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1 1 Jarrett B. Brown, Michael Dropmann, Rene Laufer, Christoph Montag and Truell W. Hyde Characterization of Piezo Sensors particles on the order of microns can also cause mission AbstractPiezoelectric materials produce an electric signal in affecting damage, and particles of these sizes have not been response to mechanical deformation. Using this property, Piezo reliably modeled in the near Earth environment [6][5]. Being Sensors offer a way to detect debris in space, and volcanic ash in the atmosphere. By detecting these particles, the threat of able to detect impacts from particles of this size would allow damage to important instruments, and aircraft engines, can be for satellites and probes to collect information on their reduced. To assess the viability of this use, the electrical response distribution to improve current models. These improved of such sensors over the relevant impulse range must be models will allow for the risks posed by micron sized debris to determined. This was done by applying band-pass filters in be more accurately assessed and handled. series, to eliminate frequencies not associated with the signal On Earth, volcanic ash particles can erode the exterior of an produced in response to the impact. The decay of the magnitude of this signal was analyzed, by looking at local maxima, and using aircraft, while acidic aerosol particles cause acid etching on a smoothing function to eliminate oscillation. Finally, a decaying the aircraft's surfaces [7][8]. These particles can be as small as exponential was fit to the smoothed signal. The electrical a micron, and are primarily silicate materials [9]. Due to the response was not linearly dependent on the strength of the small size of the ash particles, ash clouds cannot be detected impulse for the entire impulse range examined. The electrical by aircraft radar, or seen by pilots [10]. Jet engines operate at signals were detectable, and distinguishable for each impulse temperatures of nearly 2000 C, while the melting point of the which was examined. The Soft Piezo Sensor is superior for detecting isolated particle impacts with impulses from 3.416 and glassy silicate rock in the ash cloud is about 7001100 C , at 172.7Ns. these temperatures silica can melt and accumulate in the turbine [11]. This can block off airflow to the engine, and Index TermsCurve fitting, Digital filters, Piezoelectric cause the engine to stop [7]. By detecting the ash particles, devices, Sensor phenomena and characterization pilots would be able to alter course to prevent potentially catastrophic damage, ensuring the safety of passengers, and helping to maintain the functional integrity of the aircraft. I. INTRODUCTION Ash colliding with aircraft, and debris collisions with O VER the course of space flights collisions with debris poses a serious threat to the integrity of mission necessary instruments. This risk will only increase, with spacecraft occur over a similar range of impulses [12][13]. Piezo plates can be used to detect impacts with particles that produce impulses over these ranges [13]. Piezoelectric increased traffic in Low Earth Orbit, as an increase in the materials, like those found in Piezo plates, exhibit the number of satellites makes the collision of satellites more piezoelectric effect. The piezoelectric effect is the likely. Due to collisions, the debris field will grow over the electromechanical reaction between the electrical and next 200 years, even without further launches, posing a greater mechanical states of crystalline materials. This electrical threat to future missions [1]. China's destruction of a satellite signal can be used to identify the impulse of the colliding in orbit, in 2007, produced over 2,000 trackable debris particle, which can then be used to determine whether fragments, further increasing risk of satellite collisions [2]. In instruments, or turbine operations, are in danger. 2009 there was a collision between two satellites, Iridium 33 Impact from particles is not the only source of deformation and Cosmos 2251, which produced thousands of debris to Piezo plates being used to detect debris, or ash. There are fragments [3]. In near earth orbit, ground-based radar can be many other sources of deformation, which create electrical used to track particles larger than 10 cm [4]. In the near Earth signals. These electrical signals, caused by factors other than environment dust and debris models provide sufficient the impact of particles, produce background noise in the information for particles larger than 1 mm [5]. However, electric signal that is produced by the Piezo plate at the time of the impact of a particle. This noise comes from various Manuscript received August 6, 2013. This work was supported in part by the National Science Foundation under Program Solicitation NSF 13-542. sources, including: deformation from the mounting of the J. B. Brown is with CASPER, Baylor