Low-Frequency Modal Equalization Of Loudspeaker-Room Responses
Aki Mäkivirta, Poju Antsalo1, Matti Karjalainen1 and Vesa Välimäki1
Genelec Oy,
Olvitie 5, FIN-74100 Iisalmi, Finland
aki.makivirta@genelec.com
1
Helsinki University of Technology,
Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland
ABSTRACTIn a room with strong low-frequency modes the control of excessively long decays is problematic or impossible
with conventional passive means. In this paper we present a systematic methodology for active modal equalization able to correct the modal decay behavior of a loudspeaker-room system. Two methods of modal equalization
are proposed. The first method modifies the primary sound such that modal decays are controlled. The second
method uses separate primary and secondary radiators and controls modal decays with sound fed into the secondary radiator. Case studies of the first method of implementation are presented.
INTRODUCTION
A loudspeaker installed in a room acts as a coupled system where
the room properties typically dominate the rate of energy decay. At
high frequencies, typically above a few hundred Hertz, passive
methods of controlling the rate and properties of this energy decay
are straightforward and well established. Individual strong reflections are broken up by diffusing elements in the room or trapped inabsorbers. The resulting energy decay is controlled to a desired levelby introducing the necessary amount of absorbance in the acousticalspace. This is generally feasible as long as the wavelength of soundis small compared to dimensions of the space.
As we move toward low frequencies passive means of controlling
reverberant decay time become more difficult because the physicalsize of necessary absorbers increases and may become prohibitively
large compared to the volume of the space, or absorbers have to bemade narrow-band. Related to this, the cost of passive control ofreverberant decay greatly increases at low frequencies. Methods for
optimizing the response at a listening position by finding suitablelocations for loudspeakers have been proposed [1] but cannot fully
solve the problem. Because of these reasons there has been an increasing interest in active methods of sound field control at low
frequencies, where active control becomes feasible as the wavelengths become long and the sound field develops less diffuse [2-6].
Modal resonances in a room can be audible because they modify themagnitude response of the primary sound or, when the primarysound ends, because they are no longer masked by the primarysound [7,8]. Detection of a modal resonance appears to be veryMÄKIVIRTA ET AL.

MODAL EQUALIZATION

dependent on the signal content. Olive et al. report that 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].
Olive et al. report detection thresholds for resonances both for continuous broadband sound and transient discontinuous sound. At low
Q values antiresonances (notches) are as audible as resonances. As
the Q value becomes high audibility of antiresonances reduces dramatically for wideband continuous signals [8]. Detectability ofresonances reduces approximately 3dB for each doubling of the Q
value [7,8] and low Q resonances are more readily heard with zeroor minimal time delay relative to the direct sound [7]. Duration of
the reverberant 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.
In this paper, we present methods to actively control low-frequency
reverberation. We will first present the concept and two basic typesof modal equalization. A target for modal decay time versus frequency will be discussed based on existing recommendations forhigh quality audio monitoring rooms. Methods to identify andparametrize modes in an impulse response are introduced. Modal
equalizer design for an individual mode is discussed with examples.
Several case studies of both synthetic modes and modes of realrooms are presented. Finally, synthesis of IIR modal equalizer filters
is discussed.

Type I Modal Equalization
Type I implementation (Fig. 1) modifies the audio signal fed into
the primary loudspeaker to compensate for room modes. The totaltransfer function from the primary radiator to the listening position
represented in z-domain is

H ( z) = G ( z ) H m ( z)(3)where G(z) is the transfer function of the primary radiator from theelectrical input to acoustical output and Hm(z) = B(z)/A(z) is thetransfer function of the path from the primary radiator to the listening position. The primary radiator has essentially flat magnitude
response and small delay in our frequency band of interest, and cantherefore be neglected in the following discussion,G( z) = 1

(4)

G
Hm

Hc
primary

listening
position

THE CONCEPT OF MODAL EQUALIZATIONroom

The present work is restricted to frequencies below 200Hz andenvironments where sound wavelength relative to dimensions of aroom is not very small. We are not interested in global control in aroom, but in introducing a change at the primary listening position.

Fig. 1. Type I modal equalization using the primary sound source.

