Low Distortion of Noise Filter Realization with 6.34 V/S Fast Slew Rate and 120 Mvp-P Output Noise Signal

sensors

Letter Low Distortion of Noise Filter Realization with 6.34 V/µs Fast Slew Rate and 120 mVp-p Output Noise Signal

Fang-Ming Yu 1,* , Kun-Cheng Lee 2, Ko-Wen Jwo 2, Rong-Seng Chang 2 and Jun-Yi Lin 2

1 Department of Electrical Engineering/Information and Communication Engineering, St. John’s University, New Taipei City 25135, Taiwan 2 Department of Optics and Photonics, National Central University, Taoyuan City 32001, Taiwan; [email protected] (K.-C.L.); [email protected] (K.-W.J.); [email protected] (R.-S.C.); [email protected] (J.-Y.L.) * Correspondence: [email protected]

Abstract: In order to reduce Gaussian noise, this paper proposes a method via taking the average of the upper and lower envelopes generated by capturing the high and low peaks of the input signal. The designed fast response filter has no cut-off frequency, so the high order harmonics of the actual signal remain unchanged. Therefore, it can immediately respond to the changes of input signal and retain the integrity of the actual signal. In addition, it has only a small phase delay. The slew rate, phase delay and frequency response can be confirmed from the simulation results of Multisim 13.0. The filter outlined in this article can retain the high order harmonics of the original signal, achieving a slew rate of 6.34 V/µs and an almost zero phase difference. When using our filter to physically test the input signal with a noise level of 3 Vp-p Gaussian noise, a reduced noise signal of 120 mVp-p is obtained. The noise can be suppressed by up to 4% of the raw signal. Keywords: low distortion noise filter; Gaussian noise; high order harmonics; fast response; Citation: Yu, F.-M.; Lee, K.-C.; small phase delay Jwo, K.-W.; Chang, R.-S.; Lin, J.-Y. Low Distortion of Noise Filter Realization with 6.34 V/µs Fast Slew

Rate and 120 mVp-p Output Noise 1. Introduction Signal. Sensors 2021, 21, 1008. Electronic signals are widely used in our daily lives. In fact, all the information in https://doi.org/10.3390/s21031008 the environment, such as sounds heard, images seen, and even the electrical signals of the human body, can be converted into electronic signals in different ways for various Academic Editor: Iren E. Kuznetsova applications. In general, these signal sources comprise two kinds of signals: the ﬁrst one Received: 30 November 2020 Accepted: 28 January 2021 is the real signal that we want to obtain, and the second one is the noise signal. Note Published: 2 February 2021 that the primary physiological signals obtained by biomedical signal measurement systems are mainly low-frequency signals, such as electrocardiography (ECG) measurement,

Publisher’s Note: MDPI stays neutral electroencephalography (EEG) measurement, electromyography (EMG) measurement, with regard to jurisdictional claims in electroneurography (ENG) measurement [1], etc. Physiological signals obtained by these published maps and institutional affil- measurement systems can reflect and aid in diagnosing physical conditions [2–4]. With iations. such relevant application to the medical field, the importance of eliminating noise in low-frequency signals is clear. The dedicated work of several medical studies [5–8] also points toward further work in finding ways to both efficiently and cost-effectively reduce noise. However, in current methods of noise reduction for low-frequency signals, the most common signal processing method is to use a low-pass filter to retain low-frequency Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. information. If the noise in the input signal has a lower frequency than the high order This article is an open access article harmonics of the real signal, a general low-pass filter is not sufficient, as only a cut-off distributed under the terms and frequency lower than the noise can be given. While eliminating the noise, the high order conditions of the Creative Commons harmonics will also be eliminated, resulting in a low slew rate (SR) and severe signal Attribution (CC BY) license (https:// distortion. On the other hand, when the cut-off frequency is set higher than the noise, it creativecommons.org/licenses/by/ essentially invalidates the filter, so there is the dilemma. 4.0/).

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The purpose of this research is to design a noise filter that can eliminate noise and achieve low phase distortion, fast response, and preserve the original signal characteristics. The presented method is effective for noise elimination in cases where the noise meets the criteria of symmetry and possesses random properties. This is a new methodology for noise filtering that uses the characteristics of Gaussian distribution of noise, meaning relevant academic texts are extremely limited or dated. However, this research builds on current methods for analyzing the distribution characteristics of noise [9,10] to design a low distortion noise filter. The simulation results of the general second-, third-, and fourth-order low-pass filters show that there are phase distortions, indicating the inability to retain high order harmonics information [11–13]. In our experimental proposal, the simulation is performed by using Multisim 13.0 with a sine wave and square wave as the signal source. The first step is to mix the real signal with the noise. Then we input the mixed signal into the general second-, third-, fourth order low-pass filter and the designed low-distortion noise filter. With the data we gathered, we proceeded to compare the designed low-distortion noise filter with the general second-, third-, and fourth-order low-pass filters, quantizing the attenuation amplitude of the noise, observing the frequency response and analyzing the slew rate to compare the ability of each filter in noise elimination, phase distortion, response speed, and the ability of real signal retention. When compared with use of the average peaks method [14], the presented method can achieve a fast slew rate of 6.34 V/µs and the output noise can be suppressed up to 4%, which is better than the slew rate of 472 V/ms and the 9% suppressed noise of the previous work [14]. This low-distortion filter can directly capture the high and low peak values of the input signal, using RC charge and discharge principle to generate the signal’s upper and lower envelopes and then take their average to reduce the noise. This design can handle noise easily without complex digital signal processor (DSP) processing [15–17], and the circuit design is simple and features low power consumption.

2. Mathematics Background The main purpose of this study is to design a noise ﬁltering circuit. As mentioned in the previous section, thermal noise and shot noise are commonly seen in various types of sensors, and these noise types are considered Gaussian distribution of noise. Figure1 shows an estimate of Gaussian random variable distribution of the noise, which is generated by Multisim Function Generator and its distribution is similar to Gaussian distribution [12]. However, Gaussian distribution of noise has an important feature, that is, the Gaussian distribution of noise is a noise with random probability distribution, six standard deviations can contain 99.9997% of the entire noise distribution. Therefore, since the signal is random, in this article, we will use the Gaussian distribution characteristics as well as use the RC charge and discharge characteristics to capture the upper and lower envelopes of the input signal with noise and average them to achieve the elimination of the noise. In this research, the overall system architecture, as shown in Figure2, can be divided into three parts; buffer, capturing of upper and lower envelopes, and the operations of the signal. In Figure2, the voltage value of the input signal vi is the voltage value of the original signal vs joins with the voltage value of the noise vn and vo is the voltage value of the system output. From Figure2, it is found that by using the characteristics of Gaussian distribution to capture the high and low envelopes and then averaging them to maintain the integrity of the true signal and to effectively suppress the noise. The upper and lower envelopes are generated by RC charging and discharging circuits. An example is given in Figure3, using a sine wave with noise, the edge signals are captured from the upper and lower edges of the signal for data analysis. As shown in Figure4, it can be observed that the captured upper and lower envelopes can show a Gaussian distribution, and it is expected that the corresponding peaks can be averaged. This ﬁgure shows that the noise standard deviation of the input signal is 0.1183, and the captured upper and lower envelopes can reduce the standard deviation to 0.028, which is about 23% of the raw signal, proving that this method can indeed attenuate the noise. Sensors 2021, 21, x FOR PEER REVIEW 3 of 18 Sensors 2021, 21, x FOR PEER REVIEW 3 of 18

Sensors 2021, 21, 1008 3 of 17 Sensors 2021, 21, x FOR PEER REVIEW 3 of 18 Sensors 2021, 21, x FOR PEER REVIEW 3 of 18

Figure 1. An estimate of Gaussian random variable distribution. Figure 1. An estimate of Gaussian random variable distribution.

Upper Upper envelope envelope

Buffer Average Buffer Buffer Average Buffer 𝑉 𝑉 𝑉 𝑉 Lower Lower envelope FigureFigure 1. 1.An An estimate estimate of ofenvelope Gaussian Gaussian random random variable variable distribution. distribution. Figure 1. An estimate of Gaussian random variable distribution. Figure 2. System architecture diagram. Figure 2. System architectureUpper diagram. envelopeUpper envelope An example is given in Figure 3, using a sine wave with noise, the edge signals are Buffer Average Buffer captured from the upper and lowerAverage edges of the signal for data analysis. As shown in captured푉𝑖 Buffer from the upper and lower edges of Bufferthe signal 푉for표 data analysis. As shown in 𝑉Figure 4, it can be observed that the captured upper and𝑉 lower envelopes can show a Figure 4, it can be observedLower that the captured upper and lower envelopes can show a Gaussian distribution,envelopeLower and it is expected that the corresponding peaks can be averaged. Gaussian distribution,envelope and it is expected that the corresponding peaks can be averaged. This figure shows that the noise standard deviation of the input signal is 0.1183, and the Figurecaptured 2. System upper architecture and lower diagramenvelopes. can reduce the standard deviation to 0.028, which is Figure 2.2. System architecture diagram.diagram. about 23% of the raw signal, proving that this method can indeed attenuate the noise. An example is given in Figure 3, using a sine wave with noise, the edge signals are An example is given in Figure 3, using a sine wave with noise, the edge signals are captured from the upper and lower edges of the signal for data analysis. As shown in captured from the upper and lower edges of the signal for data analysis. As shown in Figure 4, it can be observed that the captured upper and lower envelopes can show a Figure 4, it can be observed that the captured upper and lower envelopes can show a Gaussian distribution, and it is expected that the corresponding peaks can be averaged. Gaussian distribution, and it is expected that the corresponding peaks can be averaged. This figure shows that the noise standard deviation of the input signal is 0.1183, and the This figure shows that the noise standard deviation of the input signal is 0.1183, and the captured upper and lower envelopes can reduce the standard deviation to 0.028, which is captured upper and lower envelopes can reduce the standard deviation to 0.028, which is about 23% of the raw signal, proving that this method can indeed attenuate the noise. about 23% of the raw signal, proving that this method can indeed attenuate the noise.

