Page 1

On the subtraction method for in-situ reflection

and diffusion coefficient measurements

Philip Robinson and Ning Xiang

Graduate Program inArchitecturalAcoustics, Rensselaer Polytechnic Institute, 110 8th Street,Troy, NewYork 12180

robinp@rpi.edu, xiangn@rpi.edu

Abstract:

tant acoustic measurements. It involves subtracting a reference measurement

including only direct sound from one with direct sound and a reflection, to

isolate the reflection. The process is very sensitive to environmental condi-

tions, such as changes in temperature, air movement, and microphone posi-

tioning.These variations cause small time differences between the reference

and reflection measurements, which prevent complete subtraction of the di-

rect sound; the residual direct sound then pollutes analysis of the isolated

reflection. This work evaluates methods to compensate for differences to

achieve minimal interference from the residual direct sound.

© 2010 Acoustical Society ofAmerica

PACS numbers: 43.58.Bh, 43.58.Vb, 43.55.Ev [MS]

Date Received: November 4, 2009

DateAccepted:

The subtraction method is a technique critical to several impor-

January 4, 2010

1. Introduction

The subtraction method is an important step in several acoustic measurements that are impor-

tant to acoustic practitioners in the field.The first is the in-situ absorption/reflection coefficient

measurement. This method was suggested byYuzawa1as early as 1975 and further developed

byMommertz.2Nocke3andGarai4alsoutilizedthistechnique.ISOStandard13472(Ref.5)for

the measurement of the absorption of roadway surfaces incorporates the subtraction technique

andhasbeenappliedtoothersurfacessuchasgrassandturf.6Themeasurementinvolvestaking

animpulseresponseinthefreefield,andanotherwiththemicrophoneclosetothesurfaceunder

test. The free field measurement is subtracted from the test measurement to isolate the reflec-

tion, which is then compared to the free field direct sound to deduce the reflection coefficient

fromthesurface.Thismeasurementtechniqueisofhighpracticalsignificancebecauseitallows

testing of installed materials without the need to remove material to a laboratory for reverbera-

tion chamber7or impudence tube8measurements.

The second measurement that requires this technique is the measurement of diffusion

coefficients.9Diffusion coefficients are measured by placing a semicircular or hemispherical

array of microphones around the sample to be measured, a source outside of the microphone

array, and taking impulse responses for each microphone. The polar response of the panel is

then distilled into a diffusion coefficient using an auto-correlation function as in Eq. (2). In

attaining the polar response from an architectural wall panel for many source incidences, it is

often necessary to place the source at grazing incidence to the panel with the microphone op-

posite. Figure 1 (right) illustrates this condition. This results in the direct sound and panel re-

flection arriving at the microphone in close succession. Since only the reflection is of interest, a

second measurement is taken without the panel present to attain a reference that can be sub-

tracted from the measurement to isolate the panel reflection.This measurement is a simple and

practical method to measure surfaces such as those described by Olson and Bradley,10Dadiotis

et al.,11and D’Antonio,12among others. Both of these measurements require taking either two

measurements with the same microphone or taking one measurement with two different micro-

phones. In either case, there will be slight differences between the measurements.These differ-

encesarisefromdifferencesintheresponseofthemicrophones,nomatterhowcloselymatched

P. Robinson and N. Xiang: JASA Express Letters

?DOI: 10.1121/1.3299064?

Published Online 11 February 2010

J. Acoust. Soc. Am. 127 ?3?, March 2010© 2010 Acoustical Society of AmericaEL99

Page 2

they are, temperature changes and air movement between measurements, and variations in the

response of the measurement equipment due to temperature or electrical deviations.This work

examines methods to compensate for these small changes.

2. Experiment description

This experiment uses impulse responses obtained from diffusion coefficient measurements. A

measurement as shown in Fig. 1 (left) without the diffuser panel present yields the reference

measurement; one with the panel present provides the reflection measurement. Subtracting the

former from the latter eliminates the direct sound and leaves only the reflection. This paper

examines a variety of methods to correct for minor time misalignments between the two signals

to attain the optimal result.The subtraction results from each method are then used to calculate

the diffusion coefficient,13and variations are analyzed. Figure 1 illustrates the measured im-

pulse responses and subtraction results.

3. Subtraction methods

The first scheme examined is the direct subtraction method.This method assumes that environ-

mental conditions are constant between measurements and measurements can be repeated ex-

actly. If this assumption is correct, direct subtraction should produce complete elimination of

the direct sound and leave only the isolated reflection. If this is not effective, which is often the

case, further processing is necessary.

Several steps may be taken to improve the results.All of these steps require comparing

different features of the two measurements.To make this comparison more accurate it is desir-

able to oversample both signals so corrections of less than one sampling point can be made;

after subtraction the signals are downsampled to their original sampling frequency.