These limitations lead into a problem formulation where the modal
behavior of the listening space can be modeled by a distinct numberof modes such that they can be individually controlled. Each mode
is modeled by an exponential decay function

We now design a pole-zero filter Hc(z) having zero pairs at theidentified pole locations of the modal resonances in Hm(z). Thiscancels out existing room response pole pairs in A(z) replacing themwith new pole pairs A´(z) producing the desired decay time in themodified transfer function H'm(z)hm (t ) = Am e −τ m t sin(ω m t + φ m )(1)Here Am is the initial envelope amplitude of the decaying sinusoid,τm is a coefficient that denotes the decay rate, ωm is the angularfrequency of the mode, and φm is the initial phase of the oscillation.
We define modal equalization as a process that can modify the rateof a modal decay. The concept of modal decay can be viewed as acase of parametric equalization, operating individually on selectedmodes in a room. A modal resonance is represented in the z-domaintransfer function as a pole pair with pole radius r and pole angle θH m ( z) =1
(1 − re jθ z −1 )(1 − re − jθ z −1 )(2)

The closer a pole pair is to the unit circle the longer is the decay
time of a mode. To shorten the decay time the Q-value of a resonance needs to be decreased by moving poles toward the origin. We
refer to this process of moving pole locations as modal equalization.
Modal decay time modification can be implemented in several ways
– either the sound going into a room through the primary radiator ismodified or additional sound is introduced in the room with one ormore secondary radiators to interact with the primary sound. The
first method has the advantage that the transfer function from asound source to a listening position does not affect modal equalization. In the second case differing locations of primary and secondary
radiators lead to different transfer functions to the listening position,and this must be considered when calculating a corrective filter. We
will now discuss these two cases in more detail drawing some conclusions on necessary conditions for control in both cases.

' ( z) = H ( z) H ( z) =HmcmA( z ) B ( z ) B ( z )=A' ( z ) A( z ) A' ( z )(5)This leads to a correcting filter

H c ( z) =A( z )(6)

A' ( z )The new pole pair A'(z) is chosen on the same resonant frequency
but closer to the origin, thereby effecting a resonance with a decreased Q value. In this way the modal resonance poles have been
moved toward the origin, and the Q value of the mode has been
decreased. The sensitivity of this approach will be discussed laterwith example designs.
Type II Modal Equalization
Type II method uses a secondary loudspeaker at appropriate position in the room to radiate sound that interacts with the sound field
produced by the primary speakers. Both speakers are assumed to be
similar in the following treatment, but this is not required for practical implementations. The transfer function for the primary radiator
is Hm(z) and for the secondary radiator H1(z). The acoustical summation in the room produces a modified frequency response H'm(z)with the desired decay characteristics

H 'm ( z ) =AES 111TH CONVENTION, NEW YORK, NY, USA, 2001 SEPTEMBER 21–24

B( z )A' ( z )= H m ( z ) + H c H1 ( z )(7)2MÄKIVIRTA ET AL.

MODAL EQUALIZATIONTARGET OF MODAL EQUALIZATION
primary

The in-situ impulse response at the primary listening position ismeasured using any standard technique. The process of modal
equalization starts with the estimation of octave band reverberation
times between 31.5 Hz – 4 kHz. The mean reverberation time at mid
frequencies (500Hz – 2kHz) and the rise in reverberation time isused as the basis for determining the target for maximum lowfrequency reverberation time.

Hm

Hc
H1

listening
position

The target allows the reverberation time to increase at low frequencies. Current recommendations [9–11] give a requirement for average reverberation time Tm in seconds for mid frequencies (200Hz to4kHz) that depends on the volume V of the room

secondaryroom

1Fig. 2. Type II modal equalization using a secondary radiator.
This leads to a correcting filter Hc(z) where Hm(z) and H'm(z) differ
by modified pole radii

H 'm ( z ) − H m ( z )H1 ( z )A ( z ) B( z ) A( z ) − A' ( z )= 1
B1 ( z ) A( z )A' ( z )(8)andB ( z)
H1 ( z ) = 1
A1 ( z )(9)Note that if the primary and secondary radiators are the same
source, Equation 8 reduces into a parallel formulation of a cascaded
correction filter equivalent to the Type I method presented above