Figure 3. Upper/lower envelopes captured schematic. FigureFigure 3. 3.Upper/lower Upper/lower envelopesenvelopes capturedcaptured schematic.schematic.

Histogram of V(i), Upper envelopes, Lower envelopes, V(o) Histogram of V(i), Upper envelopes, Lower envelopes, V(o) 2500 Variable 2500 Variable V(i) V(i) Upper envelopes Upper envelopes Lower envelopes Lower envelopes 2000 V(o) 2000 V(o) Mean StDev N Figure 3. Upper/lower envelopes captured schematic. Mean StDev N 0.9789 0.1183 14513 Figure 3. Upper/lower envelopes captured schematic. 0.9789 0.1183 14513 1500 1.105 0.02874 14513 1.105 0.02874 14513 1500 0.8358 0.02839 14513 0.8358 0.02839 14513 0.9705 0.02852 14513 Histogram of V(i), Upper envelopes, Lower envelopes, V(o)0.9705 0.02852 14513 1000 Histogram of V(i), Upper envelopes, Lower envelopes, V(o) 25001000 Variable Number of times 2500 Variable

Number of times V(i) V(i)Upper envelopes Upper envelopes 500 Lower envelopes 2000500 LowerV(o) envelopes 2000 V(o)

Mean StDev N s

e 0.9789Mean StDev0.1183 14513 N

m 0 0.97891.105 0.02874 0.1183 1451314513 i 1500 t 0 0.50 0.75 1.00 1.25 1.50 f 1500 0.83581.105 0.028740.02839 1451314513

o 0.50 0.75 1.00 1.25 1.50 0.83580.9705 0.028390.02852 1451314513

r Voltage e Voltage 0.9705 0.02852 14513 b m 1000

N 1000

FigureNumber of times 4. Normal distribution of vi, vo and upper/lower envelopes. 500 3. Design500 Method 3.1. Circuit Design: Retention of the Real Signal 0 In0 circuit0.50 design,0.75 the transistor1.00 is used1.25 as a buffer,1.50 and the RC loop formed by resistor and capacitor0.50 is used0.75 to captureVoltage1.00 the upper1.25 and lower1.50 envelopes information. Finally, a Voltage

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Sensors 2021, 21, x FOR PEER REVIEW 5 of 18 voltage divider circuit formed by a resistor is used for averaging the upper and lower envelope signals. The overall designed noise ﬁlter (named BJT) circuit is proposed as shown in Figure5.

𝑣

𝑣 𝑣

𝑣

Figure 5. The designed noise ﬁlter (named BJT) circuit. Figure 5. The designed noise filter (named BJT) circuit. In Figure5, the signal is transmitted from the input to the output. Since it does not need too much attenuation, two transistors of PNP and NPN forms are set. When the signalIn is transmittedtheory, the to forward the ﬁrst stage bias of of the each transistors transistor (as a buffer) will Qbe3 and the Q same,4, the signal but there are still slightis transmitted differences from thein practica base to thel components, emitter. There willand be the a superpositionvoltage drift of between the forward components will alsovoltage occurvbe, sodepending the signal of on the the upper operating and lower temper envelopesature. can be Nevertheless, listed as: the main purpose of the filter design is to eliminate the noise. To avoid the voltage drift problems, we chose Upper envelope : vEh = vi + vbe3 (1) PNP & NPN packaged in one group of two components carriers, Q3 and Q1 are in one = − group, Q2 and Q4 are anotherLower group. envelope : vEl vi vbe4 (2)

whereInv i this= vs way,+ vn; v wes is thecan voltage see that value 𝑣 of the minus original 𝑣 signal, (𝑣 andv−𝑣n is the) voltage, and valuenegative 𝑣 plus of the noise. 𝑣 (−𝑣 +𝑣) can approach zero, and the temperature drift voltage can be avoided due Thento the the different signal of v temperaturei is transmitted toof theeach second-stage component. of transistors Thus, the Q1 andfinal Q 2expression. In of the addition to being a buffer, the function here is to eliminate the forward bias superposed by output voltage can be listed as: the ﬁrst-stage transistors. By the upper RC circuit, the upper envelope of signal vh will be (v + v + v ) − v ; v is the upper envelope of v . And by the lower RC circuit, the s nh be3 be1 nh 𝑣 +𝑣 n lower envelope of signal v will be (v −𝑣 v =− v ) + v ; v is the lower envelope of v . l s nl be24 be2 nl n Normally, vnh is equal to vnl (vnh ≈ vnl). Therefore, the upper envelope of vh and lower envelope of v can be expressed𝑣 +𝑣 as follows: +𝑣 − 𝑣 +𝑣 −𝑣 −𝑣 +𝑣 l = (6) 2 Upper envelope : vh = vs + vnh + vbe3 − vbe1 (3) 𝑣 +𝑣 = =𝑣 Lower envelope : vl = vs −2vnl − vbe4 + vbe2 (4)

InTherefore, Equations (1)–(4),from Equation the vbe1, vbe 2(6),, vbe 3itand is demovbe4 arenstrated the forward that bias the between average the baseof the upper and lowerand emitter envelopes of Q1,Q can2,Q achieve3 and Q4 ,the normally elimination inﬂuenced of noise by the and temperature the retention (T), Vcc ofand the real signal.

3.2. Circuit Design: Capturing of Upper and Lower Envelopes As shown in Figure 6, the main method of the circuit design for noise attenuation is to use the high and low peaks of the noise, then by charging and discharging the capacitors and resistors of R14, R15, C4, and C5 to capture the high and low envelopes, but this requires special attention to the matching of resistance and capacitance.

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Vss, we can name as vbe1(T, Vcc,Vss), vbe2(T, Vcc,Vss), vbe3(T, Vcc,Vss) and vbe4(T, Vcc,Vss). The voltage value of vbe can be expressed as follows [18]:

v = kT ln Ic be q Is −23 −1 (5) = 1.3805×10 JK ln Ic 1.602×10−19C Is

where J is energy (Joule), K is absolute temperature, C is unit of charge (Coulombs), Ic is the collector current (mA) and Is is the reverse saturation current (pA). In theory, the forward bias of each transistor will be the same, but there are still slight differences in practical components, and the voltage drift between components will also occur depending on the operating temperature. Nevertheless, the main purpose of the ﬁlter design is to eliminate the noise. To avoid the voltage drift problems, we chose PNP & NPN packaged in one group of two components carriers, Q3 and Q1 are in one group, Q2 and Q4 are another group. In this way, we can see that vbe3 minus vbe1 (vbe3 − vbe1), and negative vbe4 plus vbe2 (−vbe4 + vbe2) can approach zero, and the temperature drift voltage can be avoided due to the different temperature of each component. Thus, the ﬁnal expression of the output voltage can be listed as:

vh+vl vo = 2

vs+vnh+vbe3− vbe1+vs−vnl −vbe4 +vbe2 = 2 (6) vs+vs = 2 = vs Therefore, from Equation (6), it is demonstrated that the average of the upper and lower envelopes can achieve the elimination of noise and the retention of the real signal.

3.2. Circuit Design: Capturing of Upper and Lower Envelopes As shown in Figure6, the main method of the circuit design for noise attenuation is to Sensors 2021, 21, x FOR PEER REVIEWuse the high and low peaks of the noise, then by charging and discharging the capacitors 6 of 18 and resistors of R14,R15,C4, and C5 to capture the high and low envelopes, but this requires special attention to the matching of resistance and capacitance.