University, Waco, TX 76798 USA plate, thermal noise, electromagnetic noise, and vibrations due (e-mail: [email protected]). to the engine or turbulence [14]. This paper accounts for noise M. Dropmann was with the University of Stuttgart, Stuttgart, Germany. due to mounting, other noise sources must be identified and He is now with CASPER, Baylor University, Waco, TX 76798 USA (e-mail: [email protected]). accounted for depending on the environment of the sensor. R. Laufer was with the University of Stuttgart, Stuttgart, Germany. He There are two different types of Piezos that will be is now with CASPER, Baylor University, Waco, TX 76798 USA (e-mail: considered in this paper: Hard Piezos, and Soft Piezos. Hard [email protected]). C. Montag is with the Technical University of Dresden, Dresden, and Soft Piezos are distinguished by their damping. Hard Germany. He was with CASPER, Baylor University, Waco, TX 76798 USA Piezos have a high damping value, while Soft Piezos have a (e-mail: [email protected]). low damping value. Because of the larger damping value, T. W.Hyde is with CASPER, Baylor University, Waco, TX 76798 USA (e-mail: [email protected]). signals from Hard Piezos oscillate less than Soft Piezos. Thus

2 2 noise is a more significant problem in Soft Piezos, as the Butterworth Band-stop filter centered at 63kHz with a 2kHz initial signal produced by some deformation due to noise will width; a Butterworth Band-stop filter centered at 52.2kHz with oscillate more. a 2kHz width; a Butterworth Band-stop filter centered at The goal of this paper is to characterize the response of both 50.2kHz with a 1kHz width; a High-pass Chebysev Type 1 Hard and Soft Piezo Plates over a range of impulses. It is filter with a 3dB point at 200 Hz. The 200 Hz Chebysev filter expected that the Piezo Sensors will display a linear is applied because the period associated with this frequency is dependence, at least over some impulse range, of the the time length of the entire signal. The noise which dominates magnitude of their electrical signals on the impulse of the the signal has many full periods over the course of the decay impact which produces the electrical signal. Through the of the impulse signal, so attenuating these low frequencies is characterization of the electric signals produced by Piezo justified. Sensors, in response to impacts of a wide range of impulses, The vector is then transformed to the frequency domain, the actual form of the relationship between the magnitude of using a Fast Fourier Transform (FFT). This can be seen at the the electrical signal and the impulse of the particle can be bottom of Fig. 1., for comparison, a FFT of the unfiltered determined. Additionally, by characterizing both Hard and signal is shown at the top of Fig. 1. Soft Piezos, it should be possible to determine which type of Piezo is preferable under different circumstances. This was accomplished by the use of computational methods, to filter, and analyze the electrical signals produced by the impact of a particle over a range of impulses. II.METHODS A. Data Collection The experiments which produced the data that was used in the analysis found in this paper, were designed and run by Montag.[13] The experiment consisted of dropping aluminum, and iron spheres of various radii, from a height of 60mm, and 19.5mm, onto a mounted Piezo Plate. By varying the height of the drop, and the radius and material of the sphere, different impulses were achieved. Montag's experiments used two different set ups, direct impact of falling particles on the Piezo Plate, and indirect impact of falling particles on the Piezo Plate. All data in this paper was taken from experiments using the direct impact method. Both Hard and Soft Piezo Plates were used to detect particles at every impulse. For each type of Piezo, and each impulse, at least 10 drops were made, and Fig. 1. Magnitude as a function of Frequency. Raw Signal (top). Signal filtered as described in Section B, in order to determine the noise frequency the resulting signals recorded, as a vector of the magnitude of (bottom). Signals are from an impact of 6.583Ns on a Hard Piezo. Shows the signal, in mV, over time, in microseconds. how filtering out certain peaks helps in identifying the noise frequency. B. Identifying the Noise Frequency The index at which the FFT vector has the greatest absolute For each dataset with distinct parameters, all vectors from value is identified, and the frequency corresponding to this signals with these parameters are loaded into MATLAB. index is considered to be the noise frequency. Signals have a duration of 5ms, with a time resolution of 2s. The maximal voltage reached in a signal is between 10V, and C. Identify the Characteristic Frequency 2mV, depending on the impulse of the particle that struck the A Butterworth Band-stop filter is constructed to attenuate Piezo Plate, and whether the Piezo Plate is Hard or Soft. Each the noise, centered at the noise frequency, with