H 'm ( z ) = H m ( z ) ( 1 + H c ( z ) )≠ 0 , forHm( f )

where the reference room volume Vo of 100m3 yields a reverberation time of 0.25s. Below 200Hz the reverberation time may linearly
increase by 0.3s as the frequency decreases to 63Hz. Also a maximum relative increase of 25% between adjacent 1/3-octave bands as
the frequency decreases has been suggested [10,11]. Below 63Hz
there is no requirement. This is motivated by the goal to achieve
natural sounding environment for monitoring [11]. An increase inreverberation time at low frequencies is typical particularly in rooms
where passive control of reverberation time by absorption is compromised, and these rooms are likely to have isolated modes withlong decay times.
We define the target decay time relative to the mean T60 in midfrequencies (500Hz – 2kHz), increasing (on a log frequency scale)linearly by 0.2s as the frequency decreases from 300Hz down to
50Hz.
Measured(10)

A necessary but not sufficient condition for a solution to exist is thatthe secondary radiator can produce sound level at the listening location in frequencies where the primary radiator can, within the frequency band of interest

H1 ( f )

(13)

≠0

(11)

At low frequencies where the size of a radiator becomes small relative to the wavelength it is possible for a radiator to be located suchthat there is a frequency where the radiator does not couple well into
the room. At such frequencies the condition of Equation 11 may not
be fulfilled, and a secondary radiator placed in such location will
not be able to affect modal equalization at that frequency. Becauseof this it may be advantageous to have multiple secondary radiatorsin the room. In the case of multiple secondary radiators, Equation 7
is modified into form

H 'm ( z ) = H m ( z ) + ∑ H c, n ( z ) H1, n ( z )

(12)

N

where N is the number of secondary radiators.
After the decay times of individual modes have been equalized inthis way, the magnitude response of the resulting system may becorrected to achieve flat overall response. This correction can be
implemented with any of the magnitude response equalization
methods.
In this paper we will discuss identification and parametrization ofmodes and review some case examples of applying the proposed
modal equalization to various synthetic and real rooms, mainly
using the first modal equalization method proposed above. The useof one or more secondary radiators will be left to future study.

Reverberation time

H c ( z) =

 V 3
Tm = 0.25  Vo 

TargetFrequency
Fig. 3. Reverberation time target and measured octave band reverberation time.
MODE IDENTIFICATION AND PARAMETER ESTIMATION
After setting the reverberation time target, transfer function of theroom to the listening position is estimated using Fourier transform
techniques. Potential modes are identified in the frequency responseby assuming that modes produce an increase in gain at the modal
resonance. The frequencies within the chosen frequency range (f <200Hz) where level exceeds the average mid-frequencies level(500Hz to 2kHz) are considered as potential mode frequencies.
The short-term Fourier transform presentation of the transfer function is employed in estimating modal parameters from frequency
response data. The decay rate for each detected potential room modeis calculated using nonlinear fitting of an exponential decay + noisemodel into the time series data formed by a particular short-term
Fourier transform frequency bin. A modal decay is modeled by an
exponentially decaying sinusoid (Equation 1 reproduced here forconvenience)

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3MÄKIVIRTA ET AL.

MODAL EQUALIZATIONhm (t ) = Am e − τ m t sin(ω m t + φ m )(14)

where Am is the initial envelope amplitude of the decaying sinusoid,τm is a coefficient defining the decay rate, ωm is the angular frequency of the mode, and φm is the initial phase of modal oscillation.
We assume that this decay is in practical measurements corrupted
by an amount of noise nb(t)nb (t ) = An n(t )2A m e − 2τ t + An

2

(16)

The optimal values A n, τm and Am are found by least-squares fitting
this model to the measured time series of values obtained with ashort-term Fourier transform measurement. The method of nonlinear modeling is detailed in [12]. Sufficient dynamic range ofmeasurement is required to allow reliable detection of room modeparameters although the least-squares fitting method has beenshown to be rather resilient to high noise levels. Noise level estimates with the least-squares fitting method across the frequency
range provide a measurement of frequency-dependent noise levelA(f) and this information is later used to check data validity.
Modal Parameters
The estimated decay parameters τm(f) across the frequency range areused in identifying modes exceeding the target criterion and in
calculating modal equalizing filters. It can be shown that the spectral peak of a Gaussian-windowed stationary sinusoid calculatedusing Fourier transform has the form of a parabolic function [13].
Therefore the precise center frequency of a mode is calculated byfitting a second-order parabolic function into three Fourier transform bin values around the local maximum indicated by decay
parameters τm(f) in the short-term Fourier transform dataG( f ) = a f 2 + b f + c(17)

The frequency where the second-order function derivative assumes
value zero is taken as the center frequency of the mode

∂G ( f )=0∂fConsider now a second-order all-pole transfer function having poleradius r and pole angle θH ( z) =

⇒

f =−

b
2a

(18)

In this way it is possible to determine modal frequencies more precisely than the frequency bin spacing of the Fourier transform presentation would allow.

