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">t</figure-callout>}}{s_{}}^{}{{}_3}^2{t}^2-6e^{s_{3}t}{s}_{3}t)/{{\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">s</figure-callout>}}_3}^4---\left(18\right)$

Unit-step response

$y\left(t\right)={\&Integral;}_0^{t}h\left(x\right)\mathrm{dx}---\left(19\right)$

According to formula (18) and formula (19), it is as shown in Figure 8 to map.

Can know that by Fig. 8 the cascaded system unit impulse response of 4 grades of second-order systems and unit-step response family curve very approach the resonse characteristic of Gauss system.

The physics realization of 3 Gauss systems

For the electronic system of geometric sense multimetering system, sensor signal need be passed through processing such as amplification, detection, filtering, sampling usually, and what influence the system responses characteristic mainly is the realization of wave filter.Typical sensor signal processing system schematic diagram is as shown in Figure 9.

For first-order system, approach the Gauss system that realizes with cascades such as sensor, hardware filtering device, software filter and amplifying circuit with first-order system characteristic, detecting circuits; For second-order system, approach the Gauss system that realizes with cascades such as sensor, hardware filtering device, software filter and amplifying circuit with second-order system characteristic, detecting circuits.

Realize that with first-order system the Gauss system is an example, on the basis that the principle that the cascade of Gauss system is realized is studied, consider to influence the key component of system performance with the realization of RC wave filter.The RC wave filter is a low-pass filter commonly used in the instrument, a joint RC active filter ^[92]Shown in figure 10.

For a joint RC filtering circuit, its ssystem transfer function

$\mathrm{<figure-callout\; id="1"\; label="H\; f"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">H</figure-callout>}{f}_{}{}_1=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}=\frac{\mathrm{jRC\&omega;}}{1+\mathrm{jRC\&omega;}}$

Wherein,

R and C represent resistance and electric capacity respectively.

The amplitude-frequency function square do

${\left|\mathrm{<figure-callout\; id="1"\; label="H\; f"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">H</figure-callout>}{f}_{}{}_1\right|}^2=\frac{1}{1+{({\mathrm{\&omega;}}_c/\mathrm{\&omega;})}^2}$

2RC filter circuit construction form is shown in figure 11.

For the 2RC filtering circuit, its ssystem transfer function

${\mathrm{<figure-callout\; id="2"\; label="Hf"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">Hf</figure-callout>}}_2=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}={\mathrm{<figure-callout\; id="1"\; label="Hf"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">Hf</figure-callout>}}_1^2={\left(\frac{\mathrm{jRC\&omega;}}{1+\mathrm{jRC\&omega;}}\right)}^2$

Wherein,

R and C represent resistance and electric capacity respectively.

The amplitude-frequency function square do

${\left|{\mathrm{<figure-callout\; id="2"\; label="Hf"\; filenames="HSA00000675174500011.png,HSA00000675174500012.png"\; state="\{\{state\}\}">Hf</figure-callout>}}_2\right|}^2=\frac{1}{1+{({\mathrm{\&omega;}}_c/\mathrm{\&omega;})}^2}$

For n joint 2RC filtering circuit, have

${\mathrm{Hf}}_n=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}={\mathrm{Hf}}_1^n={\left(\frac{\mathrm{jRC\&omega;}}{1+\mathrm{JRC\&omega;}}\right)}^n$

The amplitude-frequency function square do

${\left|H{f}_n\right|}^2={\left[\frac{1}{1+{({\mathrm{\&omega;}}_c/\mathrm{\&omega;})}^2}\right]}^n$

According to formula (1), the n level cascade of then available 2RC wave filter approaches the realization Gaussian filter.Gaussian filter can approach realization through the cascade of multistage Butterworth filter, so just changes into our the very design of familiar Butterworth filter to the design of the Gaussian filter of complicacy ^[90]Then, the cascade of sensing system, amplification, detection, filtering system according to formula (1), can realize having the system of gaussian characteristics, and it is simpler to use the system architecture that this method obtains, and is prone to realize, and is shown in figure 12.

According to above-mentioned analysis, in the practical application, can obtain satisfied effect with 4 joint 2RC wave filter cascades.The application cascade approaches the sensor and the electronic system that realize the Gauss system and in many cover multimetering instruments, is used widely, and is the basis of supporting geometric sense multimetering technical application.

Claims (1)

1. the implementation method of a Gauss system is characterized in that: exponential function e ^xExpansion

$e^x=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}+...,x\&Element;(-\&infin;,+\&infin;),$

Have in view of the above

$e^{x^2}=\underset{n=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\left(x^2\right)}^n}{n!}=1+x^2+\frac{{\left(x^2\right)}^2}{2!}+...+\frac{{\left(x^2\right)}^n}{n!}+...,x\&Element;(-\&infin;,+\&infin;),$

The different power items of the above-mentioned function of intercepting

expansion are as Gaussian function

approximating function formula; There is following relational expression to set up for object function

(1) for first-order system

$\underset{n\&RightArrow;\&infin;}{\lim }{\left[\frac{1}{1+{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2}\right]}^n=e^{-{(\mathrm{\&alpha;\&Omega;}/{\mathrm{\&Omega;}}_c)}^2},$

The cascades such as sensor, amplifying circuit, detecting circuit, hardware filtering device and software filter that employing has the first-order system characteristic approach the Gauss system that realizes;

(2) for second-order system

$\underset{n\&RightArrow;\&infin;}{\lim }{\left[\frac{1}{1+{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^2+\frac{{({\mathrm{\&beta;}}_n\mathrm{\&Omega;}/{\mathrm{\&Omega;}}_c)}^4}{2}}\right]}^n=e^{-{(\mathrm{\&alpha;\&Omega;}/{\mathrm{\&Omega;}}_c)}^2},$

The cascades such as sensor, amplifying circuit, detecting circuit, hardware filtering device and software filter that employing has the second-order system characteristic approach the Gauss system that realizes.

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