Experimental Investigation of Deviations in Sound Reproduction

Abstract

Sound reproduction is the electro-mechanical re-creation of sound waves using analogue and digital audio equipment. Although sound reproduction implies that repeated acoustical events are close to identical, numerous fixed and variable conditions affect the acoustic result. To arrive at a better understanding of the magnitude of deviations in sound reproduction, amplitude deviation and phase distortion of a sound signal were measured at various reproduction stages and compared under a set of controlled acoustical conditions, one condition being the presence of a human subject in the acoustic test environment. Deviations in electroacoustic reproduction were smaller than ±0.2 dB amplitude and ±3 degrees phase shift when comparing trials recorded on the same day (Δt < 8 h, mean uncertainty $u$ = 1.58%). Deviations increased significantly with greater than two times the amplitude and three times the phase shift when comparing trials recorded on different days (Δt > 16 h, $u$ = 4.63%). Deviations further increased significantly with greater than 15 times the amplitude and the phase shift when a human subject was present in the acoustic environment ($u$ = 24.64%). For the first time, this study shows that the human body does not merely absorb but can also cause amplification of sound energy. The degree of attenuation or amplification per frequency shows complex variance depending on the type of reproduction and the subject, indicating a nonlinear dynamic interaction. The findings of this study may serve as a reference to update acoustical standards and improve accuracy and reliability of sound reproduction and its application in measurements, diagnostics and therapeutic methods.

1. Introduction

The electro-mechanical reproduction of sound waves by analogue and digital audio equipment is generally considered to provide for repeated identical acoustic events. Numerous applications rely on accurate reproducibility of sound, such as acoustic engineering, room acoustic measurement and audio equipment manufacturing. Sound recording and reproduction systems are also used for various imaging techniques in medical diagnostics, such as ultrasonography, obstetric, abdominal, pelvic and cardiac imaging, and medical therapeutic uses, including the promotion of bone and soft tissue regeneration, destruction of kidney stones and tumor ablation [1,2]. Beyond medicine, precise acoustic acquisition is also vital in geophysical prospecting, where the development of nodal acoustic technology represents a significant leap in recording fidelity [3] and advanced acoustic prediction methods are now successfully used to accurately predict formation properties [4]. In all these areas, the quality and accuracy of sound reproduction may be of critical importance, and the conditions of the environment have to be carefully considered [5].

Numerous conditions are known to affect sound reproduction. These conditions may be categorized into two groups of factors: (i) fixed conditions, such as the acoustics of the room, loudspeaker placement and orientation, source material quality, amplification process, connections and cables used to interconnect the devices; and (ii) variable conditions, such as atmospheric changes, environmental noise, and the presence and positioning of human subject(s) [6,7].

1.1. Fixed Conditions Affecting the Sound Field

The fixed acoustic conditions of an enclosed area are known to greatly influence the propagation of sound waves [7]. However, accurate prediction of room acoustics is considered incomplete due to the complexity of sound fields in closed areas, i.e., the large number of degrees of freedom that need to be taken into consideration, and the acoustic quality of a room is ultimately proven by subjective judgement [8,9]. Careful consideration of source placement in relation to reflecting surfaces is used to achieve a uniformly mixed reflected sound field [6], while acoustic treatment of the room and the use of loudspeakers and their placement in the room can determine and optimize sound pressure and frequency balance and widen or narrow the effective sound field [8]. Effective sound field reproduction in a reverberant enclosure and invertibility of the room impulse response function using microphones and loudspeakers is generally constrained [10,11] due to the very high density and volume of loudspeakers required to reproduce a sound field consistently [8,11].

1.2. Variable Conditions Affecting the Sound Field

The maximum permitted uncertainty of a sound reproduction system can be calculated by taking into account the specification of various random error components in a measurement chain. Measurement uncertainty and its application to acoustic standards is considered not to exceed a maximum of 2.17% (0.19 dB) for typical loudspeaker reproduction. The components of uncertainty within the maximum permitted range comprise generator accuracy (0.0043 dB), measurement microphone (0.05 dB), microphone preamplifier (0.01 dB), detector accuracy (0.05 dB) and positioning (0.04 dB) [12].

Environmental noise, i.e., ambient noises such as traffic or machinery, can affect and interfere with sound reproduction by occluding frequency regions that make accurate detection of the sound source more difficult [5,7,13]. Atmospheric conditions affect sound reproduction as sound intensity is proportional to the density of the medium (i.e., the air pressure) and the speed of sound in air is proportional to the temperature and humidity [14,15,16]. Temperature and humidity were also found to be positively correlated to increased reverberation time [17]. The challenge of accurately quantifying moisture impact is not unique to acoustics but is a critical factor in various engineering disciplines, such as the assessment of unbound pavement layers [18].

