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An Acoustical Evaluation of the QRD Diffractal in Finney Chapel
Patrick Landreman Bruce Richards (Advisor) Honors Thesis: Oberlin College April 2008
1
Table of Contents Abstract 2 Introduction 4
1.1 Architectural Acoustics 4 1.2 Acoustic Diffusers 5 1.3 The Schroeder Diffuser 6 1.4 RPG Diffusor Systems, Inc., and the Diffractal 7 1.5 Finney Chapel 8 1.6 Purpose of Experiment 8
Theory 9 2.1 Scattering of pressure waves 9 2.2 The Quadratic Residue Sequence 10 2.3 Periodicity Lobes 10 2.4 Design Equations 11 2.5 Critical Frequencies 11 2.6 The Diffractal 12 2.7 Sound Absorption Within Wells 13
Experimental Method 14 3.1 Signal pulses 14 3.2 Microphone Placement 15 3.3 Data Collection 16
Data Analysis and Results 19 4.1 Mathematica 19 4.2 Fourier Analysis 20 4.3 Low-Pass Filtering 20 4.4 Basic Behavior and Expectations 22 4.5 Determining Polar Response 25 4.6 Response Plots 27 4.7 Interpretation of Response Graphs 28
Conclusion 30 5.1 Discussion of Experimental Method 30 5.2 Future Work 31 5.3 Conclusions about Reflection Phase Grating Type Diffusers 32
Appendices 34 A – Mathematica Scripts 35 B – Table of Filtered Audio Data 39 C – Architectural Drawings of Finney Chapel 49
Acknowledgements 51 References 52
2
Abstract Acoustic diffusers are an important component in enhancing the quality of room
acoustics. In this paper, a method is proposed to quantitatively evaluate the effectiveness
of an acoustic diffusion panel which has already been installed under the balcony in
Finney Chapel at Oberlin College. The diffuser in this example is a QRD Diffractal,
produced by RPG Diffusor Systems, Inc. The shape of the diffuser is obtained using a
quadratic residue sequence, an idea first proposed by Manfred Schroeder in 1975.
The polar response of the diffuser was measured by reflecting short, sine wave
pulses off of the diffuser and simultaneously recording the resulting reflections in two
microphones. One microphone served as a reference to determine when the initial sound
wave had arrived, while the second microphone was repositioned at varying angles about
the center of the diffuser. A control experiment was performed by covering the diffuser
with panels of plywood.
The location of the reflection from the diffuser was identified by combining an
estimated value of the speed of sound with the geometry of the test setup, as well as by
comparison of microphone response aimed at the diffuser versus directly away from the
diffuser. Polar response graphs were generated by taking the ratio of incident amplitude
to reflected amplitude as a function of angle.
3
Polar response results were generally consistent with expectations. In particular,
scattered energy in the specular direction was substantially dissipated by the presence of
the diffuser. Results from opposite 90° arcs about the diffuser were inconsistent,
suggesting asymmetric behavior, which disagreed with existing literature results.
4
Chapter 1: Introduction 1.1 Architectural Acoustics
Everyone has at some point in his or her life experienced an echo. When a sound
wave encounters a large, flat surface, it reflects and propagates such that the angle of
reflection is equal to the angle of incidence. If the reflection is large enough relative to
the ambient noise and is heard sufficiently later than the original noise, we interpret that
sound as an echo.
Unfortunately, we as humans have a tendency to build structures with large, flat
surfaces. In certain cases, the resulting acoustic effects of flat walls can be detrimental to
the function of the space. It is particularly important to control the reflection of sound in
rooms for music listening. Echoes can be distracting for performers and audience
members alike. Uncontrolled reflections can interfere and produce irregular frequency
response, which colors sound and distorts timbre. Most importantly, no one will pay to
hear a concert in a room with poor acoustics.
There are two approaches to eliminating echoes in a room1. The first is to convert
the acoustical energy of the propagating wave into some other form of energy, usually
heat. This method is known as absorption. The second method is to break the echo into
many reflections as it leaves the surface of reflection. Instead of hearing a single burst of
5
sound at high intensity, the sound is distributed to other surfaces of the room, arriving at
the listener in rapid succession. The sound is then interpreted as reverberation, rather
than an echo. The process of dispersing sound while preserving the total acoustic energy
is called diffusion.
1.2 Acoustic Diffusers
There are a variety of materials commonly used to achieve diffusion in rooms.
These devices are called acoustic diffusers. Diffusers may range from flat panels hung
over the stage, called clouds, to very complicated surfaces generated by computer
optimization programs. Convex surfaces are a simple means of scattering sound waves
and are frequently found in music halls around the world.2
The ideal diffuser is one for which the distribution of reflected energy is
independent of angle, though in special cases certain directions may be preferred (stage
clouds and bandshells are examples of this case, where the diffuser is in actuality a plane
surface which redirects all sound towards the more remote regions of the audience).
One means of visualizing the effect of a diffuser is through a polar response plot,
which displays the ratio of reflected to incident sound intensity as a function of angle
about the diffuser being tested. An example of such a plot is shown in Fig. 1 which
compares the response of a plane surface to an array of pyramidal structures.
In examining literature regarding methods of producing acoustic diffusion, one
might be perplexed to find that both diffuser and diffusor appear as accepted spellings of
the same word. This confusion is due to Schroeder and D’Antonio wanting to distinguish
acoustic diffusers from their optical counterparts1. Due to the interconnectedness of
optical and acoustic wave theory and the wish to avoid confusing the academic
Fig. 1 – Polar response plot by RPG Diffuser Systems, Inc. comparing the reflection off a plane surface (thin line) to that of an array of pyramidal diffusers (thick line). The pyramids remove energy from the specular reflection and create increased reflections near 30°, resulting in the appearance of large spikes.
6
community with excessive vocabulary, the term diffuser is preferred in this paper.