a width of signal is centered at 0 mV, by subtracting the mean of the 2kHz. The filter has order 2. The order of a filter affects the vector, from each element of the vector. amplitude response, and sharpness of a filter. A filter's order Six filters were applied in series, to attenuate the signal over determines the power with which the attenuation changes frequency ranges which have magnitudes on the same order as between the pass and stop bands. The filters in this paper, are the noise frequency. These frequencies are not considered to specified by their 3dB points, so the extent of attenuation is be the primary noise frequency, because their relatively high controlled by the width of the filter, and its order. The wider a magnitude occurs over a short impulse range, and in most filter, the greater the maximal attenuation can be, since cases is not the highest magnitude at these frequencies. The additional width provides a greater frequency range over filters which are applied are: a Butterworth Band-stop filter which attenuation can be increased. Furthermore, by centered at 188kHz with a 1kHz width; a Butterworth Band- increasing the order of the filter, the attenuation can change stop filter centered at 118kHz with a 1kHz width; a more rapidly over a shorter frequency range. This allows for

3 3 the creation of filters with the same width, and different that it was observed that the greatest magnitude occurs at the attenuations. Since the order of the filter controls the power by second multiple of the characteristic frequency in some which the magnitude response of the filter changes, it also signals. In a continuation of this analysis, it would be controls the sharpness of the filter. A filter with a low order necessary to allow for the dominant frequency to be any changes its magnitude response slowly, and consequently is multiple of the characteristic frequency, but this was comparatively shallow; a filter with a high order changes its unnecessary for the data sets analyzed in this paper. magnitude quickly, and consequently is nearly constant in D. Filtering magnitude response above the 3dB point, then drops sharply at around the 3dB. Butterworth Band-pass filters, with widths of 10kHz, are Additional filters are applied, a Butterworth Band-stop filter applied in series to the original signal. The first is centered at of order 200 centered at 63kHz, with a width of 14kHz, and a the characteristic frequency, and every subsequent filter is High-pass Chebysev Type 1 filter with a 3dB point at centered at an integer multiple of the frequency. The last filter 48.4kHz. These filters, as well as the filter centered at the is centered at the greatest integer multiple, such that the entire noise frequency, are applied to the original signal. In the passband lies within the frequency range. The maximal frequency domain, this results in the signal shown at the frequency, which can be determined from the FFT, is the bottom of Fig. 2., at the top of Fig. 2. an FFT of the raw signal Nyquist Frequency, which is determined by: is shown for comparison. The Chebysev filter attenuates low 1 f N = frequencies, the reason this is necessary, is that for some 2T S (1) impulses, among the lower frequencies there are many small where fN is the Nyquist Frequency, and TS is the sampling peaks. Sharper peaks occur at higher frequencies, but they period. may have lower magnitudes, than the band of peaks at low In this case, the maximal frequency is 250 kHz, because the frequencies. Applying a high-pass filter reduces the sampling rate is 2 microseconds. Fig. 3. Voltage as a function of time. Raw Signal (top). Signal filtered with band-pass filters in series (bottom). Signals are from an impact of 3.416Ns on a Soft Piezo. Shows how applying band-pass filters in series, at multiples Fig. 2. Magnitude as a function of frequency. Raw Signal (top). Signal with of the characteristic frequency, reduces noise in the signal. a band-stop filter applied at noise frequency, as well as other frequencies (bottom). Signals are from an impact of 3.416Ns on a Soft Piezo. Shows how filtering noise frequency helps in identifying the characteristic An additional Butterworth Band-stop filter of order 200 frequency. centered at 63kHz with a width of 1kHz, is applied to the signal. A High-pass Chebysev Type 1 filter is applied with a magnitudes at low frequencies, allowing for the sharp peaks at 3dB point at 975 Hz below the characteristic frequency. higher frequencies to be selected, rather than the shallow If the characteristic frequency is over 1kHz greater or less peaks at low frequencies. than 59kHz, then an additional Butterworth Band-stop filter, Now, the index at which this filtered vector has its maximal centered at 118kHz is applied. This filter has a width of absolute value, taking into account both imaginary and real 1.2kHz. The signal that results from the application of these parts, is identified, and the frequency associated