1(1 − re=(15)

and that this noise is uncorrelated with the decay. Statistically the
decay envelope of this system isa (t ) =Discrete-Time Representation of a Mode

jθ −1

z

)(1 − re − jθ z −1 )1

1 − 2r cos θ z −1 + r 2 z − 2(20)Taking the inverse z-transform yields the impulse response of thissystem ash( n ) =r n sin(θ (n + 1) )u ( n)sin θ(21)

where u(n) is a unit step function.
The envelope of this sequence is determined by the term rn. To
obtain a matching decay rate to achieve T60 we require that the
decay of 60dB is accomplished in N60 steps given a sample rate fs,

20 log(r N 60 ) = −60,

N 60 = T60 f s(22)

We can now solve for the pole radius r

r = 10−3
T60 f s(23)

Using the same approach we can also determine the desired polelocation, by selecting the same frequency but a modified decay time
T60 and hence a new radius for the pole.
Some error checking of the identified modes is necessary in order todiscard obvious measurement artifacts. A potential mode is rejected
if the estimated noise level at that modal frequency is too high,
implying insufficient signal-to-noise ratio for reliable measurement.
Also, candidate modes that show unrealistically slow decay or no
decay at all are rejected because they usually represent technicalproblems in the measurement such as mains hum, ventilation noiseor other unrelated stationary error signals, and not true modal resonances.
MODAL EQUALIZER DESIGN
For sake of simplicity the design of Type I modal equalizer is presented here. This is the case where a single radiator is reproducing
both the primary sound and necessary compensation for the modal
behavior of a room. Another way of viewing this would be to saythat the primary sound is modified such that target modes decay
faster.

Estimation of modal pole radius can be based on two parameters,
the Q-value of the steady-state resonance or the actual measurementof the decay time T60. While the Q-value can be estimated for isolated modes it may be difficult or impossible to define a Q-value formodes closely spaced in frequency. On the other hand the decay
time is the parameter we try to control. Because of these reasons weare using the decay time to estimate the pole location.

A pole pair z = F(r,θ ) models a resonance in the z-domain based on
measured short-term Fourier transform data while the desired resonance Q-value is produced by a modified pole pair zc = F(rc ,θc ).
The correction filter for an individual mode presented in Equation 5
becomes

The 60-dB decay time T60 of a mode is related to the decay time
constant τ by

H c ( z) =1
6.908
T60 = − ln(10 −3 ) ≈
τ
τ

(19)

The modal parameter estimation method employed in this work [12]provides us an estimate of the time constant τ. This enables us tocalculate T60 to obtain a representation of the decay time in a form
more readily related to the concept of reverberation time.

=A ( z)A' ( z )(1 − re jθ z −1 )(1 − re − jθ z −1 )(1 − rce jθ c z −1 )(1 − rc e − jθ c z −1 )(24)

To give an example of the correction filter function, consider asystem defined by a pole pair (at radius r = 0.95, angular frequency
ω = ±0.18π) and a zero pair (at r = 1.9, ω = ±0.09π). We want toshift the location of the poles to radius r = 0.8. To effect this we use
the Type I filter of Equation 24 with the given pole locations, hav-

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4MÄKIVIRTA ET AL.

MODAL EQUALIZATION

ing a notch-type magnitude response (Fig. 4). This is because numerator gain of the correction filter is larger than denominator gain.
As a result, poles at radius r = 0.95 have been cancelled and newpoles have been created at the desired radius (Fig. 5). Impulse responses of the two systems (Fig. 6) verify the reduction in modal
resonance Q value. The decay envelope of the impulse response(Fig. 7) now shows a rapid initial decay.
The quality of a modal pole location estimate determines the success of modal equalization. The estimated center frequency determines the pole angle while the decay rate determines the pole distance from the origin. Error in these estimates will displace thecompensating zero and reduce the accuracy of control. For example,
an estimation error of 5% in the modal pole radius (Fig. 7) or poleangle (Fig. 8) greatly reduces control, demonstrating that precise
estimation of correct pole locations is paramount to success of modal equalization.