Sound attenuation is also affected by the atmospheric conditions. Sound attenuation is proportional to the frequency where high frequencies dissipate faster than low frequencies and is further influenced by temperature, humidity and air pressure [19,20,21]. Air currents moving in the same or opposite direction as sound waves can either increase or decrease the speed of sound but have no notable effects on sound propagation across short distances [22,23].

Propagating waves with a different velocity, such as electromagnetic waves, are generally considered not to affect sound waves, although the interaction between these types of waves has not been examined extensively [24,25].

1.3. Effects of the Human Subject on the Sound Field

The human body is reported to absorb 0.1–10% of the mid and high frequencies found from measurements of small body surface areas to estimate the total acoustic energy absorbed [26,27]. Drawbacks of such generalization have been reported, emphasizing the importance of where the energy is absorbed and how it is distributed once it enters the body [28].

Another factor is the positioning of the human body in the sound field. Previous studies have explored how sound absorption depends on the occupation density of the position, in the range between 4.5% and 12% depending on the frequency [29]; the clothing of the human subjects, where absorption coefficients of clothing were found many times greater than those of the human skin and with differences based on the type of clothing [29,30,31,32]; and the posture, whether the subject is standing or seated [29], where standing yielded a 30–40% increase in sound absorption compared to prostration for different setups [32]. Various studies also investigated the effects of body edges on the sound absorption coefficient, i.e., the edge effect [31,33,34], including reported deviations of ISO measurements [35,36].

While human bodies are generally absorbers, they also act as diffractors or scatterers that can locally increase sound pressure levels through constructive interference or cavity resonance effects and induce significant frequency-dependent phase shifts due to the body acting as an acoustic barrier [37,38,39]. While diffraction and scattering occurs when audible sound waves encounter obstacles smaller than the wavelength, transmission and refraction of sound waves occurs when the wavelength is small compared to the structure. When waves pass through the human body, they are refracted depending on different body tissues with varying densities and refractive indices. This causes a phase shift when the wave undergoes a change in its position relative to a reference point. For example, acoustic phase distortions were detected in human skull bones by using a non-invasive ultrasound imaging method [40]. Similarly, in cases of hearing loss, the refraction of waves in the human body can cause a phase shift that leads to a distorted projection onto the cochlea with loss of the perceived sound as a result [41].

It has been shown that both the type of reproduction, i.e., the amount and placement of loudspeakers, and the presence of human subjects influence the reproduced sound field in an enclosed environment. However, their simultaneous and combined effects on deviations in sound reproduction have not been specifically investigated in prior studies. The purpose of this study is to experimentally establish to what extent and in which way the sound frequency, the type of reproduction, and the presence of a human subject affect deviations in sound reproduction. We investigated the magnitude and direction of amplitude deviation and phase distortion across repeated trials reproduced in an identical manner in a controlled acoustic environment. Deviations in audio signals are measured and compared at three stages: (i) digital reproduction of the audio signal, i.e., the system output before digital-to-analogue (D/A) conversion; (ii) electroacoustic reproduction without a human subject present in the acoustic environment; and (iii) electroacoustic reproduction with a human subject present in the acoustic environment.

2. Materials and Methods

2.1. Acoustic Test Environment

Acoustic measurements were obtained inside the Sphere, a sound-proof, anechoic spherical enclosure (Figure 1) with an integrated configuration of omnidirectional loudspeakers and microphones (Figure 2A,B). A custom-designed chair to support human subjects in a seated position is centered inside the Sphere (Figure 3). Environmental noise levels are reduced by −30 dB when the door of the Sphere is closed. The temperature and relative humidity inside the Sphere are monitored and regulated during all trials at 20 °C ± 0.2 and 50% ± 2, respectively. The absorption coefficient of the inner shell of the Sphere is $\alpha$ = 0.99 (f > 200 Hz) and of the chair $\alpha$ = 0.90 (f > 500 Hz).

2.2. Technical Setup

A CPU processes the audio input signal (AIS) and distributes it to eight loudspeaker channels, comprising the audio signal components of the audio output signal (AOS). Acoustic recordings are obtained using six microphones. The total fixed latency from source (CPU) to destination (loudspeakers) is 10 ms and another 10 ms from receivers (microphones) back to source (Figure 4).