with this filters can be seen at the bottom of Fig. 3., the top of Fig. 3. index is taken to be the characteristic frequency. The shows the signal before the application of these filters. frequency is additionally constrained to be at least 500 Hz E. Isolating Decaying Portion of Signal away from any integer multiple of the noise frequency. If the frequency identified by this method lies between the 0 th and 2nd A new vector is produced, using the signal filtered as multiple of the noise frequency, it is taken to be the described in D. An inverse fast Fourier transform is applied to characteristic frequency. If it is instead between the 2 nd and 4th the FFT of this signal, with an additional parameter, specifying multiple of the noise frequency, it is taken to be the second a new extended length, such that the new vector's length is 40 multiple of the characteristic frequency. The reason for this, is times the original length. Interpolation is used to assign new

4 4 values between those in the original vector, so that the vector amplitude, the index and value of every peak in the smoothed has the proper length. The duration of the signal in time is signal is found, to calculate the average period of oscillation. unchanged, however its time resolution has increased. Then Another smoothing function is applied to this signal, again the first part of the vector, until the maximal value is reached, with a moving average, with a width that is three times the is removed from the vector. average period of oscillation of the signal, or 101 indexes if The index and value of every peak, or local maximum, is the average period is less than 34 indexes. The resulting plot found. Then two new vectors are made. The first contains the can be seen in blue in Fig. 4., with the filtered signal in green value of each peak, and the second contains the time at which in Fig. 4., to illustrate the results of analyzing, and smoothing each peak occurs. the peaks of this signal. For impulses between 11.55 and 59.54Ns , the FFT of The purpose of the second smoothing function is to signals from Hard Piezos have a number of significant peaks eliminate low frequency oscillations, which may correspond to in close frequency proximity. This occurs when a single peak beats if any exists in the signal. This leaves us with a signal splits into two peaks, and ceases when only one peak remains. which has very little oscillation, and can be best fit by an The exact nature of this splitting is unknown, and beyond the exponential decay. scope of this paper. Due to the close proximity, the non- F. Fitting Exponential Decay to Signal characteristic peaks are not significantly attenuated by the band-pass filters. As a result, the time-domain signal shows The smoothed signal is fit to a function of the form: t interference of these peaks leading to an amplitude oscillation, f = Ae +c (2) so the local maximal values of the signal are modeled by an where A is the magnitude of the exponential, is the decay exponentially decaying sine wave, rather than a simple time, t is time that has passed since impact, and c is the offset. exponential decay. An initial value for A is determined by the value of the A smoothing function, with a moving average, and a width smoothed vector at the first index. An initial value for is of 53 indexes, is applied to the first vector. This eliminates determined by the time at which the value of the smoothed high frequency noise in the signal, leaving only low frequency vector first falls below 1/e times the initial value for A. An oscillations, which corresponds to frequency of the beats initial value for c is determined by the mean value of the formed in the signal, from peak interference, if any such beats second half of the smoothed vector. exist These initial values, and the smoothed vector are fed to a If the signal exhibits significant oscillations in amplitude, function which uses a least squares algorithm to minimize the which were determined to occur in Hard Piezos at impulses differences between the given, vector, and the function fitting from 11.55 to 59.54Ns, then the time and voltage values of it. There are absolute bound placed on each parameter, for A the local maxima are taken, and used as the smoothed signal in and the bounds are the initial guess plus or minus 50 percent Part F. This way amplitude oscillations don't influence the fit, of the initial guess. For c it is between 0 and twice the initial since every point being fit occurs at a local maxima, the only guess. For every iteration of the least squares algorithm, there change in magnitude is due to the exponential decay. are 40 values, evenly spaced between the bounds established If the signal doesn't exhibit significant oscillations in for that iteration, and every combination of values is tested. If the sum of the squares of the differences between the vector and a fit function with the parameters of the most recent iteration, is less than the previously