Fig. 6. Impulse responses of original and mode-equalized system.

Fig. 7. Original and corrected Hilbert decay envelope with exact anderroneous mode pole radius.
Fig. 4. Effect of mode pole location relocation on the example
system and the magnitude response of modal equalizer filter.

Fig. 8. Original and corrected Hilbert decay envelope with exact anderroneous mode pole angle.

CASE STUDIES

Fig. 5. Poles (mark x) and zeros (mark o) of the mode-equalized
system.

Case studies in this section demonstrate the modal equalization
process. These cases contain artificially added modes and responsesof real rooms equalized with the proposed method.
The waterfall plots in Figs. 9-15 are computed using a sliding rectangular time window of length 1 second. The purpose is to maximize spectral resolution. The problem of using a long time window
is the lack of temporal resolution. Particularly, the long time window causes an amount of temporal integration, and noise in impulse
response measurements affects level estimates. This effectively
produces a cumulative decay spectrum estimate [15], also resembling Schroeder backward integration [16].
Cases I and II use an impulse response of a two-way loudspeakermeasured in an anechoic room. The waterfall plot of the anechoic
impulse response of the loudspeaker (Fig. 9) reveals short reverberant decay at low frequencies where the absorption is no longer sufficient to fulfill free field conditions. Dynamic range of the waterfall
plots of cases I and II is 60dB, allowing direct inspection of the

AES 111TH CONVENTION, NEW YORK, NY, USA, 2001 SEPTEMBER 21–24

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MODAL EQUALIZATION

decay time. Case III is based on impulse response measured in a realroom.
Cases with Artificial Modes
Case 1 attempts to demonstrate the effect of the developed mode
equalizer calculation algorithm. It is based on the free field responseof a compact two-way loudspeaker measured in an anechoic room.
An artificial mode with T60 = 1 second has been added to the data at
f = 100Hz and an equalizer has been designed to shorten the T60 to0.26 seconds. The room mode increases the level at the resonant
frequency considerably (about 30dB) and the long decay rate isevident (Fig. 10). After equalization the level is still higher (about
15dB) than the base line level but the decay now starts at a lower
level and has shortened to the desired level of 0.26 s (Fig. 11).
Case II uses the same anechoic two-way loudspeaker measurement.
In this case five artificial modes with slightly differing decay times
have been added. See Table 1 for original and target decay times
and center frequencies of added modes. For real room responses, thetarget decay time is determined by mean T60 in mid-frequencies,
increasing linearly (on linear frequency scale) by 0.2s as the frequency decreases from 300Hz down to 50Hz. For the synthetic CaseII the target decay time was arbitrarily chosen as 0.2 seconds. Again
we note that the magnitude gain of modal resonances (Fig. 11) isdecreased by modal equalization (Fig. 12). The target decay times
have been achieved except for the two lowest frequency modes(50Hz and 55Hz). There is an initial fast decay, followed by a slow
low-level decay. This is because the center frequencies and decay
rates were not precisely identified, and the errors cause the controlof the modal behavior to deteriorate.

Fig. 9. Anechoic waterfall plot of a two-way loudspeaker response
used in case examples I and II.

Table 1. Case II artificial modes center frequency f, decay time T60,
and target decay time T’60.
mode nof [Hz]T60 [s]T’60 [s]

1
2
3
4
5

50
55
100
130
180

1.4
0.8
1.0
0.8
0.7

0.30
0.30
0.26
0.24
0.20

Fig. 10. Case I, free field response of a compact two-way loudspeaker with an added artificial room mode at f = 100Hz.

Cases with Real Room Responses
Case III is a real room response. It is a measurement in a hardwalled approximately rectangular meeting room with about 50m2
floor area. The target decay time specification is the same as in CaseII.
In Case III the mean T60 in mid frequencies is 0.75s. 20 modes wereidentified with decay time longer than the target decay time. The
mode frequency fm, estimated decay time T60 and target decay time
T'60 are given in Table 2.
The waterfall plot of the original impulse response (Fig. 14) and themodally equalized impulse response (Fig. 15) show some reductionof modal decay time. A modal decay at 78Hz has reduced significantly from the original 2.12s. The fairly constant-level signals
around 50Hz are noise components in the measurement file. Also
the decay rate at high mode frequencies is only modestly decreased
because of imprecision in estimating modal parameters. On theother hand, the decay time target criterion relaxes toward low frequencies, demanding less change in the decay time.

