
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00007259/00001
Material Information
 Title:
 The Effects of bandwidth reduction on crosscorrelation computations in the analyses of recorded gunshot sounds
 Creator:
 Lacey, Douglas Scott
 Place of Publication:
 Denver, CO
 Publisher:
 University of Colorado Denver
 Publication Date:
 2019
 Language:
 English
Thesis/Dissertation Information
 Degree:
 Master's ( Master of science)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Music and Entertainment Industry Studies, CU Denver
 Degree Disciplines:
 Recording arts
 Committee Chair:
 Grigoras, Catalin
 Committee Members:
 Smith, Jeffrey M.
Maher, Robert C.
Notes
 Abstract:
 Forensic audio examiners often use quantitative measures, such as crosscorrelation computations, of recorded gunshot sounds in an attempt to assess the number of different firearms that were fired and to determine which gunshot events are consistent with having been fired by the same firearm. When used in conjunction with ballistics evidence gathered at the scene, conclusions drawn from such analyses can assist in establishing a timeline of events and answer questions such as "who fired first?â€ Forensic recordings are typically made in uncontrolled environments and are of low quality compared to recordings made in controlled environments (such as recording studios) using highquality microphones and uncompressed audio formats with high sampling rates and wide dynamic range. The relatively poor quality, limited bandwidth, and lossy compression artifacts in forensic recordings, combined with uncontrolled acoustic conditions, can negatively affect the reliability of quantitative analyses. This thesis examines the effects of bandwidth reduction on crosscorrelation computations of recorded gunshot sounds captured in a controlled environment with a highquality recording system.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 Copyright Douglas Scott Lacey. Permission granted to University of Colorado Denver to digitize and display this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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Full Text 
THE EFFECTS OF BANDWIDTH REDUCTION ON CROSSCORRELATION COMPUTATIONS
IN THE ANALYSES OF RECORDED GUNSHOT SOUNDS
by
DOUGLAS SCOTT LACEY B.S., University of Miami, 1996
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Recording Arts Program
2019
Â©2019
DOUGLAS SCOTT LACEY
ALL RIGHTS RESERVED
This thesis for the Master of Science degree by
Douglas Scott Lacey has been approved for the Recording Arts Program by
Catalin Grigoras, Chair Jeffrey M. Smith Robert C. Maher
Date: December 14, 2019
Lacey, Douglas Scott (M.S., Recording Arts Program)
The Effects of Bandwidth Reduction on CrossCorrelation Computations in the Analyses of Recorded Gunshot Sounds
Thesis directed by Associate Professor Catalin Grigoras
ABSTRACT
Forensic audio examiners often use quantitative measures, such as crosscorrelation computations, of recorded gunshot sounds in an attempt to assess the number of different firearms that were fired and to determine which gunshot events are consistent with having been fired by the same firearm. When used in conjunction with ballistics evidence gathered at the scene, conclusions drawn from such analyses can assist in establishing a timeline of events and answer questions such as "who fired first?" Forensic recordings are typically made in uncontrolled environments and are of low quality compared to recordings made in controlled environments (such as recording studios) using highquality microphones and uncompressed audio formats with high sampling rates and wide dynamic range. The relatively poor quality, limited bandwidth, and lossy compression artifacts in forensic recordings, combined with uncontrolled acoustic conditions, can negatively affect the reliability of quantitative analyses. This thesis examines the effects of bandwidth reduction on crosscorrelation computations of recorded gunshot sounds captured in a controlled environment with a highquality recording system.
The form and content of this abstract are approved. I recommend its publication.
Approved: Catalin Grigoras
IV
I dedicate this thesis to my wife, Holly Breault, who continually motivates and supports me in my professional endeavors and my personal life. I owe my Beautiful an unrepayable debt of gratitude for making me strive for greater things. 143.
And to my mom, Donna Lacey, who along with my late dad, Richard Lacey, provided a nurturing environment which enabled me to pursue my educational and career goals, and who patiently stuck with me when those goals changed over time.
v
ACKNOWLEDGEMENTS
I would like to thank Dr. Robert C. Maher for his ongoing and highly informative research into gunshot acoustics and analysis. Of particular importance to this thesis is his U.S. Department of Justicefunded research project titled "Advancing Audio Forensics of Gunshot Acoustics." The database of gunshot recordings utilized in this thesis was produced by Dr. Maher and his team as part of this research project and was kindly made available publicly through the Office of Justice Programs' National Criminal Justice Reference Service.
I am also indebted to my mentor, colleague, and friend Bruce Koenig, for his guidance and professional partnership over the past 23 years. When approaching complicated problems, he encouraged me to think more like a scientist and less like an engineer. In the words of (Dirty) Harry Callahan, "A man's got to know his limitations."
Steve Beck also receives my sincere thanks for letting me bend his ear on a number of occasions regarding his research into recorded gunshot analysis and for his continuing interest and ongoing research in the field.
Lastly, I would be remiss if I didn't thank Dr. Catalin Grigoras, Jeff Smith, Cole Whitecotton, and Leah Haloin at the University of Colorado Denver, for their encouragement throughout the Master's program and their tireless efforts with keeping the program (and students) running smoothly. Cole receives special recognition for running massive amounts of computational data for me in the 11th hour.
VI
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION.................................................................1
Gunshot Analysis.............................................................1
Prior Research...............................................................2
Gunshot Acoustics.......................................................2
CrossCorrelation Computations..........................................5
Limitations..................................................................7
Research Focus...............................................................8
II. MATERIALS...................................................................10
Recorded Gunshot Database...................................................10
Firearms and Shots.....................................................10
Microphone Setup......................................................11
Recording Characteristics..............................................13
III. METHODOLOGY.................................................................14
General.....................................................................14
Audio File Preparation......................................................14
Extraction of Independent Channels.....................................14
Direct Current Offset Removal..........................................17
Bandwidth Reduction Through Resampling......................................19
CrossCorrelation Computations..............................................20
vii
Statistical Calculations
27
IV. RESULTS.............................................................28
V. CONCLUSIONS.........................................................52
VI. FUTURE RESEARCH.....................................................55
REFERENCES...............................................................56
APPENDIX.................................................................58
Case Example........................................................58
viii
LIST OF TABLES
TABLE
1 Summary of the qualitative and quantitative results from [7]...................6
2 Firearmtomicrophone distances (Range) and azimuth angles relative to the line of fire
for the six (6) firearm recording configurations [8]...............................7
3 Firearm and shot information [11, 12]...............................................10
4 Summary of the number of crosscorrelation computations made successively for each
firearm at each angle (Tfirearm_angie), across all angles (Tfirearm_aii_angies), and the total across all sampling rates (Tfirearm_totai)..................................................22
5 Average maximum crosscorrelation values and their corresponding standard deviation values for the intrafirearm comparisons for sampling rates 500, 250,192, 125, 96, 88.2,
62.5, 48 and 44.1 kHz......................................................................31
6 Average maximum crosscorrelation values and their corresponding standard deviation
values for the intrafirearm comparisons for sampling rates 32, 31.25, 24, 22.05, 16, 15.625,12, 11.025, and 8 kHz.........................................................32
7 Average maximum crosscorrelation values and their corresponding standard deviation values for the interfirearm comparisons for sampling rates 500, 250,192, 125, 96, 88.2,
62.5, 48 and 44.1 kHz........................................................................33
8 Average maximum crosscorrelation values and their corresponding standard deviation values for the interfirearm comparisons for sampling rates 32, 31.25, 24, 22.05, 16,
15.625,12, 11.025, and 8 kHz
34
9
Average maximum crosscorrelation values and their corresponding standard deviation
values for the four (4) recorded gunshots in the case example for sampling rates 44.1,
32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz...........................58
x
LIST OF FIGURES
FIGURE
1 Timealigned waveforms for a 2channel recording of a supersonic bullet fired from a
.308 caliber rifle, illustrating the basic acoustical elements of the gunshot [4]....3
2 Comparisons of the shock wave geometry for a bullet traveling at Mach 1.05 and Mach
3 [4]...............................................................................4
3 Detail of a recording of an "N" wave caused by a supersonic bullet passing the
diaphragm of a microphone [2]........................................................4
4 Illustration of the microphone rig for the database collection process [10]...........12
5 Image depicting the shooter positioned in the center of the microphone rig during the
database capture process [11]....................................................12
6 Waveform displays for shot #1 of the SIG Sauer P239 (.357) from angle 1 (0Â°) at the top
to angle 12 (180Â°) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.15
7 Waveform displays for shot #1 of the .308 caliber rifle from angle 1 (0Â°) at the top to
angle 12 (180Â°) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.........16
8 Waveforms of signals X and Y, where Y is equal to X but with a 300 quantization level
shift. The crosscorrelation values for X/X and X/Y from lags 50 to +50 are given. The maximum crosscorrelation value (at lag 0) dropped from +1 to +0.70108 with the introduction of DC offset........................................................18
XI
9
The intrafirearm crosscorrelation comparisons for shots #1 and #2 for angle 1 (0Â°) of
the 12gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum crosscorrelation value for the respective comparison; both values are provided in the title of each plot..........24
10 The intrafirearm crosscorrelation comparisons for shots #1 and #3 for angle 1 (0Â°) of
the 12gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum crosscorrelation value for the respective comparison; both values are provided in the title of each plot..........25
11 The intrafirearm crosscorrelation comparisons for shots #2 and #3 for angle 1 (0Â°) of
the 12gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum crosscorrelation value for the respective comparison; both values are provided in the title of each plot..........26
12 Average maximum crosscorrelation results vs. sampling rate for all firearms. Solid blue
plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot........................35
13 Average maximum crosscorrelation results vs. sampling rate for the 12gauge shotgun.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................36
14 Average maximum crosscorrelation results vs. sampling rate for the .22 caliber rifle.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................37
XII
15
Average maximum crosscorrelation results vs. sampling rate for the .308 caliber rifle.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................38
16 Average maximum crosscorrelation results vs. sampling rate for the AR14 M4 Carbine.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................39
17 Average maximum crosscorrelation results vs. sampling rate for the Colt handgun. Solid
blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot........................40
18 Average maximum crosscorrelation results vs. sampling rate for the Glock 19 handgun.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................41
19 Average maximum crosscorrelation results vs. sampling rate for the Glock 23 handgun.
Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot................42
20 Average maximum crosscorrelation results vs. sampling rate for the Ruger SP101 handgun (.357). Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot... 43
21 Average maximum crosscorrelation results vs. sampling rate for the Ruger SP101
handgun (.38). Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.........44
XIII
22 Average maximum crosscorrelation results vs. sampling rate for the SIG Sauer P239
handgun. Solid blue plot is the intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.....45
23 Percent changes in the average maximum crosscorrelation values for sampling rates
500 kHz down to 15.625 kHz in downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm values...46
24 Percent changes in the average maximum crosscorrelation values for sampling rates
192 kHz down to 12 kHz in downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm values.........47
25 Percent changes in the average maximum crosscorrelation values for sampling rates
88.2 kHz down to 11.025 kHz in downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm values...48
26 Percent changes in the average maximum crosscorrelation values for sampling rates 32
kHz down to 8 kHz in downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm values.........49
27 Percent changes per kHz in the average maximum crosscorrelation values for the intrafirearm comparisons in successive sampling rate steps..................................50
28 Percent changes per kHz in the average maximum crosscorrelation values for the interfirearm comparisons in successive sampling rate steps..................................51
29 Waveform display of the last four (4) recorded intrafirearm gunshots from the case
example, over a onesecond time span and with sampling rate of 44.1 kHz................60
XIV
30 Average maximum crosscorrelation results vs. sampling rate for the last four (4) recorded gunshots in the case example. Standard deviation bars shown for each plot. 61
31 Percent changes per kHz in the average maximum crosscorrelation values for the four
(4) recorded gunshots in the case example in successive sampling rate steps...........62
xv
INTRODUCTION
Gunshot Analysis
The forensic analysis of recorded gunshot sounds, while requested less frequently than audio enhancement and audio authentication, can provide critical information during an investigation of criminal activity or of actions related to civil litigation. With the proliferation of mobile devices and law enforcement body cameras, and the widespread adoption of home and business video surveillance systems, the likelihood that gunshots occurring in urban and rural environments will be recorded has increased. And with that increase comes greater opportunity for analysis.
Requests for recorded gunshot analysis typically center on one (1) or more of the following questions [1]:
â€¢ Are these sounds gunshots?
â€¢ How many gunshots were there?
â€¢ How many firearms were there?
â€¢ How many and which gunshots did each firearm discharge?
â€¢ Who fired first?
â€¢ What are the firearm types/calibers?
â€¢ Where was each shooter positioned?
â€¢ What is the timing between gunshots?
Various techniques may be employed to analyze the recorded audio and to draw conclusions to address the questions posed above. These techniques may include preprocessing/filtering of signals, critical listening, timedomain (waveform) analysis,
1
energy/envelope analysis, frequencydomain analysis, crosscorrelation computations, and time
difference of arrival (TDOA) [1]. The focus of the present research and thesis is on the use of crosscorrelation computations in the analysis of recorded gunshot sounds and does not directly address the other listed techniques.
Prior Research
Gunshot Acoustics
The mechanisms of firearms and the acoustical characteristics of their discharges have been covered by several research papers and presentations aimed at the audio forensics and signal processing fields. Many of these papers/presentations have resulted from the work of Dr. Robert C. Maher (Montana State University, Department of Electrical and Computer Engineering) and his colleagues.
Maher and Shaw [24] have previously discussed the principle mechanics of a gunshot and placed these elements in context with the acoustical signals which are produced by the event. They also identified limitations of capturing gunshots in "real world" conditions with lessthanideal microphone and recording systems.
Figure 1 provides an acoustical, timedomain overview of a .308 caliber rifle firing a supersonic bullet (i.e., faster than the speed of sound) and recorded by two (2) professionalquality microphones at different locations in a controlled environment [4]. The supersonic bullet produces a shock wave which is followed by its ground reflection, both of which arrive at the microphones prior to the muzzle blast, which is traveling at the speed of sound. A ground reflection of the muzzle blast then ends the sequence. In the case of a bullet traveling at less than the speed of sound, no shock wave (or reflected shock wave) would be present.
2
Figure 1  Timealigned waveforms for a 2channel recording of a supersonic bullet fired from a .308 caliber rifle, illustrating the basic acoustical elements of the gunshot [4].
The shock wave expands in a conical fashion behind the bullet, and the angle at which the shock wave front propagates is relative to the bullet's speed divided by the speed of sound, a value referred to as the Mach Number [2, 3]. The higher the Mach Number, the shallower the angle of the shock wave front is relative to the bullet's trajectory, as depicted in Figure 2 [4]. As the shock wave passes the microphone diaphragm, it causes a positive overpressure maximum followed by a corresponding underpressure minimum, which forms an "N" shape in the waveform; this "N" shape can be seen in the Figure 1 waveforms and is provided in more detail in Figure 3 [2].
3
Figure 2  Comparisons of the shock wave geometry for a bullet traveling at Mach 1.05 and
Mach 3 [4].
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time [milliseconds]
Figure 3  Detail of a recording of an "N" wave caused by a supersonic bullet passing the
diaphragm of a microphone [2].
