# Filtering Method Based on Symmetrical Second Order Systems

## Abstract

**:**

## 1. Introduction

_{0}s + 1)

_{0}= 1/ T

_{0}of about ω/ω

_{0}times. A filtering device represented by a low-pass filter of second order with the transfer function

_{0}

^{2}s

^{2}+ 2ξT

_{0}s + 1)

_{0})

^{2}times for an alternating signal with angular frequency ω > ω

_{0}= 1/T

_{0}.

_{0}

^{2}s

^{2}+ 1)

_{0})

^{2}times (where ω

_{0}= 1/T

_{0}). By integrating these undamped symmetrical oscillations on the time interval (0, 2πT

_{0}), a supplementary attenuation of about (2πω/ω

_{0}) is obtained. Moreover, in case of a unity-step input the quantity to be integrated equals zero at the end of the integration period, (see [1] for details).

_{0})

^{3}for an alternating input of angular frequency ω on the time interval 2πT

_{0}= 2π/ω

_{0}.

## 2. Materials and Methods

_{0})

_{0}, the value of y(t) being two (double as related to the amplitude of the step input). Therefore, a robust sampling procedure could act at this time moment.

_{0})

^{2}lower than the amplitude of the alternating input (according to partial fraction decomposition applied to generated output within Laplace transform theory).

_{0}is also generated (corresponding to proper oscillation of the undamped second order system), we must take care that this alternating component should be added at the sampling time moment to the attenuated output of angular frequency ω. In fact, two alternating terms sin(ω

_{0}t) and cos(ω

_{0}t) of angular frequency ω

_{0}are generated, but sin(ω

_{0}t) equals zero for ω

_{0}t = π (the suggested sampling moment of time). For null initial conditions, its amplitude corresponds to the opposite value of the oscillating output of angular frequency ω at zero moment of time. Since this has been attenuated (ω/ω

_{0})

^{2}times (as shown before), the sum of these two alternating components (the attenuated component of angular frequency ω and the cosine term cos(ω

_{0}t) of the proper oscillation required for compensating this component at zero moment of time) cannot be greater than two times the amplitude of the attenuated component of angular frequency ω.

_{0})

^{2}/2.

## 3. Results

_{0}

^{2}f′′(t) + f(t) = u(t)

_{alt}(t) is represented by an alternating function denoted as f

_{alt}(t). This means

_{0}

^{2}f

_{alt}′′(t) + f

_{alt}(t) = A × cos(ωt + φ)

_{alt}(t) for this standard linear differential equation will be represented by

_{alt}(t) = {A × cosφ/[1 − T

_{0}

^{2}ω

^{2}]} cos(ωt) − {A × sinφ/[1 − T

_{0}

^{2}ω

^{2}]} sin(ωt) −

{A × cosφ/[1 − T

_{0}

^{2}ω

^{2}]} cos(ω

_{0}t) + (ω/ω

_{0}) {A × sinφ/[1 − T

_{0}

^{2}ω

^{2}]} sin(ω

_{0}t)

_{S}is chosen at half-period. This means

_{0}t

_{S}= π, cos(ω

_{0}t

_{S}) = −1, sin(ω

_{0}t

_{S}) = 0

_{alt}(t) generated by the second order oscillating system for t = t

_{S}would be

_{alt}(t

_{S}) = {A × cosφ/[1 − T

_{0}

^{2}ω

^{2}]} cos(ωt

_{S})

− {A × sinφ/[1 − T

_{0}

^{2}ω

^{2}]} sin(ωt

_{S}) + {A × cosφ/[1 − T

_{0}

^{2}ω

^{2}]}

_{0}t

_{S}) = −1, sin(ω

_{0}t

_{S}) = 0 were substituted in Equation (7) according to Equation (8)).

_{S}) − sinφ sin(ωt

_{S}) = cos(ωt

_{S}+ φ)

_{alt}(t

_{S}) can be written as

_{alt}(t

_{S}) = {A/[1 − T

_{0}

^{2}ω

^{2}]} {cos(ωt

_{S}+ φ) − cosφ}

_{S}+ φ)|≤ 1, cosφ ≤ 1

_{S}+ φ) − cosφ| ≤ 2

_{0}

^{2}ω

^{2}]} {cos(ωt

_{S}+ φ) − cosφ }| ≤ 2|{A/[1 − T

_{0}

^{2}ω

^{2}]}|

_{alt}(t

_{S})| = |{A/[1 − T

_{0}

^{2}ω

^{2}]} {cos(ωt

_{S}+ φ) − cosφ}| ≤ |{A/[1 − T

_{0}

^{2}ω

^{2}]} cos(ωt

_{S}+ φ)| + | {A × cosφ / [1 − T

_{0}

^{2}ω

^{2}]}|≤ 2 |{A/[1 − T

_{0}

^{2}ω

^{2}]}|

^{2}) times less than input amplitude (the validity of Equation (15) being confirmed).

_{0}. This means

_{0}

^{2}f′′(t) + f(t) = A

_{0}

_{0}(t) generated by this step signal is represented by

_{0}(t) = A

_{0}{1 − cos(ω

_{0}t)}

_{0}= 1/T

_{0}.