Figure 6. Sinusoidal wave with noise and the Upper/Lower envelopes. (vnh is the upper envelope of vn; vnl is the lower envelope of vn; vn is the noise signal). Figure 6. Sinusoidal wave with noise and the Upper/Lower envelopes. (𝑣 is the upper envelope of 𝑣Figure; 𝑣 7 is[ the7] shows lower the envelope ripple voltage of 𝑣 generated; 𝑣 is the by noise a full-wave signal). rectiﬁer with smoothing capacitor. As shown in Figure7, the value of RC not only affects the charge and discharge time,Figure but also affects7 [7] the shows smoothness the ofripple the captured voltage envelope. generated In fact theby highera full-wave the value rectifier with smoothing capacitor. As shown in Figure 7, the value of RC not only affects the charge and discharge time, but also affects the smoothness of the captured envelope. In fact the higher the value of RC, the better the smoothing. In Figure 7, the time constant of 𝜏 and output voltage of 𝑣 can be expressed as following: 𝜏=𝑅𝐶 (7)

𝑣(𝑡) = 𝑉 𝑒 =𝑉𝑒 (8)

Figure 7. RC filter showing the ripple voltage.

In Figure 7, some symbols are defined as follows:

• VM is the maximum output voltage and VL is the minimum output voltage. • Vr is the ripple voltage. • 𝑡 is the time for discharging and 𝑇 is the discharging period. • Tp is the period of the ripple.

3.3. Circuit Design: Fast Charge and Discharge Design As shown in Figure 8, the larger the value of time constant 𝜏, the longer the charge and discharge time will be, and the captured envelope will thus be smooth. The opposite occurs under a smaller 𝜏 value. If the value of 𝜏 is increased, during the rapid changes of the input signal, the distortion will be caused by insufficient charge and discharge time. Figure 9 shows the output of the designed circuit caused by only using RC charging and discharging circuit under the input signal of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise. The noise is generated by the Multisim Function Generator Interface.

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Figure 6. Sinusoidal wave with noise and the Upper/Lower envelopes. (𝑣 is the upper envelope of 𝑣; 𝑣 is the lower envelope of 𝑣; 𝑣 is the noise signal). Figure 7 [7] shows the ripple voltage generated by a full-wave rectifier with smoothing capacitor. As shown in Figure 7, the value of RC not only affects the charge and discharge time, but also affects the smoothness of the captured envelope. In fact the Sensors 2021, 21, 1008 6 of 17 higher the value of RC, the better the smoothing. In Figure 7, the time constant of 𝜏 and output voltage of 𝑣 can be expressed as following:

of RC, the better the smoothing. In Figure7, the time𝜏=𝑅𝐶 constant of τ and output voltage of (7) vo can be expressed as following: = 𝑣τ(𝑡)RC = 𝑉 𝑒 =𝑉𝑒 (7) (8) 0 0 − t − t vo(t) = VMe τ = VMe RC (8)

Figure 7. 7.RC RC ﬁlter filter showing showing the ripple the ripple voltage. voltage. In Figure7, some symbols are deﬁned as follows:

• VInM isFigure the maximum 7, some output symbols voltage are and definedVL is the as minimum follows: output voltage. • • VVr Mis is the the ripple maximum voltage. output voltage and VL is the minimum output voltage. • t0 is the time for discharging and T0 is the discharging period. • Vr is the ripple voltage. • Tp is the period of the ripple. • 𝑡 is the time for discharging and 𝑇 is the discharging period. 3.3.• CircuitTp is Design:the period Fast Charge of the and ripple. Discharge Design As shown in Figure8, the larger the value of time constant τ, the longer the charge and3.3. dischargeCircuit Design: time will Fast be, andCharge the capturedand Discharge envelope Design will thus be smooth. The opposite occurs under a smaller τ value. If the value of τ is increased, during the rapid changes of the inputAs shown signal, the in distortionFigure 8, will the be larger caused the by insufﬁcientvalue of time charge constant and discharge 𝜏, the time. longer the charge Figureand discharge9 shows the time output will of thebe, designedand the circuitcaptured caused envelope by only using will RCthus charging be smooth. and The opposite Sensors 2021, 21, x FOR PEER REVIEW 7 of 18 dischargingoccurs under circuit a smaller under the 𝜏 input value. signal If the of 10value KHz, of 1.5 𝜏 Vpis squareincreased, wave during mixed with the rapid changes 5of Vp-p the noise.input The signal, noise isthe generated distortion by the will Multisim be caused Function by Generator insufficient Interface. charge and discharge time. Figure 9 shows the output of the designed circuit caused by only using RC charging and discharging circuit under the input signal of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise. The noise is generated by the Multisim Function Generator Interface.

FigureFigure 8. Transient 8. Transient response response analysis analysis of charging of time.charging time.

Figure 9. The output waveform distortion of the designed noise filter for only using the RC charging and discharging circuit, without using D1–D6 Schottky diodes. (𝑣: red line; 𝑣: green line).

As shown in Figure 9, the output waveform distortion of the designed noise filter is caused when only using the RC charging and discharging circuit without using D1–D6 Schottky diodes. In order to achieve high noise removal ability and fast response speed as well as keep the signal undistorted, in the designed noise filter of Figure 5, we set the Schottky diodes between the base and emitter of Q1 and Q2, and between 𝑣 and 𝑣, using their forward bias conduction characteristics as a fast discharge circuit switch. In other words, when the instantaneous change of the voltage value of the input signal 𝑣 (∆𝑣) is greater than the voltage value of the diode forward bias of 𝑉 (∆𝑣 𝑉), a fast discharge is performed. It’s important to note that the Schottky diode is used in our design, mainly because of its characteristics of fast switching and low forward bias voltage. In general, the forward bias of a typical diode is about 0.7~1.7 volts, while the forward bias of a Schottky diode is about 0.15 to 0.45 volts. In our circuit design, we use two Schottky diodes connected in series, so that when (∆𝑣 0.4𝑉), fast discharge is performed. Using the designed noise filter and given the same input signal of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise as in Figure 9, the output result can show that the waveform distortion caused by RC charging and discharging is no longer visible as shown in Figure 10. However, if without using D5 and D6 Schottky diodes, the output result is shown in Figure 11. Sensors 2021, 21, x FOR PEER REVIEW 7 of 18

푅퐶1 < 푅퐶2 < 푅퐶3 < 푅퐶4

Sensors 2021, 21, 1008 7 of 17 Figure 8. Transient response analysis of charging time.

FigureFigure 9. The 9. The output output waveform waveform distortion ofdistortion the designed of noise the filter designed for only noise using the filter RC chargingfor only using the RC andcharging discharging and circuit, discharging without using circuit D1,–D without6 Schottky using diodes. D (v1–o:D red6 Schottky line; vi: green diodes line).. (푣표: red line; 푣𝑖: green line). As shown in Figure9, the output waveform distortion of the designed noise filter is caused when only using the RC charging and discharging circuit without using D1–D6 SchottkyAs diodes. shown In in order Figure to achieve 9, the highoutput noise waveform removal ability distortion and fast of response the designed speed noise filter is ascaused well as when keepthe only signal using undistorted, the RC incharging the designed and noisedischarging filter of Figure circuit5, wewithout set using D1–D6 theSchottky Schottky diodes diodes. betweenIn order the to baseachieve and emitterhigh noise of Q1 removaland Q2, and ability between andv lfastand response speed vash, usingwell theiras keep forward the biassignal conduction undistorted, characteristics in the designed as a fast discharge noise filter circuit of switch. Figure 5, we set the In other words, when the instantaneous change of the voltage value of the input signal Schottky diodes between the base and emitter of Q1 and Q2, and between 푣푙 and 푣ℎ, vi (∆vi) is greater than the voltage value of the diode forward bias of VD (∆vi > VD), ausing fast discharge their forward is performed. bias conduction It’s important characteristics to note that the Schottkyas a fast diode discharge is used incircuit switch. In ourother design, words, mainly when because the ofinstantaneous its characteristics change of fast of switching the voltage and low value forward of the bias input signal 푣𝑖 voltage. In general, the forward bias of a typical diode is about 0.7~1.7 volts, while the (∆푣𝑖) is greater than the voltage value of the diode forward bias of 푉퐷 (∆푣𝑖 > 푉퐷), a fast forward bias of a Schottky diode is about 0.15 to 0.45 volts. In our circuit design, we use discharge is performed. It’s important to note that the Schottky diode is used in our de- two Schottky diodes connected in series, so that when (∆vi > 0.4 V), fast discharge is performed.sign, mainly Using because the designed of its noisecharacteristics filter and given of fast the switching same input signaland low of 10 forward KHz, bias voltage. 1.5In Vpgeneral square, wavethe forward mixed with bias 5 Vp-pof a noisetypical as indiode Figure i9s, theabout output 0.7 result~1.7 volts can show, while the forward that the waveform distortion caused by RC charging and discharging is no longer visible Sensors 2021, 21, x FOR PEER REVIEWbias of a Schottky diode is about 0.15 to 0.45 volts. In our circuit design, we use8 of two 18 asSchottky shown in diodesFigure 10 connected. However, if in without series, using so Dthat5 and when D6 Schottky (∆푣 > diodes,0.4 푉 the), fast output discharge is per- result is shown in Figure 11. 𝑖 formed. Using the designed noise filter and given the same input signal of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise as in Figure 9, the output result can show that the waveform distortion caused by RC charging and discharging is no longer visible as shown in Figure 10. However, if without using D5 and D6 Schottky diodes, the output result is shown in Figure 11.