4
CrossCorrelation Computations
Crosscorrelation computations provide for a quantitative measure indicating the similarity between two (2) signals and is defined by the following equation [5, 6]:
sNm1
Rxy{m) =
I
71 = 0
Xn+myn
m > 0
(1)
V Ryx(~m) m < 0
In equation (1), "x" and â€œy" refer to the input signals of sample length â€œN", and â€œm" is the displacement (or lag) in samples as â€œx" and â€œy" are slid over each other while the crosscorrelation computations are performed. Normalization of the output, such that the crosscorrelation value computed of a signal aligned sampleforsample with itself (i.e., autocorrelation) will be +1, is achieved by dividing the output of equation (1) by the product of the norms of "x" and "yâ€, as follows [5, 6]:
Normalized Rxy(m) =
Rxy(m)
+ xl + â€”h *2
n) (Vyi + yf +  + vn)
(2)
The resulting normalized crosscorrelation values will be constrained between 1 and +1, with +1 being the autocorrelation result (as indicated above) and 1 being the autocorrelation result with one (1) of the signals being 180Â° out of phase. The normalized crosscorrelation value will approach 0 for two (2) signals that are completely uncorrelated (e.g., true white noise).
Koenig et al. [7] explored the application of crosscorrelation computations to the forensic analysis of recorded gunshot sounds through a collection of gunshots fired on an outdoor firing range by five (5) firearms at four (4) different positions, relative to the locations
of nine (9) recording/sensing devices which simultaneously recorded the shots. The
5
recording/sensing devices ranged from consumer to professionalgrade and included law enforcementspecific devices. Nearly all the recording/sensing systems were analog, and all were commonly encountered by forensic audio examiners at the time that the research was conducted. Crosscorrelation computations were run, in part, for shots from the same firearm ("cross shot"), and were compared with visual, qualitative assessments of the corresponding waveforms. The general hierarchy given in Table 1 summarizes the correspondence of the qualitative, visual observations of the waveforms with the quantitative crosscorrelation results for the "cross shot" events.
Table 1  Summary of the qualitative and quantitative results from [7].
Visual Observation Average Correlation Correlation Range
Excellent 0.920 0.6450.997
Good 0.834 0.6100.976
Fair 0.686 0.3640.942
Poor 0.498 0.2530.692
As an extension to the research conducted by Koenig, et al. [7], a new set of gunshots was recorded using digital audio recorders [16bit pulse code modulation (PCM) encoding; 96,000 samples per second or 96 kilohertz (kHz)] and four (4) B&K model 4136 microphones with wide frequency response (flat from 4 Hz to 70 kHz) and dynamic range [greater than 172 decibels (dB)]. The microphones were arranged in six (6) different configurations of distance and angle, relative to the position of the firearm, as given in Table 2 [8].
6
Table 2  Firearmtomicrophone distances (Range) and azimuth angles relative to the line of fire for the six (6) firearm recording configurations [8].
Configuration Range (m) Azimuth angle (deg)
1 1.5, 3, 6, 30 3
2 1.5, 3, 6, 30 90
3 3 3, 30, 60, 90
4 30 3, 30, 60, 90
5 3 90, 120, 150, 180
6 30 90, 120, 150, 180
Seven (7) different firearms were utilized in [8], some firing multiple types of ammunition, and crosscorrelation computations arrived at similar results to [7]. Namely, "successiveshot correlations with source, environment, and receiver variations held constant are very high", and "[correlations between waveforms from different angles and different distances are typically lower than those between successive shots."
Limitations
An overriding observation that pervades much of the prior research conducted in the field of recorded gunshot analysis is that there are many factors that affect the ability to answer the common questions listed above and to otherwise draw meaningful conclusions. These factors include, but are not limited to, the following [79]:
â€¢ Microphone type
â€¢ Distance between the microphone and the firearm
â€¢ Relative angle between the microphone and firearm
â€¢ Recorder settings
â€¢ Acoustical environment
â€¢ Type of firearm discharged
7
â€¢ Differences in ammunition
â€¢ Muzzle blast size
Because these factors affect how a gunshot is ultimately recorded, they also impact the quantitative results that are derived from those recordings.
Research Focus
While many factors come into play when analyzing recorded gunshot sounds, such as those listed above, this thesis focuses on how reductions in the audio bandwidth affect the quantitative results arrived at through the application of crosscorrelation. In realworld cases, the forensic examiner does not typically have the benefit of receiving highquality, controlled recordings, nor multiple simultaneous recordings of the same series of events. The utilization of a controlled database of gunshot recordings for this thesis (discussed below in the "MATERIALS" chapter) allowed for wide flexibility regarding the production of reduced bandwidth recordings of the same gunshot event, thereby permitting observations to be made of crosscorrelation computations as the bandwidth is reduced.
With the reduction of the recorded bandwidth comes the removal of highfrequency components within the recorded gunshots, which is expected to lead to fewer distinctive features between intra and interfirearm gunshots (i.e., the recorded gunshot sounds will appear more alike as the bandwidth is reduced). Accordingly, the central hypotheses that were tested for this thesis are as follows:
â€¢ As the bandwidth of an audio recording is decreased, the corresponding crosscorrelation results for intrafirearm comparisons will increase.
8
â€¢ As the bandwidth of an audio recording is decreased, the corresponding cross
correlation results for interfirearm comparisons will increase.
â€¢ The ability to statistically distinguish between recorded gunshot sounds from different firearms may be compromised as the bandwidth of an audio recording is decreased.
9
MATERIALS
Recorded Gunshot Database
The recorded gunshot database that arose from Maher and Routh [10, 11], and subsequently made publicly available online [12], was used as the basis for the research conducted for this thesis. This database was collected anechoically (i.e., without early sound reflections) in an outdoor environment in Montana, USA, and under conditions which were designed to be scientifically reliable and repeatable. The creation of this database was unique in several ways, as discussed below, and provided recorded data that was tailormade for exploring the effect of frequency bandwidth reduction (through downsampling) on crosscorrelation computations.
Firearms and Shots
Table 3 provides a listing of the ten (10) firearm/caliber scenarios that were used during the database collection process, with two (2) different calibers of ammunition (.38 and .357) fired by the RugerSPlOl handgun.
Table 3  Firearm and shot information [11,12].
Scenario Firearm Caliber # of Shots
1 Glock 23 handgun .40 10
2 Glock 19 handgun 9mm 10
3 SIG Sauer P239 handgun .357 10
4 Colt handgun .45 10
5 Ruger SP101handgun .38 9
6 .357 10
7 Rifle .22 20
8 Rifle .308 10
9 Remington shotgun 12ga 3
10 AR14 M4 Carbine 5.56x45mm 10
TOTAL # OF SHOTS 102
10
A total of 21 shots were fired by the .22 rifle, comprised of a set of ten (10) followed by
a set of eleven (11), the latter of which was recorded with a 20decibel (dB) amplification of the input levels. However, the amplified recordings of shot #6 were determined to be unusable, as they featured no discernible gunshot sounds.
Microphone Setup
Twelve (12) GRAS Sound & Vibration A/S type 46DP microphone sets were utilized for the capture process. Each microphone set consisted of a type 40DP 1/8" Externally Polarized Pressure Microphone, a type 26TC%" preamplifier, and type 12AAand 12AG power modules providing the 200volt polarization and 120volt preamplifier power. The microphones provided for a Â±2 dB frequency response out to 140 kHz, with a dynamic range specified between 46 dB (lower limit) and 178 dB (upper limit), resulting in an overall dynamic range of 132 dB [10].
The twelve (12) microphone sets were arranged in a semicircular pattern along a semioctagonal, aluminum rig having a threemeter radius. The shooting position was located at the center of the rig from an elevated position, and the microphone sets were positioned three (3) meters above the ground at 0Â°, 16.4Â°, 32.7Â°, 49.1Â°, 65.5Â°, 81.8Â°, 98.2Â°, 114.5Â°, 130.9Â°, 147.3Â°, 163.6Â°, and 180Â°, relative to the angle of fire. For purposes of this paper, these angles will be referred to as angle #1 through angle #12, respectively. Figure 4 illustrates the characteristics of the microphone rig and the relative location of the shooting position [10], while Figure 5 shows the marksman in the shooting position within the microphone rig during the capture process [11]
11
Figure 4  Illustration of the microphone rig for the database collection process [10].
Figure 5  Image depicting the shooter positioned in the center of the microphone rig during the
database capture process [11].
12
Recording Characteristics
The twelve (12) microphone channels for each shot were recorded simultaneously using a National Instruments Nl PXIe1071 chassis equipped with a Nl PXIe8840 Core processor and Nl PXIe6358 data acquisition card. Each channel was recorded with 16bit PCM encoding and with a sampling rate of 500 kHz, providing a recorded bandwidth of 250 kHz per the Nyquist sampling theorem [13]. The recorded audio for each shot was saved as a MATLAB data file (".mat"), with the twelve (12) columns in the array corresponding to the separate microphone channels from the 0Â° position (column 1) to the 180Â° position (column 12). The data values within the ".mat" files consist of the decimal equivalents of the 16bit quantization values for each audio sample, meaning that the values range from (215) or 32,768 to (2151) or 32,767.
The lengths of the recording from each angle of a shot was identical, but the lengths were not identical across all the shots. The recordings were each a multiple of one (1) second, meaning that their lengths in samples were divisible by 500,000, except for shot #8 of the SIG Sauer P239 which has a length of 2,000,001 samples (4.000002 seconds at a sampling rate of 500 kHz). The total range of lengths across the recordings was from three (3) seconds (shot #4 of the Glock 19 handgun and shots #2 and #3 of the Ruger SP101 handgun firing .38 caliber ammunition) to fifteen (15) seconds (shot #1 of the .308 caliber rifle).
13
METHODOLOGY
General
The overall methodology devised for this thesis can be broken down into the following phases:
1. Audio File Preparation
2. Bandwidth Reduction Through Resampling
3. CrossCorrelation Computations
4. Statistical Calculations
Audio File Preparation Extraction of Independent Channels
The first steps for preparing the ".mat" files for use in this research were to extract each column of data (i.e., each microphone channel) as a separate vector, normalize the vector's sample values to decimal values relative to the maxima of 16bit quantization, and then save that vector to a monaural PCM wavefile with a sampling rate of 500 kHz. With this process, a PCM wavefile was produced for each recorded angle for each shot; for the total of 102 shots, this equated to a total of 1,224 PCM wavefiles (12 angles per shot x 102 shots). Figure 6 and Figure 7 show the twelve (12) timealigned waveform displays for shot #1 of the SIG Sauer P239 (.357) and shot #1 of the .308 caliber rifle, respectively, from 0Â° (top) to 180Â° (bottom). Note in Figure 7 that the ballistic shockwave from the supersonic bullet is seen clearly in the first three angles as the "N"shaped signal preceding the higheramplitude muzzle blast. As the angle between the direction of fire and the microphone increases, the time differential between the ballistic shockwave and the onset of the muzzle blast decreases.
14
sig239_357_shot01_angle01 wav
Figure 6  Waveform displays for shot #1 of the SIG Sauer P239 (.357) from angle 1 (0Â°) at the top to angle 12 (180Â°) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.
15
Figure 7 Waveform displays for shot #1 of the .308 caliber rifle from angle 1 (0Â°) at the top to angle 12 (180Â°) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.
16
Direct Current Offset Removal
Direct current (DC) offset can occur in an audio recording when one (1) or more components (e.g., microphone, microphone preamplifier) induce DC voltage into the audio signal, manifesting itself as a vertical shift of the audio samples away from the xaxis [14, 15]. From review and measurement of the waveforms extracted from the source ".mat" files, it was discovered that the DC offsets of the recorded signals varied across the microphones, with the signals recorded at angles 9 through 12 exhibiting the largest offsets in the order of300 quantization levels at 16bit, or 0.9%. While this percentage may not be large, the crosscorrelation results will be affected by the presence of the DC offset. Crosscorrelation computations are immune to a scalar change of amplitude across the sample values (e.g., reducing the amplitude of an entire signal by a fixed value), but shifting the DC offset of one (1) of the signals will reduce the maximum computed crosscorrelation value, especially when the overall amplitudes are lower.
To exemplify this, consider Signal X (monaural, 16bit PCM, 8 kHz sampling rate, onesecond length) which is created by frequency modulating a 60 Hz sine wave with white noise, both with peak amplitudes of40 dB. Signal Y is created by shifting Signal X by300 quantization levels. By definition, the autocorrelation of Signal X results in a value of +1 at lag 0 (i.e., Signal X is aligned sampleforsample with itself when the crosscorrelation computation is run). A crosscorrelation computation is then run of Signals X and Y, and the result is +0.70108 at a lag of 0. The presence of DC offset reduced the maximum crosscorrelation value from +1 to +0.70108. Figure 8 summarizes this example and includes graphs of the crosscorrelation values versus lag values from 50 to +50.
17
Figure 8  Waveforms of signals X and Y, where Y is equal to X but with a 300 quantization level shift. The crosscorrelation values for X/X and X/Y from lags 50 to +50 are given. The maximum crosscorrelation value (at lag 0) dropped from +1 to +0.70108 with the introduction
of DC offset.
Because of the negative impact that the presence of DC offset can have on the crosscorrelation computations and the fact that DC offset is a channel artifact that does not convey any signaldependent information, the wavefiles extracted from the ".mat" files were each processed to remove any DC offset present in them by performing mean subtraction and saving the results separately as new wavefiles. Mean subtraction was conducted in MATLAB R2019b using the following command, where "xâ€ is the input signal, â€œxdc" is the DCcorrected signal, and "n" is the total number of samples in "x":
xdc = x ~ mean(x) (3)
y>}_ x.
mean(x) = 11 1 (4)
n
18
Bandwidth Reduction Through Resampling
The DC offsetcorrected wavefiles were downsampled from their native 500 kHz to the following sampling rates, which are all factors of 500 kHz: 250 kHz, 125 kHz, 62.5 kHz, 31.25 kHz, and 15.625 kHz. Additional downsampled wavefiles were produced at the following sampling rates, which are commonly used in professional and consumer recording systems:
192 kHz, 96 kHz, 88.2 kHz, 48 kHz, 44.1 kHz, 32 kHz, 24 kHz, 22.05 kHz, 16 kHz, 12 kHz, 11.025 kHz, and 8 kHz. In total, seventeen (17) sets of downsampled wavefiles were produced for each firearm/shot/angle recording.
The resampling processes were performed using the "resamp" function within MATLAB R2019b. The basic syntax for the "resamp" function is as follows [16]:
y = resamp(x,p,q[,n]) (5)
"y" is the resampled output signal, "xâ€ is the input signal, â€œp/q" is the factor by which the signal is resampled, "n" is an optional variable that affects the order of the antialiasing finite impulse response (FIR) lowpass filter (utilizing Kaiser windowing) employed during the resampling process, as follows [16,17]:
Filter order = 2 x n x [max(p, q)] (6)
Generally, "p" is the value of the output file's sampling rate, and â€œq" is the value of the input file's sampling rate; however, any values which satisfy the same ratio can be used. For example, to downsample a 500 kHz signal (x) to a 250 kHz signal (y), either of the following functions could be used:
y = resamp(x, 250000,500000) (7)
y = resamp(x, 1,2) (8)
19
For the resampling processes performed in this research, the source signals were always the
native 500 kHz DCcorrected wavefiles, meaning that the value of â€œq" was always 500,000. The default value of "n" in MATLAB R2019b is ten (10), and that value was utilized for this research [16].