1.3 The Schroeder Diffuser
In 1975, Manfred R. Schroeder published a
paper3 which is widely credited for creating a new
family of acoustic diffusers1,4,5. In his brief
publication, Schroeder proposed that introducing a
phase shift to sound that reflects off certain regions
of the surface could make an ideal diffuser. The
phase shift is described by a reflection factor. If
the reflection factor as a function of position had
the property that its Fourier transform were
constant, then the reflected energy distribution
would be independent of angle. Such diffusers are called reflection phase gratings
(RPGs), similar to the phase gratings in optics.
Schroeder proposed maximum-length sequences for the surface function in his
first paper. In a later publication, quadratic residue sequences were given as a preferred
alternative.6 The phase shift was achieved by carving rectangular grooves into a flat
surface, such that waves propagating into different grooves would experience a different
path length and thus reemit out of phase.
Fig. 3 – Cross-sectional schematic from Reference 6 of a QRD diffuser as proposed by Schroeder. This drawing shows two periods of the residue sequence generated using N = 17.
Fig. 2 – A Schroeder diffuser built using a quadratic residue sequence.
7
Further contributions to the theory of RPGs were made by Gerlach7, Berkhout8,
Strube9, D’Antonio, Cox, and many others. In practice, Schroeder’s model did not
produce even energy distribution at all angles, but a pattern of energy lobes at regular
intervals, each with equal amplitude. These lobes were found to be a result of the
periodic nature of the sequences used to generate the diffusers. Research has been done
to examine the effects of arranging multiple Schroeder diffusers from different generating
sequences in modulation to eliminate this lobing effect.
Ultimately, Schroeder’s diffuser has been determined to be not ideal. Other
diffuser models have gained popularity, particularly due to the limiting aesthetic
constraints of the Schroeder diffuser.1 However, Schroeder’s concept of a reflection
phase grating diffuser is still being applied in critical listening spaces, and such diffusers
are available commercially today.
1.4 RPG Diffusor Systems, Inc., and the Diffractal
In 1983 Dr. Peter D’Antonio founded a company named after the reflection phase
grating diffuser.10 RPG Diffusor Systems, Inc. patented Schroeder’s quadratic residue
diffuser (QRD) in 1987, and operates acoustic laboratories that have contributed to the
present understanding of diffusers, including Schroeder-type phase gratings.11
In 1994, RPG presented a paper in the Journal
of the Audio Engineering Society to promote a new
line of phase grating diffuser called the QRD
Diffractal12. A standard QRD is modified by
replacing the bottom surface of each well with a
similar QRD texture (see Fig. 4). In doing so, the
expected effective bandwidth of the diffuser would be
greater, since wave components above the design
frequency of the large diffuser (which normally would
be unaffected) are scattered by the smaller scale
diffuser within the well. The term ‘diffractal’ results
because the geometry of the diffuser is repeated within
itself on a smaller scale, forming fractal geometry. Fig. 4 - Visual description of QRD Diffractal (courtesy of RPG Diffusor Systems, Inc.)
8
The recursion may be repeated several times, and diffractals containing up to three
“generations” may be ordered to custom dimensions.13
1.5 Finney Chapel
In 1999, Oberlin College installed a new organ in Finney Chapel, a historic venue
used for religious services, musical performance and visiting academic speakers. To
accompany the installation of the new instrument, Dana Kirkegaard was hired to provide
an acoustical consultation to enhance the sound quality of the space. In his report to the
college, Kirkegaard recommended the installation of a QRD Diffractal.14 “Reduced
clarity and tonal distortion” and degraded tone quality due to an echo from the lower rear
wall were cited as motivation for the upgrade. According to Kirkegaard’s report, the rear
wall echo was “10dB above other reflections.” Following Kirkegaard’s
recommendations, Oberlin College purchased a diffractal, which was installed in the
chapel on the rear wall, below the balcony. The diffractal is split into three sections –
one behind each of the columns of seats. Representatives of the Oberlin Conservatory of
Music have been very pleased with the performance of the diffractal in Finney.
Complaints of a pronounced “slapback” echo from instruments with a sharp attack, such
as piano, brass or high strings have been assuaged.15
1.6 Purpose of Experiment
The goal of this project was to empirically quantify the effect of the QRD
Diffractal on reflections off the lower rear wall of Finney Chapel. The experiment was
novel in that most measurements of acoustic properties are performed in controlled
environments, such as anechoic chambers or reverberation rooms. Finney posed an
interesting problem in that the inherent acoustics of the space could not be eliminated,
and thus a method had to be developed to isolate reflections due to the rear wall. In
addition, the diffuser was fixed to the chapel wall using both screws and glue, and could
not be removed for experimental purposes. This created a challenge to design a method
of controlling the experiment for the presence of the diffuser.
9
Chapter 2: Theory
2.1 Scattering of pressure waves
D’Antonio and Cox have summarized the theoretical work done on phase grating
diffusers to date.1 The pressure at a point in space r due to scattered waves may be
approximated by
!
ps r( ) = "ik
8# 2e"ik r+r
o( )sinc
kb
r
$
% &
'
( ) cos* +1[ ] R rs( )eikxs sin* dxs
"a
a
+ (2.1)
Here, ro is the position of the sound source, r is the position of the receiver, rs is the
location of a point on the surface of the reflecting surface, i is the square root of -1, k is
the wavenumber of the propagating wave, a is
one-half the length of the diffuser in the x-
direction, b is one-half the length of the
diffuser in the z-direction, and θ is the angle
between r and the diffuser normal. R(x) is a
function which describes the reflection factor
at position x across the diffuser’s length and is
the source of the phase shifting for a Schroeder
diffuser. Fig. 5 – Definition of variables for Equation 2.1. The vectors ro and r correspond to the location of the sound source and receiver, respectively.