2.3. Audio Signal Reproduction

The AIS comprises 14 pure tone frequencies within the range of 50–6000 Hz: 73 Hz; 110 Hz; 220 Hz; 330 Hz; 440 Hz; 880 Hz; 990 Hz; 1385 Hz; 1770 Hz; 1860 Hz; 2200 Hz; 2910 Hz; 4200 Hz; and 5600 Hz; reproduced at an equal sound pressure level (SPL) of ~70 dB and with a 15 s duration of each trial. Trials were reproduced in two conditions (C1 and C2). In C1, the AOS is distributed as a stereo projection, i.e., one channel left (L) and one channel right (R), where the destination of L = Ls2 and R = Ls3 (see Figure 2A,B). In C2, the AOS is a spatial projection, i.e., distributed across 8 loudspeaker channels.

2.4. Audio Analysis

Recorded trials were exported as stand-alone WAV files with identical time length, encoded at a resolution of 32-bit depth and a sample rate of 48 kHz. The maximum amplitude $L_0$ (in decibels full scale (dBFS) with FS = 0) was logged for each frequency over the full duration of each trial.

Let two recorded signals with the same frequency ($s_1$,$s_2$) be expressed as

$$\begin{gathered} s_1=A_1 sin(\omega t-{\varphi }_1) \\ {s}_2=A_2 sin(\omega t-{\varphi }_2) \end{gathered}$$

(1)

where the amplitude $A= {10}^{ {\displaystyle \frac{L_{0 }}{20}}}$. The amplitude deviation $A_D$ (in dB) between the two signals is determined as

$$A_D=L_1-L_2$$

(2)

and the phase shift between the two signals is determined as

$$\psi ={\varphi }_1-{\varphi }_2$$

(3)

The second signal is then shifted by 180 degrees and added to the first signal, i.e., their difference is determined as

$$s_D=s_1-s_2=A_1 sin(\omega t-{\varphi }_1)-A_2 sin(\omega t-{\varphi }_2)$$

(4)

This function is harmonic with the same frequency but a new amplitude $\alpha$ and phase $\beta$ as follows:

$$s_D=\alpha sin(\omega t-\beta )$$

(5)

The parameters $\alpha$ and $\beta$ can be calculated from the original parameters $A_1$, $A_2$ and ${\varphi }_1,$ ${\varphi }_2$ as

$$\alpha =\sqrt{A_1^2+A_2^2-2{ A}_1{ A}_2 cos(\psi )}$$

(6)

and

$$\beta =arctan \left({\displaystyle \frac{A_1 cos({\varphi }_1)-A_2 cos({\varphi }_2)}{\alpha }},{\displaystyle \frac{A_{1 }sin({\varphi }_1)-{ A}_{2 }sin({\varphi }_2)}{\alpha }}\right)$$

(7)

Since ${\varphi }_1$ and ${\varphi }_2$ are not known, the above relationship can be inverted and $\psi$ is determined as

$$\psi =arccos {\displaystyle \frac{A_1^2+A_2^2-{\alpha }^2}{2 A_1 A_2}}$$

(8)

This definition requires that $|{ A}_1-A_{2 }|\le \alpha \le A_1+A_2$ and this is equivalent with the argument of $arccos$ being between −1 and 1. It should be noted that $arccos$ returns a value for $\psi$ between 0 and $\pi$; thus, the sign of $\psi$ cannot be determined this way.

To determine the sign of $\psi$, the second signal is shifted by 90 degrees and added to the first signal, and their difference is taken a second time as

$$s_{D 2}=s_1-s_2=A_1 sin(\omega t-{\varphi }_1)-A_{2 }sin(\omega t-{\varphi }_2+{\displaystyle \frac{\pi }{2}})$$

(9)

After logging the resulting amplitude ${\alpha }_2$, ${\psi }_2$ is determined as

$${\psi }_2=arccos {\displaystyle \frac{A_1^2+A_2^2-{\alpha }_2^2 }{2 A_1 A_2}}$$

(10)

and the sign of $\psi$ is determined as

$$\begin{gathered} -\psi \mathrm{if} {\psi }_2>{\displaystyle \frac{\pi }{2}} \\ +\psi \mathrm{if} {\psi }_2<{\displaystyle \frac{\pi }{2}} \end{gathered}$$

(11)

Statistical significance of the differences between sets of recorded trials are determined with a two-sample independent t-test and reported as two-tailed p-values with a significance level of α = 0.05. To obtain the t-test, the difference of means is determined for two samples (n₁ and n₂) and the pooled standard deviation of the samples is used to estimate the standard error of the difference between the two means.

The uncertainty of audio signal reproduction is determined by converting the amplitude deviation to a percentage

$$u=100\cdot ({10}^{{\displaystyle \frac{\sqrt{A_D^{ 2 }}}{20}}}-1) \mathrm{\%}$$

(12)

where 100% uncertainty equals a difference of ±6 dB in the audio signal reproduction. Statistical significance of the difference between the sample mean and the standard uncertainty are determined with a one-sample t-test and reported as two-tailed p-values with a significance level of α = 0.05. To obtain the t-test, the standard deviation of the sample mean is used to estimate the standard mean error.