Downsampling the DCcorrected, 500 kHz recordings was chosen as the process for bandwidth reduction in lieu of applying a lowpass filter. This decision was made primarily to expedite the subsequent crosscorrelation computation processes. With lowpass filtering, the sampling rate of the files would remain at 500 kHz, even though the bandwidth of the recorded signal would be bandlimited; lowpassfiltered files would have required a greater number of crosscorrelation computations compared to a downsampled version of the same file. As indicated above, the downsampling process inherently includes an antialiasing lowpass filter, but the resulting files do not contain the extraneous data between the cutoff frequency of the lowpassfiltered version and the original Nyquist frequency (250 kHz).
CrossCorrelation Computations
Crosscorrelation computations were run of all shots within each firearm within each angle (intrafirearm, intraangle) and for all shots across firearms within each angle (interfirearm, intraangle), with the process being repeated for each sampling rate from 500 kHz down to 8 kHz. For example, the three (3) recorded shots from the Remington 12gauge shotgun were crosscorrelated to each other within each angle and for each sampling rate, and then were crosscorrelated with the other nine (9) firearm/caliber scenarios within each angle and for each sampling rate. No interangle crosscorrelations were considered for this research; interangle comparisons have been shown to result in lower crosscorrelation values and
20
greater variance because of "[a]ngular dependence on blast size, internal ballistics, nonlinear
spreading, and ground reflections" [8].
For a given number of recorded shots (n), the formula for the number of pairs of unique intrafirearm combinations (i.e., n items taken two at a time with no repetitions) for each firearm (Tintra) is as follows [18]:
fn\ n! n!
Tintra = = 2!(n2)! = 2(n  2)1 ^
These combinations exclude the autocorrelations, which are the crosscorrelations of each shot/angle recording with itself. The total number of unique, interfirearm comparisons (Tinter) at each angle/sampling rate is given as the following, where 102 is the total number of shots in this study and "n" is the number of shots from the individual firearm (see Table 3):
Tinter = <102  n) (10)
Hence, the total number of comparisons for each firearm at each angle/sampling rate (Tfirearmangle) is the SUITI of Tintra and Tinter, and the total number of comparisons for each firearm across all angles (Tfirearm_aii_angies) for a given sampling rate is 12 times Tfirearm_angieâ– Lastly, the total number of comparisons for each firearm across all angles and across all sampling rates
(Tfirearm total) is 18 times Tfirearm_all_angles
21
Table 4Summary of the number of crosscorrelation computations made successively for each firearm at each angle (Tfirearm_angie), across all angles (Tfirearm_aii_angies), and the total across all
sampling rates (Tfirearm_totai).
Firearm #shots in) Tintra "Winter Tfirearm_angle Tfirearm_ all_ an gles Tfirearm_total
Glock 23 handgun 10 45 920 965 11,580 208,440
Glock 19 handgun 10 45 920 965 11,580 208,440
SIG Sauer P239 handgun 10 45 920 965 11,580 208,440
Colt handgun 10 45 920 965 11,580 208,440
RugerSPlOl handgun (.357) 10 45 920 965 11,580 208,440
Ruger SP101 handgun (.38) 9 36 837 873 10,476 188,568
Rifle (.22) 20 190 1,640 1,830 21,960 395,280
Rifle (.308) 10 45 920 965 11,580 208,440
Remington shotgun 3 3 297 300 3,600 64,800
ARM M4 Carbine 10 45 920 965 11,580 208,440
TOTALS 102 544 9,214 9,758 117,096 2,107,728
The crosscorrelation computations were made using the "xcorr" function within MATLAB R2019b. The basic syntax for the "xcorr" function utilized in this research was as follows [5]:
CC = xcorr{a,bf coeff) (11)
"a" and "b" are the input wavefiles for the crosscorrelation analysis, and â€œCC" is the output array containing the results of the computations, â€œcoeff refers to the method of normalization which results in the values being scaled between 1 and +1, where +1 is the autocorrelation of a signal with itself at lag 0 and 1 would be the same but with the phase of one (1) of the input signals inverted.
Normalization of the results is optional, but when a method is specified, the input signals must be of the same length. Accordingly, because the recordings in the database were not all of the same length, the maximum length of the recordings within a given set of computations was first determined, and any recordings within that set which were shorter than
22
that maximum length were zeropadded with the appropriate number of samples. The cross
correlation computations were then carried out with pairs of files having the same length.
From the output array ("CC") of a given pair of wavefiles, the maximum positive crosscorrelation value was identified and documented in a spreadsheet for each angle and for each sampling rate. Additionally, the corresponding lag positions for the maximum positive crosscorrelation values were similarly documented in a separate set of spreadsheets by angle and sampling rate. As example sets of crosscorrelation comparisons, Figure 9 displays the 500 kHz and 8 kHz intrafirearm comparisons of shot #1 with shot #2 for the 12gauge shotgun at angle 1 (0Â°), aligned at the lag values which resulted in the maximum crosscorrelation values and shown with a time range of twenty (20) milliseconds (i.e., 10,000 samples at 500 kHz, 160 samples at 8 kHz). Similarly, Figure 10 and Figure 11 display the same data for the shot #l/shot #3 and shot #2/shot #3 comparisons, respectively. The ground reflection in each waveform is present approximately eleven (11) milliseconds (i.e., 5,500 samples at 500 kHz, 88 samples at 8 kHz) following the onset of the respective muzzle blast.
23
Shot 1 vs. 2 (500 kHz) [max CC=0.95265, lag=475549]
Shot 1 vs. 2 (8 kHz) [max CC=0.96602, lag=7609]
Figure 9  The intrafirearm crosscorrelation comparisons for shots #1 and #2 for angle 1 (0Â°) of the 12gauge shotgun at the 500
kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross
correlation value for the respective comparison; both values are provided in the title of each plot.
Shot 1 vs. 3 (500 kHz) [max CC=0.93768, lag=826796]
Shot 1 vs. 3 (8 kHz) [max CC=0.95019, lag=13229]
Figure 10  The intrafirearm crosscorrelation comparisons for shots #1 and #3 for angle 1 (0Â°) of the 12gauge shotgun at the
500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross
correlation value for the respective comparison; both values are provided in the title of each plot.
Shot 2 vs. 3 (500 kHz) [max CC=0.9524, lag=351247]
Shot 2 vs. 3 (8 kHz) [max CC=0.97544, lag=5620]
Figure 11  The intrafirearm crosscorrelation comparisons for shots #2 and #3 for angle 1 (0Â°) of the 12gauge shotgun at the
500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross
correlation value for the respective comparison; both values are provided in the title of each plot.
Statistical Calculations
Using the maximum crosscorrelation value spreadsheets, averages and standard deviation values were calculated for all intrafirearm/intraangle comparisons and all interfirearm/intraangle comparisons.
For example, average and standard deviation values were calculated for the set containing all the maximum crosscorrelation values for shots #1 through #3 of the Remington 12gauge shotgun for all intraangle comparisons (e.g., shots #1 and #2 at angle 1, shots #1 and #3 at angle 1, shots #2 and #3 at angle 1, shots #1 and #2 at angle 2, ..., shots #2 and #3 at angle 12). From Table 4, there were three (3) intraangle comparisons made for the Remington 12gauge shotgun for each angle, or a total of 36 comparisons across the twelve (12) angles. Following that, similar average and standard deviation calculations were calculated for Remington 12gauge shotgun shots #1 through #3 against all the intraangle comparisons made with the other firearms. Again from Table 4, there were 297 interangle comparisons made for the Remington 12gauge shotgun for each angle, or a total of 3,564 comparisons across the twelve (12) angles.
27
RESULTS
Table 5 and Table 6 list the computed averages and standard deviations for the maximum crosscorrelation values from the intrafirearm comparisons for sampling rates 500 kHz to 44.1 kHz and 32 kHz to 8 kHz, respectively. The results are listed by firearm and for a set containing all firearms. Similarly, Table 7 and Table 8 list the same computations for the interfirearm comparisons.
The following figures display the average maximum crosscorrelation values versus sampling rate plots for the intrafirearm/intraangle (solid blue) and interfirearm/intraangle (dashed orange) comparisons, as detailed below:
â€¢ Figure 12  for all firearms
â€¢ Figure 13  for the Remington 12gauge shotgun
â€¢ Figure 14  for the .22 caliber rifle
â€¢ Figure 15  for the .308 caliber rifle
â€¢ Figure 16  for the AR14 M4 Carbine
â€¢ Figure 17  for the Colt handgun
â€¢ Figure 18  for the Glock 19 handgun
â€¢ Figure 19  for the Glock 23 handgun
â€¢ Figure 20  for the Ruger SP101 handgun (firing .357 caliber ammunition)
â€¢ Figure 21  for the Ruger SP101 handgun (firing .38 caliber ammunition)
â€¢ Figure 22  for the SIG Sauer P239 handgun
Standard deviation bars are provided for each data point and are colored accordingly (blue for the intrafirearm data points and orange for the interfirearm data points). The intrafirearm
28
standard deviation bars are capped with an arrow, while the interfirearm standard deviation bars are capped with a solid oval, to more easily distinguish the bars that overlap.
The percent changes for each set of sampling rates related by sequential, downward octaves (i.e., halving of the sampling rate) for all firearms are given in the following figures for both intra and interfirearm comparisons:
â€¢ Figure 23  for sampling rates 500 kHz down to 15.625 kHz
â€¢ Figure 24  for sampling rates 192 kHz down to 12 kHz
â€¢ Figure 25  for sampling rates 88.2 kHz down to 11.025 kHz
â€¢ Figure 26  for sampling rates 32 kHz down to 8 kHz These percent change values were calculated per the following equation:
% change
AvgMaxCC(SR/2) â€” AvgMaxCC^SR2j AvgMaxCC(SR)
x 100%
(12)
â€œAvgMaxCCsR" is the average maximum crosscorrelation value at a given sampling rate ("SRâ€), and "AvgMaxCC(sR/2)" is the average maximum crosscorrelation value at half the given sampling rate (i.e., one octave down). The average maximum crosscorrelation values are taken from the "All" rows of Table 5 through Table 8. For example, the percent change for the average maximum crosscorrelation values from 500 kHz to 250 kHz for the intrafirearm comparisons was calculated as follows (average maximum crosscorrelation values taken from Table 5):
% change =
AvgMaxCC(250 kHz)  AvgMaxCC(500 kHz) AvgMaxCCiS00kHz)
x 100% =
0.6790  0.5989 0.5989
X 100% = 13.37%
(13)
29
Lastly, the percent changes per kHz for each successive sampling rate interval were
calculated for all firearms (intra and interfirearm comparisons separately) as follows:
/AvgMaxCC(SR2) â€” AvgMaxCC(SR1)\ % change \ AvgMaxCC(SR1) )
kHz
SR1SR2
x 100%
(14)
â€œAvgMaxCCsRi" is the starting average maximum crosscorrelation value at a given sampling rate ("SRI", in kHz), and â€œAvgMaxCCsR2n is the average maximum crosscorrelation value at the ending sampling rate ("SR2", in kHz). As with equation (12) above, the average maximum crosscorrelation values are taken from the "All" rows of Table 5 through Table 8. For example, the percent change per kHz for the intrafirearm, 500 kHz ("SRI") to 250 kHz (â€œSR2") interval comparison was derived as follows:
(AvgMaxCC(2S0kHz)  AvgMaxCC(S00kHz)\
% change \
AvgMaxCC(S00kHz)
kHz
500  250 '0.6790  0.5989
x 100% =
0 5989 / _ %
/ x 100% = 0.053
250
kHz
(15)
The percent change per kHz results are given in Figure 27 (intrafirearm comparisons) and Figure 28 (interfirearm comparisons).
30
Table 5  Average maximum crosscorrelation values and their corresponding standard deviation values for the intrafirearm
comparisons for sampling rates 500, 250,192,125, 96, 88.2, 62.5, 48 and 44.1 kHz.
INTRAFIREARM COMPARISONS Sampling Rate (kHz)
Firearm Sample size Value 500 250 192 125 96 88.2 62.5 48 44.1
All 6,528 AVG 0.5989 0.6790 0.6963 0.7196 0.7335 0.7370 0.7490 0.7556 0.7580
SD 0.2985 0.2372 0.2287 0.2188 0.2129 0.2114 0.2061 0.2033 0.2027
12ga shotgun 36 AVG 0.7766 0.7917 0.7967 0.8086 0.8148 0.8172 0.8215 0.8249 0.8258
SD 0.1223 0.1121 0.1096 0.1052 0.1015 0.1013 0.0983 0.0968 0.0950
Rifle (.22) 2,280 AVG 0.2741 0.4332 0.4646 0.5014 0.5221 0.5280 0.5489 0.5613 0.5611
SD 0.1534 0.1453 0.1525 0.1605 0.1646 0.1660 0.1715 0.1757 0.1754
Rifle (.308) 540 AVG 0.7404 0.7694 0.7769 0.7921 0.8021 0.8043 0.8120 0.8161 0.8190
SD 0.2137 0.1839 0.1789 0.1698 0.1628 0.1614 0.1565 0.1542 0.1502
AR14 M4 Carbine 540 AVG 0.7578 0.7891 0.7979 0.8162 0.8283 0.8307 0.8393 0.8441 0.8486
SD 0.2069 0.1738 0.1676 0.1549 0.1458 0.1438 0.1383 0.1343 0.1294
Colt 540 AVG 0.7618 0.8022 0.8129 0.8286 0.8382 0.8407 0.8472 0.8497 0.8542
SD 0.2035 0.1714 0.1657 0.1549 0.1460 0.1446 0.1392 0.1367 0.1305
Glock 19 540 AVG 0.7291 0.7866 0.8002 0.8192 0.8312 0.8336 0.8422 0.8455 0.8504
SD 0.2101 0.1655 0.1600 0.1481 0.1384 0.1373 0.1313 0.1298 0.1232
Glock 23 540 AVG 0.7621 0.8131 0.8251 0.8409 0.8511 0.8535 0.8597 0.8636 0.8672
SD 0.2003 0.1596 0.1551 0.1445 0.1362 0.1350 0.1301 0.1267 0.1213
Ruger SP101 (.357) 540 AVG 0.8326 0.8541 0.8613 0.8747 0.8831 0.8856 0.8917 0.8950 0.8984
SD 0.1320 0.1115 0.1067 0.0973 0.0912 0.0899 0.0859 0.0837 0.0794
Ruger SP101 (.38) 432 AVG 0.7611 0.8067 0.8184 0.8371 0.8490 0.8513 0.8597 0.8639 0.8680
SD 0.1875 0.1488 0.1431 0.1324 0.1239 0.1222 0.1170 0.1142 0.1091
SIG Sauer P239 540 AVG 0.8389 0.8668 0.8740 0.8868 0.8952 0.8966 0.9021 0.9040 0.9074
SD 0.1447 0.1172 0.1113 0.1010 0.0938 0.0924 0.0877 0.0861 0.0813
Table 6  Average maximum crosscorrelation values and their corresponding standard deviation values for the intrafirearm
comparisons for sampling rates 32, 31.25, 24, 22.05,16,15.625,12,11.025, and 8 kHz.
