Current standards for the measurement of impulse noise (e.g., MIL-STD-1474E) recommend using a sampling rate of at least 200 kHz in order to accurately estimate the risk of hearing damage. The given motivation for this high sampling rate is to ensure a temporal resolution in the impulse waveform fine enough to accurately capture the peak pressure. However, the Nyquist-Shannon sampling theorem specifies that a sampled signal can accurately reconstruct both the amplitude and phase information of a signal given the sampling rate is at least twice the highest frequency present in the original signal. Thus, it is possible to reconstruct a band limited signal with the same temporal resolution as one captured at a higher sampling rate if the contributions of energy above the Nyquist rate can be ignored. In this study, resampling techniques are applied to a signal sampled at 48 kHz to extract A-weighted sound pressure energy estimates within 0.1 dB of those obtained at a higher sampling rate. Our results suggest sampling rates for impulsive noise should be based on the range of frequencies expected to make a contribution to injury risk rather than on concerns about temporal resolution.

To estimate the risk of noise exposure on hearing health, acoustic sound pressure waves are measured using a microphone and are converted into a damage metric through a mathematical formula. This process typically first relies on the digitization of sound pressures through sampling. To ensure that a digitized (i.e., discrete-time) signal can be used to perfectly reconstruct the amplitude and phase present in the original, continuous-time sound pressure waveform, the Nyquist-Shannon sampling theorem specifies that the sampling rate should be twice the highest frequency present in the original analog signal (Nyquist, 1928; Oppenheim , 1997; Shannon, 1949).

Impulse noise is characterized by a very fast rise in pressure lasting tens of microseconds, followed by an exponential decay lasting milliseconds (Davis and Clavier, 2017) depending on the source. When generated from weapons fire, impulse noise is often thought to require a sampling rate of at least 100–200 kHz or more, which is fast enough to capture frequency components well beyond the limits of human hearing (about 20 kHz) (Rosen and Howell, 2011). MIL-STD-1474E specifies a sampling rate of at least 192 kHz (U.S. Department of Defense, 2015), with similar recommendations made by the National Institute for Occupational Safety & Health (NIOSH) (Kardous , 2005) and ANSI S12.7–1986 (ANSI/ASA, 1986).

A primary concern with undersampling (i.e., using too low of a sampling rate) an impulse waveform is “missing” the peak, or underestimating its value in between two samples of the digital waveform. The peak sound pressure level (SPL) is often used as a proxy to characterize an impulse (e.g., “a 165 dB” rifle impulse noise usually refers to the peak). Lobarinas (2016) as well as Meinke (2016) have found that traditional sound level meters designed for continuous (non-impulsive) noise may underestimate the peak SPL. Possible reasons for these underestimates could include peak level clipping due to microphone saturation, insufficient microphone frequency response, or waveform peak picking from a signal with an insufficient sampling rate to capture the peak level of the impulse noise. Distortions of measured signals that occur due to limitations in the dynamic range or bandwidth of the microphone are not easily resolved. However, with regard to digital sampling, the Nyquist-Shannon Theorem says that as long as the signal is band limited to a frequency range equal to half the sampling rate, the true peak pressure, and the entire continuous time waveform, can be perfectly reconstructed. It is important to note, however, that accurate reconstruction of the temporal features of the waveform is only possible if the signal is reconstructed by upsampling it to a high enough frequency to resolve those features. We believe that it may be possible to improve the estimate of damage risk metrics (e.g., the peak pressure level) for “undersampled” impulse noise when first upsampling to a higher sampling rate prior to analysis.

Of course, in the real world, there are no signals that are truly “band limited,” so there will always be some energy that will be excluded from an impulse noise measurement no matter how high the sampling rate. The high-frequency response of the noise recording is inherently limited by the bandwidth of the sensor, so the high-pass characteristic of the sensor or microphone always sets a de facto maximum frequency where energy will be present in an impulse noise recording. There are many cases where the actual waveform of a blast or impulse noise has most of its energy concentrated in the low-frequency portion of the spectrum. For example, Fig. 1 illustrates a prototypical Friedlander blast pressure waveform with an A-duration of 1 ms. In the time domain, the Friedlander waveform is characterized by a sharp rise in pressure followed by an exponential decay, overshoot, and recovery to baseline. In the frequency domain, the magnitude of the Friedlander spectrum decays exponentially with increasing frequency, meaning that most of the energy in the waveform is in the low-frequency portion of the spectrum that could be captured with a relatively low sampling rate.

