Modal Equalization by Temporal Shaping ofRoom Response
Matti Karjalainen1 , Poju Antsalo1 , and Aki Mäkivirta2
1

Helsinki Univ. of Tech., Lab. of Acoustics and Audio Signal Processing, P.O. Box 3000, FIN-02015 HUT, Finland

2

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

Correspondence should be addressed to Matti Karjalainen (matti.karjalainen@hut.fi)

ABSTRACTThe low-frequency behavior of sound reproduction in listening rooms is often problematic due to long-ringing
modes that are diﬃcult and expensive to control by acoustic means. Modal equalization has been proposed
recently to correct the low-frequency problems by shortening the decay times of problematic modes throughmodiﬁcation of transfer function poles. While the previous methods were based on the estimation of isolatedmodes and their parameters, the new method proposed here is a technique to change the time-domain
response more directly. It is an advanced windowing technique where the temporal shaping of a given
impulse response can be done in a frequency-dependent manner. The method is compared with previous
modal equalization techniques.

1 INTRODUCTION
In a listening room it is desirable to have a balanced reverberation time behavior at all frequencies.
Low-frequency modes, however, are often not wellcontrolled, showing too slow decay times compared
to mid- and high-frequency reverberation time, although some increase toward lowest frequencies is allowed [1, 2]. Designing and constructing a carefully
damped listening room is acoustically challenging,
expensive, and demands much space. Particularly
frequencies around and below 100 Hz are problematic in this sense.
Decay time correction by signal processing in theelectronic reproduction channel is an attractive alternative to acoustical improvement of a listening
space, particularly at low frequencies where thewavelength is large enough. Traditional magnitude
response equalization, however, cannot control thereverberation time nor the decay time of modes indetail. Also, DSP equalization by the inverse ﬁlter

derived from the measured response, while possible
at low frequencies, may shorten decay times too aggressively and may also lead to system overload byboosting notched frequency areas.
We have recently proposed modal equalization of
low-frequency behavior in listening room conditions[3, 4]. Contrary to conventional equalization, where
the aim is to ﬂatten the magnitude response, modal
equalization attempts to balance the decay time of
low-frequency modes to correspond to the reverberation time at mid-frequencies.
The previously proposed methods of modal equalization are realized as follows. First, the system
response at a receiving point or area is measured.
Then the modal decay time has to be estimated as afunction of frequency. The maximum allowed decay
time is determined according to the reverberation
time at mid frequencies, allowing a slight increase
towards the lowest frequencies. A problematic modeis equalized by designing a pole-zero correction ﬁlter

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KARJALAINEN ET AL.
whenever the measured decay time exceeds the target for any low-frequency mode (typically below 200
Hz). This means using a zero pair to cancel out apole pair associated with a physical mode that is too
close to the unit circle, and to insert a new pole pairwith the desired faster decay characteristics. This isiterated for problematic modes until the decay time
is below the allowed limit value at all frequencies, oruntil a predeﬁned maximum number of modes has
been corrected.
Two methods were proposed for the critical taskof modal parameter estimation. The ﬁrst one [3]is based on time-frequency analysis (such as shorttime Fourier analysis) to track the envelope trajectory of each prominent mode. The second method[4] is based on parametric estimation through AR
or ARMA modeling whereby the overall responseof a selected frequency band is modeled as an allpole or pole-zero ﬁlter. Both of these methods
work well when the problematic modes to be equalized are clearly isolated in frequency. The AR and
ARMA techniques are more powerful than the timefrequency analysis when the modal density increases,
but they also show problems with a high modal
density. The new technique proposed in this paper
shows more promise in such cases.
The new method is based on frequency-dependent
windowing. The measured impulse response of the
reproduction channel is ﬁrst analyzed to obtain an
estimate of the decay time as a function of frequency.
The allowed upper limit is determined as describedabove. Next the impulse response is windowed by
an exponentially decaying window at those frequencies where the decay time has to be shortened, andat the rest of frequencies the response remains in
its original decay shape. The frequency-dependent
windowing can be realized in diﬀerent ways, as described in [5]. The equalization ﬁlter is obtained bydeconvolution of the desired and the measured response. It is inherently of the FIR type and of relatively high order, but in a multirate implementation
it is computationally eﬃcient.
The relevance of modal equalization at low frequencies is dependent on the perceptual eﬀects of modal
behavior. Dips and peaks in the magnitude response properties of sound reproduction have been
discussed for example in [6, 7]. The eﬀects of tem-

TEMPORAL SHAPING . . .