INTRAFIREARM COMPARISONS Sampling Rate (kHz)
Firearm Sample size Value 32 31.25 24 22.05 16 15.625 12 11.025 8
All 6,528 AVG 0.7727 0.7732 0.7775 0.7789 0.7820 0.7823 0.7835 0.7834 0.7815
SD 0.2043 0.2039 0.2017 0.2011 0.1993 0.1992 0.1988 0.1985 0.1987
12ga shotgun 36 AVG 0.8388 0.8393 0.8422 0.8449 0.8464 0.8427 0.8454 0.8460 0.8461
SD 0.0870 0.0873 0.0847 0.0839 0.0810 0.0833 0.0845 0.0828 0.0835
Rifle (.22) 2,280 AVG 0.5531 0.5541 0.5620 0.5645 0.5704 0.5710 0.5741 0.5737 0.5726
SD 0.1750 0.1749 0.1773 0.1778 0.1807 0.1809 0.1850 0.1841 0.1877
Rifle (.308) 540 AVG 0.8431 0.8437 0.8466 0.8483 0.8516 0.8524 0.8545 0.8564 0.8585
SD 0.1236 0.1235 0.1208 0.1205 0.1157 0.1163 0.1130 0.1122 0.1088
AR14 M4 Carbine 540 AVG 0.8793 0.8796 0.8836 0.8851 0.8901 0.8898 0.8952 0.8925 0.8962
SD 0.0956 0.0949 0.0906 0.0899 0.0846 0.0849 0.0807 0.0808 0.0753
Colt 540 AVG 0.8840 0.8843 0.8860 0.8866 0.8868 0.8866 0.8875 0.8850 0.8802
SD 0.0926 0.0924 0.0893 0.0887 0.0840 0.0832 0.0808 0.0788 0.0770
Glock 19 540 AVG 0.8855 0.8865 0.8902 0.8902 0.8922 0.8921 0.8912 0.8926 0.8900
SD 0.0787 0.0790 0.0743 0.0749 0.0704 0.0710 0.0674 0.0672 0.0645
Glock 23 540 AVG 0.8934 0.8930 0.8939 0.8943 0.8925 0.8938 0.8924 0.8943 0.8866
SD 0.0867 0.0864 0.0838 0.0830 0.0802 0.0807 0.0784 0.0789 0.0749
Ruger SP101 (.357) 540 AVG 0.9172 0.9175 0.9189 0.9203 0.9224 0.9221 0.9208 0.9221 0.9185
SD 0.0595 0.0590 0.0564 0.0553 0.0528 0.0531 0.0517 0.0530 0.0537
Ruger SP101 (.38) 432 AVG 0.8981 0.8983 0.9014 0.9018 0.9047 0.9034 0.9011 0.9022 0.8999
SD 0.0713 0.0707 0.0677 0.0672 0.0633 0.0640 0.0630 0.0627 0.0629
SIG Sauer P239 540 AVG 0.9285 0.9285 0.9303 0.9303 0.9298 0.9300 0.9287 0.9274 0.9235
SD 0.0530 0.0526 0.0506 0.0499 0.0469 0.0469 0.0467 0.0452 0.0469
Table 7  Average maximum crosscorrelation values and their corresponding standard deviation values for the interfirearm
comparisons for sampling rates 500, 250,192,125, 96, 88.2, 62.5, 48 and 44.1 kHz.
INTERFIREARM COMPARISONS Sampling Rate (kHz)
Firearm Sample size Value 500 250 192 125 96 88.2 62.5 48 44.1
All 110,568 AVG 0.4895 0.5395 0.5478 0.5596 0.5670 0.5690 0.5752 0.5791 0.5799
SD 0.2815 0.2543 0.2524 0.2508 0.2498 0.2495 0.2482 0.2473 0.2478
12ga shotgun 3,654 AVG 0.4000 0.4281 0.4320 0.4378 0.4414 0.4424 0.4456 0.4473 0.4479
SD 0.1830 0.1651 0.1634 0.1616 0.1607 0.1603 0.1593 0.1585 0.1588
Rifle (.22) 19,680 AVG 0.2013 0.2756 0.2853 0.2964 0.3035 0.3057 0.3134 0.3186 0.3164
SD 0.0902 0.0941 0.0959 0.1017 0.1067 0.1079 0.1122 0.1155 0.1134
Rifle (.308) 11,040 AVG 0.5236 0.5619 0.5683 0.5786 0.5853 0.5870 0.5924 0.5957 0.5968
SD 0.2629 0.2361 0.2338 0.2308 0.2288 0.2284 0.2268 0.2257 0.2255
AR14 M4 Carbine 11,040 AVG 0.5497 0.5917 0.5993 0.6118 0.6200 0.6221 0.6286 0.6325 0.6341
SD 0.2702 0.2420 0.2394 0.2364 0.2342 0.2337 0.2318 0.2306 0.2303
Colt 11,040 AVG 0.5762 0.6240 0.6329 0.6456 0.6535 0.6555 0.6615 0.6651 0.6668
SD 0.2748 0.2433 0.2405 0.2368 0.2341 0.2335 0.2312 0.2296 0.2295
Glock 19 11,040 AVG 0.5597 0.6153 0.6256 0.6399 0.6489 0.6512 0.6582 0.6624 0.6643
SD 0.2664 0.2319 0.2293 0.2255 0.2228 0.2222 0.2202 0.2189 0.2186
Glock 23 11,040 AVG 0.5719 0.6252 0.6347 0.6480 0.6564 0.6585 0.6649 0.6687 0.6705
SD 0.2674 0.2339 0.2314 0.2278 0.2253 0.2248 0.2227 0.2214 0.2210
Ruger SP101 (.357) 11,040 AVG 0.5500 0.5850 0.5915 0.6012 0.6076 0.6092 0.6142 0.6173 0.6183
SD 0.2817 0.2588 0.2568 0.2548 0.2534 0.2530 0.2518 0.2508 0.2510
Ruger SP101 (.38) 10,044 AVG 0.5402 0.5916 0.6004 0.6131 0.6212 0.6232 0.6296 0.6336 0.6352
SD 0.2610 0.2279 0.2252 0.2217 0.2194 0.2189 0.2171 0.2159 0.2155
SIG Sauer P239 11,040 AVG 0.5930 0.6330 0.6406 0.6519 0.6591 0.6609 0.6664 0.6696 0.6712
SD 0.2748 0.2485 0.2462 0.2437 0.2420 0.2416 0.2400 0.2389 0.2392
Table 8  Average maximum crosscorrelation values and their corresponding standard deviation values for the interfirearm
comparisons for sampling rates 32, 31.25, 24, 22.05,16,15.625,12,11.025, and 8 kHz.
INTERFIREARM COMPARISONS Sampling Rate (kHz)
Firearm Sample size Value 32 31.25 24 22.05 16 15.625 12 11.025 8
All 110,568 AVG 0.5891 0.5894 0.5925 0.5934 0.5966 0.5968 0.5992 0.5998 0.6020
SD 0.2514 0.2513 0.2503 0.2500 0.2490 0.2490 0.2486 0.2487 0.2478
12ga shotgun 3,654 AVG 0.4531 0.4535 0.4546 0.4551 0.4560 0.4562 0.4569 0.4566 0.4572
SD 0.1610 0.1609 0.1602 0.1599 0.1593 0.1591 0.1585 0.1583 0.1581
Rifle (.22) 19,680 AVG 0.3069 0.3074 0.3118 0.3130 0.3174 0.3175 0.3206 0.3209 0.3239
SD 0.1054 0.1057 0.1082 0.1090 0.1127 0.1128 0.1168 0.1177 0.1211
Rifle (.308) 11,040 AVG 0.6069 0.6072 0.6097 0.6105 0.6132 0.6133 0.6155 0.6159 0.6174
SD 0.2235 0.2234 0.2222 0.2218 0.2205 0.2203 0.2197 0.2195 0.2178
AR14 M4 Carbine 11,040 AVG 0.6475 0.6478 0.6511 0.6522 0.6557 0.6559 0.6589 0.6596 0.6632
SD 0.2294 0.2293 0.2280 0.2276 0.2262 0.2262 0.2254 0.2255 0.2244
Colt 11,040 AVG 0.6818 0.6821 0.6847 0.6856 0.6878 0.6880 0.6899 0.6904 0.6917
SD 0.2289 0.2288 0.2270 0.2266 0.2245 0.2244 0.2228 0.2226 0.2203
Glock 19 11,040 AVG 0.6814 0.6817 0.6853 0.6864 0.6901 0.6902 0.6932 0.6937 0.6963
SD 0.2169 0.2168 0.2156 0.2152 0.2136 0.2135 0.2124 0.2125 0.2105
Glock 23 11,040 AVG 0.6859 0.6862 0.6888 0.6894 0.6918 0.6917 0.6935 0.6944 0.6954
SD 0.2194 0.2192 0.2179 0.2173 0.2159 0.2157 0.2150 0.2149 0.2132
Ruger SP101 (.357) 11,040 AVG 0.6284 0.6286 0.6312 0.6320 0.6347 0.6350 0.6374 0.6378 0.6407
SD 0.2512 0.2511 0.2501 0.2497 0.2485 0.2483 0.2476 0.2473 0.2464
Ruger SP101 (.38) 10,044 AVG 0.6486 0.6489 0.6522 0.6533 0.6574 0.6578 0.6612 0.6624 0.6654
SD 0.2136 0.2136 0.2123 0.2121 0.2112 0.2112 0.2112 0.2113 0.2100
SIG Sauer P239 11,040 AVG 0.6849 0.6851 0.6878 0.6885 0.6909 0.6913 0.6931 0.6935 0.6948
SD 0.2405 0.2402 0.2390 0.2387 0.2372 0.2373 0.2365 0.2366 0.2360
ALL FIREARMS  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
Figure 12  Average maximum crosscorrelation results vs. sampling rate for all firearms. Solid blue plot is the intrafirearm
computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
12ga Shotgun  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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Sampling rate (kHz)
Figure 13  Average maximum crosscorrelation results vs. sampling rate for the 12gauge shotgun. Solid blue plot is the intra
firearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
.22 Caliber Rifle  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
Figure 14  Average maximum crosscorrelation results vs. sampling rate for the .22 caliber rifle. Solid blue plot is the intrafirearm
computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
LO
00
.308 Caliber Rifle  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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Figure 15  Average maximum crosscorrelation results vs. sampling rate for the .308 caliber rifle. Solid blue plot is the intra
firearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
AR14 M4 Carbine  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
0.2
0.1
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11.025
Figure 16  Average maximum crosscorrelation results vs. sampling rate for the AR14 M4 Carbine. Solid blue plot is the intra
firearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
Colt Handgun  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
0.2
0.1
0
500 250 192 125 96 88.2 62.5 48 44.1 32 31.25 24 22.05 16 15.625 12 11.025 8
Sampling rate (kHz)
Figure 17  Average maximum crosscorrelation results vs. sampling rate for the Colt handgun. Solid blue plot is the intrafirearm
computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
Glock 19 Handgun  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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11.025 8
Figure 18  Average maximum crosscorrelation results vs. sampling rate for the Glock 19 handgun. Solid blue plot is the intra
firearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
Glock 23 Handgun  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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500 250 192 125 96 88.2 62.5 48 44.1 32 31.25 24 22.05 16 15.625 12 11.025 8
Sampling rate (kHz)
Figure 19  Average maximum crosscorrelation results vs. sampling rate for the Glock 23 handgun. Solid blue plot is the intra
firearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each plot.
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Ruger SP101 Handgun (.357)  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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Sampling rate (kHz)
Figure 20  Average maximum crosscorrelation results vs. sampling rate for the Ruger SP101 handgun (.357). Solid blue plot is the
intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each
plot.
Ruger SP101 Handgun (.38)  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
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11.025 8
Figure 21  Average maximum crosscorrelation results vs. sampling rate for the Ruger SP101 handgun (.38). Solid blue plot is the
intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each
plot.
â– p*
Ui
SIG Sauer P239 Handgun  AVERAGE MAXIMUM CROSSCORRELATION VALUES VS. SAMPLING RATE
0.2
0.1
0
500 250 192 125 96 88.2 62.5 48 44.1 32 31.25 24 22.05 16 15.625 12 11.025 8
Sampling rate (kHz)
Figure 22  Average maximum crosscorrelation results vs. sampling rate for the SIG Sauer P239 handgun. Solid blue plot is the
intrafirearm computations, and dashed orange plot is the interfirearm computations. Standard deviation bars shown for each
plot.
Figure 23  Percent changes in the average maximum crosscorrelation values for sampling rates 500 kHz down to 15.625 kHz in
downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm
values.
Figure 24  Percent changes in the average maximum crosscorrelation values for sampling rates 192 kHz down to 12 kHz in
downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm
values.
Figure 25  Percent changes in the average maximum crosscorrelation values for sampling rates 88.2 kHz down to 11.025 kHz in
downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm
values.
Figure 26  Percent changes in the average maximum crosscorrelation values for sampling rates 32 kHz down to 8 kHz in
downward octave steps. Solid black plot represents the intrafirearm values, and dashed red plot represents the interfirearm
values.
Figure 27  Percent changes per kHz in the average maximum crosscorrelation values for the intrafirearm comparisons in
successive sampling rate steps.
Figure 28  Percent changes per kHz in the average maximum crosscorrelation values for the interfirearm comparisons in
successive sampling rate steps.
CONCLUSIONS
The results of the research conducted for this thesis support the hypotheses that as the bandwidth of an audio recording is decreased, the corresponding maximum crosscorrelation values will increase for both intra and interfirearm comparisons.
Except for the transition from 16 kHz to 8 kHz for the intrafirearm condition, all the percent changes in the average maximum crosscorrelation computations for the octaveinterval results were positive (see Figure 23, Figure 24, Figure 25, and Figure 26). The greatest percent change was observed in the 500 kHz to 250 kHz transition for both intra and interfirearm comparisons. The percent changes generally decreased as the sampling rates decreased; however, there were two (2) instances in the interfirearm percent changes where successive values slightly increased (from 2.12% to 2.32% for the transition between 96 kHz/48 kHz and 48 kHz/24 kHz, and from 1.92% to 2.33% for the transition between 88.2 kHz/44.1 kHz and 44.1 kHz/22.05 kHz).
The results of the successive sampling rate percent changes per kHz revealed positive results for all the intra and interfirearm comparison intervals, except for the last two (2) transitions of the intrafirearm results (0.005%/kHz for the 12 kHz/11.025 kHz transition and 0.081%/kHz for the 11.025 kHz to 8 kHz transition). Both the intra and interfirearm results exhibit a noticeable peak at the 44.1 kHz to 32 kHz transition, the reason for which is not readily apparent.