3. Results

3.1. Uncertainty in Digital Reproduction

Deviations in the digital reproduction of recorded trials were examined to determine the accuracy of the generator reproduction (CPU). To enable harmonic distortion measurements to 0.1% accuracy, the generator distortion must be <0.0043 dB, equivalent to a standard uncertainty of 0.05% [12]. The AOS was recorded over DANTE before D/A conversion (see Figure 4) and measurements were obtained by averaging the extracted parameters from the AOS components.

Trials (n = 584) were recorded for 14 frequencies in two conditions (C1 and C2) in a randomized order and at random times during the day between 10 am and 6 pm. Trials of the same frequency and condition were evaluated using multiple pairwise comparisons between recorded trials ($s_1$,${ s}_2$) and significance was adjusted using a Bonferroni correction. The results are shown in

Table 1. Magnitude of amplitude deviation and phase distortion in digital signal reproduction. Data are mean ± standard deviation (SD), unless otherwise noted. A_D is indicated as dB. $\psi$ is indicated as 0–180 degrees. Significant differences between groups after Bonferroni correction are indicated with *.

	C1 Mean ± SD	C2 Mean ± SD	Mean Difference ± SD	p-Value
AD (dB)	3.47 × 10−7 ± 7.79 × 10−7	1.00 × 10−6 ± 1.44 × 10−6	−6.55 × 10−7 ± 1.29 × 10−6	<0.00001 *
$\psi$ (degrees)	1.16 × 10−5 ± 2.33 × 10−5	2.96 × 10−4 ± 3.31 × 10−4	−2.84 × 10−4 ± 2.82 × 10−4	<0.00001 *

.

The mean uncertainty of the generator ($u$ = 9.44 × 10⁻⁶%) was determined to be significantly smaller than the standard uncertainty (−0.05% ± 1.52 × 10⁻⁵, p < 0.00001).

3.2. Uncertainty in Electroacoustic Reproduction

Deviations in the electroacoustic reproduction of recorded trials were examined to determine the mean uncertainty of the acoustic test environment. The uncertainty budget for a typical loudspeaker measurement includes accuracy of the generator, measurement microphones, amplifiers and analysis system, adding up to a standard uncertainty of 2.17% (0.19 dB) [12]. Acoustic measurements were obtained by six microphones at fixed positions inside the acoustic test environment (see Figure 2A,B) and averaging the extracted parameters from the obtained microphone signal components.

Trials (n = 612) were recorded for 14 frequencies in two conditions (C1 and C2) in a randomized order and at random times during the day between 10 am and 6 pm. Trials of the same frequency and condition were evaluated using multiple pairwise comparisons between recorded trials ($s_1$,${ s}_2$) and significance was adjusted using a Bonferroni correction. The results are shown in

Table 2. Magnitude of amplitude deviation and phase distortion in electroacoustic signal reproduction. Data are mean ± standard deviation (SD), unless otherwise noted. A_D is indicated as dB. $\psi$ is indicated as 0–180 degrees. Significant differences between groups after Bonferroni correction are indicated with *.

	C1 Mean ± SD	C2 Mean ± SD	Mean Difference ± SD	p-Value
AD (dB)	0.16 ± 0.22	0.36 ± 0.57	−0.20 ± 0.44	<0.00001 *
$\psi$ (degrees)	3.50 ± 6.03	6.25 ± 9.24	−2.75 ± 7.93	0.00002 *

.

The mean uncertainty of the acoustic test environment ($u$ = 3.13%) was found to be significantly higher than the standard uncertainty (0.96% ± 5.30, p = 0.00001). To further investigate the possible cause of this difference, we stratified the data by comparing trials of the same frequency and condition that were reproduced on the same day (Δt < 8 h: n = 492) and trials of the same frequency and condition that were reproduced on different days (Δt > 16 h; n = 516). The results are shown in

Table 3. Magnitude of amplitude deviation and phase distortion in electroacoustic signal reproduction stratified for time difference (Δt). Data are mean ± standard deviation (SD), unless otherwise noted. AD is indicated as dB. $\psi$ is indicated as 0–180 degrees. Significant differences between groups after Bonferroni correction are indicated with *.

	Δt < 8 h Mean ± SD	Δt > 16 h Mean ± SD	Mean Difference ± SD	p-Value
AD (dB)	0.14 ± 0.30	0.39 ± 0.53	−0.26 ± 0.43	<0.00001 *
$\psi$ (degrees)	2.30 ± 5.21	7.46 ± 9.28	−5.15 ± 7.57	<0.00001 *

.