Fig. 1: Basic setup for modal equalization.
poral response properties have not been studied until recently. In an investigation on the incremental
improvement of using modal equalization in addition to conventional magnitude equalization it was
shown that a detailed modal equalization is only of
secondary importance and often of no noticeable difference [8]. However, careful control of modal behavior guarantees that equalization is well done andmay contribute to perceived sound quality in criticalconditions.
This paper is organized as follows. Section 2 is a
short description of the modal equalization principle
and its functioning. Section 3 introduces the theoryof the new method, the frequency-dependent windowing of impulse responses. In Section 4 we show
how the new method works for a real room response,and ﬁnally in Section 5 we include a summary andconclusions.
2 MODAL EQUALIZATION
Figure 1 depicts the basic setup for equalizing loudspeaker reproduction in a room, in the present casefor modal equalization. Another setup is to use asecondary loudspeaker channel to produce the equalizing sound to the listening space [3], but here we will
discuss only the case of Fig. 1 with primary loudspeaker equalization.
The total transfer function from the signal input tothe listening position in Fig. 1, represented in thez-domain, is
Ht (z) = Hc (z) G(z) Hm (z)(1)

where Hc(z) is the equalizing ﬁlter, G(z) is the transfer function of the sound radiator from electric input
to acoustic output, and Hm (z) is the transfer func-

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TEMPORAL SHAPING . . .

tion of the path from the sound radiator to the listening position. When a desired target response Ht(z)is speciﬁed, the response for the equalizer Hc (z) isachieved by
Ht (z)
Hc(z) =
(2)
G(z) Hm (z)Now modal equalization is deﬁned as a process thatmodiﬁes the rate of modal decay in a room. Roommodes are created by standing wave resonances. If
there is only little damping, which is typical at low
frequencies in a hard-walled room, the energy of themodal resonance decays slowly. The idea of modal
equalization is to make the decay faster1 by modifying the overall transfer function of the sound reproduction channel. This means that the (passive)
acoustics of the room is not changed, but the reproduced sound behaves in a more controlled way.
A simpliﬁed explanation on how the decay is madefaster is that after the onset of a modal vibration
the loudspeaker starts to radiate sound properly in
opposite phase, thus actively reducing modal soundenergy in the room.
To understand the modal equalization from a systemic point of view, the behavior of a single modecan be considered. Each mode can be understood asa time function of exponential decayhm (t) = Am e−τm t sin(ωm t + φm )(3)

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 themode, and φm is the initial phase of oscillation. The
modal resonance is represented in the z-domain by atransfer function having a pole pair with pole radiusr and pole angle θHm(z) =

1
(1 − rejθ z −1 )(1 − re−jθ z −1 )(4)

To shorten the decay time, i.e., to increase the decayrate and to decrease the Q-value of the resonance,
the pole pair re±jθ related to the mode must be
moved closer to the origin by reducing radius r.
The impulse response of the acoustic path measured
from the loudspeaker to the listening position is
1 We may also lengthen the modal decay, or even create
new artiﬁcial modes, but such boosting must be made withcaution to avoid overloading the reproduction system.