FigureFigure 10. 10.The The output output waveform waveform distortion distortion is improved is improved using the designed using noisethe designed ﬁlter circuit, noise with filter circuit, with RCRC circuit circuit and and D1–D D61Schottky–D6 Schottky diodes. diodes (vo: red. line;(푣표: vredi: green line; line). 푣𝑖: green line).

Figure 11. The output waveform of the designed noise filter, without D5 and D6 Schottky diodes. (푣표: red line; 푣𝑖: green line).

4. Experiment Results 4.1. Circuit Simulations The filtering effect of the designed noise filter circuit is verified by Multisim 13.0 software simulation. The verified items include circuit frequency response, phase response, noise filtering capability, response speed, and Fourier amplitude spectrum. In circuit simulations, the cut-off frequency 푓푐 of 20 KHz will be used as the verification control group. Meanwhile, the verification method uses an input signal obtained from the pre-stage mixing circuit of an original signal of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise generated by Multisim Function Generator Interface.

4.1.1. Low-Pass Filter In this section, the traditional low-pass filters will be introduced. The circuits of second-, third-, and fourth-order low-pass Butterworth filters are shown in Figure 12. Figure 12a is the RC network of the second-order low-pass filter, composed of two groups of 푅1, 퐶1, and 푅2, 퐶2, and the cut-off frequency 푓푐 in this filter can be expressed as follows: 1 푓푐 = (9) 2휋 × √푅1퐶1푅2퐶2 Figure 12b shows a third-order low pass filter. This filter is composed of a second-order filter and a first-order filter connected in series, the cut-off frequency 푓푐 can be expressed as: 1 푓푐 = 3 (10) 2휋 × √푅1푅2푅3퐶1퐶2퐶3 Figure 12c shows a fourth-order low-pass filter. This filter is composed of two second-order filters connected in series, the cut-off frequency 푓푐 can be expressed as:

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Sensors 2021, 21, 1008 8 of 17 Figure 10. The output waveform distortion is improved using the designed noise filter circuit, with RC circuit and D1–D6 Schottky diodes. (푣표: red line; 푣𝑖: green line).

FigureFigure 11. 11.The The output output waveform waveform of the designed of the designed noise filter, noise without filter, D5 andwithout D6 Schottky D5 and diodes. D6 Schottky diodes. (v(o푣:표 red: red line; line;vi: green 푣𝑖: green line). line). 4. Experiment Results 4.1.4. CircuitExperiment Simulations Results 4.1.The Circuit filtering Simulations effect of the designed noise filter circuit is verified by Multisim 13.0 software simulation. The verified items include circuit frequency response, phase response, Sensors 2021, 21, x FOR PEER REVIEW The filtering effect of the designed noise filter circuit is verified9 ofby 18 Multisim 13.0 noise filtering capability, response speed, and Fourier amplitude spectrum. In circuit simulations,software thesimulation. cut-off frequency The verifiedfc of 20 KHz items will include be used circuit as the verification frequency control response, phase re- group.sponse, Meanwhile, noise filtering the verification capability, method response uses an input speed, signal and obtained Fourier from amplitude the pre- spectrum. In stagecircuit mixing simulations, circuit of an originalthe cut signal-off frequency of 10 KHz, 1.5 푓푐 Vp of square 20 KHz wave will mixed be with used 5 Vp-p as the verification noise generated by Multisim Function Generator1 Interface. control group. Meanwhile,𝑓 = the verification method uses an input signal(11) obtained from 2𝜋 × 𝑅 𝑅 𝑅 𝑅 𝐶 𝐶 𝐶 𝐶 4.1.1.the Low-Passpre-stage Filter mixing circuit of an original signal of 10 KHz, 1.5 Vp square wave mixed FigurewithIn 13 this5 Vpshows section,-p noise the the frequencygenerated traditional response by low-pass Multisim filtersof the Function willsecond-, be introduced. Generator third-, and The Interface fourth-order circuits. of low-passsecond-, Butterworth third-, and filters. fourth-order low-pass Butterworth filters are shown in Figure 12. 4.1.1. Low-Pass Filter In this section, the traditional low-pass filters will be introduced. The circuits of second-, third-, and fourth-order low-pass Butterworth filters are shown in Figure 12. Figure 12a is the RC network of the second-order low-pass filter, composed of two groups of 푅1, 퐶1, and 푅2, 퐶2, and the cut-off frequency 푓푐 in this filter can be expressed as follows: 1 푓푐 = (9) (a) 2휋 × √(b푅)1 퐶1푅2퐶2 Figure 12b shows a third-order low pass filter. This filter is composed of a second-order filter and a first-order filter connected in series, the cut-off frequency 푓푐 can be expressed as: 1 푓푐 = 3 (10) 2휋 × √푅1푅2푅3퐶1퐶2퐶3 Figure 12c shows a fourth-order low-pass filter. This filter is composed of two second-order filters connected in series, the cut-off f requency 푓 can be expressed as: (c) 푐

Figure 12.Figure (a) Second-order 12. (a) Second-order low pass low Butterworth pass Butterworth filter; filter; (b) (Third-orderb) Third-order low low pass pass Butterworth Butterworth filter;filter; (c(c)) Fourth-order Fourth-order low-pass Butterworth filter. low-pass Butterworth filter.

Figure 13. Frequency response of the second-, third-, and fourth-order low-pass Butterworth filters (𝑓 = 𝑓).

In general, for an n-th order Butterworth low-pass filter, the gain 𝐺(𝜔) can be expressed as: 𝐺 𝐺 ( ) 𝐺 𝜔 = 𝜔 = (12) 1+( ) 𝑓 𝜔 1+( ) 𝑓 where • n = the order of the filter. • 𝜔 = the radian frequency which is equal to 2πf. • 𝜔 = cut off radian frequency. • 𝑓 = cutoff frequency = frequency at which the gain drops to −3 dB. • 𝐺 is the DC gain (zero frequency gain).

Sensors 2021, 21, x FOR PEER REVIEW 9 of 18

1 𝑓 = (11) 2𝜋 × 𝑅𝑅𝑅𝑅𝐶𝐶𝐶𝐶 Figure 13 shows the frequency response of the second-, third-, and fourth-order low-pass Butterworth filters. Sensors 2021, 21, 1008 9 of 17

Figure 12a is the RC network of the second-order low-pass filter, composed of two groups of R1, C1, and R2, C2, and the cut-off frequency fc in this filter can be expressed as follows: 1 fc = √ (9) 2π × R C R C 1 1 2 2 (a) Figure 12b shows a third-order low pass filter. This filter( isb)composed of a second-order filter and a first-order filter connected in series, the cut-off frequency fc can be expressed as:

1 = √ fc 3 (10) 2π × R1R2R3C1C2C3 Figure 12c shows a fourth-order low-pass filter. This filter is composed of two second- order filters connected in series, the cut-off frequency fc can be expressed as:

1 f = √ (11) c 4 2π × R1R2R3R4C1C2C3C4 (c) Figure 13 shows the frequency response of the second-, third-, and fourth-order Figure 12. (a) Second-order low pass Butterworth filter; (b) Third-order low pass Butterworth filter; (c) Fourth-order low-pass Butterworth filter.low-pass Butterworth ﬁlters.

FigureFigure 13. 13.Frequency Frequency response response of of the the second-, second-, third-, third-, and and fourth-order fourth-order low-pass low-pass Butterworth Butterworth ﬁlters filters (𝑓 = 𝑓). ( f−3db = fc).

InIn general,general, forfor anan nn-th-th orderorder ButterworthButterworth low-passlow-pass filter,filter, thethe gaingain G𝐺(𝜔)(ω) can can bebe ex-ex- pressedpressed as: as: 2 2 G0 G0 G2(ω) = 𝐺 = 𝐺 ( ) 2n 2n (12) 𝐺 𝜔 1=+ ( ω )𝜔 = f (12) 1+(ωc ) 1 + ( )𝑓 𝜔 1+(fc ) 𝑓 where where • n = the order of the filter. • • ωn == the radianorder of frequency the filter. which is equal to 2πf. • 𝜔 = the radian frequency which is equal to 2πf. • ωc = cut off radian frequency. • 𝜔 = cut off radian frequency. • fc = cutoff frequency = frequency at which the gain drops to −3 dB. • 𝑓 = cutoff frequency = frequency at which the gain drops to −3 dB. • G0 is the DC gain (zero frequency gain). • 𝐺 is the DC gain (zero frequency gain). In the low-pass Butterworth filters of Figure 12, the operational amplifier is connected as a voltage follower, so at zero frequency, its gain is 1, that is G0 = 1 and the frequency response will be attenuated to 0dB. Therefore, if f = 10 fc, then Equation (12) can be expressed as follows:

2 2 G2(w) = G0 = G0 ω 2n f 2n 1+( ω ) 1+( ) c fc −2n (13) ∼ 1 f −2n −n 2 = 2n = ( f ) = 10 = (10 ) ( f ) c fc Sensors 2021, 21, x FOR PEER REVIEW 10 of 18

