_________________________________
Audio Engineering Society

Convention Paper 5844
Presented at the 114th Convention
2003 March 22–25 Amsterdam, The Netherlands

This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or considerationby the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request
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All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.

_________________________________
Comparison of Modal Equalizer
Design Methods
Poju Antsalo1, Matti Karjalainen1, Aki Mäkivirta2 and Vesa Välimäki1
1Helsinki University of Technology, Lab. of Acoustics and Audio Signal Processing,
P.O. Box 3000, FIN-02015 HUT, Finland
2

Genelec Oy, Olvitie 5, FIN-74100 Iisalmi, Finland

ABSTRACT
Modal equalization of low-frequency room modes has recently been proposed as a method to improve sound reproduction in spaces where modal decay time is too long. Modal equalization is achieved by signal processing reducing
the pole radii of problematic modes in the overall transfer function. In this paper, we compare the performance oftwo proposed methods for designing modal equalizers. Comparison includes a preliminary subjective listening test
indicating a possible marginal improvement by modal equalization over conventional magnitude equalization.
1. INTRODUCTION
A loudspeaker installed in a room acts as a coupledsystem where the room properties typically dominate
the rate of energy decay. The characteristics of thisenergy decay are well known.
Passive methods of controlling the decay rate andproperties of this energy decay are also straightforward and well established. At low frequencies thesemethods are unpractical or very expensive to implement because the physical size and cost of necessary
absorbers increases rapidly with decreasing frequency.
Active control becomes feasible as the wavelengths
become long and the sound field develops less diffuse [1-5]. Methods to optimize the response at alistening position by selecting suitable loudspeakerlocations have been proposed [6] but cannot fully
solve the problem. Because of these reasons, there
has been an increasing interest in active methods ofsound field control at low frequencies.
Modal resonances in a room can become audible as
timbre changes because they modify the magnitude
response of the primary sound or, when the primarysound ends, because they are no longer masked [7,8].
The ability to detect a modal resonance appears to bevery dependent on the signal content. Olive et al.
report that for mid and high frequencies low-Q resonances are more readily audible with continuous signals containing a broad frequency spectrum while
high-Q resonances become more audible with transient discontinuous signals [8]. They also report detection thresholds for resonances for both continuous
broadband sound and transient discontinuous sound.

Antsalo et al.

Modal Equalization

At low Q values, antiresonances (notches) are as
audible as resonances. Audibility of antiresonances
reduces dramatically for wideband continuous signals
as the Q value becomes high [8]. Detectability ofresonances reduces by approximately 3 dB for eachdoubling of the Q value [7,8] and low Q resonances
are more readily heard with zero or minimal timedelay relative to the direct sound [7]. Duration of thereverberant decay in itself appears an unreliable indicator of the audibility of the resonance [7] as audibility seems to be more determined by frequency domain characteristics of the resonance.

Fig. 1. A simple set-up for modal equalization.
An impulse response of the acoustic path measured
from the loudspeaker to the listening position isneeded in order to design a modal equalization filter.
Modal equalization concentrates to frequencies below 200 Hz and within a restricted listening area. The
process of modal equalization starts with the estimation of octave band reverberation times between
500 Hz and 2 kHz to calculate a mean reverberation
time at mid frequencies, which is used as the basis
for determining the target for maximum allowed lowfrequency reverberation time. We define the target
decay time relative to the mean decay time T60 in
mid-frequencies, increasing for example by 0.2 s as
the frequency decreases from 300 Hz down to 50 Hz.

We have previously proposed two methods for theestimation of modal parameters and using the estimated mode information to design filters that reducethe modal decay time [9-12]. In this paper we present
a series of experiments where we have studied theapplicability and performance of the proposed modal
equalization techniques with case studies and measured listening room response data.
2. MODAL EQUALIZER DESIGNWe define modal equalization as a process that modifies the rate of modal decay. A modal resonance isrepresented in the z-domain transfer function as apole pair with pole radius r and pole angle θH m ( z) =1
(1 − re z )(1 − re − jθ z −1 )jθ

−1We have previously developed two methods to design modal equalizer filters [9-11]. The first method(later called AMK) attempts to directly identify mode
frequencies in the magnitude response and then to
obtain the decay rate τm and mode frequency ωm byusing time-frequency presentation of the impulse
response at frequencies below 200 Hz [9-12]. The
decay rate for each identified mode frequency is calculated using a nonlinear fitting technique modelingthe data as a sum of an exponential decay and background noise. Mode frequencies are estimated directly in the magnitude spectrum using a special detection function [12] for those frequency bins thatshow slow decay. The modal equalizer filter is thendesigned using the mode parameter data. The longest
decay time is corrected by filtering with an equalizer,
and the process is continued iteratively until all decay
rates are within the desired bounds.