Fig. 11. Case I, mode-equalized artificial room mode at f = 100Hz.

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6MÄKIVIRTA ET AL.

Fig. 12. Case II, five artificial modes added to an impulse responseof a compact two-way loudspeaker anechoic response.

MODAL EQUALIZATION

Fig. 15. Case III, mode-equalized room 1 measurement.
Table 2. Case III, equalized mode frequency fm, original and target
decay rate T60.

Fig. 13. Case II, mode-equalized five-mode case.

fm
[Hz]

T60
[s]

T'60
[s]

44
60
64
66
72
78
82
106
109
116
120
123
128
132
142
155
161
165
171
187

2.35
1.38
1.57
1.66
1.51
2.12
1.32
1.31
1.40
1.57
1.32
1.15
1.06
1.17
0.96
1.06
1.08
1.24
0.88
0.89

0.95
0.94
0.94
0.94
0.93
0.93
0.92
0.90
0.90
0.90
0.89
0.89
0.89
0.88
0.88
0.87
0.86
0.86
0.85
0.84

IMPLEMENTATION OF MODAL EQUALIZERS
Type I Filter Implementation
To correct N modes with a Type I modal equalizer, we need anorder-2N IIR transfer function. The most immediate method is to
optimize a second-order filter, defined by Equation 24, for eachmode identified. The final order-2N filter is then formed as a cascade of these second-order subfilters
Fig. 14. Case III, real room 1, original measurement.

H c ( z ) = H c,1 ( z ) ⋅ H c,2 ( z ) ⋅  ⋅ H c, N ( z )(26)Another formulation allowing design for individual modes is servedby the formulation in Equation 10. This leads naturally into a parallel structure where the total filter is implemented as

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H c ( z ) = 1 + ∑ H c, k ( z )MODAL EQUALIZATION

N

(27)

N

Asymmetry in Type I Equalizers
At low angular frequencies the maximum gain of a resonant system
may no longer coincide with the pole angle [14]. Similar effects also
happen with modal equalizers, and must be compensated for in thedesign of an equalizer.
Basic Type I modal equalizer (see Equation 24) becomes increasingly unsymmetrical as angular frequency approaches ω = 0. A caseexample in Figure 16 shows a standard design with pole and zero atωp,z = 0.01 rad/s, zero radius rz = 0.999 and pole radius rp = 0.995.
There is a significant gain change for frequencies below the resonant frequency. This asymmetry may cause a problematic cumulative change in gain when a modal equalizer is constructed along theprinciples in Equations 26 and 27.
It is possible to avoid asymmetry by decreasing the sampling frequency in order to bring the modal resonances higher on the discrete
frequency scale.
If sample rate alteration is not possible, we can symmetrize a modal
equalizer by moving the pole slightly downwards in frequency (Fig.
16). Doing so, the resulting modal frequency will shift slightly
because of modified pole frequency, and the maximal attenuation of
the system may also change. These effects have to be accounted forin symmetrizing a modal equalizer at low frequencies. This can behandled by an iterative fitting procedure with a target to achieve
desired modal decay time simultaneously with a symmetrical response.

Fig. 16. A modified Type I modal equalizer with symmetrical gain
having zero radius r = 0.999 at ω = 0.01 rad/s and pole radius r =0.995 at ω = 0.0087 rad/s (solid), and a standard Type I modal
equalizer having both a pole and zero at ω = 0.01 rad/s (dash-dot).
Type II Filter Implementation
Type II modal equalizer requires a solution of Equation 8 for eachsecondary radiator. The correcting filter Hc(z) can be implemented
by direct application of Equation 8 as a difference of two transfer
functions convolved by the inverse of the secondary radiator transfer function, bearing in mind the requirement of Equation 11. A
more optimized implementation can be found by calculating thecorrecting filter transfer function Hc(z) based on measurements, andthen fitting an FIR or IIR filter to approximate this transfer function.
This filter can then be used as the correcting filter. Any filter design
technique can be used to design this filter.
In the case of multiple secondary radiators the solution becomesslightly more convoluted as the contribution of all secondary radiators must be considered. For example, solution of Equation 12 forthe correction filter of the first secondary radiator isH c,1( z ) =H ' m ( z ) − H m ( z ) − ∑ H c, n ( z ) H1, n ( z )n=2