y
x
z
ro
r ψ
θ
a b
rs
10
Several important assumptions are needed to obtain this equation. The sound
wave is assumed to be at normal incidence to the diffuser (for oblique incidence the
sin(θ) term is replaced by a sin(θ) + sin(ψ) term where ψ is the angle between ro and the
diffuser normal – see Reference 1, Equation 9.8). The conditions for Fraunhofer
diffraction must hold, essentially requiring the distance between source and diffuser to be
much larger than the wavelength of the reflected pressure wave.16
The advantage of making these assumptions is that Equation 2.1 can be viewed as
a Fourier transform of the reflection function, R(x). The Fourier transform of R(x)
determines the amplitude of scattered waves as a function of angle3. Specifically, if R(x)
is chosen such that it has a flat power spectrum, then sin(θ) + sin(ψ) is a constant. Note
that this disagrees with Schroeder’s first-order approximation, in which the amplitude is
constant as a function of angle.6
2.2 The Quadratic Residue Sequence
The nth element in a quadratic residue sequence is given by the equation
!
sn
= n2modN (2.2)
where N is a prime number. Notice that the sequence is symmetric and periodic with
period equal to N. As an example, consider the case where N = 17, beginning with n = 0:
s = {0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, 2, 8, 16, 9, 4, 1; 0, 1, etc.}.
Consider a plane surface with a sequence of wells, whose depths, d, are given by
!
dn
="
Nsn (2.3)
In this case, the reflection factor for the nth well becomes
!
Rn
= e
i2"
Nsn (2.4)
for which the sequence Rn has a constant power spectrum.6 Thus, Schroeder’s first-order
approximation predicts that the quadratic residue sequence produces an ideal diffuser.
2.3 Periodicity Lobes
D’Antonio asserts that instead of uniform energy distribution, the QRD produces
a set of discrete lobes of equal energy at particular angles (see Fig. 6). The number and
sharpness of lobes for a given frequency are functions of the ratio of wavelength to total
11
diffuser length17 and the
number of periods of the
sequence present in the
diffuser. A smaller ratio
will result in more lobes,
and increasing the number
of periods will increase the
sharpness of the lobes.
2.4 Design Equations
The effective
bandwidth of a QRD can be controlled based on a small set of parameters. This property
is why the QRD is relatively important in the field of acoustic treatment. The minimum
wavelength for which the diffuser will scatter predictably is
λmin = 2w (2.5)
where w is the well width of the diffuser. Shorter wavelengths will not demonstrate pure
plane wave propagation within the wells of the diffuser, although some complicated
scattering due to the shape of the surface is still expected. The maximum wavelength is a
function of both the maximum well depth, and the period width of the diffuser, Nw. The
equation
!
dn
=sn"
o
2N (2.6)
is used to determine the well depth based on the desired design, or maximum wavelength
λo. If the period width is too narrow, then the diffuser will not produce as many lobes at
low frequency as would be considered ideal.
2.5 Critical Frequencies
For a given diffuser, there will be certain frequencies for which each well re-
radiates in phase with the other wells. At these frequencies, the diffuser effectively
becomes a plane surface. When designing a diffuser, the parameters should be chosen so
that the lowest critical frequency lies above the target bandwidth of the diffuser.
Fig. 6 – Measurements by RPG of the polar response of an N=7 QRD at 3kHz. The number of periods from left to right were 1, 6, and 50.
12
2.6 The Diffractal
The diffractal uses the same principles that govern the standard QRD. The depth
of the hth well in a two-generation diffractal is determined using12
!
dh
= h /N" #modM
2 $M
2M+ h
modM( )mod N
2 $N
2N. (2.7)
Here, λM and λN are the design wavelengths of the low-frequency diffuser and high-
frequency diffuser, respectively, where M and N are the prime numbers used to produce
the quadratic residue sequence. The function
!
x" # represents the integer floor function.
The surface created by this equation is clever, in that it effectively superimposes a
small QRD designed for high frequency scattering into each well of a larger QRD
designed for low frequency scattering, and simultaneously the surface reflection function
R(x) produced from this sequence of wells satisfies the same conditions as a single QRD.
Thus, uniform lobing is expected over an increased bandwidth.
Solving Equation 2.7 using M = N = 7 and well depths measured with a ruler, the
Finney Chapel diffractal has design frequencies of roughly 1300Hz and 4840Hz. The
period width of the low-frequency diffuser is 67 ± 0.5cm, resulting in an expected low-
frequency effectiveness of 509Hz. For the upper frequency limit, a well width of 1.0 ±
0.1cm results in effective bandwidth up to 17.2kHz.
Fig. 7 shows data produced at RPG laboratories demonstrating the polar response
of the high- and low-frequency diffusers, independent from each other.18 Theoretical
Fig. 7 – Polar responses of an N = 7 QRD (500 and 1000Hz) and FlutterFree QRD (5kHz). The standard QRD was of comparable dimensions to the low-frequency component of the Finney diffractal, and the FlutterFree diffuser was similar to that of the high-frequency component (FlutterFree is trademark name used by RPG for their high-frequency QRD). In each plot, three periods were used to make the sample diffuser. The black line shows the response from the diffuser, while the grey line shows the response from a plane surface of identical width.
13
data using computational methods have shown that combining the diffusers does not
degrade the quality of the high-frequency diffuser (see Fig. 8).
2.7 Sound Absorption Within Wells
It has been reported that deep, narrow wells can cause absorption of sound. Since
the QRD is based on a series of finitely wide wells, it is expected that some absorption
will occur. Such behavior has been observed experimentally19. While this absorption is
of particular concern to anyone trying to adjust the acoustics of a room, it is not an
impediment to this experiment. Any observed absorption will be included in the polar
response of the diffuser, and will appropriately reflect the influence of the diffuser on the
chapel acoustics.
Fig. 8 – Polar plots taken from Reference 12 showing computer-generated theoretical responses of a standard QRD (a) and a diffractal which modulates that diffuser into another low-frequency diffuser (b).