(1)

To shorten the decay time the Q value of the resonance must be decreased by moving the pole paircloser to the origin. Each mode is modeled by an
exponential decay function

hm (t ) = Am e −τ mt sin(ω m t + φ m )(2)where Am is the initial envelope amplitude of the decaying sinusoid, τm is a coefficient that denotes the
decay rate, ωm is the angular frequency of the mode,and φm is the initial phase of the oscillation.
One method of implementing modal equalization is
to modify the audio signal fed into a loudspeaker(Fig. 1). The total transfer function from the soundradiator to the listening position represented in zdomain is

H ( z ) = G( z) H m ( z)The second method (later called ARMA) [10,11]
identifies the pole and zero pairs describing a modal
resonance by fitting a pole-zero least-squares model
directly to the room impulse response. Similarly to
the AMK method, the longest decay rate detected byfinding the pole closest to the unit circle is compensated by designing an equalizer filter for it, and themethod is then iteratively applied until all decay rates
are within the desired bound. The second method
does not require the intermediate stage of determining values for the decay rate of a mode, and is better
able to model closely spaced mode resonances.

(3)

where G(z) is the transfer function of the sound radiator from the electrical input to acoustical output
and Hm(z) is the transfer function of the path from thesound radiator to the listening position.

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Modal Equalization

2.1. A simple case with synthetic modes

tion. Synthetic cases were created to observe how
accurately the algorithms estimate the mode frequency and decay time values.

The case study presented here clarifies the idea ofmodal equalization. Figure 2 shows the waterfall plotof a loudspeaker response measured in an anechoic
chamber, with five synthetic modes added at frequencies 50, 55, 100, 130, and 180 Hz. The corresponding decay times are 1.4, 0.8, 1.0, 0.8, and 0.7 s.
Now, we set up a modal equalizer design target to
reduce these decay times to 0.30, 0.30, 0.26, 0.24,
and 0.20 s. After processing the synthetic responsewith the equalizer designed by the AMK algorithm,
the decay times have been reduced (Fig. 3). However, the decay at 50 and 55 Hz continues with theoriginal rate after an initial rapid decay of 15-20 dB.
This shows that the method is able to control the decay rate of individual modes, but its performance islimited when modes are close to each other in frequency.

3.1. Synthetic responses
There are three different cases each representing an
increasing “degree of difficulty” in the sense of how
close the modal frequencies f are to each other (Tables 1-3). The decay time t for each mode is thesame.
Table 1. Modal frequencies and decay times for synthetic response 1.
f (Hz)t (s)Mode 150.4

0.6

Mode 2100.4

0.6

Mode 3140.4

0.6

Table 2. Modal frequencies and decay times for synthetic response 2.
f (Hz)t (s)Mode 190.4

0.6

Mode 2100.4

0.6

Mode 3116.4

0.6

Table 3. Modal frequencies and decay times for synthetic response 3.

Fig. 2. Waterfall plot of five synthetic modes.

f (Hz)t (s)Mode 199.4

0.6

Mode 2100.4

0.6

Mode 3102.4

0.6

3.2. Mode parameter estimation errors
The estimation error of mode frequency ∆f and theerror of the decay rate ∆t are calculated using Eqs. 4
and 5,

Fig. 3. Waterfall plot of five synthetic modes aftermodal equalization.
3. OBJECTIVE EVALUATION

∆f =

f −f'
f

∆t =

t −t'
t

(4)

(5)

where the actual mode frequency is denoted by f anddecay time by t, and the estimated mode frequency
by f ' and decay time by t'.

The performance of AMK and ARMA equalizer design algorithms was studied with objective evalua-

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When the modes are sparsely spaced in frequency,
both equalizer design methods show similar performance in estimating parameters.

Table 6. Synthetic case 3, estimation errors percentages for the mode frequency and decay time.
Equalizer design method

Estimation errors (Tables 4-6) tend to increase asmodes move closer to each other, especially in thecase of decay time estimated by the AMK design
method. For closely spaced modes (response 3), the
ARMA design method performs well, significantly
better than the AMK method. This is due to the decay
envelope fluctuation (beating) in level estimates ofclosely spaced modes producing energy-time curves
more complex than simple exponential decays, causing an error when fitting data assuming an exponential decay model as in the AMK method.