needed in order to design a modal equalization ﬁlter. In practice modal equalization concentrates tofrequencies below 200 Hz and within a restricted listening area.
The process of modal equalization starts with theestimation of mid-range reverberation time, for octave bands between 500 Hz and 2 kHz, by calculating the mean of them. This mean decay time T60
at mid-frequencies is the basis for maximum allowed
mode decay time at low-frequencies, but allowing
the decay time to increase maximally by for example 0.2 seconds as the frequency decreases from 300
Hz down to 50 Hz.
We have previously developed two methods to design modal equalizer ﬁlters [3, 4]. The ﬁrst method(called AMK) attempts to directly identify mode frequencies in the magnitude response and then to obtain the decay rate τm and mode (angular) frequency
ωm by using a time-frequency presentation of theimpulse response at frequencies below 200 Hz [9, 3].
The decay rate for each identiﬁed mode frequency is
calculated using a nonlinear ﬁtting technique modeling the data as a sum of an exponential decay
and background noise. The modal equalizer ﬁlter isthen designed using the mode parameter data. The
longest decay time is ﬁrst corrected by ﬁltering withthe equalizer ﬁlter, and the process is then iterated
until all decay rates are within desired bounds.
The second method (called ARMA) [4] identiﬁes thepole and zero pairs describing a modal resonanceby ﬁtting a pole-zero least-squares model directly tothe room impulse response. Similarly to the AMK
method, the longest decay rate detected by ﬁnding
the pole closest to the unit circle is compensated bydesigning an equalizer ﬁlter for it, and the method
is then iteratively applied until all decay rates arewithin the desired bound. This second method does
not require the explicit stage of ﬁtting to the decayrate of a mode, and is better able to estimate theparameters of closely spaced modes.
The new method proposed in this paper deviatesfrom the two previous ones in the sense that there
is no need to deal with individual modes directly,
but to shape the entire time-frequency properties ofthe measured impulse response. Before introducing
this frequency-dependent windowing method, a synthetic example of modal equalization is shown to

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TEMPORAL SHAPING . . .
reduced, see Fig. 3. However, the decay at 50 and
55 Hz continues with the original 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 is limited when modes are close
to each other in frequency.

Fig. 2: Waterfall plot of a time-frequency responsewith ﬁve synthetic modes.

The ARMA method performs better in such cases,
but for a dense distribution of modes, often foundin real rooms, both the AMK and ARMA methods
are problematic. Only the beginning of the decay
is then accelerated through modal equalization, but
soon the decay rate starts to follow the unequalized
rate, as it occured in Fig. 3 for the modes at 50 and
55 Hz.
3 THEORY OF FREQUENCY-DEPENDENT WINDOWING
Signal windowing is one of the most common operations in signal processing. It is motivated by thefact that ﬁnite support (span, range) is needed either due to limited processing capacity or most often
due to non-stationarity of signals whereby they haveto be processed frame by frame to obtain temporally
localized representations.

Fig. 3: Waterfall plot of the ﬁve synthetic mode caseafter modal equalization.
demonstrate the eﬀect achieved by modal equalization.
2.1

Synthetic example of modal equalization

A case study with synthetic modes presented here
clariﬁes the idea of modal equalization. Figure 2
shows the waterfall plot of 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 response with an equalizer designedby the AMK algorithm, the decay times have been

Windowing in its traditional form is a multiplicative operation (·) in the time domain and thus corresponds to convolution () in the frequency domain,i.e.,y(t) = w(t)x(t)(5)
Y (ω) = W (ω)  X(ω)
where x(t) is a signal to be windowed, w(t) is a windowing function, and y(t) is the windowed signal.
Upper case symbols denote Fourier transforms, respectively.
The form of a windowing function w(t) can be any,
but most often it is monotonically decreasing towards positive and negative time directions fromits maximum value point, and for practical reasonsachieves value 0 outside a speciﬁc span. Among mostcommon symmetrical window functions are Hamming, Hann(ing), Blackman, Kaiser, and rectangular (or boxcar) window [10]. An asymmetrical window may be for example an exponentially decaying(and truncated) window. The selection of window
type and possible parameters associated with it depends on the criteria of each speciﬁc application at
hand.

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TEMPORAL SHAPING . . .