In the low-pass Butterworth filters of Figure 12, the operational amplifier is connected as a voltage follower, so at zero frequency, its gain is 1, that is 𝐺 =1 and the frequency response will be attenuated to 0dB. Therefore, if 𝑓 = 10𝑓, then Equation (12) can be expressed as follows: 𝐺 𝐺 𝐺 (𝑤) = 𝜔 = 1+( ) 𝑓 𝜔 1+( ) 𝑓 (13) 1 𝑓 Sensors 2021, 21, 1008 ≅ =( ) =10 =(10) 10 of 17 𝑓 𝑓 ( ) 𝑓 In this case, |𝐺(𝜔)| will be 10, that is |𝐺(𝜔)| ≅10, and the frequency re- | ( )| −n | ( )| =∼ −n sponse at 𝑓In = this10𝑓 can case, expressedG ω willas follows: be 10 , that is G ω 10 , and the frequency response at f = 10 fc can expressed as follows: 20 𝑙𝑜𝑔|𝐺(𝜔)| =20𝑙𝑜𝑔10 =−20×𝑛 dB (14) −n As Figure 13 indicates, the slope20 logof the|G( frequencyω)| = 20log response10 = diagram−20 × ncandB be described (14) as −20 × n dB/decade when the frequency exceeds 𝑓. In Figure 13, we can find once the frequency exceedsAs Figure 𝑓 , the 13 slopesindicates, of second-, the slope third-, of the and frequency fourth-order response low-pass diagram Butter- can be described − × worth filtersas are20 −40n dB,dB/decade −60 dB, and when −80 thedB frequencyrespectively. exceeds fc. In Figure 13, we can ﬁnd once the frequency exceeds fc, the slopes of second-, third-, and fourth-order low-pass Butterworth 4.1.2. Frequencyﬁlters are Response−40 dB, Analysis−60 dB, and −80 dB respectively. As shown4.1.2. Frequencyin Figure 14a, Response the traditional Analysis second-, third-, and fourth-order low-pass filters have reduced the gain to be −40~−80 dB at the frequency of 10 times of 𝑓. How- ever, in FigureAs 14a, shown we can in Figuresee the 14BJTa, desig the traditionalned noise filter second-, has a third-, wider and frequency fourth-order re- low-pass sponse range.filters Figure have reduced 14 and theTable gain 1 toshow be − the40~ frequency−80 dB at response the frequency of second-, of 10 times third-, of fc. However, fourth-order,in Figure and the 14a, BJT we designed can seethe noise BJT filter. designed Unlike noise general filter filters has a of wider cut-off frequency frequency response range. limitation,Figure the proposed 14 and Table noise1 showfilter thecan frequencyoperate at response a wider offrequency second-, third-,response fourth-order, range and the normallyBJT within designed 1 MHz. noise filter. Unlike general filters of cut-off frequency limitation, the proposed noise filter can operate at a wider frequency response range normally within 1 MHz.

(a)

(b)

f Figure 14. (a) Frequency response at f = 10 fc: (Log = 1);(b) Frequency response from 1 Hz f−3db to 1 GHz. (BJT is the designed noise ﬁlter, fc = f−3db = 20 KHz).

Table 1. The dB values of frequency response at 1 KHz, 10 KHz, 200 KHz, and 1 MHz. ( fc = f−3db = 20 KHz).

Frequency BJT 2 Order 3 Order 4 Order 1 KHz 0 dB 0 dB 0 dB 0 dB 10 KHz 0 dB 0 dB 0 dB 0 dB 200 KHz 0 dB −40 dB −60 dB −80 dB 1 MHz −1 dB −42 dB −78 dB −84 dB Sensors 2021, 21, x FOR PEER REVIEW 11 of 18 Sensors 2021, 21, x FOR PEER REVIEW 11 of 18

푓 Figure 14. (a) Frequency response at f = 10푓 : (Log ( 푓 ) = 1); (b) Frequency response from 1 Hz 푐 푓 Figure 14. (a) Frequency response at f = 10푓푐: (Log ( −3db) = 1); (b) Frequency response from 1 Hz 푓−3db to 1 GHz. (BJT is the designed noise filter, 푓푐 = 푓−3db = 20 KHz). to 1 GHz. (BJT is the designed noise filter, 푓푐 = 푓−3db = 20 KHz). Table 1. The dB values of frequency response at 1 KHz, 10 KHz, 200 KHz, and 1 MHz. (푓 = 푓 = Table 1. The dB values of frequency response at 1 KHz, 10 KHz, 200 KHz, and 1 MHz. (푓푐 = 푓−3db = 20 KHz). 푐 −3db 20 KHz). Frequency BJT 2 Order 3 Order 4 Order Frequency BJT 2 Order 3 Order 4 Order 1 KHz 0 dB 0 dB 0 dB 0 dB 1 KHz 0 dB 0 dB 0 dB 0 dB 10 KHz 0 dB 0 dB 0 dB 0 dB 10 KHz 0 dB 0 dB 0 dB 0 dB 200 KHz 0 dB −40 dB −60 dB −80 dB Sensors 2021, 21, 1008 200 KHz 0 dB −40 dB −60 dB −80 dB11 of 17 1 MHz −1 dB −42 dB −78 dB −84 dB 1 MHz −1 dB −42 dB −78 dB −84 dB 4.1.3. Square and Sine Wave Response 4.1.3.4.1.3. SquareSquare andand SineSine WaveWave Response Response As shown in Figure 15, similarly, a 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p As shown in Figure 15, similarly, a 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noiseAs generated shown in by Figure Multisim 15, similarly, Function a Generator 10 KHz, 1.5 is Vpused square as the wave input mixed signal with. In unit 5 Vp-p scale noise generated by Multisim Function Generator is used as the input signal. In unit scale noiseof Time/Div generated = 50 by μs Multisim, it can be Function observed Generator that the performance is used as the of input this designed signal. In circuit unit scale has a of Time/Div = 50 μs, it can be observed that the performance of this designed circuit has a offast Time/Div response = speed, 50 µs, itand can there be observed is no waveform that the performancedistortion caused of this by designed the RC charg circuiting has and a fast response speed, and there is no waveform distortion caused by the RC charging and fastdischarg responseing time. speed, At and the there same is time, no waveform in the simulation distortion results caused of by Fig theure RC 16, charging it can also and be discharging time. At the same time, in the simulation results of Figure 16, it can also be dischargingobserved that time. there At is the no same sine time,wave inphase the simulationshift caused results by the of delay Figure of 16RC, it charging can also and be observed that there is no sine wave phase shift caused by the delay of RC charging and observeddischarging that time there when is no using sine wavethe BJT phase noise shift filter caused. by the delay of RC charging and dischargingdischarging timetimewhen when usingusing thethe BJTBJT noisenoise ﬁlter. filter.

Figure 15. Square wave response. FigureFigure 15.15. SquareSquare wavewave response.response.

Figure 16. Sinusoidal wave response. FigureFigure 16.16. SinusoidalSinusoidal wavewave response.response. 4.1.4. Noise Attenuation Analysis 4.1.4.4.1.4. NoiseNoise AttenuationAttenuation AnalysisAnalysis As described in Section 4.1.3., using the same input signal of 10 KHz, 1.5 Vp square AsAs describeddescribed inin SectionSection 4.1.34.1.3.,, using using the the same same input input signal signal of of 10 10 KHz, KHz, 1.5 1.5 Vp Vp square square wave mixed with 5 Vp-p noise, the 5 Vp-p noise is generated by Multisim Function wavewave mixedmixed with with 55 Vp-p Vp-p noise,noise, the the 55 Vp-pVp-p noisenoise is is generated generated by by Multisim Multisim Function Function Generator Interface. Figure 17 and Table 2 show the designed circuit noise filtering be- Sensors 2021, 21, x FOR PEER REVIEWGeneratorGenerator Interface. Interface. Figure Figure 17 17 and and Table Table2 show 2 show the designed the designed circuit circuit noise ﬁlteringnoise filtering behavior12 of be-18 havior and effect. In Table 2, the one standard deviation (one sigma; 1휎) of the input andhav effect.ior and In effect. Table2 In, the Table one standard2, the one deviation standard (one deviation sigma; 1(oneσ) of sigma; the input 1휎 noise) of the signal input is noise signal is 81 mVrms. It can obtain 19 mVrms of output noise signal. This can show 81noise mVrms. signal It canis 81 obtain mVrms 19. mVrms It can obtain of output 19 m noiseVrms signal. of output This cannoise show signal. that theThis amplitude can show that the amplitude of the noise in one sigma (1휎) is attenuated by 76.5%. Because the ofthat the the noise amplitude in one sigma of the (1 noiseσ) is attenuatedin one sigma by 76.5%.(1휎) is Becauseattenuated the noiseby 76.5 distribution%. Because is the a coverednoise distribution with six standard is a Gaussian deviations distribution (six sigma;, a range 6𝜎). Inof Figure99.97% 17 noise and amplitudeTable 2, the can six be Gaussiannoise distribution distribution, is a a Gaussian range of99.97% distribution noise, amplitude a range of can 99.97% be covered noise withamplitude six standard can be standarddeviations deviations (six sigma; (six6σ ).sigma; In Figure 6𝜎) 17of andthe Tableinput2 ,noise the six signal standard is 0.486 deviations Vrms, resulted (six sigma; in the6σ) output of the input noise noise signal signal of 0.114 is 0.486 Vrms. Vrms, It resultedshows that in the the output overall noise noise signal amplitude of 0.114 is Vrms. also attenuatedIt shows that by the76.5%. overall noise amplitude is also attenuated by 76.5%.