AMK

Equalizer design method
ARMA
∆t

1

0.16%

2.28%

0.07%

3.8%

2

0.0082%0.32%

0.005%

0.58%

3

0.017%

2.9%

0.0034%

0.60%

1

0.53%

19%

0.057%

0.077%

2

0.12%

86%

0.036%

3.2%

3

0.27%

15%

0.020%

0.55%

The two rooms are similar in size to typical professional audio monitoring spaces, and are referred tolater as Room G and Room S. The response magnitude and waterfall plots of these rooms show distinct
room modes (Fig. 4). The volume of room G is about
102 m3 and room S about 90 m3, and mid-frequency
RT60 values for these rooms are 0.40 s and 0.77 s,
respectively.

Table 5. Synthetic case 2, estimation error percentages for the mode frequency and decay time.
Equalizer design method
AMK∆t

To facilitate the subjective listening tests impulse
responses of two real rooms were measured using anomnidirectional microphone and both ears of adummy head1. The omnidirectional microphonemeasurement was used later in the design of theequalizer filters. The dummy head equalized impulse
responses were convolved with audio test samples to
produce subjective test items for headphone listening.

Table 4. Synthetic case 1, estimation error percentages for the mode frequency and decay time.

∆f∆f

4.1. Room response measurements

The responses of two representative rooms were
measured and equalizers were designed for eachroom. Audio material with energy content at low∆t∆t

Subjects were asked to grade on a continuous scale
two properties of the sound samples, the temporaldefinition and the spectral balance.

The perceptual benefit of modal equalization wasstudied with listening tests. The aim of the listening
test was to demonstrate the incremental improvements provided by conventional magnitude equalization and modal equalization.

∆f∆f

Test subjects listened to audio test items over headphones. The reference was the sound sample convolved with the original room response. This wascompared to the sound sample processed with a magnitude equalizer, or a combination of a modal equalizer design either by the AMK method or the ARMA
method and a magnitude equalizer.

4. SUBJECTIVE LISTENING TEST

Mode

Mode

frequencies were chosen and convolved with theequalizer filters to produce the audio test items.

With true room responses the performance of bothdesign methods is limited mainly by the complexityof the modal characteristics, measurement anomalies
and noise.

AMK

ARMA

4.2. Design of room equalizers

ARMA

Mode

∆f

∆t

∆f

∆t

1

0.19%

8.2%

0.015%

0.80%

2

0.17%

16%

0.008%

0.16%

3

0.075%

5.7%

0.001%

0.096%

Three room equalizers were designed for each room
(Table 7). All equalizers were designed for the target
frequency band of 20 Hz … 200 Hz using omnidirectional microphone measurements of the rooms.
1

Manufacturer Cortex Electronics GmbH, model
Manikin Mk2/NCF1.

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magnitude [dB]magnitude [dB]

Antsalo et al.

Modal Equalization
As a reference to current state-of-the-art, a magnitude
equalizer was designed as the inverse of the thirdoctave smoothed magnitude response within the target frequency band. Such an equalizer ignores individual modes, but corrects the magnitude response
unbalance within the target frequency range. Thismagnitude equalizer is later referred to with an acronym ‘Ampleq’ (Table 7).

(a)

20
0
−20
−40
0

50

100
150
frequency [Hz]

200

50

100
150
frequency [Hz]200

The modal equalizers affect primarily the decay
times, but do not necessarily provide magnitude
equalization similar to Ampleq. Because of this, andbecause it has been recognized that broadband
equalization is necessary after modal equalization[9,10], third-octave magnitude equalization was applied to both the AMK and the ARMA modal equalized systems. In this way the subjective listening test
detects the incremental improvement of using modal
equalization before magnitude equalization.

(b)

20
0
−20
−40
0

Equalizer magnitude responses of all three equalizer
designs for rooms G and S are presented in Figures
9-11. Tables of poles and zeros (sampling frequency
11025 Hz) are given for each modal equalizer design(excluding the magnitude equalizers) in Tables 1417 in appendix 1. Figures 5-8 show these poles andzeros visually.

(c)level [dB]

20
0
−20
−40
−60
0

0

Table 7. Descriptions of equalizers used in the subjective listening test.

0.5100
frequency [Hz]200 1time [s]Equalizer

Description

Ampleq

FIR inverse filter of the thirdoctave smoothed response within
20…200Hz

AMKType AMK modal equalizer followed by type ‘Ampleq’ magnitude
equalizer designed on the AMK
filtered data

ARMA

Type ARMA modal equalizer followed by type ‘Ampleq’ magnitude
equalizer designed on the ARMA
filtered data(d)level [dB]

20
0
−20
−40
−60
0

0
0.5
100
frequency [Hz]200 1

AMK design method attempts to assign multiple
notches to equalize certain types of modes, such as
closely spaced modes and modes with nonexponential decay. Also, if the frequency or decay
time of a mode is erroneously estimated, the AMK
method compensate for this failure by assigning multiple equalizer notches close to such mode. The
ARMA design method seems to be more successful
in modeling and correcting these problematic mode
structures.

time [s]Fig. 4. Responses of rooms used in the subjective
listening test, from top to bottom, (a) magnitude responses of Room G and (b) Room S, (c) waterfall
responses of Room G and (d) Room S.