minimal sum of the squares of differences between the vector and any fit function used so far, then the fit function with the new parameters is considered to be the best fit function so far. Fig. 4. Voltage as a function of time. Filtered Signal (green). Exponential decay of the Filtered Signal's Magnitude, produced from the Filtered Signal, as described in section E (blue). Signals are from an impact of 3.416Ns on a Soft Piezo. Shows how looking at local maxima, and smoothing to eliminate oscillation in the signal, results in a close approximation of the exponential decay of the signal. Fig. 5. Voltage as a function of time. Smoothed Signal, with its exponential fit. Signals are from an impact of 3.416Ns on a Soft Piezo. Shows how well the smoothed signal fits to an exponential decay

5 5 For a fit function to be selected as the best fit to the input noise ratio to get a value and corresponding uncertainty for vector, the sum (over time values) of the squares of the each value at that impulse. difference between the vector and the fit function (applied at the same time values) must be less than the same sum for the III. RESULTS previously optimal fit function. So finding the optimal The Decay Time of Soft Piezos, Fig. 6., increases from the function simply amounts to finding the parameters which lowest to the second lowest impulse, and then is in agreement minimize the squares of the differences between the vector, for the remainder of the impulse range, at a value of 290 s and the fit function. This is equivalent to minimizing the with an uncertainty of 6s . deviation of the fit from the vector, which is the same as The Decay Time of Hard Piezos, Fig. 6., is in agreement for improving the fit of the fitting function. After every the four lowest impulses, at a value of 717s with an combination of parameters within the established bounds is uncertainty of 47s. The decay time then increases, before tried, if this is the 1st or 2nd iteration over parameter values then decreasing at higher impulses. The exact form, and location of another iteration over parameter values, is made, where new the peak can't be determined without further data. bounds are established, if this is the 3 rd iteration, then the parameters are returned as the parameters of the optimal fit function. For each iteration, new bounds for each parameter are determined by the maximal tested value for the parameter which is below the current best value for the parameter, and the minimal tested value for the parameter which is above the current optimal value for the parameter. The resulting fit function can be seen in red in Fig. 5., plot against the smoothed function in blue. Having found optimal values for the parameters of the fit function, and we have found the value of the decay time, offset and magnitude for the decay of the signal. In addition to the use of these values alone, we can use them to calculate the signal to noise ratio for the signal. This can be done using the function: c+ A R SN = c (3) where c is the offset from equation 2, and A is the magnitude from equation 2. We have additionally found the characteristic frequency of Fig. 7. Magnitude as a function of impulse. Shows strength of signal in the signal, in part C. For each impulse, we can find the mean response to particles of different impulses. and standard deviation of the values for decay time, magnitude, and characteristic frequency, and the signal to The Magnitude of Hard Piezos, Fig. 7., increases to a peak, which occurs between 17.47 and 172.7Ns, before decreasing. The Magnitude of the Soft Piezo, Fig. 7., increases with amplitude at low impulses, however its response at high impulses can't be well determined, except that there is no marked increase as at low impulse. The maxima for the Soft Piezos also occurs at a fewer number of standard deviations from the points about the maxima. An additional feature of the magnitude is a peak at the second lowest impulse, for the Soft Piezo. In the Hard Piezo, the uncertainty among the lowest impulses is too large to be able to discern whether the peak occurs here as well. It's important to note that the magnitude referred to is the magnitude of the filtered, and smoothed signal, not the raw signal. The magnitude of the filtered signal is orders of magnitude smaller than the original signal. The characteristic frequency of Hard Piezo, Fig. 8., remains approximately constant for low impulses ranging from 58.7 to 59kHz, for impulses between 3.416 and 59.54Ns. At an impulse of 17.27, the characteristic frequency is found to be 51.9 kHz with an uncertainty of 800Hz. Similarly, the characteristic frequency of Soft Piezos, Fig. Fig. 6. Decay Time as a function of impulse. Shows time it takes for the signal produced due to the impact of a particle to dampen, and for the signal to become indistinguishable from a signal where no particle impacted the sensor. This is displayed over a variety of impulses for the colliding particle.