Traditional signal windowing works in a frequencyindependent manner. This is a straightforward solution, and with proper overlap-add or concatenation
processing yields perfect reconstruction from consecutive windowed signal slices. In audio and acoustics applications, however, it is often useful to apply nonuniform time-frequency resolution. In thispaper we discuss ways of controlling resolution, particularly methods that are based on time-frequency
shaping of system responses.
The two diﬀerent basic methods of frequencydependent windowing are: (a) frequency-domainand (b) time-domain techniques. The ﬁrst one modiﬁes the convolution rule of Eq. (5). The second one
applies warping techniques, and it can be done either
through frequency-domain resampling or throughwarped mappings.
3.1 Frequency-domain formulation of generalized windowing
According to the frequency-domain version of windowing [5] in Eq. (5), the Fourier transform of awindowed signal is obtained by (complex) convolution. When this is rewritten in the discrete Fouriertransform case for frequency bin m by
Y (m) =N−1

Wm (k)X(m − k)

(6)

k=0

where indices are modulo N, it becomes obvious that
if we wish to obtain frequency-dependent windowing, the windowing term Wm (k) cannot be a ﬁxed
vector but should be made bin-dependent. Each bin
m can be treated separately by diﬀerent ’window
spreading’ vector, designed according to the desired
window for that speciﬁc bin. These vectors can becomposed into an N × N matrix W for operationy = Wx(7)

where x and y are the original and the windowed
DFT vectors, respectively. Furthermore, the DFT
and IDFT transforms can be formulated as matrix
operations q = Fp and p = Gq where p is a signal
vector, q is a spectrum vector, and F and G are
the transform and its inverse transform composed ofcomplex exponentials, respectively. Thus the wholechain of Fourier transform of signal s, frequencydependent windowing, and inverse transform to windowed signal t, can be formulated [5] as

2
(a)
1
0
-1
-2
0

100

200

300

400

500

600

700

800

900 1000

1

(b)

.5

0
0

100

200

300

400

500

600

700

800

900 1000

2
(c)

1
0
-1
-2
0

100

200

300

400

500

600

700

800

900 1000

Fig. 4: Frequency-dependent windowing of a sum of
two sinusoids: (a) sum of sinusoids, (b) two Hann
windows (solid line for high frequencies, dashed line
for low frequencies), and (c) result of frequencydependent windowing.
t = GWFs = Ms

(8)

where M is an N × N matrix when the length of signal span s to be windowed is N , and the frequencydomain window matrix W is also N ×N . For a traditional frequency-independent window, M reduces toa diagonal matrix, i.e., to a sample-by-sample product of signal and window.
Due to the circularity of DFT and IDFT, frequencydependent windowing may cause folding problems if
not applied properly.
An example of using this method is given in Fig. 4
where the sum of two sinusoids is windowed so thata Hann window applied is shorter for the higher
frequency signal component than for the lower frequency one.
3.2 Generalized
windowing
frequency warping

by

time-

The idea of time-frequency warping [11] can be characterized as follows. A sinusoid is scaled (expanded

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TEMPORAL SHAPING . . .

or compressed) in time by factor β(f) depending on
frequency f, which operation in the frequency domain corresponds to remapping of frequencies (resampling of frequency to keep uniform bin distribution):

Notice that warping is not a shift-invariant operation, thus the selection of time origin is of special
importance. For impulse responses, which we areinterested in here, time origin t = 0 is an inherently
determined moment.

-50-60
Level [dB]

A sin(2πf[β(f)t] + ϕ) ↔ A sin(2π[β(f)f]t + ϕ) (9)

-40

-70
-80
-90
-100
0

0
0.2

50

0.4
100

0.6
0.8

150
1

Time-frequency warping can be realized in two ways,(a) by frequency-domain resampling or (b) by analytical mappings. Frequency resampling [12] of signal x(t) is realized by a sequence of operations

Time [s]

200

1.2
1.4

Frequency [Hz]250

Fig. 5: Waterfall plot of the measured and preprocessed impulse response for the case study room.