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Poles and zeros: AMK, Room G

Plots of power spectra demonstrate the lowfrequency energy content in the test signals (Fig. 12).

50

100
150
Frequency (kHz)200

RadiusFig. 5. Poles (×) and zeros (o) for the modal equalizerof AMK (room G).
Poles and zeros: AMK, Room S

Radius1magnitude [dB]

0.998

20
0
−20
−40

magnitude [dB]0.999

(10th-order Butterworth), and in order to limit infrasound energy, first-order Butterworth highpass filter
is applied at 27 Hz.

1

20
0
−20
−40

0

0.999
0.998

50

100
150
Frequency (kHz)200

Fig. 6. Poles (×) and zeros (o) for the modal equalizerof AMK (room S).

(a)

100
150
frequency [Hz]

200

50

100
150
frequency [Hz]

200

(b)

0

Poles and zeros: ARMA, Room G

50

0.998

50

100
150
Frequency (kHz)200

Fig. 7. Poles (×) and zeros (o) for the modal equalizer
of ARMA (room G).
Poles and zeros: ARMA, Room S

magnitude [dB]Fig. 9. Magnitude equalizer (Ampleq) responses, (a)magnitude responses for Room G and (b) for Room
S.

0.999

20
0
−20
−40
−60

magnitude [dB]

Radius

1

20
0
−20
−40
−60

(a)
0

50

100
150
frequency [Hz]

200

50

100
150
frequency [Hz]200

Radius

1
0.999
0.998

50

100
150
Frequency (kHz)200

Fig. 8. Poles (×) and zeros (o) for the modal equalizer
of ARMA (room S).

(b)0

Fig. 10. Combined modal equalizer and magnitude
equalizer responses (AMK), (a) magnitude responses
for Room G and (b) for Room S.

4.3. Selection of audio material for the test
Five test signals were chosen so that various signal
types are covered (Table 8). Test signals 1-3 are 5 to
7-second clips of tracks 72, 30 and 5, respectively,
taken from [15]. Signal 4 is a 0.5-second burst ofwhite noise followed by silence, bandpass filtered at20-200 Hz with a fourth-order Butterworth filter.
Signal 5 is an impulse lowpass filtered twice at 200
Hz (first-order Butterworth) and once at 500 Hz

4.4. Production of audio test items
The room impulse response measurements from thedummy head were used to produce listening test
stimuli. Each dummy head impulse response measurement contains a binaural impulse response, i.e.

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Modal Equalization

two responses, one for each ear in the dummy head.
Both impulse responses were filtered with the equalizers to provide the equalized impulse responses.

research project. One of the subjects was the designerof the listening test.

The audio material for the subjective test was convolved with the four impulse response pairs (theoriginal binaural room response, and the three equalized binaural room responses) for rooms G and S.

PSD of the left channel in sound 1(dB/Hz)−40

Stereophonic audio data was resampled to 44.1 kHz
sampling frequency for convolution. Convolution
processing of the audio material with binaural impulse responses was performed in floating point andthe resulting data was rounded with dithering to 16
bit fixed point presentation.

−60
−80

−100
−120 1
10

2

10

3

10
Frequency (Hz)4

10

PSD of the left channel in sound 2

20
0
−20
−40
−60

(dB/Hz)magnitude [dB]

0
−50

−100

(a)
0

−150 1
10

50

100
150
frequency [Hz]

2

10

200

3

10
Frequency (Hz)4

10

PSD of the left channel in sound 3

20
0
−20
−40
−60

(dB/Hz)magnitude [dB]

0

−100

(b)
0

−50

50

100
150
frequency [Hz]−150 1
10

200

2

10

3

10
Frequency (Hz)4

10

PSD of the left channel in sound 4

Fig. 11. Combined modal equalizer and magnitude
equalizer responses (ARMA), (a) magnitude responses for Room G and (b) for Room S.