14
Chapter 3: Experimental Method
3.1 Signal pulses
In order to have a reproducible sound source, sinusoidal wave pulses at 500Hz,
1kHz and 5kHz were generated in Mathematica (see Appendix A). These tones were
selected to sample the range of frequencies which would commonly be heard in a spoken
or musical performance. These values also have the valuable property that their
corresponding wavelengths are not integer multiples of the spacing between the pews in
Finney. Otherwise, standing waves between pews could produce unwanted resonance
and make it harder to isolate reflections. Finally, these frequencies fall within the range
over which the diffuser is expected to be effective.
Based on the architectural plans of Finney Chapel (Appendix C), and using an
estimated speed of sound of one foot per millisecond, it was determined that the incident
test pulses should be less than 3ms to maintain distinguishable reflections. The 500Hz
pulse was set at 4ms to allow for more than one period of the wave. Files were exported
in .WAV format and imported into Stineberg Cubase LE for playback.
15
3.2 Microphone Placement
Two microphones were used to simultaneously record each sound pulse. The first
microphone was used to record a reference signal. The reference microphone was placed
along the centerline of the chapel, approximately 5m away from the diffuser, and was
aimed at the sound source to detect the arrival of the incident sound pulse. The time
necessary to travel from the reference microphone to the rear wall and then to the second
microphone could be predicted, allowing identification of the desired reflection. The
second microphone was positioned at the same point, but was aimed at the center of the
central diffuser to record the resulting reflection. This “test” microphone was then
repositioned at several positions in a circular arc about the center of the diffuser to
determine angular dependence of the diffuser.
The floor of Finney Chapel is
raked, and so precautions were taken
to ensure that all measurements were
taken within a fixed horizontal plane.
A HeNe laser was attached to a ring
stand near the south wall of the
chapel. The ring stand was made
vertical using adjustment screws at
the base and levels. The laser was
attached to the stand at approximately
Fig. 9 – Test pulses for 500, 1000 and 5000Hz after being imported into Stineberg Cubase LE at a sample rate of 44.1kHz.
Fig. 10 – The reference microphone is necessary to calibrate the time scale for each measurement. As the second microphone is brought near to the diffuser, reflections from the wall arrive in at different times, but the arrival time relative to the incident pulse as seen by the reference microphone is constant.
16
half the height of the diffuser. A mark was made at the same height on the north wall of
the chapel, measuring from the floor with a measuring tape. The laser was aimed at this
mark to establish a horizontal beam. The laser was then rotated horizontally to the
middle of the rear wall, establishing the intersection of the test plane with the diffuser.
This intersection was marked with tape. Afterwards, the laser was rotated within the
same horizontal plane to the location of the reference microphone to ensure that the
microphone was also in the plane. Finally, the beam was retargeted at the mark on the
north wall.
The radius of the measurement arc was established by extending a measuring tape
between the tape mark on the diffuser and the tip of the reference microphone, making
sure that the measuring tape intersected the beam of the laser. All further radial
measurements were performed in this fashion, using the laser and the tape mark on the
diffuser to ensure that the test microphone lay in the experimental plane.
Fig. 11 shows the experimental setup and definitions of various variables. To
position the test microphone, the distance from the centerline for each angle of
measurement, x, was calculated using x = rsinθ. The microphone was moved to an
approximated position, aligned within the test plane at radial distance r, and pointed
towards the center of the diffuser. The actual value of x was then measured using a
measuring tape held at the centerline, extended perpendicularly to the microphone, and a
plumb bob suspended from the tip of the microphone.
3.3 Data Collection
For each frequency being tested, a Cubase project file was created. The
corresponding sine wave pulse was placed at 10ms from the start of the project on a
soloed track. Two mono input tracks were record primed, one for each microphone.
Loudspeakers were centered on the stage at the lip, and aimed directly at the rear wall.
The distance from the stage to the back of the chapel was deemed sufficiently large to
approximate plane wave incidence. The computer audio was output through the Finney
sound system. All recording was performed at 16 bits and 44,100 samples per second.
The microphones were first calibrated by placing the test microphone directly
adjacent to the reference microphone, both facing the loudspeakers. Signals were
17
recorded for each frequency being tested, so that the relative input gain of the
microphones could be normalized.
The test microphone was then reversed to face the rear wall. Measurements were
taken in intervals of 10° from normal reflection to 70° off-axis. The room temperature
was noted using a standard mercury thermometer so that the speed of sound could be
calculated.
Data was collected in three separate sessions on 16 February, 27 February, and 6
March, 2008. The goal of the first session was to record the effects of the diffuser in
place. Measurements were performed on the southern half of the building. The second
run was designed as a control. Three 8’x4’ ¼” plywood sheets were taped to the front of
the diffuser to approximate replacing the diffuser with a plane surface. Measurements
were again taken on the southern side of the chapel. The final run was to confirm
symmetrical behavior of the diffuser. The diffuser was uncovered, and measurements
were taken on the north side of the building.
18
Audio System Block Diagram
1 Reference Microphone – DBX RTA-M, omnidirectional, +48V Phantom Powered
2 Test Microphone – Earthworks M30, omnidirectional, +48V Phantom Powered
3 QRD Diffractal
4 Apple PowerBook G4 – 1.33 GHz PowerPC G4, 256 MB Built-in memory, Mac OS X 10.3.9
5 Presonus Firebox Firewire Audio Interface – input and main output gains set to maximum
6a DBX 480 Digital Drive Rack – House master EQ and crossover
6b Amplifiers – Lab Gruppen fP 2200 (High/Mid) and fP 3400 (Low)
7 Loudspeakers – 1x EAW KF300 and 1x EAW KF330 stacked vertically with the 300 on top
8 HeNe Laser – used for alignment. Not part of audio system, but very useful nonetheless.
(1)
(2)
(3)
(4) (5)
(6a)
(7)
(6b)
x r
θ
(8)
Fig. 11 – Experimental setup and equipment list.