As indicated above in the "Research Focus section", the primary reason for the increases in the maximum crosscorrelation values is likely the systematic removal of the highfrequency variations in the recorded gunshots as the sampling rate (and therefore, bandwidth) is reduced.
52
The cumulative effect of the differences in these highfrequency variations results in minor but
quantified differences in the corresponding crosscorrelation values.
For the individual firearm computations, the only firearm which exhibited a clear separation between the intra and interfirearm plots of the average maximum crosscorrelation values (i.e., no overlap of their standard deviation ranges) was the Remington 12gauge shotgun (see Figure 13). The intra and interfirearm plots for all other individual firearms and for the set of all firearms overlap within one (1) standard deviation. The mechanisms by which the shotgun discharges and the differences in its ammunition type, relative to the handguns and rifles, likely led to its shots being more distinctive among the set of tested firearms.
It was noted that the intrafirearm results for the .22 caliber rifle never exceeded 0.6 (not including the standard deviation range), which was relatively poor compared to the other firearms which always exceeded 0.7. Similarly, the interfirearm results for the .22 caliber rifle were lower overall than the other firearms' results, with the maximum being 0.3239 at 8 kHz; whereas the other firearms ranged from 0.4572 (Remington 12gauge shotgun) to 0.6963 (Glock 19) for the interfirearm results at 8 kHz. These results may have resulted from the inclusion of the initial set of ten (10) shots from the .22 caliber rifle, which exhibited poorer signaltonoise than the subsequent set of eleven (11) shots with the 20dB amplification.
The standard deviation ranges for the intrafirearm computations for all firearms generally decreased as the sampling rate decreased, with the .22 caliber rifle and Remington 12gauge shotgun exhibiting the lowest rates of change. For the interfirearm computations, the differences in the standard deviation ranges also decreased but were not as significant as
53
the intrafirearm results, which may be due to the inclusion of different firearm classes (e.g.,
handguns, rifles, shotgun) in the test set.
The hypothesis regarding the decreases in bandwidth compromising the ability to statistically distinguish between recorded gunshot sounds from different firearms is not supported by the data in this research. As observed in Figure 12 through Figure 22, the overlaps in the intra and interfirearm plots and their standard deviation ranges generally decrease as the sampling rate/bandwidth decreases, indicating that discrimination between the two sets (intra and inter) becomes greater.
54
FUTURE RESEARCH
As noted above, this research utilized highquality recordings in a controlled environment; hence, conducting the same or similar research using recordings captured in nonanechoic but semicontrolled conditions (e.g., same microphone rig) and/or "real world" cases with known circumstances would likely shed more light on the results in conditions commonly encountered by forensic audio examiners. A simple, intrafirearm case example with known circumstances is presented in the Appendix.
Producing similar databases of controlled recordings using a wider array of firearms/ammunition would also improve the breadth of the data presently available and enable more detailed comparisons between firearm classes and specific models/ammunition.
The effects of bandwidth reduction on other quantitative measures, such as mean quadratic difference, and utilizing power data in lieu of waveforms in the same workflow could also be explored.
55
REFERENCES
1. Begault, D.R., S.D. Beck, and R.C. Maher, Overview of Forensic Audio Gunshot Analysis Techniques, in 2019 AES International Conference on Audio Forensics. 2019: Porto, Portugal.
2. Maher, R.C., Modeling and Signal Processing of Acoustic Gunshot Recordings, in 2006 IEEE 12th Digital Signal Processing Workshop & 4th IEEE Signal Processing Education Workshop. 2006. p. 257261.
3. Maher, R.C., Acoustical Characterization of Gunshots, in 2007 IEEE Workshop on Signal Processing Applications for Public Security and Forensics. 2007. p. 15.
4. Maher, R.C. and S.R. Shaw, Deciphering Gunshot Recordings, in 33rd International Conference: Audio Forensics. 2008: Denver, USA.
5. The Mathworks Inc. MATLAB R2019b "xcorr"function, [cited 2019 November 6]; Available from: https://www.mathworks.com/help/matlab/ref/xcorr.html.
6. Lacey, D.S., B.E. Koenig, and C.E. Reimond, The Effect of Sample Length on CrossCorrelation Comparisons of Recorded Gunshot Sounds, in 54th International Conference: Audio Forensics. 2014: London, UK.
7. Koenig, B.E., S.M. Hoffman, H. Nakasone, and S.D. Beck, Signal Convolution of Recorded FreeField Gunshot Sounds. J. Audio Eng. Soc, 1998. 46(7/8): p. 634653.
8. Beck, S.D., H. Nakasone, and K.W. Marr, Variations in recorded acoustic gunshot waveforms generated by small firearms. J Acoust Soc Am, 2011. 129(4): p. 17481759.
9. Maher, R.C. and E. Hoerr, Forensic Comparison of Simultaneous Recordings of Gunshots at a Crime Scene, in 147th Convention of the Audio Engineering Society. 2019: New York, USA.
10. Maher, R.C. and T. Routh, Advancing Forensic Analysis of Gunshot Acoustics, in 139th Convention of the Audio Engineering Society. 2015: New York, USA.
56
11. Maher, R.C., Advancing Audio Forensics of Gunshot Acoustics. 2018, National Criminal Justice Reference Service.
12. Maher, R.C. Example recorded data. 2018 [cited 2019 November 6]; Available from: http://www.montana.edu/rmaher/gunshots/gunshot data.html.
13. Pohlmann, K.C., Principles of digital audio. 6th ed. 2011, New York: McGrawHill.
14. Koenig, B.E. and D.S. Lacey, The Average Direct Current Offset Values for Small Digital Audio Recorders in an Acoustically Consistent Environment. J Forensic Sci, 2014. 59(4): p. 960966.
15. Koenig, B.E., D.S. Lacey, C. Grigoras, S.G. Price, and J.M. Smith, Evaluation of the Average DC Offset Values for Nine Small Digital Audio Recorders. J Audio Eng Soc, 2013. 61(6): p. 439448.
16. The Mathworks Inc. MATLAB R2019b "resamp"function, [cited 2019 November 6]; Available from: https://www.mathworks.com/help/signal/ref/resample.html.
17. The Mathworks Inc. Resampling, [cited 2019 November 6]; Available from: https://www.mathworks.com/help/signal/ug/resampling.html.
18. Mendenhall, W. and T. Sincich, Statistics for engineering and the sciences. 5th ed. 2007, Upper Saddle River, New Jersey: Prentice Hall, Inc.
57
APPENDIX
Case Example
As a simple case example, fourteen (14) shots were fired from a handgun (Glock 22, .40 caliber) in an outdoor environment at night, near a microphone mounted within a law enforcement vehicle. The microphone signal was recorded onto one channel of the hifi stereo audio track of a VHS tapebased dashboard camera recording system. The individual firing the handgun was panning slightly from their left to right over the first ten (10) shots but was relatively still for the last four (4) shots, as observed in the video recording of a second dashboard camera recording system. Figure 29 displays the waveforms for these last four (4) recorded shots, as digitized at a sampling rate of 44.1 kHz from the VHS hifi audio track.
The methodology described in this thesis was applied to these last four (4) shots, each segmented into separate 175millisecond WAV files, with the downsampling processes performed using the DC offsetcorrected, 44.1 kHz digitized file segments. The results are provided in Table 9 and are displayed graphically in Figure 30. Additionally, the percent changes per kHz in the average maximum crosscorrelation values are shown in Figure 31.
Table 9  Average maximum crosscorrelation values and their corresponding standard deviation values for the four (4) recorded gunshots in the case example for sampling rates 44.1, 32, 31.25, 24, 22.05, 16,15.625, 12, 11.025, and 8 kHz.
OJ _D ns > u u Sampling Rate (kHz)
(0 44.1 32 31.25 24 22.05 16 15.625 12 11.025 8
AVG 0.7175 0.7174 0.7176 0.7171 0.7176 0.7165 0.7165 0.7203 0.7191 0.7237
SD 0.1356 0.1356 0.1358 0.1355 0.1361 0.1345 0.1346 0.1366 0.1342 0.1346
58
From this case example, it is evident that bandwidth reduction had little impact on the
average maximum crosscorrelation values and corresponding standard deviations, but the results are generally consistent with those obtained from the controlled database recordings utilized in this research for the same 44.1 kHz to 8 kHz range (see Table 5 and Table 6 and the corresponding figures).
Both the average and standard deviation values from the case example were highly consistent across the sampling rates, with the overall average/standard deviation range being 0.7183Â±0.1353. The corresponding percent change per kHz results oscillated above and below 0%/kHz, with the values for the lowest sampling rate transitions (15.625 kHz/12 kHz, 12 kHz/11.025 kHz, and 11.025 kHz/8 kHz) having the greatest deviations.
As a general observation, the waveforms of the recorded gunshots in this case example (Figure 29) are noticeably different than those captured in the controlled database (as exemplified in Figure 9 through Figure 11). Whereas the controlled database recordings exhibit quick acoustic decay and return to the ambient noise level within approximately eleven (11) milliseconds, the recordings in the case example have much longer acoustic decay patterns and more complex signatures following the onset of the muzzle blasts. These differences are due in large part to the effects of the nonoptimal microphone and recording system employed in the dashboard recording system.
59
Figure 29  Waveform display of the last four (4) recorded intrafirearm gunshots from the case example, over a onesecond time
span and with sampling rate of 44.1 kHz.
0.9
Case Example (intrafirearm, four consecutive shots)  AVERAGE MAXIMUM CROSSCORRELATION
VALUES VS. SAMPLING RATE
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Figure 30  Average maximum crosscorrelation results vs. sampling rate for the last four (4) recorded gunshots in the case
example. Standard deviation bars shown for each plot.
Figure 31  Percent changes per kHz in the average maximum crosscorrelation values for the four (4) recorded gunshots in the
case example in successive sampling rate steps.

Full Text 
PAGE 1
THE EFFECTS OF BANDWIDTH REDUCTION ON CROSS CORRELATION COMPUTATIONS IN THE ANALYSES OF RECORDED GUNSHOT SOUNDS by DOUGLAS SCOTT LACEY B.S., University of Miami, 1996 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Recording Arts Program 2019
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ii 2019 DOUGLAS SCOTT LACEY ALL RIGHTS RESERVED
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iii This thesis for the Master of Science degree by Douglas Scott Lacey has been approved for the Recording Arts Program by Catalin Grigoras, Chair Jeffrey M. Smith Robert C. Maher Date: December 14 , 2019
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iv Lacey, Douglas Scott (M.S., Recording Arts Program) The Effects of Bandwidth Reduction on Cross Correlation Computations in the Analyses of Recorded Gunshot Sounds Thesis directed by Associate Professor Catalin Grigoras ABSTRACT Forensic audio examiners often use quantitative measures, such as cross correla tion computations, of recorded gunshot sounds in an attempt to assess the number of different firearms that were fired and to determine which gunshot events are consistent with having been fired by the same firearm . When used in conjunction with ballistics evidence gathered at the scene, conclusions drawn from such analyses can assist in establishing a tim eline of events and answer questions such as "who fired first? Forensic recordings are typically made in uncontrolled environments and are of low quality compared to recordings made in controlled environments (such as recording studios) using high quality microphones and uncompressed audio formats with high sampling rates and wide dynamic range. The relatively poor quality, limited bandwidth, and lossy compression artifacts in forensic recordings, combined with uncontrolled acoustic conditions, can negativ ely affect the reliability of quantitative analyses. This thesis examines the effects of bandwidth reduction on cross correlation computations of recorded gunshot sounds captured in a controlled environment with a high quality recording system. The form an d content of this abstract are approved. I recommend its publication. Approved: Catalin Grigoras
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v I dedicate this thesis to my wife, Holly Breault, who continually motivates and supports me in my professional endeavors and my personal life . I owe m y Beautiful an unrepayable debt of gratitude for making me strive for greater things. 143 . And to my mom, Donna Lacey, who along with my late dad, Richard Lacey, provided a nurturing environment which enabled me to pursue my educational and career goals, and who patiently stuck with me when those goals changed over time .
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vi ACKNOWLEDGEMENTS I would like to thank Dr. Robert C. Maher for his ongoing and highly informative research into gunshot acoustics and analysis. Of particular importance to this thesis is his U.S. Department of Justice Acousti cs The database of gunshot recordings utilized in this thesis was produced by Dr. Maher and his team as part of this research project and was kindly made available publicly ce. I am also indebted to my mentor, colleague, and friend Bruce Koenig, for his guidance and professional partnership over the past 23 years. When approaching complicated problems, he encouraged me to think more like a scientist and less like an engineer. In the words of (Dirty) A man's got to know his limitations Steve Beck also receives my sincere thanks for letting me bend his ear on a number of occasions regarding his research into recorded gunshot analysis and for his continuing int erest and ongoing research in the field. Whitecotton, and Leah Haloin at the University of Colorado Denve r, for their encouragement nd their tireless efforts with keeping the program (and students) running smoothly. Cole receives special recognition fo r running massive amounts of computational data for me in the 11 th hour.