FigureFigure 17. 17. DesignedDesigned circuit circuit noise noise filtering ﬁltering behavior. behavior.

Table 2. Designed circuit noise filtering effect.

1𝝈 𝟑𝝈 𝟔𝝈

V 0.081 V 0.241 V 0.486 V V 0.019 V 0.057 V 0.114 V

4.1.5. Response Speed Analysis As shown in Figure 18 and Table 3, the fastest response speed of this designed noise filter can reach 0.27 μs, it is the time from 10% Vp-p rise to 90% Vp-p. If the voltage variation is taken into consideration, the fast slew rate of 6.34 volts per microsecond (6.34 V/μs) can be obtained, that is the slope of the voltage change in 0.27 μs, which is much greater than the 0.07 volts per microsecond (0.07 V/μs) of the conventional filter. In Table 3, the ∆𝑇 and ∆𝑇 are defined as follows:

1. ∆𝑇 = Time from 10% Vp-p rise to 90% Vp-p. 2. ∆𝑇 = Time from 90% Vp-p fall to 10% Vp-p.

Figure 18. Response speed of the designed noise filter.

Table 3. The data of the designed noise filter response speed.

Items ∆𝑻𝒖𝒑 (𝛍𝐬) ∆𝑻𝑫𝒐𝒘𝒏 (𝛍𝐬) BJT 0.27 0.27 2 order 25.48 25.47 3 order 31.57 31.53 4 order 33.69 33.63

4.1.6. Fourier Amplitude Spectrum Analysis

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covered with six standard deviations (six sigma; 6휎). In Figure 17 and Table 2, the six standard deviations (six sigma; 6휎) of the input noise signal is 0.486 Vrms, resulted in the output noise signal of 0.114 Vrms. It shows that the overall noise amplitude is also attenuated by 76.5%.

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Figure 17. Designed circuit noise filtering behavior. Table 2. Designed circuit noise ﬁltering effect. Table 2. Designed circuit noise filtering effect. 1σ 3σ 6σ 1흈 ퟑ흈 ퟔ흈 Vi−rms 0.081 V 0.241 V 0.486 V Vi−rms 0.081 V 0.241 V 0.486 V Vo−rms 0.019 V 0.057 V 0.114 V Vo−rms 0.019 V 0.057 V 0.114 V

4.1.5. Response Speed Analysis 4.1.5. Response Speed Analysis As shown in Figure 18 and Table3, the fastest response speed of this designed noise As shown in Figure 18 and Table 3, the fastest response speed of this designed noise filter can reach 0.27 µs, it is the time from 10% Vp-p rise to 90% Vp-p. If the voltage variation filter can reach 0.27 μs, it is the time from 10% Vp-p rise to 90% Vp-p. If the voltage vari- is taken into consideration, the fast slew rate of 6.34 volts per microsecond (6.34 V/µs) can ation is taken into consideration, the fast slew rate of 6.34 volts per microsecond (6.34 be obtained, that is the slope of the voltage change in 0.27 µs, which is much greater than V/μs) can be obtained, that is the slope of the voltage change in 0.27 μs, which is much the 0.07 volts per microsecond (0.07 V/µs) of the conventional filter. In Table3, the ∆T greater than the 0.07 volts per microsecond (0.07 V/μs) of the conventional filter. In Taupble and ∆TDown are defined as follows: 3, the ∆푇푢푝 and ∆푇퐷표푤푛 are defined as follows: 1. ∆Tup = Time from 10% Vp-p rise to 90% Vp-p. 1. ∆푇푢푝 = Time from 10% Vp-p rise to 90% Vp-p. 2. ∆TDown = Time from 90% Vp-p fall to 10% Vp-p. 2. ∆푇퐷표푤푛 = Time from 90% Vp-p fall to 10% Vp-p.

FigureFigure 18. 18.Response Response speed speed of of the the designed designed noise noise ﬁlter. filter.

Table 3. The data of the designed noise filter response speed. Table 3. The data of the designed noise ﬁlter response speed.

Items ∆푻풖풑 (훍퐬) ∆푻푫풐풘풏 (훍퐬) Items ∆T (µs) ∆T (µs) BJT up0.27 Down0.27 2BJT order 0.2725.48 0.2725.47 23 order order 25.4831.57 25.4731.53 34 order order 31.5733.69 31.5333.63 4 order 33.69 33.63 4.1.6. Fourier Amplitude Spectrum Analysis 4.1.6. Fourier Amplitude Spectrum Analysis Figure 19 illustrates the Fourier amplitude spectrum of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise of input signal. It can be seen that the noise intensity is similar and

scattered in each spectrum. However, as the waveform of the main signal is a square wave, only odd harmonics exist in the frequency space. A Fourier series can be used to represent an ideal square wave, this Fourier series will be represented by an inﬁnite polynomial as follows [19]: ∞ ( ) = 4 sin((2k−1)2π f t) xsquare t π ∑ (2k−1) k=1 (15) 4 1 1 = π sin(2π f t) + 3 sin(6π f t) + 5 sin(10π f t) + ... Sensors 2021, 21, x FOR PEER REVIEW 13 of 18