When comparing trials reproduced on the same day, i.e., with a time difference < 8 h between compared trials ($u$ = 1.58%), the mean uncertainty was significantly smaller than the standard uncertainty (−0.59% ± 3.47, p < 0.00013). When comparing trials reproduced on different days, i.e., with a time difference >16 h between compared trials ($u$ = 4.63%), the mean uncertainty was significantly greater than the standard uncertainty (2.46% ± 6.25, p < 0.00001).

3.3. Uncertainty in Electroacoustic Reproduction with a Human Subject

Deviations in the electroacoustic reproduction of recorded trials with a human subject in the acoustic test environment were examined to determine how the presence of the human subject affects the uncertainty of sound reproduction.

A total of 50 adult subjects (21 male; 29 female) participated in the study. Subjects provided informed consent prior to enrollment. Trials with human subjects were conducted on site at The Works Research Institute in Budapest, Hungary, in accordance with the Declaration of Helsinki and approved by The United Ethical Review Committee for Research in Psychology (EPKEB). Subjects were seated in an identical position supported by the chair inside the Sphere (see Figure 3) and were requested not to move during the trial. Unforeseen movements and/or noise produced by the subjects were logged during each trial and the corresponding time fractions in the recordings were eliminated prior to analysis.

Acoustic measurements were obtained by six microphones at fixed positions inside the acoustic test environment (see Figure 2A,B) and averaging the extracted parameters from the obtained microphone signal components. A total of n = 2933 trials were recorded, with multiple trials for each subject for 14 frequencies in two conditions (C1 and C2) in a randomized order and at random times during the day between 10 am and 6 pm. Each trial with a human subject in the acoustic environment ($s_1$) was compared to a reference trial of the same frequency and condition without a subject reproduced on the same day ($s_2$ = REF). The results are shown in

Table 4. Magnitude of amplitude deviation and phase distortion in electroacoustic signal reproduction with a human subject. Data are mean ± standard deviation (SD), unless otherwise noted. A_D is indicated as dB. $\psi$ is indicated as 0–180 degrees.

	C1 Mean ± SD	C2 Mean ± SD	Mean Difference ± SD	p-Value
AD (dB)	1.97 ± 2.59	1.76 ± 1.97	−0.2 ± 2.32	0.23645
$\psi$ (degrees)	26.41 ± 36.91	25.98 ± 36.62	−0.43 ± 36.77	0.87722

.

The mean uncertainty of sound reproduction across all trials with a human subject was u = 24.64% with no significant differences in uncertainty between C1 and C2.

A significant positive correlation was found between the magnitude of amplitude deviation and magnitude of phase shift per frequency with a power series (R = 0.68, p < 0.00001), as shown in Figure 5. A significant positive correlation was also found between the magnitude of amplitude deviation and frequency (R = 0.55, p < 0.00001) and between the magnitude of phase shift and frequency (R = 0.78, p < 0.00001). The variance in frequency explains 30% of the magnitude of amplitude deviation and 60% of the magnitude of phase shift.

3.4. Within-Subject Variability of Deviations

We assessed the within-subject variability of deviations throughout the duration of each trial with a human subject. The variability represents the average difference of values from each measured time fraction $t_i = 1 s$ to the next $t_{i+1}$ as

$$Variability= {\displaystyle \frac{\sqrt{ {\sum }_{i=1}^n{ (t_i-t_{i+1})}^2}}{n}}$$

(13)

The mean variability of uncertainty within trials with a human subject was u = 3.74%. A significant positive correlation was found between the variability of amplitude deviation and variability of phase shift per frequency with a power series (R = 0.93, p < 0.00001), as shown in Figure 6. Significant positive correlations were also found between the magnitude of amplitude deviation and frequency (R = 0.72, p < 0.00001) and the magnitude of phase shift and frequency (R = 0.80, p < 0.00001). The variance in frequency explains 52% of the variability of amplitude deviation and 64% of the variability of phase shift.

3.5. Condition-Dependent Deviations with a Human Subject

We compared trials with a human subject in C1 and C2 for magnitude and directionality of deviations to assess whether the presence of the human subject caused a positive amplitude difference (amplification) or a negative amplitude difference (attenuation) per frequency compared to REF and whether the phase shift per frequency was positive (leading) or negative (lagging) compared to REF. The results are shown in Figure 7A,B.

When taking the directionality of deviations into account, a very weak negative correlation was found between the amplitude deviation and phase shift per frequency (R = −0.19, p < 0.00001). Amplitude deviation and phase shift were not significantly correlated to frequency, i.e., the increase in frequency did not show a linear relationship with the increase in attenuation and/or amplification, nor with the increase in leading and/or lagging of the sound waves.