One of the primary reasons for making high-level impulse noise recordings is to process the recordings with a model that estimates the risk of hearing damage (Wall , 2019), and most of these models also incorporate some bandwidth limitations that primarily focus on frequencies within the range of human hearing. While the accuracy of the peak pressure is often of high concern by those taking noise measurements, it is known to be less predictive of audiometric shifts, as well as cochlear hair cell loss in a chinchilla model as compared to energy-based methods (e.g., L Aeq) (Murphy and Kardous, 2012; Murphy , 2011). Furthermore, the peak level, which was used in the now retired MIL-STD-1474D, has been superseded by MIL-STD-1474E, which offers two options: the A-weighted sound pressure energy, and a mathematical model of the ear referred to as the Auditory Hazard Assessment Algorithm for Humans (AHAAH) (Price and Kalb, 1991; U.S. Department of Defense, 2015). The A-weighting function in particular has a low-pass cut-off frequency around 13 kHz, so the necessity of extremely high sample rates is questionable at best if L Aeq 8 hr is the damage risk metric, since high frequency information is filtered out during computation. The AHAAH model uses a lumped element network model to capture the middle ear frequency response, and thus also has a low-pass affect above 20 kHz, and as low as 4 kHz when the middle ear contraction is present (Price and Kalb, 2018; Song, 2010).

While a simple guideline for impulse noise measurement could be to always oversample out of caution, field studies of weapon system noise must assess the trade-off of the size, weight, power, and cost requirements of achieving an ideal sampling with the portability of personalized measurements. Traditional personalized noise monitoring for industrial environments is not suitable for impulse noise > 140 dB SPL due to microphone, peak sound pressure, and sampling rate limitations (Kardous and Willson, 2004). More recently, attempts have been made to develop on-body systems that overcome these limitations through the use of high-SPL microphones (Davis , 2019; Smalt , 2017; Smalt , 2022), but achieving the required sampling rate is still a challenge. Because of this trade-off between high sampling rate and portability, past field measurements of weapon system noise have a wide range of sampling rates, from as low as 48 kHz (Maher, 2007; Murphy and Tubbs, 2007) to 1 MHz or higher (Nakashima , 2017). Studies aimed at meeting the requirements in MIL-STD-1474E often use 200 kHz (Rasband , 2018) or even 800 kHz for laboratory studies (Rasmussen , 2010).

In this study, we parametrically investigate the effect of sampling rate on damage risk criteria. Our hypothesis is that, if proper signal reconstruction techniques are employed, extremely high sampling rates (i.e., >100 kHz) should not be necessary for characterizing auditory damage risk. This is especially true when using metrics such as the A-weighted energy filter out those higher frequencies. The sections that follow outline both our simulation and experimental approaches to testing the effect of sampling rate on relevant impulse noise damage risk metrics.

Our approach to studying the effect of sampling rate on impulse noise measurement was twofold: First, we simulated the effect of lower sampling rates through decimation of a digitally sampled pressure waveform, and second, we simultaneously digitized an analog impulse noise signal on two independent data acquisition systems at two different sampling rates. In each case, the signal was reconstructed to a 200 kHz sampling using resampling (i.e., the resample function in matlab®). The resample function was called using the format Y = resample(X,P,Q,N), where X is the signal, P is the new lower sampling rate, Q is the original sampling rate, and N =512. According to the documentation, the function uses a weighted sum of 2 * N * max ( 1 , Q / P ) samples of X to compute each sample of Y, where the length of the FIR filter is proportional to N. The FIR filter is designed using the least squares method (firls).

A single, impulse-noise waveform of a shoulder-fired, recoilless rifle was recorded at a 400 kHz sampling rate (Fig. 2). The recording was obtained in a highly reverberant environment with multiple peak pressures due to reflections. To simulate the effect of various sampling rate frequencies, we first randomly jittered the waveform in time using interpolation, then downsampled to one of six sampling rates between 20 and 200 kHz. This process was repeated 50 times for each sampling rate to simulate a variety of jitters (i.e., varying the exact times that the continuous-time pressure waveform is discretized into a digital signal). This waveform was chosen in particular for its high initial sampling rate, and to present a potential challenge to capturing the peak exposure during downsampling.