19
Chapter 4: Data Analysis and Results
4.1 Mathematica
All data analysis was performed in Mathematica by Wolfram Research, Inc. An
assortment of functions was designed to process the data in this experiment. All
necessary Mathematica code may be found in Appendix A.
Using the import command, audio files were converted into lists of sample values.
The list entries were converted into pairs including the sample amplitude and time
position of the sample. Plotting signal amplitude versus time led to graphs similar to
those in Fig. 12.
Fig. 12 - .WAV recording of a 1kHz test signal imported into Mathematica. The vertical axis depicts the pressure at the microphone converted into an electrical signal, shown as a function of time.
20
4.2 Fourier Analysis
Fig. 13 shows the result of a Fourier Transform of the incident pressure wave of a
1kHz signal pulse as recorded by the test microphone. The signal is overwhelmingly
characterized by 1kHz as expected. The width of the peak is due to the finite number of
terms in the Fourier Transform, as well as the fact that the signal pulse is not infinite in
length. However, the localization of energy at 1kHz means that measurements performed
using this signal pulse were demonstrative of behavior at that frequency, and were not
being influenced by unintended frequency content.
4.3 Low-Pass Filtering
The imported audio data was squared and run through a low-pass filter to remove
the sinusoidal oscillations of the signal itself. The resulting data produced graphs that
visualize the overall sound energy, E, present at the microphone as a function of time.
The script for the filter was modeled after the method for low-pass filtering in
Hamming20, pp. 127-9. The samples in the wave data are passed through the filter given
by
!
yn = ckun"kk="N
N
# (4.1)
where yn are the filtered data values, uk are the wave data values, and ck are found from
the Fourier series of the transfer function, H(ω):
!
ck
=1
"H #( )cos k#( )d#, 0 < k
0
"
$
ck
=2
"H #( )cos k#( )d#, 0 = k
0
"
$
%
&
' '
(
' '
(4.2)
For a low-pass filter, H(ω) is a step function:
!
H "( ) =1, " # f
o
0, " > fo
$ % &
(4.3)
After filtering using a cutoff of fo = 300Hz, N = 100, and sample rate = 44.1kHz,
the waveform appeared as in Fig. 14. This same filter was used for all subsequent data
processing.
21
Fig. 14 (Below) – The same 1kHz recording shown over a longer time domain (1st graph). The wave data is filtered using the transfer function H[x] (2nd graph) to produce a graph of the energy level as a function of time (bottom graph).
Fig. 13 (Left) – A graphical representation of incident 1kHz pulse in Mathematica and the resulting Fourier Transform when taken over the displayed time interval. Some width in the peak is expected because the signal is not a pure sine wave.
22
4.4 Basic Behavior and Expectations
The filtered data provide a convenient means of locating the arrival of reflections
at a particular microphone. Each graph contains a roughly 250ms region of no signal,
broken by a large peak corresponding to the arrival of the direct sound from the
loudspeakers. The subsequent activity is due to reflections from other surfaces in the
room.
A comparison of the reference microphone data for all trials of a particular
frequency on a particular date (see Fig. 15) reveals that the data are mostly identical,
except that the time axes are shifted up to 28ms from each other, and the amplitude of the
signal varies between trials at different angles. These discrepancies are likely dependent
on inconsistent response from the audio electronics. For example, Cubase may produce
some variable lag between the point when it begins entering information into a new
.WAV file and when it begins replaying the project audio track containing the signal
pulse, variables in the amplifiers and loudspeakers might result in slight differences in the
amplitude of the signal output, and so forth. The time shift cannot be due to changes in
the speed of the propagation of the signal pulse because such a change would alter the
resulting series of reflections, and would ultimately produce a very different-looking
reference signal. Ultimately,
the similarity between
reference signals suggests
that the recording
environment on a given day
was consistent from angle to
angle, except for some slight
fluctuation in amplitude
(which can be eliminated by
normalization – see §4.5
below).
As the test microphone is moved away from 0°, it moves simultaneously closer to
the rear wall and further from the sound source. It was expected that this would be seen
in the data by a shift in the incident sound pulse occurring later, and
Fig. 15 – Two 500Hz filtered reference signals recorded on 2/16. The first was recorded with the test microphone at 10°, the second at 30°. There is no appreciable difference between the graphs except for a shift in the time-axis.
23
2/27 1kHz 0°
2/27 1kHz 40°
2/27 1kHz 70°
Fig. 16 – A comparison of data recorded using a 1kHz pulse shows a trend in the positions of the incident and reflected peaks. The incident sound arrives later as the test microphone is moved closer to the wall, and thus farther from the sound source. The reflection arriving from the rear wall appears sooner.
24
a growing reflection from the rear wall occurring sooner than in the reference
microphone data. These trends are apparent in Fig. 16. It is worth noting that these
earlier reflected peaks were created by sound reflecting from the rear wall at the point
closest to the microphone, and not necessarily from the point at the center of the wall.
Comparing corresponding data from the three different frequencies tested
demonstrates another expected behavior, that peaks at 500Hz are substantially less
isolated than those at 1kHz or
5kHz. 500Hz has a longer
period; thus the test pulse for
that frequency was necessarily
longer to include more than a
single period.
Of particular
importance is the presence of a
substantial increase in signal
in the test microphone relative to the reference microphone at a peak approximately 30-
35ms after the arrival of the incident sound pulse for some test/reference pairs of the
same audio playback at 0°. Since the only appreciable difference between the
microphones in that configuration was the direction they were facing (one towards the
diffuser, one away), the
change in the test
microphone was due to a
reflection from the diffuser
(despite being
omnidirectional, the
Earthworks microphone
exhibits a drop in sensitivity
from behind). From estimates
of the speed of sound and the
dimensions of our
Fig. 17 – Comparison of test microphone signals at 0° with the diffuser uncovered. The upper graph shows the response for 1kHz, while the lower graph shows 500Hz. The greater length of the 500Hz test pulse causes reflections peaks to be less distinct.