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vii TABLE OF CONTENTS CHAPTER I. INTRODUCTION ................................ ................................ ................................ ................ 1 Gunshot Analysis ................................ ................................ ................................ .............. 1 Prior Research ................................ ................................ ................................ .................. 2 Gunshot Acoustics ................................ ................................ ................................ .. 2 Cross Correlation Computations ................................ ................................ ............ 5 Limitations ................................ ................................ ................................ ........................ 7 Research Focus ................................ ................................ ................................ ................ 8 II. MATERIALS ................................ ................................ ................................ ..................... 10 Recorded Gunshot Database ................................ ................................ ......................... 10 Firearms and Shots ................................ ................................ ............................... 10 Microphone Set up ................................ ................................ ............................... 11 Recording Charac teristics ................................ ................................ ..................... 13 III. METHODOLOGY ................................ ................................ ................................ ............. 14 General ................................ ................................ ................................ ........................... 14 Audio File Preparation ................................ ................................ ................................ ... 14 Extraction of Independent Channels ................................ ................................ .... 14 Direct Current Offset Removal ................................ ................................ ............. 17 Bandwidth Reduction Through Resampling ................................ ................................ .. 19 Cross Correlation Computations ................................ ................................ ................... 20
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viii St atistical Calculations ................................ ................................ ................................ ... 27 IV. RESULTS ................................ ................................ ................................ .......................... 28 V. CONCLUSIONS ................................ ................................ ................................ ................ 52 VI. FUTURE RESEARCH ................................ ................................ ................................ ......... 55 REFERENCES ................................ ................................ ................................ ............................. 56 APPENDIX ................................ ................................ ................................ ................................ . 58 Case Example ................................ ................................ ................................ ................. 58
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ix LIST OF TABLES TABLE 1 Summary of the qualitative and quantitative results from [7]. ................................ .......... 6 2 Firearm to microphone distances (Range) and azimuth angles relative to the line of fire for the six (6) firearm recording config urations [8]. ................................ ........................... 7 3 Firearm and shot information [11, 12]. ................................ ................................ ............ 10 4 Summary of the number of cross correlation computations made successively for each firearm at each angle (T firearm_angle ), across all angles (T firearm_all_angles ), and the total across all sampling rate s (T firearm_total ). ................................ ................................ ......................... 22 5 Average maximum cross correlation values and their corresponding standard deviatio n values for the intra firearm comparisons for sampling rates 500, 250, 192, 125, 96, 88.2, 62.5, 48 and 44.1 kHz. ................................ ................................ ................................ ....... 31 6 Average maximum cross correlation values and their corresponding standard deviation values for the intra firearm comparisons f or sampling rates 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz. ................................ ................................ .......................... 32 7 Average maximum cross correlation values and their corresponding standard deviation values for the inter firearm comparisons for sampling rates 500, 250, 192, 125, 96 , 88.2, 62.5, 48 and 44.1 kHz. ................................ ................................ ................................ ....... 33 8 Average maximum cross correlation values and their corresponding standard deviation values for the inter firearm comparisons for sampling rates 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz. ................................ ................................ .......................... 34
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x 9 Average maximum cross correlation values and their corresponding standard deviation values for the four (4) recorded gunshots in the cas e example for sampling rates 44.1, 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz. ................................ .................. 58
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xi LIST OF FIGURES FIGURE 1 Time aligned waveforms for a 2 channel recording of a supersonic bullet fired from a .308 caliber rifle, illustrating the basic acoustical elements of the gunshot [4]. ............... 3 2 Comparisons of the shock wave geometry for a bullet traveling at Mach 1.05 and Mach 3 [4]. ................................ ................................ ................................ ................................ .... 4 3 diaphragm of a microphone [2]. ................................ ................................ ......................... 4 4 Illustration of the microphone rig for the database collection process [10]. .................. 12 5 Image depicting the shooter positioned in the center of the microphone rig during the database capture process [11]. ................................ ................................ ........................ 12 6 Waveform displays for shot #1 of the SIG Sauer P239 (.357) from angle 1 (0) at the top to angle 12 (180) at the bottom. Normalized amplitude on the vertical axis vers us seconds on the horizontal axis, with a total displayed length of 20 milliseconds. .......... 15 7 Waveform displays for shot #1 of the .308 caliber rifle from angle 1 (0) at the top to angle 12 (180) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds. ........................ 16 8 Waveforms of signals X and Y, where Y is equal to X but with a 300 quantization level shift. The cross correlation values for X/X and X/Y from lags 50 to +50 are given. The maximum cross correlation value (at lag 0) dropped from +1 to +0.70108 with the introduction of DC offset. ................................ ................................ ................................ . 18
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xii 9 The intra firearm cross correlation comparisons for shots #1 and #2 for angle 1 (0) of the 12 gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross correlation value for the respective comparison; both values are provided in the title of each plot. .............. 24 10 The intra firea rm cross correlation comparisons for shots #1 and #3 for angle 1 (0) of the 12 gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross correlation value for the respe ctive comparison; both values are provided in the title of each plot. .............. 25 11 The intra firearm cross correlation comparisons for shots #2 and #3 for angle 1 (0) of the 12 gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Wavefo rms are aligned at the lag value which resulted in the maximum cross correlation value for the respective comparison; both values are provided in the title of each plot. .............. 26 12 Average maximum cross correlation results vs. sampling rate for all fir earms. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ................................ ......... 35 13 Average maximum cross correlation results vs. sampling rate for the 12 gauge sh otgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 36 14 Average maximum cross correlation results vs. sampling rate for the .22 caliber rifle. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 37
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xiii 15 Average maximum cross correlation results vs. sampling rate for the .308 caliber rifle. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 38 16 Average maxim um cross correlation results vs. sampling rate for the AR14 M4 Carbine. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 39 17 Average maximum cross correlation results vs. sampling rate for the Colt handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ................................ ......... 40 18 Average maximum cro ss correlation results vs. sampling rate for the Glock 19 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 41 19 Average maximum cro ss correlation results vs. sampling rate for the Glock 23 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............................ 42 20 Average maximum cross correlation results vs. sampling rate for the Ruger SP101 handgun (.357). Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. .. 43 21 Avera ge maximum cross correlation results vs. sampling rate for the Ruger SP101 handgun (.38). Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. .......... 44
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xiv 22 Average maximum cross correlation results vs. sampling rate for the SIG Sauer P239 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot. ............. 45 23 Percent changes in the average maximum cross correlation values for sampling rates 500 kHz down to 15.625 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values. ............. 46 24 Percent changes in the average maximum cross correlation values for sampling rates 192 kHz down to 12 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values . ...................... 47 25 Percent changes in the average maximum cross correlation values for sampling rates 88.2 kHz down to 11.025 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values. ............. 48 26 Percent changes in the average maximum cross correlation values for sampling rates 32 kHz down to 8 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values. ...................... 49 27 Percent changes per kHz in the average maximum cross correlation values for the intra firearm comparisons in successive sampling rate steps. ................................ .................. 50 28 Percent changes per kHz in the average maximum cross correlation values for the inter firearm comparisons i n successive sampling rate steps. ................................ .................. 51 29 Waveform display of the last four (4) recorded intra firearm gunshots from the case example, over a one second time span and with sampling rate of 44.1 kHz. .................. 60
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xv 30 Average maximum cross correlation results vs. sampling rate for the last four (4) recorded gunshots in the case example. Standard deviation bars shown for each plot. 61 31 Percent changes per kHz in the average maximum cross correlation values for the four (4) recorded gunshots in the case example in successive sampling rate steps. .............. 62
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1 INTRODUCTION Gunshot Analysis The forensic analysis of recorded gunshot sounds, while requested less frequently than audio enhancement and audio authentication, can provide critical information during an investigation of criminal activity or of acti o ns related to civil litigation. With the proliferation of mobile devic es and law enforcement body cameras, and the widespread adoption of home and business video surveillance systems, the likelihood that gunshots occurring in urban and rural environments will be recorded has increased. And with that increase comes greater op portunity for analysis. Requests for recorded gunshot analysis typically center on one (1) or more of the following questions [1] : Are these sounds gunshots? How many gunshots were there? How many firearms were there? How many and which gunshots did each firearm d ischarge ? Who fired first? What are the firearm types/calibers? Where was each shooter positioned? What is the timing between gunshots? Various techniques may be employed to analyze the recorded audio and to draw conclusions to address the questions posed above. These techniques may include pre processing/filtering of signals, critical listening, time domain (waveform) analysis,
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2 energy/envelope analysis, frequency domain analysis, cross correlation computations, and time difference of arrival (TDOA) [1] . The focus of the present research and thesis is on the use of cross correlation computations in the analysis of recorded gunshot sounds and does not directly address the other listed techniques. Prior Research Gunshot Acoustics The mechanisms of firearms and the acoustical characteristics of their discharges have been covered by several research papers and presentations aimed at the audio fore nsics and signal processing fields. Many of these papers/presentations have resulted from the work of Dr. Robert C. Maher (Montana State University, Department of Electrical and Computer Engineering) and his colleagues. Maher and Shaw [2 4] have previo usly discuss ed the principle mechanics of a gunshot and placed these elements in context with the acoustical signals which are produced by the event . They also identified less than ideal microphone and recording systems. Figure 1 provides an acoustical , time domain overview of a .308 caliber rifle firing a supersonic bullet (i.e., faste r than the speed of sound) and recorded by two (2) professional quality microphones at different locations in a controlled environment [4] . The supersonic bullet produces a shock wave which is f ollowed by its ground reflection, both of which arrive at the microphones prior to the muzzle blast, which is traveling at the speed of sound. A ground reflection of the muzzle blast then ends the sequence. In the case of a bullet traveling at less than th e speed of sound, no shock wave (or reflected shock wave) would be present.
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3 Figure 1 Time aligned waveforms for a 2 channel recording of a supersonic bullet fired from a .308 caliber rifle, illustrating the basic acoustical elements of the gunshot [4] . The shock wave expands in a conical fashion behind the bullet, and the angle at which divided by the speed of sound , a value referred to as the Mach Number [2, 3] . The higher the Mach Number, the shallower the Figure 2 [4] . As the shock wave passes the microphone diaphragm, it causes a positive overpressure maximum followed by a corresponding under the Figure 1 waveforms and is provided in more detail in Figure 3 [2] .
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4 Figure 2 Comparisons of the shock wave geometry for a bullet traveling at Mach 1.05 and Mach 3 [4] . Figure 3 diaphragm of a microphone [2] .
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5 Cross Correlation Computations Cross correlation computations provide for a quantitative measure indicating the similarity between two (2) signals and is defined by the foll owing equation [5, 6] : In equation (1), x and y refer to the input signals of sample length N and m is the displacement (or lag) in sample s as x and y are slid over each other while the cross correlation computations are performed. Normalization of the output, such that the cross correlation v alue computed of a signal aligned sample for sample with itself (i.e., autocorrelation) will be +1, is achieved by dividing the output of equation (1) by the product of the norms of x and y , as follows [5, 6] : The resulting normalized cross correlation values will be constrained between 1 and +1, with +1 being the autocorrelation result (as indicated above) and 1 being the autocorrelation result with one (1) of the signals being 180 out of phase. The normaliz ed cross correlation value will approach 0 f or two (2) signals that are completely uncorrelated (e.g., true white noise ) . Koenig et al. [7] explored t he application of cross corre lation computations to the forensic analysis of recorded gunshot sounds through a collection of gunshots fired on an outdoor firing range by five (5) firearms at four (4) different positions , relative to the locations of nine (9) recording/sensing devices which simultaneously recorded the shots. The
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6 recording/sensing devices ranged from consumer to professional grade and included law enforcement specific devices. Nearly all the recording/sensing systems were analog, and all were commonly encountered by forensic audio examiners at the time that the research was conducted . Cross correlation c omputations were run , in part, for shots from the same firearm qualitative assessments of the c orresponding waveforms. The general hierarchy given in Table 1 summarizes the correspondence of the qualitative, visual observations of the waveforms with the quantitati ve cross correlation results Table 1 Summary of the qualitative and quantitative results from [7] . Visual Observation Averag e Correlation Correlation Range Excellent 0.920 0.645 0.997 Good 0.834 0.610 0.976 Fair 0.686 0.364 0.942 Poor 0.498 0.253 0.692 As an extension to the research conducted by Koenig, et al. [7] , a new set of gunshots was recorded using digital audio recorders [16 bit pulse code modulation (PCM) encoding; 96,000 samples per second or 96 kilohertz (kHz)] and four (4) B&K model 4136 microphones with wide frequency response (flat from 4 Hz to 70 kHz) and dynamic range [greater than 172 decibels (dB) ]. The microphones were arranged in six (6) different configurations of distance and angle, relative to the position of the firearm, as given in Table 2 [8] .
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7 Table 2 Firearm to microphone distances (Range) and azimuth angles relative to the line of fire for the six (6) firearm recording configurations [8] . Configuration Range (m) Azimuth angle (deg) 1 1.5, 3, 6, 30 3 2 1.5, 3, 6, 30 90 3 3 3, 30, 60, 90 4 30 3, 30, 60, 90 5 3 90, 120, 150, 180 6 30 90, 120, 150, 180 Seven (7) differen t firearms were utilized in [8] , some firing multiple types of ammunition, and c ross correlation computations arrived at similar results to [7] . Namely, shot correlations with sourc e, environment, and receiver variations held constant distances are typically lower than those between successive shots Limitations An overriding observation that p ervades much of the prior research conducted in the field of recorded gunshot analysis is that there are many factors that affect the ability to answer the common questions listed above and to otherwise draw meaningful conclusions. These factors include, b ut are not limited to, the following [7 9] : Microphone type Distance between the microphone and the firearm Relative angle between the microphone and firearm Recorder settings Acoustical environment Type of firearm discharged
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8 Differences in ammunition Muzzle blast size Because these factors affect how a gunshot is ultimately recorded, they also impact the quantitative results that are derived from those recordi ngs. Research Focus While many factors come into play when analyzing recorded gunshot sounds , such as those listed above , this thesis focuses on how reductions in the audio bandwidt h a ffect the quantitative results arrived at through the application of cross correlation. In real world cases, the forensic examiner does not typically have the benefit of receiving high quality, controlled recordings, nor multiple simultaneous recordings of the same series of events . The utilization of a controlled database of gunshot recordings for this thesis (discussed below in the allowed for wide flexibility regarding the production of reduced bandwidth recordings of the same gunsho t event , thereby permitting observations to be made of cross correlation computations as the bandwidth is reduced . With the reduction of the recorded bandwidth comes the removal of high frequency components within the recorded gunshots, which is expected t o lead to fewer distinctive features between intra and inter firearm gunshots (i.e., the recorded gunshot sounds will appear more alike as the bandwidth is reduced). Accordingly, t he central hypotheses that were tested for this thesis are as follows: As t he bandwidth of a n audio recording is decreased, the corresponding cross correlation results for intra firearm comparisons will increase.
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9 As the bandwidth of a n audio recording is decreased, the corresponding cross correlation results for inter firearm com parisons will increase. T he ability to statistically distinguish between recorded gunshot sounds from different firearms may be compromised as the bandwidth of an audio recording is decreased.
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10 MATERIALS Recorded Gunshot Database T he recorded gunshot data base that arose from Maher and Routh [10, 11] , and subsequently made publicly available on line [12] , was used as the basis for the research conducted for this thesis. This database was collected an echoically (i.e., without early sound reflections) in an outdoor environment in Montana, USA, and under conditions which were designed to be scientifically reliable and repeatable. The creation of this database was unique in several ways , as discussed belo w, and provided recorded data that was tailor made for exploring the effect of frequency bandwidth reduction ( through down sampling ) on cross correlation computations. Firearms and Shots Table 3 provides a listing of the ten ( 10 ) firearm/caliber scenarios that were used during the database collection process , with two (2) different calibers of ammunition ( . 38 and . 357) fired by the Ruger SP101 handgun . Table 3 F irearm and shot information [11, 12] . Scenario Firearm Caliber # of Shots 1 Glock 23 handgun .40 10 2 Glock 19 handgun 9mm 10 3 SIG Sauer P 239 handgun .357 10 4 Colt handgun .45 10 5 Ruger SP101 handgun .38 9 6 .357 10 7 R ifle .22 2 0 8 Rifle .308 10 9 Remington s hotgun 12ga 3 10 AR14 M4 Carbine 5.5645mm 10 TOTAL # OF SHOTS 102
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11 A total of 21 shots were fired by the .22 rifle, comprised of a set of ten (10) followed by a set of eleven (11) , the latter of which was recorded with a 20 decibel (dB) amplification of the input levels. However, the amplified recordings of shot #6 were d etermined to be unusable, as they featured no discernible gunshot sounds. Microphone Set up Twelve (12) GRAS Sound & Vibration A/S type 46DP microphone sets were utilized for the capture process. Each microphone set consist ed of a type 40DP 1/8" Externally Polarized Pressure Microphone , a type 26TC " preamplifier, and type 12AA and 12AG power modules providing the 200 volt polarization and 120 volt preamplifier power. The microphones provide d for a dB frequency response out to 1 40 kHz, with a dynamic range specifi ed between 46 dB ( lower limit ) and 178 dB ( upper limit ), resulting in an overall dynamic range of 132 dB [10] . The twelve (12) microphone sets were arranged in a semi circular pattern along a semi octag onal, aluminum rig having a three meter radius. The shooting position was located at the center of the rig from an elevated position , and t he microphone sets were position ed three (3) meters above the ground at 0, 16.4, 32.7, 49.1, 65.5, 81.8, 98.2, 114.5, 130.9, 147.3, 163.6, and 180, relative to the angle of fire. For purposes of this paper, these angles will be referred to as angle #1 through angle #12 , respectively . Figure 4 illustrates the characteristics of the microphone rig and the relative location of the shooting position [10] , while Figure 5 shows the marksman in the shooting position within the microphone rig during the capture process [11] .