(dB/Hz)

0
−50

−100
−150

Table 8. Audio material for listening tests.
Description

Duration

1

Male voice

7s

2Drum hit 1

5s

3

Drum hit 25s4Bandpass filtered noise burst

5s

5Lowpass filtered impulse

5s−200 1
10

2

10

3

10
Frequency (Hz)4

10

PSD of the left channel in sound 5
−50
−100

(dB/Hz)

Sound

−150
−200
−250 1
10

4.5. Subjects
Temporal definition experiment had four subjects.
The spectral balance experiment had three subjects.
All subjects were male with normal hearing aquity,with ages ranging from 31 to 56 years. The listeners
had earlier experience in listening to sound properties
targeted by the study, and all had association to the

2

10

3

10
Frequency (Hz)4

10

Fig. 12. Power spectral density plots of the test signals (left channel). Right channels spectra are notshown as they are very similar to the left channel
spectra.

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4.6. Method of assessmentchanges to the timbre of the low-frequency audio, but
any timbre or coloration changes were to be ignored
for this test item.

Test signals were played binaurally through headphones using a subjective listening test automation
software2 [13,14]. Subjects were recommended alistening sound level, but allowed to initially adjust
the listening level according to their preference withthe recommendation to keep the sound level fixedthen throughout the tests.

“Spectral balance” was agreed to signify the lack ofcoloration at any discrete frequency within the target
frequency range. It was recognized that a coloration
change may be associated with a lengthening of the
decay at some frequency, but it was agreed that forthis item any changes in the temporal characteristics
were to be ignored.

The task of the subjects was to listen to two audio
signals A and B as many times as they wanted. The
order, sequence and method of comparing the two
signals were controlled entirely by the listener using
a window-based user interface (Fig. 13). Repetitionof items and change between items was allowed at
any time and as many times as the subjects wanted.
Subjects were to give a rating on a continuous grading scale between 0 and 10 for each signal usinggraphical sliders (Fig. 13). No verbal descriptors
were given for values on the grading scale. The subjects were instructed to use the entire grading scale if
possible. After rating of both items A and B clicking
the "Done" button proceeded the test to the next pairof signals.
The data stored for each signal pair evaluation were
the assessments with the precision of one decimal(three digit precision), the time taken to complete theevaluation (signified by clicking the “Done” button),
the test items, the number of switches between theitems, and the sequence number for the evaluation.
Subjects were asked to evaluate two qualities about
the low-frequency reproduction of a pair of test items
at a time. These were termed “temporal definition”
and “spectral balance”. Prior to the test the subjects
were given instructions, and all subjects discussed ina group with the instructor about the execution of thetest task and the meaning of the qualities to be tested.
A consensus opinion was reached of the properties
assigned to each of the test objective qualities “temporal definition” and “spectral balance”. Subjects
knew prior to the test that the target frequency range
for testing was frequencies below 200 Hz. The subjects were asked to ignore possible high frequency
differences in audio.

Fig. 13. Software user interface used in grading the
subjective listening test items. Buttons “A” and “B”
play the audio items to be graded. Grading is done bysetting the two sliders.
4.7. Results
There were two room responses to be evaluated,
termed Room S and Room G. Five audio test items
were used. The original room response and the three
processed alternatives were compared in a full permutation set. The processing alternatives for eachroom were magnitude equalization alone, modal
equalizer design AMK with magnitude equalization,and modal equalizer design ARMA with amplitude
equalization.

“Temporal definition” was agreed to signify the lackof detectable lengthening of the decay time at any
discrete frequency within the test frequency range. It
was recognized that such lengthening may also cause

Each subject performed 60 pair-wise comparisons in
the test. The median time of grading a signal pair wasapproximately 50 seconds.

2

The listening test automation software used in thetests is called GuineaPig 2.0 and has been developed
at the Helsinki University of Technology, Laboratoryof Acoustics and Audio Signal Processing [13,14].

The signal pairs (Table 9) were not randomized foreach subject, for the audio test item or for the comparison order of processing alternatives, but the subjects did not have a chance to discuss the individual

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test items until all subjects had finished the test. The
temporal property was assessed before the spectral
property. Some learning during the test about theorder in with the test item pairs were presented wasreported.

valuesSince no verbal descriptors for the grading scale values were given, the subjects used the grading scale inrather different ways. In order to enable pooling ofdata for different subjects, the grades given by eachsubject were scaled for the same mean and variance.
The Test results are presented in Figures 14-23. For
room G, “G0” refers to the sound sample convolved
with the room G response, “G1” magnitude equalized(Ampleq), “G2” to magnitude and modal equalizedwith a modal equalizer design using the AMK
method and “G3” to magnitude and modal equalized,
modal equalizer design using the ARMA method.
Room S is marked similarly. The male speech andnoise signals show lowest variance in gradings of thetemporal definition (Fig. 14-18). The noise signal
shows also the lowest variance for the gradings of thespectral balance (Fig. 19-23).

9
8
7
6
5
4
3
2
1
0
G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 15. Gradings of “temporal definition” for soundsample 2.