6 6 9., remains constant at 53.4kHz, for impulses between 3.416 IV. DISCUSSION and 59.54Ns. At an impulse of 172.7Ns, the characteristic These results indicate that the response of both Hard and frequency is found to be 55.1 kHz with an uncertainty of Soft behavior is not linear over a large range of impulses. 100Hz. The Decay Time for Soft Piezos is in agreement over an impulse range of 5.992 to 172.7Ns, the value at 3.416Ns, differs from this value by 2.5 standard deviations, indicating that the decay time is not constant over the entire impulse range. While a constant decay time does not imply a linear relationship, it would be easier to analyze signals with a constant decay time, as this would allow for the time resolution, and duration used in analyzing the signal to be optimized for all signals. The decay time of Hard Piezos is in agreement for an impulse range of 3.416 to 11.55Ns, with a value of 717s, and an uncertainty of 47s. Over the remaining impulses, the decay time increases, and decreases again. As with the Soft Piezo, the Hard Piezo shows that important characteristics of the signal are not independent of impulse The characteristic frequency of both Hard and Soft Piezos, remains approximately constant with impulse, until there is a major change in the characteristic frequency of both Piezos at an impulse of 172.7Ns. The non-constant, behavior of the decay time and characteristic frequency, suggests that the Piezo's response varies with impulse. Additionally, the nonlinear dependence of Fig. 8. Characteristic Frequency as a function of impulse. Shows the magnitude on impulse suggests that the response of the Piezo characteristic frequency of the signal response of the Hard and Soft Piezos. does not simply scale with the impulse. How linear the relationship between the signal's magnitude, and the impulse The Signal to Noise Ratio, Fig. 9., is higher for Soft Piezos of the impacting particle, can be measured using linear at impulses of 3.416, 17.47, 33.68, 172.7 Ns. For other regression methods on the mean values of the magnitude. This impulses, the Signal to Noise ratio is in agreement for Hard method does not take into account the uncertainties in the and Soft Piezos. values for magnitude at different impulses. Over an impulse range of 3.416 to 33.68Ns the magnitude of the signal produced by both the Hard and Soft Piezo, is approximately linearly related to the impulse of the particle impacting the sensor. The Coefficient of Determination indicates how good a linear fit to a set of data is, it ranges from 0 to 1, with 1 being a perfect linear fit, and 0 be a pessimal linear fit. For the Hard Piezo over this impulse range, the coefficient of determination is 0.9525; the coefficient of determination for the Soft Piezo over this range is 0.9812. This shows that both Hard and Soft Piezos respond linearly within this impulse range. Over the entire range of impulses, the coefficient of determination for the Hard piezo is 4.34110 -5, and for the Soft Piezo the coefficient of determination is 0.1228. Such low coefficients of determination indicate that a linear fit is very poor over this range of impulses, so we can conclude that the response of the Piezo Sensors is non-linear over some portion of the impulse range that was tested. Furthermore, since the region in which the sensors respond linearly occurs at low impulses, and the response only becomes non linear as highest tested impulses are also Fig. 9. Signal to Noise Ratio as a function of impulse. Shows the relative included in the fit, this suggests that the Piezos respond magnitude of the signal corresponding to the particle being detected, with linearly for low impulses (which was seen for observed respect to the magnitude of the signal corresponding to noise. impulses) then at some higher impulse the Piezo material becomes saturated and responds in a nonlinear manner. That the signal to noise ratio is generally larger for Soft

7 7 Piezos, indicates that with the filtering and analysis methods analysis. By analyzing the response at more impulses within utilized, Soft Piezos produce signals which more closely this region, the exact regions where these peaks occur, as well match a simple exponential decay, with no offset. as the concavity of these regions, and more generally the From these results, it can also be determined, that Soft behavior in these regions can be determined. By analyzing Piezos are better suited for detecting a single impacting regions of higher or lower impulses, it will be possible to particle with an impulse in the range of 3.416 to 172.7Ns. At determine if there are definite limits on what impulses produce least, when using the filtering and analysis methods outlined distinct, and detectable signals in the Piezos, and whether such in this paper. bounds include the desired range of impulses for detecting The Soft Piezo exhibits less impulse dependence for both space debris, and volcanic ash. characteristic frequency, and decay time. Furthermore, the Another possible area of further research is improving upon linear fit of the Soft Piezo, was an improvement on the linear the filtering and analysis methods. This can