Fig. 18 – Comparison of test microphone response at 0° when aimed at the diffuser (above) and at the sound source (below). The peak at the estimated arrival time of 30ms shows a visible change in amplitude between the two graphs.
25
experimental setup (see §4.5), the appearance of this peak is coincident with the expected
time of arrival for sound coming from the rear wall.
In data recorded at large angles, it was necessary to move the test microphone
from a region surrounded by pews into the aisle. This transition is typically accompanied
by a change in the pattern of peaks recorded by the test microphone.
4.5 Determining Polar Response
To account for sensitivity of the microphones to the particular frequency being
tested, and remove any difference in input gain, a ratio of reference microphone signal to
test microphone signal was computed using simultaneous recordings of the same signal
(see §3.3). The ratio was computed using the height of the incident sound pulse at the
frequency being tested. All test microphone data was multiplied by this value, scaling it
to match that of the reference microphone.
In order to determine the amount of acoustic energy being directed at a particular
angle, it was necessary to find the amplitude of the test microphone data at the time when
the reflection from the center of the diffuser reached the test microphone. Two methods
of finding this time were considered. The first method involved calculating the speed of
sound using a formula from Bohn21. The speed of sound is given as
!
c t( ) = 331.45 1+t
273.16 (4.4)
where t is the room temperature in degrees Celsius and c has units of meters per second.
A summary of the calculations for the speed of sound in Finney Chapel is in Table 1.
Having estimated the speed of sound, a time window for the desired reflection off the
diffuser was set by locating the position of the maximum of the incident sound pulse at
the reference microphone, and adding the time necessary to travel the radius of the test
microphone arc twice. The width
of the time window was set using
the uncertainty in the estimate of
the speed of sound.
The second method to
determine the time window for
Date Temperature (± 0.5°C)
c(t) (± 9m/s)
r (± 0.01m)
Time between incident and
reflected sound (± 1ms)
2/16 22.0 345 5.25 30 2/27 20.7 344 5.08 30 3/6 19.5 343 5.47 32
Table 1 – Calculation of the expected time between the arrival of the incident sound pulse at the reference microphone and the arrival of the reflection off the center of the diffuser at the test microphone.
26
the arrival of sound from the diffuser was to compare the filtered reference microphone
signal to the filtered, normalized test microphone signal at 0°. Often, a peak would
appear substantially higher in the test microphone signal in the vicinity of the time
predicted using the speed of sound method. The time window for the diffuser reflection
was then centered at the location of this peak, with a width equal to the half-width of the
peak. In some cases it was not possible to isolate the enlarged peak. For these situations,
the reference signal was subtracted from the normalized test signal. The resulting plot
contained a local maximum near the expected time, and this maximum was used as the
location of the diffuser’s reflection. The width of the window was set to half the width of
the corresponding peak in the difference graph.
Having obtained a time window, the relative strength of the reflection off the
diffuser was computed by averaging the filtered, normalized test microphone signal over
the time window. This quantity was then divided by the strength of the incident sound
pulse in the reference microphone data to normalize for variation in the amplitude of the
test pulse. The incident strength was taken as the average over the full width at half
height of the first peak in the reference signal. The uncertainty in the data was
determined by the percentage difference between the maximum value and the averaged
value of the direct sound peak in the reference microphone signal, about 25%. The
graphs in the next section display the results of each method of analysis.
Fig. 19 – Difference taken between test and reference microphone data for a 500Hz pulse at 0°. Data were filtered, squared and normalized before subtraction. The arrow indicates the peak at the expected time for the arrival of the reflection from the center of the diffuser. Lines on either side mark the region averaged over in determining the value of the data at that point.
27
4.6 Response Plots
500Hz
1kHz
5kHz
North Side Diffuser Covered South Side
Method of Determining Time Window
Calculated Speed of Sound
Reflection Peak Witnessed at 0°
28
4.7 Interpretation of Response Graphs
The charts above clearly demonstrate a change in the behavior of the sound field
when the diffuser was covered. For all frequencies tested the reflected energy is most
concentrated at 0° with a definitive decay at higher angles when the diffuser is covered.
Some small rise near 60° is seen, which could either be due to the transition into the aisle
region in the chapel, or could be consistent with the minor lobing seen in polar plots of
plane surfaces made in controlled acoustic environments (see Fig. 7).
The reflection at 0° is dramatically reduced when the diffuser is present, in
general agreement with the data from RPG. Measurements from this experiment at 500
and 5000Hz exhibit some rise near 30°, consistent with lobes seen in the data from RPG.
Otherwise, agreement between this new data and the existing data is not especially
astounding.
The most startling result of the polar response plots is the remarkable lack of
consistency between data taken on the north and south sides of the chapel. QRDs
demonstrate symmetric polar responses, and the chapel has no particular asymmetries.
Since the charts were produced from data taken on separate days, it is conceivable that
some variable influencing the acoustic environment had not been accounted for and was
changed from one run to the next. Using two test microphones and simultaneously
measuring the response on either half of the test arc could easily remove such a variable.
Different radii were used for the test arc during each run. However, it is unlikely that the
polar response of the diffuser is subject to significant change near a distance over five
times greater than the wavelength of the lowest frequency pulse tested.
A number of data points have a value greater than unity, suggesting that the
reflected sound was greater in amplitude than the direct sound. These data occurred in all
measurements when the diffuser was covered, and in one anomalous measurement at
1kHz on the south side of the chapel. It is highly suspect that the diffuser, which is partly
absorbing, contributed additional energy to return the reflected sound with greater
intensity. It is possible that multiple reflections arriving concurrently were summed at
the microphone. Alternatively, by averaging over the width of the incident reference
signal peak, the entire data set is raised somewhat, and so some values may have been
boosted excessively. For all the data points above unity, the corresponding error bars
29
extend below one, so that if the averaging had not occurred those points would have been
within the expected range.