8
7

no

A

B

1

room

AMK

2

room

ARMA

3

room

Ampleq

4

ARMA

AMK

5

AmpleqAMK

6AmpleqARMAvaluesTable 9. Signal pairs and their presentation order. See
Table A for equalizer abbreviations.

5
4
3
2
1
G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 16. Gradings of “temporal definition” for soundsample 3.

87values6

values

6

5
4
3
2
1

9
8
7
6
5
4
3
2
1
0

G0 G1 G2 G3 S0 S1 S2 S3filter type / room

G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 14. Gradings of “temporal definition” for soundsample 1.

Fig 17. Gradings of “temporal definition” for soundsample 4.

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8
7

8

6
5

6

7

valuesvaluesAntsalo et al.

4
3
2

5
4
3
2

1

1

G0 G1 G2 G3 S0 S1 S2 S3filter type / room

G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 18. Gradings of “temporal definition” for soundsample 5.

Fig 21. Gradings of “spectral balance” for soundsample 3.

87valuesvalues

6
5
4
3
2
1

9
8
7
6
5
4
3
2
1
0
G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 19. Gradings of “spectral balance” for soundsample 1.

Fig 22. Gradings of “spectral balance” for soundsample 4.

9
8
7
6
5
4
3
2
1
0

valuesvalues

G0 G1 G2 G3 S0 S1 S2 S3filter type / room

8
7
6
5
4
3
2
1

G0 G1 G2 G3 S0 S1 S2 S3filter type / room

G0 G1 G2 G3 S0 S1 S2 S3filter type / roomFig 20. Gradings of “spectral balance” for soundsample 2.

Fig 23. Gradings of “spectral balance” for soundsample 5.

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Table 10. Value of p for temporal definition whengrades of original and equalized signals are considered.

4.8. ANOVA analysis
Matlab function p = anova1(X) [16] performs a balanced one-way analysis of variance for comparing
the means of two or more columns of data in the mby-n matrix X, where each column represents an independent sample containing m mutually independent observations. The function returns a probability
level p for the null hypothesis that all samples in thedata set X are drawn from the same population. A
small value of p suggests that at least one sample
mean in data is not from the same population as theother sample means.

Sound

Room G

Room S1

7.0e-11

4.0e-52

1.2e-13

3.1e-123

1.6e-118.1e-124

3.3e-16

5.9e-1455.6e-110

Table 11. Value of p for temporal definition whengrades of equalized signals (Ampleq, AMK and
ARMA) are considered.

ANOVA analysis was done to confirm statisticalsignificance of differences in grade value means foreach type of equalization. Tables 10 and 11 show thevalues of p when all four cases are included, including the unequalized sound. Tables 12 and 13 show
the values of p when only equalized cases (Ampleq,
AMK and ARMA) are considered.
Visual inspection of Figs. 14-23 suggests that unequalized test sounds received lower grades thanequalized test sounds. Tables 10 and 12 confirm thatthe difference in mean is significant because p values
are small. We can conclude that any one of theequalizations used produces a significant improvement. However, the incremental improvement of using also modal equalization along with magnitude
equalization was not clearly demonstrated (Tables 11
and 13).

Sound

Room G

Room S

1

0.13

0.16

2

0.98

0.48

3

0.99

0.70

4

0.72

0.48

5

0.16

0.65

Table 12. Value of p for spectral balance whengrades of original and equalized signals are considered.

The gradings for the temporal and spectral properties
are similar. This is in agreement with the fact thatroom modes typically cause an increase in the decay
time as well as an increase in the room gain at thatparticular frequency. Since the testing was nonrandomized between subjects the present data does
not provide the possibility to assess the degree oforthogonality of the temporal and spectral properties.

Sound

Room G

Room S1

0.29e-6260e-62450e-6610e-6321e-64500e-6

4

0.27e-6

4.1e-650.27e-6

7.7e-6

Table 13. Level of p for spectral balance when gradesof equalized signals (Ampleq, AMK and ARMA) areconsidered.

When data for all audio samples is pooled together,
the result does not change. Any form of equalization
provides a significant improvement in both the temporal and spectral aspects of the low frequency reproduction. Since the third-octave smoothing basedmagnitude equalization is present in all equalization
methods evaluated, although the actual magnitude
equalizers differ for each equalization method, we
can conclude that magnitude equalization provides asignificant improvement in the quality of low frequency reproduction at frequencies below 200 Hz.
The incremental improvement potentially provided
by modal equalization remains unresolved.

Sound

Room G

Room S

1

0.55

0.35

2

0.89

0.91

3

0.70

0.90

4

0.33

0.67

5

0.23

0.99

AES 114TH CONVENTION, AMSTERDAM, THE NETHERLANDS, 2003 MARCH 22-25
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Antsalo et al.