3.6. Subject-Dependent Deviations in Electroacoustic Reproduction

We further examined how much the variance in deviations across conditions is subject-dependent, i.e., to what extent deviations in sound reproduction are associated with individual traits of a subject. To control for effects related to the type of reproduction, i.e., a stereo or spatial sound projection, mean deviations were adjusted using a general linear model by which the condition was modeled by one variable and the subject by another. For a single subject $i$, a measured deviation $Z_f = s_{i }-s_{REF}$ at frequency $f$ is explained by

$$Z_f=\alpha X+\beta Y+e$$

(14)

Removing the effect of the condition, the correlation between deviations in sound reproduction and the individual subject was assessed as

$$Z_f-\alpha X=\beta Y+e$$

(15)

A significant positive correlation was observed between the amplitude deviation and the subject (R = 0.75, p < 0.00001). Individual traits of the subject explain 56% of the amplitude deviation, as shown in Figure 8A. A significant positive correlation was also observed between the phase shift and the subject (R = 0.74, p < 0.00001). Individual traits of the subject explain 55% of the phase distortion, as shown in Figure 8B.

Six subjects (n = 6) were requested to submit a form post-participation stating self-measured weight and height. The measures for weight and height per subject were compared to the mean, minimum and maximum values of amplitude deviation and phase shift per frequency that were recorded for each individual subject, as shown in

Table 5. Self-reported weight and height, the mean, maximum and minimum of amplitude deviations, and the mean, maximum and minimum of phase shift recorded across all measured frequencies for six different subjects. A_D is indicated as ±dB. $\psi$ is indicated as degrees (0–180).

Weight (kg)	Height (cm)	Mean AD (dB)	Max AD (dB)	Min AD (dB)	$\mathbf{Mean} \mathit{\psi }$ (Degrees)	$\mathbf{Max} \mathit{\psi }$ (Degrees)	$\mathbf{Min} \mathit{\psi }$ (Degrees)
60	158	−0.86	2.45	−8.07	11.09	45.51	0.21
109	187	−3.03	2.24	−13.55	26.70	157.65	1.31
70	174	−1.06	3.2	−9.63	12.54	50.95	0.63
68	176	−1.17	2.16	−7.52	23.68	150.01	0.04
80	169	−1.43	3.79	−10.63	27.24	152.18	1.34
55	170	−0.69	2.55	−4.7	13.94	63.95	1.09

. A significant negative correlation was found between the weight of subjects and the mean amplitude deviation (R = −0.98, p = 0.00060), as shown in Figure 9. Additionally, a significant negative correlation was found between the weight of subjects and the minimum amplitude deviation (R = −0.93, p = 0.00718).

4. Discussion

In this study we investigated deviations in sound reproduction and their dependency on frequency (50–6000 Hz), the type of reproduction (stereo vs. spatial sound projection) and the presence of a human subject in the acoustic environment. To assess the uncertainty of the acoustic test environment, amplitude deviation and phase distortion were assessed at both the digital reproduction stage (signal before D/A conversion) and electroacoustic reproduction stage (microphone captured signal). All trials were recorded under controlled and identical conditions with regards to the room acoustics, atmospheric conditions, loudspeaker positioning and orientation, microphone positioning and the positioning and posture of human subjects.

We found that deviations occur consistently at every stage of sound reproduction and the magnitude of deviations increases significantly at every consecutive stage. Deviations in digital reproduction were smaller than ±1 × 10⁻⁶ dB amplitude and ±1 × 10⁻⁴ degrees phase shift, with a mean uncertainty of $u$ = 9.44 × 10⁻⁶%. Deviations in electroacoustic reproduction were smaller than ±0.2 dB amplitude and ±3 degrees phase shift when comparing trials recorded on the same day (Δt < 8 h, $u$ = 1.58%). Interestingly, we found that deviations increased greater than two times the amplitude and three times the phase shift when comparing trials recorded on different days (Δt > 16 h, $u$ = 4.63%). Deviations further increased to greater than 15 times the amplitude and the phase shift with a human subject present in the acoustic environment ($u$ = 24.64%).

4.1. Uncertainty in Digital Reproduction

The deviations measured in digital signal reproduction may be attributed to uncertainty in the digital audio processing, network speed and/or the electrical current, among other factors. They may also be subject to uncertainty of the measurement and data processing of the measured signal [8]. The measured uncertainty of the generator used for sound reproduction in this study was significantly smaller than the maximum permitted uncertainty applied to acoustic standards [12]. However, it is interesting to note that deviations in the digital signal were significantly greater in the spatial reproduction compared to the stereo reproduction. The differences in both amplitude deviation and phase distortion were found to be highly significant after Bonferroni correction (p < 0.00001). This difference may be attributed to increased workload of the generator when processing multiple audio paths at once, which may increase internal heat in the generator and lead to increased clock drift [42].