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12 Figure 4 Illustration of the microphone rig for the database collection process [10] . Figure 5 Image depicting the shooter positioned in the center of the microphone rig during the database capture process [11] .
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13 Recording Characteristics The twelve (12) microphone channels for each shot were recorded simultaneously using a National Instruments NI PXIe 1071 chassis equipped with a NI PXIe 8840 Core processor and NI PXIe 6358 data acquisition card. Each channel was recorded with 16 bit PCM encoding and with a sampling rate of 500 kHz , providing a recorded bandwidth of 250 kHz per the Nyquist sampling theorem [13] . The recorded audio for each shot was saved as a MATLAB data file , with the twelve (12) columns in the array corresponding to the separate microphone channels from the 0 position (column 1) to the 180 position (column 12). The data values mal equivalents of the 16 bit quantization values for each audio sample, meaning that the values range from (2 15 ) or 32,768 to ( 2 15 1 ) or 32,767. The lengths of the recording from each angle of a shot was identical, but the lengths were not identical acr oss all the shots. The recordings were each a multiple of one (1) second, meaning that their lengths in samples were divisible by 500,000 , except for shot #8 of the SIG Sauer P239 which has a length of 2,000,001 samples ( 4.000002 seconds at a sampling rate of 500 kHz). The total range of lengths across the recordings was from three (3) seconds (shot #4 of the Glock 19 handgun and shot s # 2 and #3 of the Ruger SP101 handgun firing .38 caliber ammunition) to fifteen (15) seconds (shot #1 of the .308 caliber ri fle).
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14 METHODOLOGY General The overall methodology devised for this thesis can be broken down into the following phases: 1. Audio File Preparation 2. Bandwidth Reduction Through Resampling 3. Cross Correlation Computations 4. Statistical Calculations Audio File Preparation Extraction of Independent Channels were to extract each column of data (i.e., each microphone channel) as a separate vector, normalize sample values to decimal value s relative to the maxima of 16 bit quantization, and then save that vector to a monaural PCM wavefile with a sampling rate of 500 kHz. With this process, a PCM wavefile was produced for each recorded angle for each shot; for the total of 102 shots, this equ ated to a total of 1,224 PCM wavefiles (12 angles per shot x 102 shots). Figure 6 and Figure 7 show the twelve (12) time aligned waveform displays for shot #1 of the SIG Sauer P239 (.357) and shot #1 of the .308 caliber rifle, respectively, from 0 (top) to 180 (bottom) . Note in Figure 7 that the ballistic shockwave from the supersonic bullet is seen clearly in the first three shaped signal preceding the higher amplitude muzzle blast. As the angle between the direction of fire and t he microphone increases, the time differential between the ballistic shockwave and the onset of the muzzle blast decreases.
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15 Figure 6 Waveform displays for shot #1 of the SIG Sauer P239 (.357) from angle 1 (0) at the top to ang le 12 (180) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.
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16 Figure 7 Waveform displays for shot #1 of the . 308 caliber rifle from angle 1 (0) at the top to angle 12 (180) at the bottom. Normalized amplitude on the vertical axis versus seconds on the horizontal axis, with a total displayed length of 20 milliseconds.
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17 Direct Current Offset Removal Direct current (DC) offse t can occur in an aud i o recording when one (1) or more components (e.g., microphone, microphone preamplifier) induce DC voltage into the audio signal, manifesting itself as a vertical shift of the audio samples away from the x axis [14, 15] . From review and measurement of the discovered that the DC offset s of the recorded signals varied across the microphones , with the signals recorded at angles 9 through 12 exhibiting the largest offse ts in the order of 300 quantization levels at 16 bit, or 0.9%. While this percentage may not be large, the cross correlation results will be affected by the presence of the DC offset. Cross correlation computations are immune to a scalar change of amplitu de across the sample values (e.g., reducing the amplitude of a n entire signal by a fixed value) , but shifting the DC offset of one (1) of the signals will reduce the maximum computed cross correlation value, especially when the overall amplitudes are lower . To exemplify this, consider S ignal X (monaural, 16 bit PCM, 8 kHz sampling rate, one second length) which is created by frequency modulating a 60 Hz sine wave with white noise, both with peak amplitudes of 40 dB. Signal Y is created by shifting Signal X by 300 quantization level s . By definition, the autocorrelation of Signal X results in a value of +1 at lag 0 (i.e., Signal X is aligned sample for sample with itself when the cross correlation computation is run). A cross correlation computation is then run of S ignals X and Y, and the result is + 0.70108 at a lag of 0 . The presence of DC offset reduced the maximum cross correlation value from +1 to +0.70108. Figure 8 summarizes this example and includes graphs of the cross correlation values versus lag values from 50 to +50.
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18 Figure 8 Waveforms of signals X and Y, where Y is equal to X but with a 300 quantization level shift. The c ross correlation values for X/ X and X/ Y from lags 50 to +50 are given. The maximum cross correlation value ( at lag 0 ) dropped from +1 to + 0. 70108 with the introduction of DC offset. Because of the negative impact that the pr esence of DC offset can have on the cross correlation computations and the fact that DC offset is a channel artifact that does not convey any signal dependent information , the wavefiles extracted from the files were each processed to remove any DC o ffset present in them by performing mean subtraction and saving the results separate ly as new wavefiles. Mean subtraction was conducted in MATLAB R2019b using the following command, where x x DC corrected signal, n x :
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19 Bandwidth Reduction Through Resampling The DC offset corrected wavefiles were downsampled from their native 500 kHz to the following sampling rates, which are all factors of 500 kHz: 250 kHz , 125 kHz , 62.5 kHz , 31.25 kHz , and 15.625 kHz . Additional downsampled wavefiles were produced at the following sampling rates, which are commonly used in professional and consumer recording systems: 192 kHz , 96 kHz , 88.2 kHz , 48 kHz , 44.1 kHz , 32 kHz , 24 kHz , 22.05 kHz , 16 kHz , 12 kHz , 11.025 kHz , and 8 kHz. In total, seventeen (17) sets of downsampled wavefiles were produced for each firearm/shot/angle recording. The resampling processes were performed using the resamp funct ion with in MATLAB R2019b. The basic syntax for the resamp function is as follows [16] : y x p / q is resampled. n that affects the order of the antialiasing finite impulse response (FIR) lowpass filter (utilizing Kaiser windowing) employed during the resampling process, as follows [16, 17] : p is q is the value of the input values which satisfy the same ratio can be used. For example, to downsample a 500 kHz signal ( x ) to a 250 kHz signal ( y ), either of the following functions could be used:
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20 For the resampling processes performed in this research, the source signals were always the native 500 kHz DC q The default value n is ten (10), and that value was utilized for this research [16] . Downsampling the DC corrected, 500 kHz recordings was chosen as the process for bandwidth reduction in lieu of applying a lowpass filter. This decision was made prima rily to expedite the subsequent cross correlation computation processes. With lowpass filtering, the sampling rate of the files would remain at 500 kHz, even though the bandwidth of the recorded signal would be bandlimited; lowpass filtered files would hav e required a greater number of cross correlation computations compared to a downsampled version of the same file. As indicated above, the downsampling process inherently includes an antialiasing lowpass filter , but the resulting files do not contain the ex traneous data between the cut off frequency of the lowpass filtered version and the original Nyquist frequency (250 kHz). Cross Correlation Computations Cross correlation computations were run of all shots within each firearm within each angle (intra firea rm, intra angle) and for all shots across firearms within each angle (inter firearm, intra angle) , with the process being repeated for each sampling rate from 500 kHz down to 8 kHz. For example, the three (3) recorded shots from the Remington 12 gauge shot gun were cross correlated to each other within each angle and for each sampling rate, and then were cross correlated with the other nine (9) firearm/caliber scenarios within each angle and for each sampling rate. No inter angle cross correlations were cons idered for this research ; inter angle comparisons have been shown to result in lower cross correlation values and
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21 linear [8] . For a given number of recorded shots ( n ), the formula for the number of pairs of unique intra firearm combinations (i.e., n items taken two at a time with no repetitions) for each firearm ( T intra ) is as follows [18] : These combinations exclude the autocorrelations, which are the cross correlations of each shot/angle recording with itself . The total number of unique , inter firearm comparisons ( T inter ) at each angle/sampling rate is given as the following , where 102 is the total number of shots in n Table 3 ) : Hence, the total number of comparisons for each firearm at each angle/sampling rate ( T firearm_ angle ) is the sum of T intra and T inter , and the total number of comparisons for each firearm across all angles ( T firearm_all_ angle s ) for a given sampling rate is 12 times T firear m_ angle . Lastly, the total number of comparisons for each firearm across all angles and across all sampling rates ( T firearm_total ) is 18 times T firearm_all_angles .
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22 Table 4 Summary of the number of cross correlation computations made successively for each firearm at each angle (T firea r m_angle ) , across all angles (T firearm_all_angles ), and the total across all sampling rates (T firearm_total ). Firearm #shots ( n ) T intra T inter T firearm_angle T firearm_all_angles T firearm_total Glock 23 handgun 10 45 920 965 11,580 208,440 Glock 19 handgun 10 45 920 965 11,580 208,440 SIG Sauer P239 handgun 10 45 920 965 11,580 208,440 Colt handgun 10 45 920 965 11,580 208,440 Ruger SP101 handgun (.357) 10 45 920 965 11,580 208,440 Ruger SP101 handgun (.38) 9 36 837 873 10,476 188,568 Rifle (.22) 20 190 1,640 1,830 21,960 395,280 Rifle (.308) 10 45 920 965 11,580 208,440 Remington shotgun 3 3 297 300 3,600 64,800 AR14 M4 Carbine 10 45 920 965 11,580 208,440 TOTALS 102 544 9,214 9,758 117,096 2,107,728 The cross correlation computations were made using the function with in MATLAB R2019b. The basic syntax for the xcorr function utilized in this research was as follows [5] : a b CC output coeff which results in the values being scaled between 1 and +1, where +1 is the autocorrelation of a signal with itself at lag 0 and 1 would be the same but with the phase of one (1) of the input signals inverted. Norma lization of the results is optional, but when a method is specified , the input signals must be of the same length. Accordingly, because the recordings in the database were not all of the same length , the maximum length of the recordings within a given set of computations was first determined, and any recordings within that set which were shorter than
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23 that maximum length were zero padded with the appropriate number of samples. The cross correlation computations were then carried out with pairs of files havin g the same length. CC correlation value was identified and documented in a spreadsheet for each angle and for each sampling rate. Additionally, the corresponding lag p ositions for the maximum positive cross correlation values were similarly documented in a separate set of spreadsheets by angle and sampling rate . As example set s of cross correlation comparisons, Figure 9 displays the 500 kHz and 8 kHz intra firearm comparisons of shot #1 with shot #2 for the 12 gauge shotgun at angle 1 (0) , aligned at the lag values which resulted in the maximum cross correlation value s and shown with a time range of twenty (20) milliseconds (i.e., 10,000 samples at 500 kHz, 160 samples at 8 kHz) . Similarly, Figure 10 and Figure 11 display the same data for the shot #1/shot #3 and shot #2/shot #3 comparisons, respectively. T he ground reflection in each waveform is present approximately eleven (11) milliseconds (i.e., 5,500 samples at 500 kHz, 88 samples at 8 kHz) following the onset of the respective muzzle blast.
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24 Figure 9 The intra firearm cross correlation comparisons for shots #1 and #2 for angle 1 (0) of the 12 gauge shotgun at the 5 00 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross correlation value for the respective comparison; both values are provided in the title of each plot.
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25 Figure 10 The intra firearm cross correlation comparisons for shots #1 and #3 for angle 1 (0) of the 12 gauge shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross correlation value for the respective comparison; both values are provided in the title of each plot.
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26 Figure 11 The intra firearm cross correlation comparisons for shots #2 and #3 for angle 1 (0) of the 12 gaug e shotgun at the 500 kHz (left) and 8 kHz (right) sampling rates. Waveforms are aligned at the lag value which resulted in the maximum cross correlation value for the respective comparison; both values are provided in the title of each plot.
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27 Stati stical Calculations Using the maximum cross correlation value spreadsheets, averages and standard deviation values were calculated for all intra firearm/intra angle comparisons and all inter firearm/intra angle comparisons. For example, average and standard deviation values were calculated for the set containing all the maximum cross correlation values for shots #1 through #3 of the Remington 12 gauge shotgun for all intra angle comparisons (e.g., shots #1 and #2 at angle 1, shots #1 and #3 at angle 1, shots # 2 and # 3 at angle 1 , shots #1 and # 2 at angle 2, 12 ). From Table 4 , there were three (3) intra angle comparisons made for the Remington 12 gauge shotgun for each angle, or a total of 36 comparisons across the twelve (12) angles. Following that, similar averag e and standard deviation calculations were calculated for Remington 12 gauge shotgun shots #1 through #3 against all the intra angle comparisons made with the other firearms. Again from Table 4 , there were 297 inter angle comparisons made for the Remington 12 gauge shotgun for each angle, or a total of 3,564 comparisons across the twelve (12) angles.
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28 RESULTS Table 5 and Table 6 list the computed averages and standard deviations for the maximum cross correlation values from the intra firearm comparisons for sampling rates 500 kHz to 44.1 kHz and 32 kHz to 8 kHz, respectively. The results are listed by firearm and for a set containing all firearms. Similarly, Table 7 and Table 8 list the same computations for the inter firearm comparisons. The following figures display the average maximum cross correlation values versus sampling rate plots for the intra firearm /intra angle ( solid blue) and inter firearm /intra angle ( dashed orange) comparisons, as detailed below: Figure 12 for all firearm s Figure 13 for the Remington 12 gauge shotgun Figure 14 for the .22 caliber rifle Figure 15 for the .308 caliber rifle Figure 16 for the AR14 M4 Carbine Figure 17 for the Colt handgun Figure 18 for the Glock 19 handgun Figure 19 for the Glock 23 handgun Figure 20 for the Ruger SP101 handgun (firing .357 caliber ammunition) Figure 21 for the Ruger SP101 handgun (firing .38 caliber am munition) Figure 22 for the SIG Sauer P239 handgun Standard deviation bars are provided for each data point and are colored accordingly (blue for the intra firearm data points and orange for the inter firearm data points). The intra firearm
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29 standard deviation bars are capped with an arrow, while the inter firearm standard deviation bars are capped with a solid oval, to more easily distinguish the bars that overlap. T he percent changes for each set of sampling rates related by sequential, downward octave s (i.e., halving of the sampling rate ) for all firearms are given in the following figures for both intra and inter firearm comparisons : Figure 23 for sampling rates 500 kHz down to 15.625 kHz Figure 24 for sampling rates 192 kHz down to 12 kHz Figur e 25 for sampling rates 88.2 kHz down to 11.025 kHz Figure 26 for sampling rates 32 kHz down to 8 kHz These percent change values were calculated per the following equation: AvgMaxCC SR correlation value at a given sampling rate SR , AvgMaxCC (SR/2) correlation value at half the given sa mpling rate (i.e., one octave down) . The average maximum cross correlation values are taken from the Table 5 through Table 8 . For example, the percent change for the average maximum cross correlation value s from 500 kHz to 250 kHz for the intra firearm comparisons was calculated as follows (average maximum cross correlation values taken from Table 5 ) :
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30 Lastly, the percent changes per kHz for each successive sampling rate interval were calculated for all firearms (intra and inter firearm comparisons separately) as follows: AvgMaxCC SR1 correlation value at a given sampling SR1 AvgMaxCC SR2 correlation value at the ending samplin ( 12 ) above, the average maximum cross Table 5 through Table 8 . For example, t he percent change per kHz for the intra SR1 SR2 comparison was derived as follows: The percent change per kHz results are given in Figure 27 (intra firearm comparisons) and Figure 28 (inter firearm comparisons).