To supplement our decimation simulation of lower sample rates, which utilized only a single impulse-noise waveform, we sought to simultaneously digitize impulse noise at two different sampling frequencies simultaneously starting from a true sound pressure voltage. A compressed air shock-tube was used to generate 34 unique impulses with peak pressures between 140 and 180 dB and A-durations of approximately 1.0–1.5 ms (Hickman , 2018; Khan , 2012). Figure 3 diagrams the experimental setup, in which the analog output of a single microphone and amplifier was connected to two independent (non-synchronized) National Instruments data acquisition systems (DAQs) with a 1/8-in. GRAS 46DP microphone (GRAS Sound and Vibration, Holte, Denmark) used as the sensor (100 kHz bandwidth). Recordings on the National Instruments (NI) PXI 4462 (NI, formerly National Instruments Corporation, Austin, TX) were made at a sampling frequency of 200 kHz. The second, independently clocked DAQ was set to one of 5 sampling frequencies: 100, 50, 25, 12, or 6 kHz.

Following data acquisition, impulse metrics were computed on each impulse waveform at the high sampling frequency (200 kHz) and compared to the lower sampling rate system following reconstruction. Metrics were calculated using the Impulsive Noise Meter software in matlab® (Zechmann, 2012).

In a secondary analysis of this data, before computing impulse metrics, the 200 kHz-sampled measurement was first low-pass filtered at the Nyquist frequency corresponding to the simultaneously measured lower-sampling-rate DAQ. This analysis serves as a baseline condition, where we would expect there to be minimal to no difference observed between the data acquisition systems since the frequency content in the two signals has been matched; only the sampling rate should differ.

Figure 4 demonstrates the effect that a reduced sample rate can have on the apparent time-domain representation of a blast waveform. The blue lines show the original shock-tube generated waveform, which was sampled at 500 kHz. The red lines show the same signal downsampled to 50 kHz, with each panel of the figure showing a different temporal offset T0 between the first sample of the 500 kHz waveform and the first sample of the 50 kHz waveform. The concern about using a lower sampling rate to determine the peak pressure is that there are some cases where the consecutive samples of the measurement will occur before and after the true peak of the waveform, resulting in an underestimate of peak pressure. This clearly occurs in the left and middle panels in the bottom row of the figure ( T 0 = 0.012 ms and T 0 = 0.014 ms), where the peak values of the 50 kHz waveforms are at least 10% lower than the peak of the 500 kHz waveform. However, the Nyquist sampling theorem states that the values of a continuous signal that occur between the individual time points of a sampled waveform can be reconstructed exactly as long as the bandwidth of the signal does not exceed the sampling rate. Thus, it should be possible to recover a low-pass filtered version of the original signal simply by digitally upsampling the 50 kHz signal by a factor of 10 and low-pass filtering with a reconstruction filter that has a cut off frequency at the Nyquist rate of the 50 kHz signal. This upsampling procedure results in the yellow line in each panel of the figure, which in all cases is virtually identical to the blue line representing the original 500 kHz waveform. In this case, where the blast wave primarily contains energy below 25 kHz, the use of upsampling to reconstruct the waveform almost perfectly captures not only the peak, but also the entire original 500 kHz waveform.

Figure 5 illustrates the changes in damage risk metrics for a single impulse as a function of downsampling (blue), and downsampling followed by upsampling (red). Peak SPL changed by less than 0.4 dB even at sampling rates as low as 48 kHz. Similarly, small changes were observed in the L Aeq 8 hr. Reconstruction of the signal using upsampling caused a slight increase in the A-weighted energy, while downsmapling only resulted in a reduction, but both were less than 0.1 dB. A-duration and the AHAAH model were both more affected by downsampling, but were not significantly affected even below 100 kHz. From a practical standpoint, the data in Fig. 5 suggest that reducing the sample rate would have at most a minor impact on the noise hazard estimates for this weapon system.

Figure 6 shows noise exposure metrics acquired on two independently clocked data acquisition systems at 200 kHz versus a reduced sample rate, as indicated by the color bar. In all cases, the reduced sample-rate signal was upsampled to 200 kHz prior to making the comparison. The results show that recording the signal at a reduced sampling rate and then upsampling the resulting signal to 200 kHz had almost no effect on peak level, L Aeq 8 hr, or AHAAH ARU. Only the time-based measurements of impulse duration and rise time were significantly changed by the reduction in sample rate. However, it is likely that these distortions were caused by the presence of high-frequency energy in the stimulus rather than by temporal distortions caused by the reduced sampling rates.