4.2. Uncertainty in Electroacoustic Reproduction

Although acoustic measurements took place in a test environment with controlled temperature and humidity and strongly reduced environmental noise levels, the deviations measured in electroacoustic reproduction may be attributed to minor fluctuations in the atmospheric conditions that happen even in a controlled environment [6,7,15,16,17]. Uncertainty in the recordings by microphones and subsequent A/D conversion and audio processing of the obtained signals may also affect the measurement. An error of 0.001 in heat ratio causes an error of ±3 × 10⁻³ dB in the sensitivity of a calibrated microphone, while a 0.15 ms⁻¹ error in the speed of sound causes the final microphone sensitivity to be in error by ±5 × 10⁻³ dB for the 3 cm³ or plane-wave coupler [16]. Calibrating a microphone by the reciprocity technique could lead to an error in the determination of the sensitivity of 0.01 dB, where the total uncertainty aimed at is ±0.05 dB or smaller [43]. However, the total measured uncertainty of the acoustic test environment used for sound reproduction in this study was significantly smaller than the standard uncertainty for a typical loudspeaker measurement [12].

Importantly, the uncertainty of the acoustic test environment became significantly greater than the standard uncertainty for loudspeaker reproduction when comparing trials recorded on different days (Δt > 16 h). It was outside the scope of this study to further investigate the nature of this observed phenomenon. It should be the subject of a future study to provide an explanation of the mechanism that could cause the observed increase in uncertainty related to increased time difference between events.

Deviations in the acoustic signal were also found to be significantly greater in the spatial reproduction compared to the stereo reproduction. The differences in both amplitude deviation and phase shift were highly significant after Bonferroni correction (p < 0.00002). This difference is likely caused by increasingly complex phase interference of the loudspeakers when the amount of sound sources is increased [44,45]. Increased uncertainty could also result from the compound of transducer manufacturing tolerances. It was found that a dispersion of a sensitivity higher than 1 dB could lead to significant misperformance in measurement deviations with multichannel loudspeaker arrays [46]. Interestingly, the difference in the magnitude of deviations between stereo and spatial sound reproduction became insignificant when a human subject was present in the acoustic test environment.

4.3. Deviations in Sound Reproduction with a Human Subject

The magnitude of amplitude deviation and phase shift in the presence of a human subject showed a significant positive correlation to frequency. This is in accordance with findings in previous studies, e.g., those that observed the degree of absorption by the human body increases for the mid and high frequencies compared to low frequencies [26,27,28,29,30,31,32,33,34,35,36], as well as those that observed the degree of phase shift [40].

The within-subject variability of amplitude deviation and phase shift confirms a significant positive correlation to frequency. The variability of uncertainty throughout a trial with the same subject ($u$ = 3.74%) was more than six times smaller than the uncertainty of sound reproduction comparing trials with and without a human subject, which indicates that the deviations caused by the presence of a human subject were immediate and relatively stable over time, comparable to the mean uncertainty of the acoustic test environment without a human subject ($u$ = 3.13%). Although subjects were monitored for body movements that produced unexpected noise during each trial, the measured variability may be attributed to the likeliness of slight head and body movements of the subjects during the trials.

We established that more than 50% of amplitude deviations and phase distortions are explained by individual traits of the subject, which may include anthropometric features such as weight. In accordance, we found a significant negative correlation between body mass and amplitude deviation. The mean attenuation per frequency was in the range of 0 and −3 dB depending on the subject, which equals a decrease in sound energy up to 30%, with a maximum recorded attenuation per frequency of up to −13.55 dB, which equals a 79% decrease in the sound energy. Attenuation of sound waves in the acoustic environment may be caused by the absorption of sound energy by the human body, as has been shown in various studies [26,27,28]. However, the measured attenuation was considerably higher than what is expected to be caused by absorption of sound energy by the human body, as reported in previous studies [26,27,28,29,30,31,32,33,34,35,36].

In this study, it was reported for the first time that the human body also causes amplification of sound energy. Our experimental results show that, while some frequencies are attenuated by the human subject in a given condition, i.e., stereo or spatial sound projection, other frequencies are amplified and that the degree of attenuation or amplification varies in dependence of the subject. The mean amplification per frequency was in the range of 0 to +4 dB, which equals an increase in sound energy up to 60%.