be done by fit of the Hard Piezo, both for the impulse range over which optimizing the properties of filters used in eliminating noise, the linear fits were good, and over the entire impulse range. and in smoothing the filtered function. While the widths and Both Soft, and Hard Piezos produce detectable, and orders of many filters were selected to exclude important distinguishable electrical signals in response to impacts from peaks and fully eliminating noise peaks, the widths and orders particles with impulses over the impulses explored in this of a number of filters which were intended to eliminate paper. However, there is a peak in the magnitude of the particular frequency ranges (not just the primary noise electrical response. Additionally the decay time is frequency, or characteristic frequency filters) were not approximately constant over the majority of the impulse range, optimized. The initial smoothing length, and the minimal for the Soft Piezo, and for the Hard Piezo, there is a peak in smoothing length were not optimized either. Another the decay time. This indicates, that the signals produced in improvement which could be made on this method of analysis response to a particle's impact, may be indistinguishable for would be to reduce the time complexity of the code. In different ranges of the particles impulse, and even for impulses particular, it would greatly reduce the run time, if each signal between 3.416 and 172.7Ns. The reason that the signals at was analyzed in parallel, using parallel computing methods, 172.7Ns, which have an impulse higher than the impulse at rather than in series. Since each run analyzes at least 10 which the peak in magnitude and decay time in Hard Piezos signals, this would decrease the run time by a factor of 10. occurs, are distinguishable from lower impulse signals is that A final possibility for further work would be to explore the the signal has a different characteristic frequency at this role that various frequency-domain peaks play in the signal. impulse. There are a number of peaks that are filtered out when Throughout the process of filtering, and analyzing the identifying the noise, or characteristic frequency, which are collected signals, there are a number of ways errors may have significant at certain impulses. It would be helpful to know been introduced. The application of filters can attenuate what effect these peaks have on the signal, whether they are frequencies that are important to the signal produced in additional noise peaks, a secondary characteristic frequency response to the impact of a particle. This problem is most exhibiting another exponential decay, or something else. Other apparent when the noise frequency is near the characteristic peaks, which do not have magnitudes comparable to the frequency, but it is also a problem if the noise or characteristic maximal magnitude of the signal, at any impulse, have frequency is near any frequency that is being filtered. Error important effects on the signal. In particular, small peaks (and introduced in this way can be reduced, but not eliminated, by large ones) near the characteristic frequency, at some optimizing the properties of all filters being applied to the impulses, produce beats in the signal. It would be beneficial to signal. Similarly, if the length of the smoothing function discover what causes these peaks, since eliminating them applied to the signal is too short for a signal, it will not makes analysis of the signal much easier. eliminate the oscillations in the signal, leading to a worse fit, and ultimately worse results for the decay time, and V. CONCLUSIONS magnitude. Another possible way errors are introduced is Both Hard and Soft Piezos behaved linearly over an impulse through the filtering of frequencies other than the noise range of 3.416 to 33.68Ns. Over an impulse range of 3.416 frequency. Filtering out these frequencies was the simplest to 172.7Ns, Hard and Soft Piezos did not exhibit a linear solution to dominant frequencies that are not the characteristic dependence of the magnitude of their signals on the impulse or noise frequency, or any multiple of these, but since it's not strength. For both the limited impulse range, and the entire well known what roles these frequencies might play in the range examined, the Soft Piezos matched a linear response signal of interest, important information could be lost by more closely. Additionally, the time delay, and characteristic filtering them. frequency varied more with impulse in the Hard Piezos, and There are a number of possible areas for future work on this the Signal to Noise Ratio is generally higher for Soft Piezos. project. The most apparent is the application of the methods of This suggests that for isolated impacts, within the range of filtering and analysis to more data, in particular at higher and impulses explored in this paper, Soft Piezos are superior to lower impulses than were analyzed in this paper. Other regions Hard Piezos. of interest, for analyzing the Piezos response, would include impulses that lie within regions where peaks occurred in this

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