This point is illustrated in Fig. 7, which is identical to Fig. 6 except that the 200-kHz signal was low-pass filtered to match the Nyquist rate of the slower A/D recording system prior to making the comparisons. In this comparison, the signals recorded at lower sampling rates produced peak, L Aeq 8 hr, AHAAH, and rise time values were almost identical to the 200 kHz recordings. Small but significant errors were observed in 3 of the 34 estimates of A-duration, and 2 of the 34 estimates of B-Duration. Overall, these results appear to confirm that all the meaningful differences between the risk criteria assessed at different sampling rates in Fig. 6 were the result of high-frequency energy in the waveform, rather than any limitation due to the temporal resolution of blast waveforms sampled at lower frequencies.

Our results suggest that the sampling rate may not have as strong of an effect on the resulting damage risk metrics as previously thought. While anecdotal evidence had suggested that the peak SPL in particular might decrease at low sampling frequencies, our simulation of lower sampling frequencies shows that a properly reconstructed noise waveform can almost perfectly measure the peak value of an impulse even at sampling rates as low as 25 kHz. Stronger evidence can be seen in the simultaneous capture of impulses, which again did not show significant changes or bias in the peak level at lower sampling frequencies.

In cases where there were differences between impulse noise metrics measured at different sampling rates, these differences were driven almost entirely by the presence of high frequency energy in the waveform. When the signal captured at 200 kHz was low-pass filtered to match the Nyquist rate of the signal captured at a lower sampling rate, virtually all differences between the impulse noise metrics disappeared. Figure 8 illustrates the effects that low-pass filtering at systematically lower frequencies has on the time domain representation of a blast waveform. No changes in the waveform are apparent until the sampling rate falls below 125 kHz, suggesting that this particular blast waveform contains some energy up to 62.5 kHz. At lower sampling rates, the limited bandwidth starts to reduce the maximum rise time of the signal, which also results in some ringing in the waveform before and after the impulse. However, the peak pressure remains consistent even at the lowest sampling rate, as shown quantitatively in Fig. 4.

Our simulations show that resampling a measurement made at a lower sampling rate (e.g., 50 kHz) up to a higher sampling rate (e.g., 200 kHz) can be effective in certain circumstances. Figure 5 showed a minimal difference for the peak level ( L Peak), L Aeq 8 hr, and AHAAH ARU measurements. If the desired result is to visualize the true peak, Fig. 4 shows near perfect reconstruction. Upsampling a low-frequency-sampled waveform may not produce the same results as a high-frequency sampled signal for time-domain measurements like stimulus duration. The sharp low-frequency filters needed for reconstruction may cause a ringing effect in the time-domain waveform that causes the distortions in A-Duration seen in Figs. 5 and 6. The ringing causes an earlier zero crossing before the impulse to be incorrectly used by the A-duration algorithm as the start of the impulse. However, it should be noted that metrics based on the locations of the zero-crossings in a time waveform may be sensitive to many types of distortions, including measurement noise floor, and that the distortions seen in Fig. 6 may not be much larger than what would be expected from test-retest errors for repeated high-frequency measurements of the same impulse noise source. Overall, we suggest that resampling (i.e., reconstruction of the waveform) could be useful for time domain analysis of impulse waveforms, but we recommend comparing the results with and without reconstruction as a means to establish a lower and upper bound for the parameter of interest.

The results of this study suggest that the use of a high sampling rate may not as critical for the assessment of impulse noise as is commonly assumed. Our analyses show that impulse rates as low as 50 kHz can accurately capture the peak level, L eqA 8, and AHAAH risk metrics associated with an impulse noise source. This was true both in simulations and in a series of measurements where an impulse noise was measured with two different A/D converters running at two different sample rates. To the extent that there were differences in the values of these metrics across different sampling rates, these differences were the result of high-frequency energy that occurred above the Nyquist rate of the A/D conversion, rather than any temporal distortions in the waveform. It may, however, be necessary to upsample a low-frequency digital waveform to a higher sample rate (in the 200 kHz range) to obtain comparable results from metrics that operate on the signal in the time domain, like the peak impulse level or the AHAAH model. Most devices designed to measure impulse levels in real time do not have the ability to perform upsampling, which may explain some of the errors that have been attributed to insufficient sampling rate in prior comparisons of different methods for measuring impulse noise (Lobarinas , 2016). These errors also may have been the result of other design factors such as frequency response or the slew rate of the analog circuitry (ANSI/ASA, 1986).