Modal Equalization

5. DISCUSSION

tion. Case studies demonstrated that the two methods
differ in their ability to model closely spaced roommodes. The equalizers designed on the basis of thereal room measurements yielded two equalizer designs with measurably differing capacity to control
modal behavior. Yet, in the subjective listening test,
we could not demonstrate a significant incremental
benefit of modal equalization over conventional
magnitude equalization.

5.1. Objective evaluation of AMK and ARMA
methods
The objective evaluation demonstrated how bothmethods of designing modal equalizers work forsparsely placed modes, but that the AMK modal
equalizer design method fails for closely spacedmodes while the ARMA design method is able todeal with these cases better. Modal equalization isable to control the rate of the initial modal decay, but
may fail to improve the decay rate at lower (later)
decay levels.

6. ACKNOWLEDGMENTS
This work is a part of the VÄRE technology program
project TAKU (Control of Closed Space Acoustics)
funded by the National Technology Agency of Finland. A part of the work was conducted in the SARA
Project funded by the Academy of Finland (project
number 201050). The authors are greateful to
D.Sc.(Tech.) Hanna Järveläinen for helpful discussions on statistical analysis.

5.2. Subjective listening test
The subjective listening test to assess the incremental
improvement of wide-band magnitude equalizationand modal equalization was unable to quantify theincremental improvement due to modal equalization.
Informal comments from subjects suggest that some
improvement was heard, but results suggest thatthird-octave magnitude equalization of low frequencies was able to produce similar level of improvement.

7. REFERENCES[1] S. J. Elliott and P. A. Nelson, “Multiple-Point
Equalization in a Room Using Adaptive Digital Filters,” J. Audio Eng. Soc., vol. 37, no. 11, pp. 899-907(1989 Nov.).
[2] R. P. Genereux, “Adaptive Loudspeaker Systems: Correcting for the Acoustic Environment,” inProc. AES 8th Int. Conf., (Washington D.C., 1990
May), pp. 245-256.
[3] J. Mourjopoulos and M. A. Paraskevas, “Poleand Zero Modelling of Room Transfer Functions,” J.
Sound and Vibration, vol. 146, no. 2, pp. 281-302(1991).
[4] J. Mourjopoulos, “Digital Equalization of
Room Acoustics,” preprint 3288, AES 92nd Conv.,(Vienna, Austria), 1992 March.
[5] S. J. Elliott, L. P. Bhatia, F. S. Deghan, A. H.
Fu, M. S. Stewart, and D. W. Wilson, “Practical Implementation of Low-Frequency Equalization Using
Adaptive Digital Filters,” J. Audio Eng. Soc., vol. 42,
no. 12, pp. 988-998 (1994 Dec.).
[6] A. G. Groh, “High-Fidelity Sound System
Equalization by Analysis of Standing Waves,” J.
Audio Eng. Soc., vol. 22, no. 10, pp. 795-799 (1974
Oct.).
[7] F. E. Toole and S. E. Olive, “The Modification of Timbre by Resonances: Perception and Measurement,” J. Audio Eng. Soc., vol. 36, no. 3, pp. 122141 (1988 March).
[8] S. E. Olive, P. L. Schuck, J. G. Ryan, S. L.
Sally, and M. E. Bonneville, “The Detection Thresholds of Resonances at Low Frequencies,” J. Audio
Eng. Soc., vol. 45, no. 3, pp. 116-127 (1997 March).

Although the subjective test lacked randomizationand the number of subjects was small, it is clear thatthe incremental benefit of modal equalization alongwith magnitude equalization is significantly smaller
than the incremental benefit of conventional magnitude response equalization to unequalized one. This
statement cannot be interpreted to mean that modal
equalization could not bring an audible improvementin situations where magnitude equalization of a listening space was to been done anyway. Such improvement may be beneficial for critical listening
environments even when the incremental improvement is small.
It must be noted that the subjective listening test of
this study is a preliminary one. Further investigations are needed, for example by using carefully
controlled synthetic room modes, in order to find thelimits to modal decay perception. On the other hand,
longer-term comparison should be done and real music signals listened to instead of isolated sounds.
5.3. Conclusion
Modal equalization is a novel technique to specifically tailor decay properties of a listening space. In
this paper we have demonstrated the efficiency of
two different methods to design modal equalizers for
low frequencies, and compared their incremental
benefit to that of conventional magnitude equaliza-

AES 114TH CONVENTION, AMSTERDAM, THE NETHERLANDS, 2003 MARCH 22-25
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Antsalo et al.