It is difficult to argue which of the methods to place the time window for the
reflection is superior. In some cases, both methods produce virtually identical results.
However, the graphs depicting the reflection of 1kHz on the north side of the chapel are
noticeably different. Preference would seem to go to the empirical method, rather than
the theoretical method, as it would avoid any imperfections in the theoretical model.
Conversely, the empirical approach requires absolute certainty in the identity of the
reflection due to the diffuser. Since such certainty is not always possible, particularly at
low frequencies, the empirical method is prone to greater uncertainty.
30
Chapter 5: Conclusion 5.1 Discussion of Experimental Method
The results of this experiment were in many ways in good agreement with
expectations. Covering the diffuser produced a much greater ratio of reflected sound to
incident sound near 0°. We can conclude that the diffractal was causing an observable
redistribution of scattered energy from the specular direction to more lateral angles. With
the diffuser present, polar responses at all frequencies showed some alternation between
high and low reflection energy, which is supportive of the lobing behavior associated
with quadratic residue diffusers.
Several factors raise questions about the effectiveness of this approach. Data
obtained symmetrically about the diffuser did not produce symmetric results, which is a
breach from the expectations. Experimental measurements made by D’Antonio are at a
higher angular resolution, such as every 2.5° rather than every 10°.1 The inherent
difficulty in positioning a microphone at a specific point in 3-dimensional space by hand
would require a substantial increase in time to increase the level of angular resolution.
Any apparatus to aid in the positioning of microphones would need to be acoustically
invisible during recording to avoid contaminating the effect of the diffuser. Without the
31
increased angular resolution, it is not possible to observe the expected periodic lobing
behavior, which can have an angular period of less than 10°.
This method is limited in its applicability to low frequencies. Even at 500Hz, it
becomes difficult to isolate specific reflections within the microphone responses. Fewer
periods may be included in the test pulse to reduce the length of each reflection, however
this will likely require more harmonic frequencies in the Fourier series of the incident
pulse, and thus the resulting polar response will be less frequency-specific. In addition,
as the test frequency is lowered, it will become necessary to lower the cutoff frequency of
the low-pass filter used to remove the oscillating behavior of the test signal. The
sharpness of the filter is reduced as the cutoff frequency is lowered, and so the filter
becomes harder to control.
5.2 Future Work
Before this method can truly be evaluated, it is necessary to repeat the experiment
in one of two ways. Either a single microphone should sweep through a complete 180°
arc about the center of the diffuser, or two microphones should simultaneously take
measurements at symmetric angles about the centerline of the chapel. The data would be
analyzed as above. If the results were still inconsistent, then it would be possible to
assert that either the diffuser has an asymmetric response, or the diffuser response has not
been isolated and some unaccounted variable is producing the asymmetry.
Assuming that it is possible to produce a reasonably symmetric polar response,
the next step would be to more directly compare results produced in Finney Chapel to
measurements made at RPG using their standard method. Some amount of graphical
manipulation would be required to produce data on similar axes. Multiplication by a
constant factor might be necessary to account for differences in normalization.
Comparison of results could then be expanded using measurements at a greater angular
resolution and at more frequencies.
There are many foundational components of this procedure that present
opportunities for future experimentation. For example, the number of periods used to
generate a pressure wave could be varied. Increasing the number of periods should result
in a sharper peak in the Fourier transform of the test signal, providing more frequency-
32
specific information about the behavior of the diffuser. Highly directional loudspeakers
or microphones could be implemented as a means of removing stray reflections from the
data. Such materials were not available during this investigation. Though it was not
expected to be significant, the low-pass filter could have affected the data. The influence
of the filtering process could be revealed by reproducing the response graphs using
different cutoff frequencies, or varying the number of terms used in the filter.
The subject of the QRD Diffractal lends itself well towards numeric computation
based on the complexity of the geometry necessary to describe wave propagation from its
surface. This presents an opportunity to take advantage of the Oberlin College Beowulf
Cluster supercomputer and its large processing power. A particularly ambitious
undergraduate could attempt to model the interior of Finney Chapel and produce a
theoretical response of the diffuser to different stimuli.
5.3 Conclusions about Reflection Phase Grating Type Diffusers
There is no question that QRDs and their kin have interesting acoustic properties,
and are worth further academic study. The existing literature on the subject, however,
leaves much to be desired. Much of the theoretical work presented in papers studying
Schroeder diffusers suffers from lack of clarity, inconsistencies or typographical errors.
Occasionally, incredible mathematical simplifications are made with dubious
justification. For instance, in Peter D’Antonio’s book1, written to be the first
comprehensive book on diffuser design, several terms in the Kirchhoff equation are
dismissed because they do not agree with theory presented by Schroeder. No explanation
is given for why such a disagreement arose.
In other instances, experimental work is poorly outlined, if any explanation is
provided at all. In Schroeder’s first paper presenting the idea of a phase grating diffuser,
a graph is produced with no description of how it was created, no units are given, nor
axes or tick marks. This investigation was unable to locate any complete description of
how RPG produced their elegant-looking polar graphs. With no ability to scrutinize
experimental method, it is impossible to compare data published from different sources.
A decibel is merely a log of a ratio – without a given reference, one cannot tell if the
decibels in one paper are comparing the same quantities as the decibels in another.
33
Ultimately, it is advised that individuals interested in enhancing the acoustic quality of
their listening space consider these diffusers based on their aesthetic qualities, and not on
their scientific merit.
34
Appendices:
A – Mathematica Scripts
B – Table of Filtered Audio Data
C – Architectural Drawings of Finney Chapel
35
Appendix A – Mathematica Scripts The following code was written in Mathematica 6.0.
