An Acoustical Evaluation of the QRD Diffractal in Finney Chapel

Patrick Landreman Bruce Richards (Advisor) Honors Thesis: Oberlin College April 2008

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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

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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

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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.

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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.

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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

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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

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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.

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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.

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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.

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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

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(2000). 11 RPG Diffusor Systems Patent Disclosure,

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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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