H1,1 ( z )(28)

It is evident that all secondary radiators interact to form the correction. Therefore the design process of these secondary filters becomes a multidimensional optimization task where all correction
filters must be optimized together. A suboptimal solution is to optimize for one secondary source at a time, such that the subsequent
secondary sources will only handle those frequencies not controllable by the previous secondary sources for instance because of poor
radiator location in the room.
DISCUSSION
Traditional magnitude equalization attempts to achieve a flat frequency response at the listening location either for the steady stateor early arriving sound. Both approaches achieve an improvement inaudio quality for poor loudspeaker-room systems, but colorations of
the reverberant sound field cannot be handled with traditional magnitude equalization. Colorations in the reverberant sound field produced by room modes deteriorate sound clarity and definition. Modal equalization is a novel approach that can specifically address
problematic modal resonances, decreasing their Q-value and bringing the decay rate in line with other frequencies.
Modal equalization decreases the gain of modal resonances thereby
affecting an amount of magnitude equalization. It is important tonote that traditional magnitude equalization does not achieve modal
equalization as a byproduct. There is no guarantee that zeros in a
traditional equalizer transfer function are placed correctly to achieve
control of modal resonance decay time. In fact, this is rather improbable. A sensible aim for modal equalization is not to achieve
either zero decay time or flat magnitude response. Modal equalization can be a good companion of traditional magnitude equalization.
A modal equalizer can take care of differences in the reverberation
time while a traditional equalizer can then decrease frequency response deviations to achieve acceptable flatness of response.
We have presented two different types of modal equalization approaches, Type I modifying the sound input into the room using the
primary speakers, and Type II using separate speakers to input themode compensating sound into a room. Type I systems are typicallyminimum phase. Type II systems, because the secondary radiator isseparate from the primary radiator, may have an excess phase component because of differing times-of-flight. As long as this is compensated in the modal equalizer for the listening location, Type II
systems also conform closely to the minimum phase requirement.
There are several reasons why modal equalization is particularlyinteresting at low frequencies. At low frequencies passive means tocontrol decay rate by room absorption may become prohibitively
expensive or fail because of constructional faults. Also, modal
equalization becomes technically feasible at low frequencies where
the wavelength of sound becomes large relative to room size and to
objects in the room, and the sound field is no longer diffuse. Local
control of the sound field at the main listening position becomes
progressively easier under these conditions.
Recommendations [9-11] suggest that it is desirable to have approximately equal reverberant decay rate over the audio range of
frequencies with possibly a modest increase toward low frequencies.
We have used this as the starting point to define a target for modal
equalization, allowing the reverberation time to increase by 0.2s as
the frequency decreases from 300Hz to 50Hz. This target may serve
as a starting point, but further study is needed to determine a psychoacoustically proven decay rate target.

AES 111TH CONVENTION, NEW YORK, NY, USA, 2001 SEPTEMBER 21–24

8MÄKIVIRTA ET AL.

SUMMARY AND CONCLUSIONSIn this paper, we introduced the principle of modal equalization,with formulations for Type I and Type II correction filters. Type I
system implements modal equalization by a filter in series with themain sound source, i.e. by modifying the sound input into the room.
Type II system does not modify the primary sound, but implements
modal equalization by one or more secondary sources in the room,
requiring a correction filter for each secondary source. Methods for
identifying and modeling modes in an impulse response measurement were presented and precision requirements for modeling andimplementation of system transfer function poles were discussed.
Several examples of mode equalizers were given of both simulated
and real rooms. Finally, implementations of the mode equalizer
filter for both Type I and Type II systems were described.
Modal equalization is a method to control reverberation in a room
when conventional passive means are not possible, do not exist or
would present a prohibitively high cost. Modal equalization is aninteresting design option particularly for low-frequency room reverberation control.
ACKNOWLEDGEMENTS
This study is part of the VÄRE technology program, project TAKU
(Control of Closed Space Acoustics), funded by Tekes (National
Technology Agency). The work of Vesa Välimäki has been financed by the Academy of Finland.
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