Figure 19 illustrates the Fourier amplitude spectrum of 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise of input signal. It can be seen that the noise intensity is similar and scattered in each spectrum. However, as the waveform of the main signal is a square wave, only odd harmonics exist in the frequency space. A Fourier series can be used to represent an ideal square wave, this Fourier series will be represented by an infinite polynomial as follows [19]: 4 𝑠𝑖𝑛(2𝑘 −) 1 2𝜋𝑓𝑡 𝑥 (𝑡) = 𝜋 (2𝑘 −) 1 (15) Sensors 2021, 21, x FOR PEER REVIEW 4 1 1 13 of 18 Sensors 2021, 21, x FOR PEER REVIEW = 𝑠𝑖𝑛(2𝜋𝑓𝑡) + 𝑠𝑖𝑛(6𝜋𝑓𝑡) + 𝑠𝑖𝑛(10𝜋𝑓𝑡) +⋯ 13 of 18 𝜋 3 5 Figure 20 is the output signal Fourier amplitude spectrum of the designed noise fil- Figure 19 illustrates the Fourier amplitude spectrum of 10 KHz, 1.5 Vp square wave ter afterFigure inputting 19 illustrates the input the signalFourier of amplitude Figure 19. spectrum It shows thatof 10 the KHz, noise 1.5 is Vp attenuated square wave and mixed with 5 Vp-p noise of input signal. It can be seen that the noise intensity is similar mixedthe high with order 5 Vp-p harmonics noise ofof inputthe main signal. signal It ca stilln be remain. seen that Comparing the noise this intensity designed is similar noise and scattered in each spectrum. However, as the waveform of the main signal is a square andfilter scattered with traditional in each filtersspectrum. of Figures However, 21–23, as wherethe waveform the noise of below the main the cut-offsignal is frequency a square wave, only odd harmonics exist in the frequency space. A Fourier series can be used to wave,of 𝑓 stillonly remains, odd harmonics but the existhigh inorder the frequencyharmonics space. of the A main Fourier signal series of canexceeding be used fre- to represent an ideal square wave, this Fourier series will be represented by an infinite representquency 𝑓 an are ideal obviously square attenuated. wave, this InFourier Figure series 20, and will Table be represented4, we can see by that an infinitethe de- polynomial as follows [19]: polynomialsigned noise as filter follows maintains [19]: the integrity of the true signal and effectively suppresses the noise. In addition, in Table 4, we can find if the quintuple harmonics Fourier amplitude 4 𝑠𝑖𝑛(2𝑘 −) 1 2𝜋𝑓𝑡 4 𝑠𝑖𝑛(2𝑘 −) 1 2𝜋𝑓𝑡 spectrum peak value of𝑥 the input(𝑡) = signal is 0.25305 V, after this designed filter, the output 𝑥(𝑡) = 𝜋 (2𝑘 −) 1 Fourier amplitude spectrum peak 𝜋value of the(2𝑘 quintuple −) 1 harmonics is 0.24055 V. In (15)this (15) case of traditional second-,4 third-, and1 fourth-order 1low-pass filters, the output Fourier = 4 𝑠𝑖𝑛(2𝜋𝑓𝑡) + 1 𝑠𝑖𝑛(6𝜋𝑓𝑡) + 1 𝑠𝑖𝑛(10𝜋𝑓𝑡) +⋯ Sensors 2021, 21, 1008 amplitude spectrum =peak𝜋 𝑠𝑖𝑛 value(2𝜋𝑓𝑡 of) the+ 3 quin𝑠𝑖𝑛(tuple6𝜋𝑓𝑡) harmonics+ 5𝑠𝑖𝑛(10𝜋𝑓𝑡 has) decayed+⋯ to be 0.04923913 of 17 V, 0.019903 V, and 0.007709𝜋 V. 3 5 Figure 20 is the output signal Fourier amplitude spectrum of the designed noise fil- Figure 20 is the output signal Fourier amplitude spectrum of the designed noise filter after inputting the input signal of Figure 19. It shows that the noise is attenuated and ter after inputting the input signal of Figure 19. It shows that the noise is attenuated and the high order harmonics of the main signal still remain. Comparing this designed noise the high order harmonics of the main signal still remain. Comparing this designed noise filter with traditional filters of Figures 21–23, where the noise below the cut-off frequency filter with traditional filters of Figures 21–23, where the noise below the cut-off frequency of 𝑓 still remains, but the high order harmonics of the main signal of exceeding fre- of 𝑓 still remains, but the high order harmonics of the main signal of exceeding frequency 𝑓 are obviously attenuated. In Figure 20, and Table 4, we can see that the de- quency 𝑓 are obviously attenuated. In Figure 20, and Table 4, we can see that the designed noise filter maintains the integrity of the true signal and effectively suppresses the signed noise filter maintains the integrity of the true signal and effectively suppresses the noise. In addition, in Table 4, we can find if the quintuple harmonics Fourier amplitude noise. In addition, in Table 4, we can find if the quintuple harmonics Fourier amplitude spectrum peak value of the input signal is 0.25305 V, after this designed filter, the output spectrum peak value of the input signal is 0.25305 V, after this designed filter, the output Fourier amplitude spectrum peak value of the quintuple harmonics is 0.24055 V. In this Fourier amplitude spectrum peak value of the quintuple harmonics is 0.24055 V. In this Figure 19. Input signalcase ofFourier traditional amplitude second-, spectrum. third-, (𝑣 isand the fourth-orderoriginal signal; low-pass 𝑣 is the filters,noise signal). the output Fourier Figure 19. Inputcase signal of Fourier traditional amplitude second-, spectrum. third-, (vs andis the fourth-order original signal; low-passvn is the noisefilters, signal). the output Fourier amplitude spectrum peak value of the quintuple harmonics has decayed to be 0.049239 amplitude spectrum peak value of the quintuple harmonics has decayed to be 0.049239 V, 0.019903Figure V,20 andis the 0.007709 output signalV. Fourier amplitude spectrum of the designed noise filter V, 0.019903 V, and 0.007709 V. after inputting the input signal of Figure 19. It shows that the noise is attenuated and the high order harmonics of the main signal still remain. Comparing this designed noise filter with traditional filters of Figures 21–23, where the noise below the cut-off frequency of fc still remains, but the high order harmonics of the main signal of exceeding frequency fc are obviously attenuated. In Figure 20, and Table4, we can see that the designed noise filter maintains the integrity of the true signal and effectively suppresses the noise. In addition, in Table4, we can find if the quintuple harmonics Fourier amplitude spectrum peak value of Figurethe 20. input Designed signal noise is 0.25305 filter output V, after signal this designedFourier amplitude filter, the spectrum. output Fourier amplitude spectrum peak value of the quintuple harmonics is 0.24055 V. In this case of traditional second-, third-, and fourth-order low-pass filters, the output Fourier amplitude spectrum peak value of the Figure 19. Input signal Fourier amplitude spectrum. (𝑣 is the original signal; 𝑣 is the noise signal). Figure 19. Input signalquintuple Fourier harmonics amplitude has spectrum. decayed (𝑣 to is be the 0.049239 original V,signal; 0.019903 𝑣 isV, the and noise 0.007709 signal). V.

Figure 20. Designed noise filter output signal Fourier amplitude spectrum. FigureFigure 20. 20. DesignedDesigned noise noise filter ﬁlter output output signal signal Fourier Fourier amplitude spectrum.

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Figure 21. Second-order low-pass filter ﬁlter output signal Fourier amplitude spectrum.

Figure 22. Third-order low-pass filter ﬁlter output signal Fourier amplitude spectrum.

Figure 23. Fourth-order low-pass filter output signal Fourier amplitude spectrum.

Table 4. Fourier amplitude spectrum peak values. (BJT is the designed noise filter).

Frequency 𝒗𝒊 BJT 2 Order 3 Order 4 Order 10 KHz 1.25896 1.20108 1.54432 1.58431 1.5003 30 KHz 0.41925 0.401337 0.212284 0.148126 0.085771 50 KHz 0.25305 0.24055 0.049239 0.019903 0.007709 70 KHz 0.180795 0.171364 0.017986 0.005179 0.001454

4.1.7. Phase Response Analysis Figure 24 and Table 5 illustrate the phase response and the lead or lag angle values in a 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise of input signal. Thanks to the utilization of the RC circuit accompanied with the Schottky diodes in this filter for capturing the upper envelope and the lower envelope, the amount of stored charge is extremely small, which results in the extremely small phase delay. In Table 5, we can see the phase delay of the proposed noise filter has little lag and less than 3.43 degrees within 100 KHz. As for the other filters, the phase shift is much larger than this value.

Figure 24. Phase response of second-, third-, fourth-order, and BJT filter. (BJT is the designed noise filter, 𝑓 =𝑓 =20 KHz).

Sensors 2021, 21, x FOR PEER REVIEW 14 of 18 Sensors 2021, 21, x FOR PEER REVIEW 14 of 18

Figure 21. Second-order low-pass filter output signal Fourier amplitude spectrum. Figure 21. Second-order low-pass filter output signal Fourier amplitude spectrum.

Sensors 2021, 21, 1008 14 of 17 Figure 22. Third-order low-pass filter output signal Fourier amplitude spectrum. Figure 22. Third-order low-pass filter output signal Fourier amplitude spectrum.

Figure 23. Fourth-order low-pass filter output signal Fourier amplitude spectrum. FigureFigure 23. 23.Fourth-order Fourth-order low-pass low-pass filter filter output output signal signal Fourier Fourier amplitude amplitude spectrum. spectrum. Table 4. Fourier amplitude spectrum peak values. (BJT is the designed noise filter). TableTable 4. 4.Fourier Fourier amplitude amplitude spectrum spectrum peak peak values. values (BJT. (BJT is is the the designed designed noise noise filter). filter). Frequency 𝒗𝒊 BJT 2 Order 3 Order 4 Order FrequencyFrequency vi 풗 BJTBJT 2 Order2 Order 3 Order3 Order 44 Order Order 10 KHz 1.25896풊 1.20108 1.54432 1.58431 1.5003 10 KHz 1.25896 1.20108 1.54432 1.58431 1.5003 1030 KHz KHz 1.25896 0.41925 1.201080.401337 1.544320.212284 0.148126 1.58431 0.085771 1.5003 30 KHz30 KHz 0.419250.41925 0.4013370.401337 0.2122840.212284 0.1481260.148126 0.0857710.085771 50 KHz 0.25305 0.24055 0.049239 0.019903 0.007709 50 KHz50 KHz 0.253050.25305 0.240550.24055 0.0492390.049239 0.0199030.019903 0.0077090.007709 7070 KHz KHz 0.180795 0.180795 0.1713640.171364 0.0179860.017986 0.0051790.005179 0.001454 70 KHz 0.180795 0.171364 0.017986 0.005179 0.001454 4.1.7. Phase Response Analysis 4.1.7.4.1.7. Phase Phase Response Response Analysis Analysis Figure 24 and Table 5 illustrate the phase response and the lead or lag angle values FigureFigure 24 24 and and Table Table5 illustrate5 illustrate the the phase phase response response and and the the lead lead or or lag lag angle angle values values in a 10 KHz, 1.5 Vp square wave mixed with 5 Vp-p noise of input signal. Thanks to the inin a a 10 10 KHz, KHz, 1.51.5 VpVp squaresquare wave mixed with with 5 5 Vp Vp-p-p noise noise of of input input signal. signal. Thanks Thanks to tothe utilization of the RC circuit accompanied with the Schottky diodes in this filter for cap- theutiliz utilizationation of ofthe the RC RC circuit circuit accompanied accompanied with with the theSchottky Schottky diodes diodes in this in this filter filter for forcap- turing the upper envelope and the lower envelope, the amount of stored charge is ex- capturingturing the the upper upper envelope envelope and and the the lower lower envelope, envelope, the the amount amount of ofstored stored charge charge is isex- tremely small, which results in the extremely small phase delay. In Table 5, we can see extremelytremely small, small, which which results results in in the extremely small phase delay.delay. InIn TableTable5 ,5, we we can can see see the phase delay of the proposed noise filter has little lag and less than 3.43 degrees thethe phase phase delay delay of of the the proposed proposed noise noise filter filter has littlehas laglittle and lag less and than less 3.43 than degrees 3.43 within degrees within 100 KHz. As for the other filters, the phase shift is much larger than this value. 100with KHz.in 10 As0 K forHz. the As other for the filters, other the filters, phase the shift phase is much shift largeris much than larger this than value. this value.