It is known that the human body partly absorbs and partly reflects the incident sound waves in the acoustic environment. The incident and reflected waves interfere and, depending on the location and the frequency, this interference can be constructive (leading to amplification) or destructive (leading to attenuation) [47]. Phenomena of amplification and attenuation have also been observed in relation to acoustic impedance of the various tissues within the human body, e.g., bone and lung tissue create a high degree of impedance mismatch, which causes sound waves to reflect. The effects of acoustic impedance have only been observed for ultrasonic frequencies and its application in medical imaging [48,49]. The natural amplification of sound frequencies can also be caused by resonance [50,51] and resonant phenomena have been reported in relation to the human body [52,53]. However, the function of the human body as a resonator of sound frequencies in the environment has not been examined in prior studies. It was outside the scope of this study to further investigate the various factors of causation of the observed amplification phenomena.

The direction of amplitude deviation and phase shift were both very weakly correlated to frequency. Instead, the degree of either attenuation or amplification per frequency varied strongly depending on both the type of reproduction and the subject. These findings suggest that the observed effects are not only related to anthropometric features of the subject in a linear way but, instead, they could result from a complex nonlinear dynamic interaction between sound frequency, the spatial conditions of sound reproduction and individual traits of the human subject. As it was outside the scope of this study to investigate the nature of these complex interactions in further detail, these should be elicited in future studies that are designed to systematically assess more variables of the spatial conditions and individual traits of human subjects.

4.4. Study Limitations

This study has several limitations that warrant attention. Data was collected for 14 discrete pure tone frequencies within a limited frequency range of 50–6000 Hz, reproduced at a fixed sound pressure level of ~70 dB SPL, and measurements of all simultaneously recorded microphone positions were aggregated. This study did not investigate variation in the sound pressure level and its effect on the resulting deviations per frequency. The variation in spatial position and distance to the subject and its effect on deviations per frequency was also not assessed. The dependency of sound wave deviations on sound pressure level and spatial position could be key aspects of further validation that contribute to a better understanding of the practical implications of deviations in sound reproduction. These aspects should be investigated in future studies.

No demographic data was collected from human subjects that participated in the study. Very limited anthropometric data was obtained from only six subjects. Due to the very small sample size, only preliminary conclusions with regard to the dependency of observed sound wave deviations to anthropometric features of the subject can be made. The study did not control for the effect of clothes of the subjects. Based on previous studies [29,30,31,32], it can be expected that the clothing of subjects had a partial influence on the observed subject-dependent deviations.

These limitations are crucial for contextualizing the results and should guide future research efforts that aim at further validation and verification of the investigated subject matter and its underlying mechanisms.

4.5. Future Research and Applications

In this study, various phenomena have been reported that were previously unknown and that require further investigation, as indicated. Although it was outside the scope of this study to illuminate further understanding on the nature of some of these phenomena, our experimental results provide first-reported evidence of its measurable impact on sound reproduction.

A better understanding of the conditions that affect deviations in sound reproduction is critical to take into account when sound reproduction systems are applied. This study shows that such deviations are ubiquitously present and can greatly affect the sound reproduction in different ways, depending on specific conditions. Collectively, the findings of this study highlight the importance of a more comprehensive integration of deviation standards in acoustic measurements, and they may support the development of new corrective measures to improve accuracy and reliability of sound reproduction for the benefit of numerous applications, including improvement of medical diagnostics and therapeutic methods relying on sound reproduction.

5. Conclusions

In this paper we have investigated the dependency of deviations in sound reproduction on frequency, types of reproduction and the presence of a human subject in the acoustic field. We found that amplitude deviations and phase distortions occur consistently across frequencies at both the digital and electroacoustic stages of sound reproduction, with a higher magnitude of deviations observed at the electroacoustic reproduction stage. These deviations can be attributed to processing and atmospheric conditions, respectively. Importantly, deviations in electroacoustic reproduction were found to be consistently greater when comparing trials recorded on different days, which requires further investigation.

When investigating the effects of the presence of a human subject in the acoustic field, it was found that both amplitude deviations and phase distortions were significantly higher and, as expected, the magnitude of deviations showed a positive correlation with frequency. Although a negative correlation between body mass and amplitude deviation was found, the measured increase in attenuation was considerably higher than what is expected to be caused by absorption of sound energy by the human body. Furthermore, it was found that, while some frequencies were attenuated, others were amplified and the degree of either attenuation or amplification per frequency varied depending on both the type of reproduction and the subject.

These findings suggest that the observed deviations cannot be attributed to anthropometric features of the subject in a linear way but, instead, they result from a nonlinear dynamic interaction between sound frequency, the spatial conditions of sound reproduction and individual traits of the human subject. Collectively, the findings of this study may serve as a reference to update acoustical standards for the benefit of applications in measurements, diagnostics and therapeutic methods.