The test signals used for the experiment were generated using the Play command. The
resulting sound object was then exported in .AIF format. The 5kHz wave was generated
using a sample rate of 98kHz to avoid aliasing issues from lower frequency sampling in
Mathematica. All audio files were reduced to 44.1kHz upon import into Cubase.
36
Digital Low-pass Filter – produces a table of coefficients, coef, to be used in subsequent
data manipulation. Lowpass produces a filter with cutoff frequency at sample rate samp,
fmax using q terms. The resulting transfer function H(w) is plotted for visual reference.
FFT Frequency Analyzer – displays the frequency content of a given time range within
an audio file, file. The sample rate of the file is given as samp, and the analysis is
performed from starttime to endtime, which are in milliseconds.
Wave reads audio data into Mathematica and displays the desired time range. The
function variables are the same as for frequency.
Power uses the coefficients generated by lowpass to remove the high frequency content
from the raw audio data. The data is first squared to produce a function of the pressure
amplitude in time.
37
Peakvalue determines the maximum value occurring in a particular data set, data,
between the times xmin and xmin, given in milliseconds.
Avalue calculates the mean value of data points from the set data lying in the range xmin
to xmax.
Incidence returns the time value in milliseconds of the maximum value of pdata in the
range xmin to xmax. This function requires having previously run an instance of power to
produce the data set pdata.
Constructdatalist is the function that was used to produce the plots of reflected energy
versus angle. The reference data file reffile is imported and filtered using power. The
incident test pulse, which appears as the maximum data value between 200 and 300ms, is
located. The arrival time of the pulse is stored as x (see incidence). The height of the
peak is found using avalue, taking the average about x, and is stored in the variable
bigpeak. The test microphone data, testfile, is then loaded and filtered. The value of the
test data at x + time is found using an average, and that average is stored as littlepeak
(time is the number of milliseconds after the arrival of the incident pulse expected for the
arrival of the reflection from the rear wall). The ratio of the reflection to the incident
pulse is calculated and appended to datalist, and the heights of both the reflected pulse
38
and incident pulse are displayed. This function was run after defining datalist as an
empty list object. Eight instances of the function were run in succession, for angles 0° to
70° in order from lowest to highest. Afterwards, datalist could be normalized and
plotted.
39
Appendix B – Table of Filtered Audio Data Presented here is a compilation of all the data collected and analyzed for this
project. The raw audio has been squared and filtered as described in §4.3. In the graphs
that follow, the time domain has been chosen to display the direct sound from the
loudspeakers and the following 50ms. The reference microphone data are represented by
the thin line, whereas the test microphone data comprise the thick line.
40
2-16 500Hz
0°
10°
20°
30°
40°
50°
60°
70°
41
2-16 1kHz
0°
10°
20°
30°
40°
50°
60°
70°
42
2-16 5kHz
0°
10°
20°
30°
40°
50°
60°
70°
43
2-27 (Diffuser Covered) 500Hz
0°
10°
20°
30°
40°
50°
60°
70°
44
2-27 (Diffuser Covered) 1kHz
0°
10°
20°
30°
40°
50°
60°
70°
45
2-27 (Diffuser Covered) 5kHz
0°
10°
20°
30°
40°
50°
60°
70°
46
3-6 (Reverse Side) 500Hz
0°
10°
20°
30°
40°
50°
60°
70°
47
3-6 (Reverse Side) 1kHz
0°
10°
20°
30°
40°
50°
60°
70°
48
3-6 (Reverse Side) 5kHz
0°
10°
20°
30°
40°
50°
60°
70°
49
Appendix C – Architectural Drawings of Finney Chapel
50
51
Acknowledgements
Special thanks to the following individuals for their contributions to this project:
• Peter D’Antonio and RPG Diffusor Systems, Inc.
• Kathy Drennan
• Jordan Gottdank
• Michael Grube
• Liz Hibbard
• Urban Landreman
• David Levin
• Pradya Martz
• Eric Michaels
• John Miess
• The Oberlin College Library
• Antonio Papania-Davis
• Carl Rosenberg and Benjamin Markham (Acentech)
• Bruce Richards
• Margaret Youngberg
• Tina Zwegat
52
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and Application. New York: Spon Press, 2006. 2 Michael Barron, Auditorium Acoustics and Architectural Design. London: E & FN
Spon, 2000. 3 M. R. Schroeder, “Diffuse Sound Reflections by Maximum-Length Sequences,” J.
Acoust. Soc. Am., 57, 149-150 (1975). 4 Trevor J. Cox and Peter D’Antonio. “Acoustic Phase Gratings for Reduced Specular
Reflection,” Applied Acoustics 60, 167-186 (2000). 5 T. J. Cox and Y. W. Lam, “Prediction and Evaluation of the Scattering From Quadratic
Residue Diffusors,” J. Acoust. Soc. Am., 95(1), 297-305 (1994). 6 M.R. Schroeder, “Binaural Dissimlarity and Optimum Ceilings for Concert Halls: More
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53
10 RPG Diffusor Systems Website, < http://www.rpginc.com/proaudio/psadvantage.htm>,
(2000). 11 RPG Diffusor Systems Patent Disclosure,
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117 (1963). 17 Warner R. Th. Ten Kate, “On the Bandwidth of Diffusors based upon the quadratic
residue sequence,” J. Acoust. Soc. Am., 98(5), 2575-2579 (1995). 18 Excel Spreadsheet of Experimentally Determined QRD Polar Responses, attachment to
personal communication from Peter D’Antonio, 6 April (2008). 19 D. E. Commins, N. Auletta and B. Suner, “Diffusion and Absorption of Quadratic
Residue Diffusers,” Proc. IoA (UK), 10(2), 223-232 (1988). 20 R. W. Hamming, Digital Filters, Third Edition, Eaglewood Cliffs, NJ: Prentice Hall
(1989). 21 Dennis A. Bohn, “Environmental Factors on the Speed of Sound,” J. Audio Eng. Soc.,
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