Figure 24. Phase response of second-, third-, fourth-order, and BJT filter. (BJT is the designed noise Figure 24. Phase response of second-, third-, fourth-order, and BJT filter. (BJT is the designed noise filter,Figure 𝑓 24.=𝑓Phase =20 response KHz). of second-, third-, fourth-order, and BJT ﬁlter. (BJT is the designed noise filter, 푓푐 = 푓−3푑퐵 = 20 KHz). ﬁlter, fc = f−3dB = 20 KHz).

Table 5. The lead or lag angle values of phase response at 1 KHz, 10 KHz, 100 KHz, and 1 MHz.

( fc = f−3dB = 20 KHz).

Frequency BJT 2 Order 3 Order 4 Order 1 KHz −0.04◦ −5.09◦ −7.63◦ −10.16◦ 10 KHz −0.36◦ −55.77◦ −79.56◦ −111.17◦ 100 KHz −3.43◦ −170.14◦ 110.90◦ 19.75◦ 1 MHz −34.36◦ 83.59◦ −22.37◦ 166.83◦

4.2. Circuit Implementation Figure 25 shows the designed noise filter PCB layout and the final product. The PCB layout design is developed by Protel 99 SE. The size of the noise filter is 12 mm × 19 mm with a cost of less than NT$100. Using this method of noise filtering, the input signal can Sensors 2021, 21, x FOR PEER REVIEW 15 of 18

Sensors 2021, 21, x FOR PEER REVIEW 15 of 18

Table 5. The lead or lag angle values of phase response at 1 KHz, 10 KHz, 100 KHz, and 1 MHz. (𝑓 =𝑓 = 20 KHz). Table 5. The lead or lag angle values of phase response at 1 KHz, 10 KHz, 100 KHz, and 1 MHz. Frequency (𝑓 =𝑓 BJT= 20 KHz). 2 Order 3 Order 4 Order 1 KHz −0.04° −5.09° −7.63° −10.16° Frequency BJT 2 Order 3 Order 4 Order 10 KHz −0.36° −55.77° −79.56° −111.17° 1 KHz −0.04° −5.09° −7.63° −10.16° 100 KHz −3.43° −170.14° 110.90° 19.75° 10 KHz −0.36° −55.77° −79.56° −111.17° 1 MHz −34.36° 83.59° −22.37° 166.83° 100 KHz −3.43° −170.14° 110.90° 19.75° 1 MHz −34.36° 83.59° −22.37° 166.83° 4.2. Circuit Implementation Sensors 2021, 21, 1008 Figure 25 shows4.2. Circuit the designedImplementation noise filter PCB layout and the final product. The PCB 15 of 17 layout design is developed by Protel 99 SE. The size of the noise filter is 12 mm × 19 mm Figure 25 shows the designed noise filter PCB layout and the final product. The PCB with a cost of less than NT$100. Using this method of noise filtering, the input signal can layout design is developed by Protel 99 SE. The size of the noise filter is 12 mm × 19 mm be directly processed through the circuits without performing complicated operations with a cost of less than NT$100. Using this method of noise filtering, the input signal can such as DSP. be directly processed through the circuits without performing complicated operations such be directly processed through the circuits without performing complicated operations Figure 26as shows DSP. the input signal with noise of 3.04 Vp-p, after the designed noise such as DSP. filter of circuit implementationFigure 26 shows test, the a input120 mVp-p signal withoutput noise noise of 3.04 signal Vp-p, can after be theobtained, designed noise ﬁlter Figure 26 shows the input signal with noise of 3.04 Vp-p, after the designed noise which can reduceof circuit the input implementation noise up to test,4% of a 120itself. mVp-p output noise signal can be obtained, which can reducefilter of the circuit input implementation noise up to 4% of test, itself. a 120 mVp-p output noise signal can be obtained, which can reduce the input noise up to 4% of itself.

(a) (b) (a) (b)

(c) Figure 25. (aFigure) Component 25. (a) sideComponent of the designed side of the noise design filtered PCB noise Layout; filter PCB (b) Soldering Layout;(c) (b side) Soldering of the designed side of the noise filter PCB designed noise filter PCB Layout; (c) The designed noise filter product. Layout; (c) The designed noiseFigure filter 25. product. (a) Component side of the designed noise filter PCB Layout; (b) Soldering side of the designed noise filter PCB Layout; (c) The designed noise filter product.

Figure 26. Test of noise filtering effect on real circuit board. FigureFigure 26. Test 26. of Test noise of ﬁltering noise filtering effect on effect real on circuit real board.circuit board.

5. Conclusions 5.1. Technique Features Signal processing and noise ﬁltering have always been very important subjects in

technology, whether used in measurement or communication. This research investigates the distribution characteristics of the noise commonly found in electronic circuits, and applies a scientiﬁc understanding of the development of noise ﬁlter technology. According to the noise distribution characteristics, this paper proposes a method to split the signal source into two parts, separately extracting the high peaks and the low peaks by individual circuit design, and using the RC charge and discharge principle to form the upper envelope signal (vh) and the lower envelope signal (vl). Further, the extracted upper and lower envelopes and the input signal all have a Gaussian distribution. Therefore, by taking the upper envelope generated by the original signal plus noise signal and the lower envelope Sensors 2021, 21, 1008 16 of 17

generated by the original signal minus noise signal, the numbers are averaged to reduce the Gaussian noise. Significantly, the noise filter of this design has no cut-off frequency, so the high order harmonics of the actual signal remain unchanged until 1 MHz. This is limited by the operation speed of the transistor. Thanks to the fast response of the noise filter, the input signal can be kept up to 6.34 V/µs and the actual signal integrity is preserved, which is better than the previous work [14] in a slew rate of 472 V/ms. Also, by utilizing the RC circuit accompanied with Schottky diodes in the circuit design, a much smaller phase delay is achieved than when a conventional filter is used. Several important features of the proposed technique are summarized as following: • High response speed • At the same input frequency signal, unlike a traditional filter, this filter can be op- erated at a slew rate (charge and discharge slope) of 6.34 V/µs. Under the same conditions, the traditional filter can only reach at a speed of 0.09 V/µs. • Minimal phase delay A traditional filter is delayed due to the charging of the RC circuit. However, the circuit in the filter designed here creates a high response speed through the combined use of the RC charging and discharging circuit and the Schottky diodes. Since the above circuits capture the upper envelope and the lower envelope, the quantity of stored electric charge is extremely small, and the resulting phase delay is correspondingly small. The phase delay is approximately 3 degrees. • Noise attenuation In the circuit simulation test, an input noise signal of 0.486 Vrms is given to this filter, achieving an output noise signal of 0.11 Vrms, with an attenuation of 25% of the raw signal. Through the physical PCB circuit test, an input signal with 3 Vp-p Gaussian Noise, after this noise filter, achieves the output noise signal of 120 mVp-p. The noise can be suppressed to 4% of the raw signal. • Maintain signal integrity Rather than following the traditional filter, the range of the passband is set at the cutoff frequency, but this filter is not affected by the cutoff frequency. After the signal is transmitted from the filter, the high order harmonics information within 1 MHz can still be retained. The Fourier amplitude spectrum peak values shown in Table4 reveal that the quintuple harmonic Fourier amplitude spectrum of the raw signal is 0.25 V, and the quintuple harmonic output from the designed filter is 0.24 V. Compared with the traditional second-order low-pass filter, the quintuple harmonic output signal has decayed to 0.049 V, while in the fourth-order low-pass filter, the output signal of quintuple harmonic is decayed even more to 0.0077 V.

5.2. Future Prospects The most important aspect of creating a noise filter is to match a detection system or a measurement system to assist the noise filtering, as to highlight its value. In the future, it is hoped that this structure can be used in measurement systems to assist in noise removal. Although the noise filter in this study can attenuate the Gaussian distributed noise, it is limited by the frequency response of using the transistors in the overall filters, the output signal gain will begin to attenuate at about 1 MHz when using transistor components. However, if the noise filter is applied to signals around 200 kHz, in square wave, 5 times of this frequency of high order harmonics will fall at the position of 1 MHz, in this case it will cause distortion. Addressing this distortion would be a valuable direction for future research to take.

Author Contributions: Proposed the main idea and performed the investigation, F.-M.Y., K.-C.L., and K.-W.J.; Developed the software, J.-Y.L. and F.-M.Y.; Wrote and revised the whole manuscript, F.-M.Y. and K.-C.L.; Lead the project, F.-M.Y. and R.-S.C. All authors have read and agreed to the published version of the manuscript. Sensors 2021, 21, 1008 17 of 17

Funding: This research was funded by the Ministry of Science and Technology, Taiwan, R.O.C., under Contract MOST108-2637-E-129-002. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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