These results have implications for design requirements of noise measuring devices. Recently there has been a lot of interest in developing wearable dosimeters capable of tracking all the impulse noise exposures experienced by an individual over an extended time period. Size, weight, power efficiency, and cost are all critical factors in the design of these devices. Storage and processing requirements generally increase linearly with the sampling rate, and power requirements generally increase with the square of the sampling rate, so there will always be a direct trade-off between the sampling rate used for a noise dosimeter and the amount of time it can record without changing the battery or downloading data. In many cases, the advantages of having smaller, lighter, cheaper dosimeters with longer battery life will outweigh the loss of fidelity that might occur from using a lower sampling rate. Based on the finding of this study, we believe it would be unwise to disregard the possibility of using a lower sampling rate in cases where it might be necessary to meet the other requirements of a measurement system. This is particularly true in cases where the high-frequency components of the signal are unimportant, either because they are not present in the source waveform or because they are not captured by the other components of the measurement system. If L Aeq is the desired risk metric, the A-weighting filter significantly attenuates any energy above 20 kHz, so we would argue that a sampling rate of 40 kHz should be enough. Impulse noise measurements made in protected locations, such as the inside of a hearing protection device, are likely to have less high-frequency energy reaching the sensor due to absorption, which may reduce the potential requirements for sample rate in a noise monitoring device (Davis , 2019; Fedele and Kalb, 2015). Some impulse noise sources may only output limited amounts of high-frequency noise. And, perhaps just as importantly, it may be the case that the very high frequency components of an impulse do not have a meaningful impact on the associated noise hazard, as is implicitly assumed when the noise waveform is A-weighted as part of the risk analysis. Our data clearly indicate that there is no advantage to sampling the impulse waveform faster than is necessary to capture the relevant bandwidth of the signal, so in cases where it is not necessary to analyze the high frequency components of an impulse signal, we believe that serious consideration should be given to the possible trade-offs and benefits that could be obtained by reducing the sampling rate of the system.

The Nyquist theorem states that both the amplitude and phase of a signal can be perfectly reconstructed from a digital waveform if it is sampled at more than half the bandwidth of the original signal, and the results of this study confirm this result both for simulations made by downsampling a pre-recorded impulse noise waveform and for new recordings made at different sample rates from the same impulse noise source. One limitation of this study is in the characteristics of the impulse waveforms evaluated. We evaluated two different types of impulses, those generated from a shock tube, and an explosive charge from a shoulder fired recoilless rifle. The weapon systems generated by shock-tube are intended to be similar in level and duration to small-arms weapons fire, such as an M4 rifle (Khan , 2012). However, the range of impulses tested do not cover all the variations in impulse noise waveform, so care should be taken to first understand the weapon system or noise source before relying on lower sampling-frequency measurements.

The choice of instrumentation beyond the sampling rate of the acquisition system should be considered for impulse noise, including the type of microphone, its orientation, diameter, frequency, response, and sensitivity (Rasmussen , 2009, 2010). Particular care should be taken to evaluate the bandwidth of the noise source of interest, as noise sources with a great deal of high frequency energy will likely require high sampling rates in order to capture all the relevant features of the time-domain waveform. However, since the noise type is often fairly static in a particular environment or site (e.g., industrial, military), we suggest that lower sampling rates could be adequate for damage risk estimation after an initial weapon system characterization has been performed with a laboratory grade system at very high sampling rates.

In conclusion, our results suggest that sampling rates as low as 100 kHz or even 50 kHz could be sufficient for personal noise dosimetry and site survey measurements if the signal is upsampled back to 200 kHz prior to analyzing the acoustic features of the waveform. This is particularly true in environments where the primary variables of interest are L Peak (peak level), L Aeq 8 hr, or the AHAAH ARU. In cases where the measurements will be analyzed to extract subtle time-domain features like the A-Duration, care must be taken to ensure that the noise source does not contain a substantial amount of high-frequency energy above the Nyquist rate of the recording system. In general, the results suggest that sampling rates for impulsive noise should be based on the range of frequencies that are expected to make a meaningful contribution to the risk of injury, rather than on any concerns about preserving enough temporal resolution to identify peaks and other features of the waveform.

The authors would like to thank William J. Murphy (NIOSH) for discussions on this topic. DISTRIBUTION STATEMENT A. Approved for public release. Distribution is unlimited. This material is based upon work supported by the Department of the Army under Air Force Contract No. FA8702-15-D-0001. The views expressed in this article are those of the authors and do not reflect the official policy of the Department of Army/Navy/Air Force, Department of Defense, or U.S. Government.

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