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31 Table 5 Average maximum cross correlation values and their corresponding standard deviation values for the intra firearm comparisons for sampling rates 500, 250, 192, 125, 96, 88.2, 62.5, 48 and 44.1 kHz.
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32 Table 6 Average maximum cross correlation values and their corresponding standard deviation values for the intra firearm comparisons for sampling rates 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz.
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33 Table 7 Average maximum cross correlation values and their corresponding standard deviation values for the inter firearm comparisons for sampling rates 500, 250, 192, 125, 96, 88.2, 62.5, 48 and 44.1 kHz.
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34 Table 8 Average maximum cross correlation values and their corresponding standard deviation values for the inter firearm comparisons for sampling rates 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz.
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35 Figure 12 Average maximum cross correlation results vs. sampling rate for all firearms. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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36 Figure 13 Average maximum cross correlation results vs. sampling rate for the 12 gauge shotgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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37 Figure 14 Average maximum cross correlation results vs. sampling rate for the .22 caliber rifle. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm comp utations. Standard deviation bars shown for each plot.
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38 Figure 15 Average maximum cross correlation results vs. sampling rate for the .308 caliber rifle. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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39 Figure 16 Average maximum cross correlation results vs. sampling rate for the AR14 M4 Carbine. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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40 Figure 17 Average maximum cross correlation results vs. sampling rate for the Colt handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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41 Figure 18 Average maximum cross correlation results vs . sampling rate for the Glock 19 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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42 Figure 19 Average max imum cross correlation results vs. sampling rate for the Glock 23 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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43 Figure 20 Average maximum cross correlation results vs. sampling rate for the Ruger SP101 handgun (.357). Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation ba rs shown for each plot.
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44 Figure 21 Average maximum cross correlation results vs. sampling rate for the Ruger SP101 handgun (.38). Solid blue plot is the intra firearm computations, and dashed orange plot is the inter f irearm computations. Standard deviation bars shown for each plot.
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45 Figure 22 Average maximum cross correlation results vs. sampling rate for the SIG Sauer P239 handgun. Solid blue plot is the intra firearm computations, and dashed orange plot is the inter firearm computations. Standard deviation bars shown for each plot.
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46 Figure 23 Percent changes in the average maximum cross correlation values for sampling rates 500 kHz down to 15.625 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values.
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47 Figure 24 Percent changes in the average maximum cross correlation values for sampling rates 192 kHz down to 12 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values.
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48 Figur e 25 Percent changes in the average maximum cross correlation values for sampling rates 88.2 kHz down to 11.025 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents th e inter firearm values.
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49 Figure 26 Percent changes in the average maximum cross correlation values for sampling rates 32 kHz down to 8 kHz in downward octave steps. Solid black plot represents the intra firearm values, and dashed red plot represents the inter firearm values.
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50 Figure 27 Percent changes per kHz in the average maximum cross correlation values for the intra firearm comparisons in successive sampling rate steps.
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51 Figure 28 Percent changes per kHz in the average maxi mum cross correlation values for the inter firearm comparisons in successive sampling rate steps.
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52 CONCLUSION S The results of the research conducted for this thesis support the hypotheses that as the bandwidth of an audio recording is decreased, th e corresponding maximum cross correlation value s will increase for both intra and inter firearm comparisons. Except for the transition from 16 kHz to 8 kHz for the intra firearm condition, all the percent changes in the average maximum cross correlation computations for the octave interval results were positive (see Figure 23 , Figure 24 , Figur e 25 , and Figure 26 ) . T he greatest percent change was observed in the 500 kHz to 250 kHz transition for both intra and inter firearm comparisons. The percent changes generally decreased as the sampling rates decreased; however, t here were two (2) instances in the inter firearm percent changes where successive values slightly increased (from 2.12% to 2.32% for the transition between 96 kHz/48 kHz and 48 kHz/24 kHz, and from 1.92% to 2.33% for the transition between 88.2 kHz/44.1 kHz and 44.1 kHz/22.05 kHz). The results of the successive sampling rate percent changes per kHz revealed positive results for all the intra and inter firearm comparison intervals, except for the las t two (2) transitions of the intrafirearm results ( 0.005%/kHz for the 12 kHz/11.025 kHz transition and 0.081%/kHz for the 11.025 kHz to 8 kHz transition ). Both the intraand interfirearm results exhibit a noticeable peak at the 44.1 kHz to 32 kHz tran sition, the reason for which is not readily apparent. he primary reason for the increases in the maximum cross correlation values is likely the systematic removal of the high frequency variations in the recorded gunshots as the sampling rate (and therefore, bandwidth) is reduced.
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53 The cumulative effect of the differences in these high frequency variations results in minor but quantified differences in the corresponding cross correlation values. For the individual firearm computations, the only firearm which exhibited a clear separation between the intra and inter firearm plots of the average maximum cross correlation values (i.e., no overlap of their standard deviation ranges) was the Re mington 12 gauge shotgun (see Figure 13 ). The intra and inter firearm plots for all other individual firearms and for the set of all firearms overlap within one (1) st andard deviation. The mechanisms by which the shotgun discharges and the differences in its ammunition type, relative to the handguns and rifles, likely led to its shots being more distinctive among the set of tested firearms. It was noted that the intra f irearm results for the .22 caliber rifle never exceeded 0.6 (not including the standard deviation range), which was relatively poor compared to the other firearms which always exceeded 0.7 . Similarly, t he inter firearm results for the .22 caliber rifle wer whereas the other firearms ranged from 0.4572 (Remington 12 gauge shotgun) to 0.6963 (Glock 19) for the inter firearm results at 8 kHz. These results may have resulte d from the inclusion of the initial set of ten (10) shots from the .22 caliber rifle, which exhibited poorer signal to noise than the subsequent set of eleven (11) shots with the 20 dB amplification. The standard deviation ranges for the intra firearm comp utations for all firearms generally decreased as the sampling rate decreased, with the .22 caliber rifle and Remington 12 gauge shotgun exhibiting the lowest rates of change. For the inter firearm computations, the differences in the standard deviation ran ges also decreased but were not as significant as
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54 the intra firearm results , which may be due to the inclusion of different firearm classes (e.g., handguns, rifles, shotgun) in the test set. The hypothesis regarding the decreases in bandwidth compromising the ability to statistically distinguish between recorded gunshot sounds from different firearms is not supported by the data in this research. As observed in Figure 12 through Figure 22 , the overlaps in the intra and inter firearm plots and their standard deviation range s generally decrease as the sampling rate/bandwidth decreases, indicating that discrimination between the two sets (intra and inter) becomes greater.
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55 FUTURE RESEARCH As noted above, this research utilized high quality recordings in a controlled environme nt; hence, conducting the same or similar research using recordings captured in non anechoic but semi controlled conditions (e.g., same microphone rig) with known circumstances would likely shed more light on the results in condit ions commonly encountered by forensic audio examiners . A simple, intra firearm case example with known circumstances is presented in the Appendix. Producing similar databases of controlled recordings using a wider array of firearms/ammunition would also im prove the breadth of the data presently available and enable more detailed comparisons between firearm classes and specific models/ammunition. The effects of bandwidth reduction on other quantitative measures, such as mean quadratic difference, and utilizi ng power data in lieu of waveforms in the same workflow could also be explored.
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56 REFERENCES 1. Begault, D.R., S.D. Beck, and R.C. Maher, Overview of Forensic Audio Gunshot Analysis Techniques , in 2019 AES International Conference on Audio Forensics . 2019: Porto, Portugal. 2. Maher, R.C., Modeling and Signal Processing of Acoustic Gunshot Recordings , in 2006 IEEE 12th Digital Signal Processing Workshop & 4th IEEE Signal Processing Education Workshop . 2006. p. 257 261. 3. Maher, R.C., Acoustical Characterization of Gunshots , in 2007 IEEE Workshop on Signal Processing Applications for Public Security and Forensics . 2007. p. 1 5. 4. Maher, R.C. and S.R. Shaw, Deciphering Gunshot Recordings , in 33rd International Conference: A udio Forensics . 2008: Denver, USA. 5. The Mathworks Inc. MATLAB R2019b "xcorr" function . [cited 2019 November 6]; Available from: https://www.mathworks.com/help/matlab/ref/xcorr.html . 6. Lacey, D.S., B.E. Koenig, and C.E. Reimond, The Effect of Sample Length on Cross Correlation Comparisons of Recorded Gunshot Sounds , in 54th International Conference: Audio Forensics . 2014: London, UK. 7. Koenig, B.E., S.M. Hoffman, H. Nakasone, and S.D. Beck, Signal Convolution of Recorded Free Field Gunshot Sounds. J. Audio Eng. Soc, 1998. 46 (7/8): p. 634 653. 8. Beck, S.D., H. Nakasone, and K.W. Marr, Variations in recorded acoustic gunshot waveforms generated by small firearms. J Acoust Soc Am, 2011. 1 29 (4): p. 1748 1759. 9. Maher, R.C. and E. Hoerr, Forensic Comparison of Simultaneous Recordings of Gunshots at a Crime Scene , in 147th Convention of the Audio Engineering Society . 2019: New York, USA. 10. Maher, R.C. and T. Routh, Advancing Forensic Analy sis of Gunshot Acoustics , in 139th Convention of the Audio Engineering Society . 2015: New York, USA.
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57 11. Maher, R.C., Advancing Audio Forensics of Gunshot Acoustics . 2018, National Criminal Justice Reference Service. 12. Maher, R.C. Example recorded data . 2018 [cited 2019 November 6]; Available from: http://www.montana.edu/rmaher/gunshots/gunshot_data.html . 13. Pohlmann, K.C., Principles of digital audio . 6th ed. 2011, New York: McGraw Hill. 14. Koenig, B.E. and D.S. Lacey, The Average Direct Current Offset Values for Small Digital Audio Recorders in an Acoustically Consistent Environment. J Forensic Sci, 2014. 59 (4): p. 960 966. 15. Koenig, B.E., D.S. Lacey, C. Grigoras , S.G. Price, and J.M. Smith, Evaluation of the Average DC Offset Values for Nine Small Digital Audio Recorders. J Audio Eng Soc, 2013. 61 (6): p. 439 448. 16. The Mathworks Inc. MATLAB R2019b "resamp" function . [cited 2019 November 6]; Available from: https://www.mathworks.com/help/signal/ref/resample.html . 17. The Mathworks Inc. Resampling . [cited 2019 November 6]; Available from: https://www.mathworks.com/help/signal/ug/resampling.html . 18. Mendenhall, W. and T. Sincich, Statistics for engineering and the sciences . 5th ed. 2007, Upper Saddle River, New Jersey: Prentice Hall, Inc.
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58 APPENDIX Case Example As a simple case example, fourteen (14) shots were fired from a handgun (Glock 22, .40 caliber) in an outdoor environment at night , near a microphone mounted within a law enforcement vehicle. The microphone signal was recorded onto one channel of t he hi fi stereo audio track of a VHS tape based dashboard camera recording system. The individual firing the handgun was panning slightly from their left to right over the first ten (10) shots but was relatively still for the last four (4) shots , as observ ed in the video recording of a second dashboard camera recording system . Figure 29 displays the waveforms for these last four (4) recorded shots, as digitized at a sampl ing rate of 44.1 kHz from the VHS hi fi audio track. The methodology described in this thesis was applied to these last four (4) shots, each segmented into separate 175 millisecond WAV files, with the downsampling processes performed using the DC offset co rrected, 44.1 kHz digitized file segments . The results are provided in Table 9 and are displayed graphically in Figure 30 . Additionally, the percent changes per kHz in the average maximum cross correlation values are shown in Figure 31 . Table 9 Average maximum cross correlation values and their corresponding standard deviation values for the four (4) recorded gunshots in the case example for sampling rates 44.1, 32, 31.25, 24, 22.05, 16, 15.625, 12, 11.025, and 8 kHz. Max CC value Sampling Rate (kHz) 44.1 32 31.25 24 22.05 16 15.625 12 11.025 8 AVG 0.7175 0.7174 0.7176 0.7171 0.7176 0.7165 0.7165 0.7203 0.7191 0.7237 SD 0.1356 0.1356 0.1358 0.1355 0.1361 0.1345 0.1346 0.1366 0.1342 0.1346
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59 From this case example, it is evident that bandwidth reduction had little impact on the average maximum cross correlation values and corresponding standard deviations, but the results are generally consistent with those obtained from the controlled database recordings utilized in this research for the same 44.1 kHz to 8 kHz range (see Table 5 and Table 6 and the co rresponding figures) . Both the average and standard deviation values from the case example were highly consistent across the sampling rates, with the overall average/standard deviation range being 0.7183.1353. The corresponding percent change per kHz res ults oscillated above and below 0%/kHz, with the values for the lowest sampling rate transitions (15.625 kHz/12 kHz, 12 kHz/11.025 kHz, and 11.025 kHz/8 kHz) having the greatest deviations. As a general observation, the waveforms of the recorded gunshots i n this case example ( Figure 29 ) are noticeably different than those captured in the controlled database (as exemplified in Figure 9 through Figure 11 ). Whereas the controlled database recordings exhibit quick acoustic decay and return to the ambient noise level within approximately eleven (11) milliseconds, the recordings in the case exampl e have much longer acoustic decay patterns and more complex signatures following the onset of the muzzle blasts. These differences are due in large part to the effects of the non optimal microphone and recording system employed in the dashboard recording s ystem.
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60 Figure 29 Waveform display of the last four (4) recorded intra firearm gunshots from the case example, over a one second time span and with sampling rate of 44.1 kHz.
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61 Figure 30 Average maximum cross correlation results vs. sampling rate for the last four (4) recorded gunshots in the case example. Standard deviation bars shown for each plot.
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62 Figure 31 Percent changes per kHz in the average maxim um cross correlation values for the four (4) recorded gunshots in the case example in successive sampling rate steps.