Modal Equalization[9] A. Mäkivirta, P. Antsalo, M. Karjalainen, andV. Välimäki, “Low-Frequency Modal Equalization ofLoudspeaker-Room Responses,” Preprint 5480, AES
111th Conv., (New York, USA), 2001 Nov/Dec.

APPENDIX 1
Table 14. Poles and zeros for the modal equalizer of
AMK (room G).

[10] M. Karjalainen, P. A. A. Esquef, P. Antsalo,
A. Mäkivirta, and V. Välimäki, “AR/ARMA Analysis and Modeling of Modes in Resonant and Reverberant Systems,” Preprint 5590, AES 112th Conv.,(Munich, Germany), 2002 May.
[11] M. Karjalainen, P. A. A. Esquef, P. Antsalo,
A. Mäkivirta, and V. Välimäki, “Frequency-Zooming
ARMA Modeling of Resonant and Reverberant Systems,” J. Audio Eng. Soc., vol. 50, no. 12, pp. 10121029 (2002 Dec.).
[12] M. Karjalainen, P. Antsalo, A. Mäkivirta, T.
Peltonen, and V. Välimäki, ”Estimation of Modal
Decay Parameters from Noisy Response Measurements,” J. Audio Eng. Soc., vol. 50, no. 11, pp.
867-878 (2002 Nov.).
[13] J. Hynninen and N. Zacharov, “GuineaPig a Generic Subjective Test System for Multichannel
Audio,” AES 106th Conv., Munich, (preprint 4871),
1999 May.
See also http://www.acoustics.hut.fi/
projects/GuineaPig2.
[14] J. Hynninen. ”A software-based system forlistening tests,” Master’s Thesis, Helsinki Universityof Technology, Espoo, Finland (2001). Available online at http://www.acoustics.hut.fi/publications/.
[15] “Dynamic Sound Power Test,” Blue Flame
CD 1, Gema 1989 Germany, LC 8593, no 40212.

Frequency (Hz)

Zero radius

Pole radius

47.8660.99956

0.99843

47.6030.99964

0.99843148.109

0.99902

0.99843

40.1200.99933

0.99843

154.2360.99924

0.99843

142.184

0.999210.99843

66.8230.99873

0.99843

66.6930.99880

0.99843

46.5340.99963

0.99843

147.784

0.99918

0.99843

141.8150.99934

0.99843

38.0610.99923

0.99843

66.441

0.998810.99843149.8620.999210.99843

76.8730.99860

0.99843

183.526

0.99902

0.99843

66.2920.99882

0.99843

76.8070.99862

0.99843

76.614

0.99865

0.99843

179.168

0.999120.99843[16] The Mathworks, Inc., “Matlab 6.1 Statistical
Toolbox User Manual,” Natick, Massachusetts, 2001.

AES 114TH CONVENTION, AMSTERDAM, THE NETHERLANDS, 2003 MARCH 22-25
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Antsalo et al.

Modal Equalization

Table 15. Poles and zeros for the modal equalizer of
AMK (room S).

Table 17. Poles and zeros for the modal equalizer of
ARMA (room S).

Frequency (Hz)

Zero radius

Pole radiusFrequency/Hz

Zero radius

Pole radius

48.9800.999390.9984338.9180.99930

0.99826

40.4130.99950

0.9984345.7480.99938

0.9982644.6500.999210.9984345.9920.99885

0.99826

40.3860.99954

0.99843

49.4730.999440.9982678.9300.99955

0.99843

68.4450.99895

0.99881

48.9220.99886

0.99843

68.4620.998990.9988170.2600.999210.99843

68.4840.99904

0.9988148.8570.999120.99843

68.5130.99911

0.9988140.1350.99959

0.99843

68.5550.999210.99881119.194

0.999220.99843

68.7680.99949

0.9988170.1540.99938

0.99843

78.8300.99967

0.99879

195.960

0.99936

0.99843

117.0560.99948

0.99873

166.308

0.99935

0.99843

122.5100.99951

0.99872168.9470.99930

0.99843133.8700.99943

0.99869

178.5390.99935

0.99843

187.518

0.99938

0.9984344.5890.99916

0.99843

188.8140.99937

0.99843

195.1960.99932

0.99843120.688

0.99926

0.99843

Table 16. Poles and zeros for the modal equalizer of
ARMA (room G).
Frequency/Hz

Zero radius

Pole radius

38.0400.999310.9982645.9010.99962

0.99826

92.1270.99965

0.99877139.6800.999410.99868

145.465

0.99887

0.99867

180.368

0.999220.99860

190.894

0.99928

0.99857

AES 114TH CONVENTION, AMSTERDAM, THE NETHERLANDS, 2003 MARCH 22-25
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