_{S}= π/ω

_{0}= π T

_{0}

_{0}t

_{s}) = cos π = −1

_{0}(t

_{S}) = 2A

_{0}

_{0}(t) possess a saddle point at this sampling time moment (its derivative presents a null value). Thus, a robust sampling procedure can be performed, the filtering and sampling structure being insensitive at small variations of the sampling moment of time.

_{0}and an alternating signal of angular frequency ω and amplitude A (the order of magnitude for A and A

_{0}being the same). An example could be represented by measurements of angular velocity based on the rectification of an alternating signal generated through electromagnetic induction (a significant alternative component being added to the constant component proportional to the angular velocity).

_{0}+ A × cos(ωt + φ)

_{0}(t) + f

_{alt}(t)

_{S}) = f

_{0}(t

_{S}) + f

_{alt}(t

_{S})

_{0}(t

_{S}) corresponds to the useful signal (proportional to the step input of amplitude A

_{0}) necessary to be sampled; the term f

_{alt}(t

_{S}) is generated by the alternating signal of angular frequency ω necessary to be filtered (its magnitude should be as less than possible).

_{alt}(t

_{S})/f

_{0}(t

_{S})| ≤ 2 |{A/[1 − T

_{0}

^{2}ω

^{2}]}|/{2A

_{0}}

_{alt}(t

_{S})/f

_{0}(t

_{S})| ≤ {A/A

_{0}} {1/[1 − T

_{0}

^{2}ω

^{2}]}|

_{0}(t

_{S})/f

_{alt}(t

_{S})| ≥ |1 − T

_{0}

^{2}ω

^{2}| {A

_{0}/A}

_{0}of this oscillating second order system would be at least several times greater than the period T of the alternating input component to be filtered, it results that

_{0}/T ≥ 1, (T

_{0}/T)

^{2}= T

_{0}

^{2}ω

^{2}≥ 1

_{0}

^{2}ω

^{2}|= |T

_{0}

^{2}ω

^{2}− 1| = (T

_{0}/T)

^{2}− 1

_{0}(t

_{S})/f

_{alt}(t

_{S})| ≥ {(T

_{0}/T)

^{2}− 1} {A

_{0}/A}

_{input}= A

_{0}/A

_{0}/T)

^{2}− 1} times so as to obtain the output signal-to noise-ratio

_{output}= |f

_{0}(t

_{S})/f

_{alt}(t

_{S})|

_{r}of

_{r}= π T

_{0}≈ 3.14 T

_{0}

## 4. Discussion

_{0}of any second order oscillating system used for filtering to be several times greater than T (it should include several periods of this oscillating input). Preliminary we denote this required number by N.

_{0}= N T, T

_{0}/T = N, so

_{0}/T)

^{2}− 1} ≈ (T

_{0}/T)

^{2}= 4 N

^{2}

_{0}(t

_{S})/f

_{alt}(t

_{S})| ≈ (4N

^{2}) {A

_{0}/A}

_{0}/T)

^{2}is much greater than unity in right-hand-side of (29)), with the response time

_{r}= π T

_{0}≈ 3.14 N T

_{0}), the working time T

_{r}

_{(i)}equals

_{r(i)}= 2πT

_{0}= N (2πT) ≈ 6.28 N T

_{0}(t

_{S})/f

_{alt}(t

_{S})|

_{(i)}≈ (2π)(ω/ω

_{0})

^{3}{A

_{0}/A}

_{0}= 1/T

_{0}, ω = 1/T in Equation (37) it results

_{0}(t

_{S})/f

_{alt}(t

_{S})|

_{(i)}≈ (2π)(T

_{0}/T)

^{3}{A

_{0}/A} = (2π) N

^{3}{A

_{0}/A}

_{r}

_{(d)}= 4T

_{0}/ξ. The time constant T

_{0}is considered to include the same number N of periods of the oscillating input (with amplitude A and angular frequency ω). Thus

_{r(d)}= 4T

_{0}/ξ = 4N T/ξ ≈ 4…6 N T

_{0})

^{2}times (according to the basic theory of second order systems for ω much greater than ω

_{0}), so the signal-to-noise ratio for a damped second order system is

_{0}(t

_{S})/f

_{alt}(t

_{S})|

_{(d)}≈ (ω/ω

_{0})

^{2}{A

_{0}/A} = (T

_{0}/T)

^{2}{A

_{0}/A}

_{0}/T = N it results

_{0}(t

_{S})/f

_{alt}(t

_{S})| ≈ N

^{2}{A

_{0}/A}

_{0}/T = 5 and A = A

_{0}= 1 (a significant alternating input) it results an alternating output (the noise) less than 1% (the same order of magnitude as for the error of the step output considered on the transient time interval of first or second order damped systems), according to Equations (33) and (34).

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Response of second order oscillating system on half-period for an input with five times greater angular frequency.

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**MDPI and ACS Style**

Toma, C.
Filtering Method Based on Symmetrical Second Order Systems. *Symmetry* **2019**, *11*, 813.
https://doi.org/10.3390/sym11060813

**AMA Style**

Toma C.
Filtering Method Based on Symmetrical Second Order Systems. *Symmetry*. 2019; 11(6):813.
https://doi.org/10.3390/sym11060813

**Chicago/Turabian Style**

Toma, Cristian.
2019. "Filtering Method Based on Symmetrical Second Order Systems" *Symmetry* 11, no. 6: 813.
https://doi.org/10.3390/sym11060813