The simplest correction is to calibrate the signals so that the amplitudes of the direct

sound peaks are the same. This ensures that differences in magnitude do not skew the results

and do not account for time misalignments. Aligning the signals in time further improves the

subtraction results.This can be done in several ways. Utilizing the peaks of the direct sounds as

a reference point and aligning them in time can help correct for overall shifts, but a small shift

in the peak may not accurately represent the shift of the whole signal. To incorporate a larger

portion of the signal into consideration, cross-correlation of a few milliseconds of the signals,

centered on the direct sound peaks, may be effective. The location of the peak of the cross-

correlation function will indicate how much to shift one signal in relation to the other to attain

Fig. 1. Diffusion coefficient measurement setup that requires the subtraction method to isolate the reflection and the

resulting impulse responses: ?a? Impulse response of reflections including the direct sound impulse. ?b? Reference

impulse response. ?c? Subtraction results.

P. Robinson and N. Xiang: JASA Express Letters

?DOI: 10.1121/1.3299064?

Published Online 11 February 2010

EL100 J. Acoust. Soc. Am. 127 ?3?, March 2010P. Robinson and N. Xiang: Subtraction optimization

Page 3

the best alignment of the direct sounds. However, the cross-correlation function is often skewed

by the presence of the reflection within the direct sound component.This is often the case when

measuring the reflection coefficient of a rigid surface when the microphone is close to the sur-

face.

4. Subtraction optimization

The optimized approach to achieve the best alignment includes all of the previous schemes,

then shifts the signals across each other several points to either side of the shift indicated by the

cross-correlation function. By subtracting the reference from the reflection measurement at

eachoftheselocations,themosteffectiveresultcanbefound.Thisisperformedatawiderange

of oversampling rates in order to have the ability to shift the signals by any fraction of a sam-

pling point. So, for an oversampling rate of 2, the signals are shifted four points in either direc-

tion; for 3, six points in either direction; etc.This is repeated up to an oversampling rate of 20 or

more, so that every fraction of shift within the four points around the peak correlation is inves-

tigated. There is some redundancy since, for example, 10/20 is the same as 1/2, but the advan-

tage is that a step of, say, 1/19 of a point is investigated.At each shift the signals are subtracted

and the residual direct sound is compared. More thorough organization of the oversampling

rates could improve calculation time by removing the redundant steps. For a limited number of

oversamplingratesandshifts,agriddedsearchtofindthemosteffectivesubtractionissufficient

to obtain optimized results; in more detailed investigations a genetic algorithm can be exploited

to improve computation time.

5. Measuring subtraction effectiveness

The process above provides many varying subtraction results; in order to choose the best result,

a metric of success is required. As the goal of the operation is to remove the direct sound,

subtractioneffectivenesscanbemeasuredbythepercentreduction,ordecibellevelreductionin

the direct sound from the measurement to the result. Specifically, the sum of the energy within

0.5 ms of either side of the direct sound can be compared before and after subtraction to find the

effective reduction. Equation (1) defines the reduction factor R.A reduction factor equal to the

peak to noise ratio of the measurement can be considered a complete subtraction, since this

would leave nothing in the area of the direct sound except the background noise.

R = 10 log10?

where ds is the time of the direct sound peak, Srefis the reference measurement, and Sresultis the

subtraction result.

Certainly, if direct subtraction provides suitable reduction in the direct sound compo-

nent, no further processing is necessary.The level of the residual direct sound in relation to the

level of the reflection should also be considered. The smaller the reflection component, the

greater influence a given residual direct sound will have. This is particularly important in the

in-situ measurement of absorptive materials and grazing source incidence diffusion coefficient

measurements. In both cases the reflection is small and also within close proximity to the direct

sound.

?

ds−0.5 ms

ds+0.5 ms

ds+0.5 ms

?Sref?2

?

ds−0.5 ms

?Sresult?2?

,

?1?

6. Results

Seventy-two sets of measurements were processed with five different subtraction schemes.The

schemes, as described above, are direct subtraction, peak amplitude matching (PAM), peak

amplitude and time matching (PATM), peak amplitude matching with cross-correlation align-

ment (PACA), and the optimized method (OM). An example of the results from two different

methods is shown in Fig. 2 (left). Calculating the reduction factor R per Eq. (1) for each mea-

P. Robinson and N. Xiang: JASA Express Letters

?DOI: 10.1121/1.3299064?

Published Online 11 February 2010

J. Acoust. Soc. Am. 127 ?3?, March 2010P. Robinson and N. Xiang: Subtraction optimization EL101

Page 4

surement and subtracting the direct method reduction factor yield the improvement offered by

each method. Table 1 lists the improvement results for the 72 measurement sets processed.

Negative values indicate results that are worse than direct subtraction.

Peak amplitude matching provides some improvement, and since this scheme is so

simple to implement, it should be used on all subtraction measurements. Note that aligning the

peaks of the direct sound provides no additional improvement. This indicates that the exact

alignment of the peaks does not correlate to the best time alignment of the entire direct sound.

Cross-correlation, used blindly, provides generally worse results than if the signals had been

subtracted directly. This is possibly due to the influence of the reflection in the correlation,

which could skew the correlation peak due to the mismatch between the reflection and back-

ground noise after the direct sound. Finally, the optimization method shows a marked improve-

ment over direct subtraction, sometimes as much as 12 dB. Where the optimization method

failed, as indicated by the −1 dB improvement, was on the rare measurement where direct sub-

traction was extremely accurate; these measurements showed direct method R factors above all

others.

7. Effect on the diffusion coefficient

Since the diffusion coefficient uses auto-correlation of many channels, per Eq. (2), it is suscep-

tible to being skewed by one faulty channel.

Fig. 2. ?Left? A comparison of the direct subtraction result ?a? and the optimized result ?b?. In this case, there was a

5 dB improvement in the reduction in the direct sound. ?Right? A polar response with an anomalous spike from failed

subtraction ?b? and the corrected response ?a?. The diffusion coefficients are 0.19 and 0.35, respectively.

Table 1. A comparison of the improvement ?Rmethod−Rdirect? provided by each subtraction scheme: PAM, PATM,

PACA, and OM.

Method

Minimum

?dB?

Mean

?dB?

Maximum

?dB?

PAM

PATM

PACA

OM

?1.6

?1.6

?29.3

?1

0.4

0.4

4.5

4.5

4.5

12.1

?2.5

2.6

P. Robinson and N. Xiang: JASA Express Letters

?DOI: 10.1121/1.3299064?

Published Online 11 February 2010

EL102 J. Acoust. Soc. Am. 127 ?3?, March 2010 P. Robinson and N. Xiang: Subtraction optimization

Page 5

D?=??

i=1

n

10Li/10?

2

−?

i=1

n

?10Li/10?2

?n − 1??

i=1

n

?10Li/10?2

,

?2?

where i is the microphone position, Liis the level at i microphone, n is number of microphones

in the array, and ? is the source incidence.

In a semicircular array of 72 microphones, if the subtraction method fails on one or

two channels, a spike can occur in the polar response.This spike is interpreted as highly direc-

tional reflection, and the diffusion coefficient will be artificially low. Figure 2 (right) shows a

polar response that has a spike from failed subtraction and the corrected response calculated

from optimized subtractions. The effect on the diffusion coefficient is pronounced.

Table 2 lists diffusion coefficients of a quadratic residue diffuser designed for 600–

1600 Hz with the source at 60° incidence for each subtraction method. The effect of failed

subtraction becomes most evident above 400 Hz where all of the non-optimized methods yield

erroneous results. Even below this threshold, the optimized method, which eliminates the

spikes, shows a higher diffusion coefficient indicating a more uniform polar response.The con-

sistencyoftheoptimizedresultsalsoindicatesthatthemethodislesssubjecttovariationsinthe

quality of the measurement from each channel of the semicircular array.

8. Concluding remarks

The subtraction method is particularly sensitive to variations in environmental conditions be-

tween the reference measurement and the reflection measurement. Aligning the signals from

these measurements in amplitude and time can eliminate the differences to provide clean sub-

traction results. This is a critical step in both in-situ reflection measurements and diffusion

coefficient measurements; if the subtraction is not effective, the results from both measure-

ments will be skewed if not entirely erroneous. This complication is amplified in diffusion co-

efficient measurements because the measurement gauges consistency across many channels.

Shifting the signals across one another at various oversampling rates to attain the highest reduc-

tion factor is an effective method to determine the optimal subtraction result.

Table 2. A comparison of the diffusion coefficients calculated from different subtraction schemes.

1

3octave band

?Hz?

Direct PAMPATMPACAOptimized

100

125

160

200

250

315

400

500

630

800

1000

1250

1600

2000

2500

0.57

0.57

0.44

0.37

0.52

0.45

0.19

0.05

0.05

0.04

0.01

0.01

0.01

0.01

0.03

0.58

0.57

0.43

0.37

0.52

0.45

0.19

0.05

0.05

0.04

0.01

0.01

0.01

0.01

0.03

0.58

0.57

0.43

0.37

0.52

0.45

0.19

0.05

0.05

0.04

0.01

0.01

0.01

0.01

0.03

0.54

0.52

0.44

0.38

0.40

0.30

0.12

0.04

0.05

0.04

0.03

0.03

0.03

0.04

0.04

0.68

0.71

0.51

0.41

0.61

0.52

0.35

0.57

0.48

0.56

0.48

0.38

0.36

0.62

0.45

P. Robinson and N. Xiang: JASA Express Letters

?DOI: 10.1121/1.3299064?

Published Online 11 February 2010

J. Acoust. Soc. Am. 127 ?3?, March 2010P. Robinson and N. Xiang: Subtraction optimization EL103