y(t) = F −1 {R{F(x(t))}}where R(·) is a resampling operator, and F (·) andF −1 (·) are the Fourier transform and the inverse
Fourier transform, respectively. While frequency resampling yields a high degree of freedom for the
warping function β(f), the method is complicatedand computationally expensive, and is not discussed
further in this paper.
The second method of warping is based on transforms that map the complex z-domain unit disk onto
itself. This approach is known from the design ofwarped digital ﬁlters [11]. Since the bilinear mapping of warped ﬁlters has only one degree of freedom,it may not be suitable in modal equalization tasks
if high frequency resolution needed. Modeling by
Kautz ﬁlters [14, 5] is a more attractive technique
for this purpose, but this approach remains outsideof the scope of this paper.
4 MODAL EQUALIZATIONWINDOWING TECHNIQUE

BY

THE

In this section we will explore the use of thefrequency-dependent windowing in modal equalization, applied to a case study of a speciﬁc room (dimensions 5.5 x 6.5 x 2.7 m3 ) with relatively longmodal decay at low frequencies.
The process starts with a measurement of the impulse response from a loudspeaker to the desired
listening position. Any modern measurement technique such as the MLS method may be applied; wehave preferred the logarithmic sweep proposed in[15] due to its good tolerance of nonlinearities. The
measured response is then preprocessed to remove
the bulk delay due to acoustic and measurement system delay, as well as applying high-pass ﬁltering to
remove possible infrasound components. Figure 5
shows a waterfall plot computed from the measured
and preprocessed impulse response that is used as acase study here.
4.1 Estimation of decay time
The next step of the new modal equalization is toestimate the decay rate in the impulse response as
a function of frequency. We have applied ﬁrst a
backward in time (to avoid lengthening of decay)
bandpass ﬁltering in frequency bands of about 10–
20 Hz width between 20 Hz and 220 Hz and then anonlinear ﬁtting of an exponential decay plus noisemodel [3] to estimate the decay time in each frequency band. There is no need to solve mode frequencies and only a moderately accurate estimate
of decay rate as a function of frequency is requiredin this approach. The solid curve in Fig. 6 depicts
the decay time vs. frequency curve of the case study
response.
4.2 Target response computation
The allowed decay time can be speciﬁed by starting from the mid-frequency reverberation time of theroom. The mean value of octave bands 500-1000 Hz
and 1000-2000 Hz is taken as a reference, and a lin-

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TEMPORAL SHAPING . . .

1.5

1amplitude

decay time [s]

0.6

0.5

0.4
0.2
0

0

0

50

100
150
frequency [Hz]

0

200

0.2

0.4

0.6
time [s]0.8

1Fig. 6: Decay time as a function of frequency forthe measured response (solid curve) and the target
maximum decay time (dashed line).

Fig. 7: Impulse response of modal equalizer computed by Eq. 13 for the case study response in thefrequency range of 0-220 Hz.

ear increase of modal decay time can be allowed at
low frequencies. In Fig. 6 the dotted line shows the
maximum allowed decay time for the response of thepresent case study. It is speciﬁed to grow from the
mid-frequency value T60 ≈ 0.35 s allowed at 200 Hz
by 0.3 s down to 30 Hz. As seen from the ﬁgure,there is need to shorten the decay time at frequencies up to 200 Hz.

as 0–220 Hz in our case. The impulse responsex(n) to be windowed is ﬁrst DFT transformed by
X(k) = DF T (x(n)) to form the vector x of Eq. 7.

4.3 Frequency-dependent exponential windowing
The next phase is to determine the frequencydependent window that is able to modify the measured response to meet the target requirements. In
each frequency bin the most natural window type isan exponential one since it retains the exponentiallydecaying character of each single mode, i.e.,
−τd t

e

−τm t −τe t=ee−(τm +τw )t

=e

(10)

where τd is the desired decay rate, τm is the measured decay rate, and τw is the decay rate of the
exponential equalization window. Now τw can becomputed byτw = τd − τm =− log(0.001) − log(0.001)
−
(11)
T60d
T60m

where T60d and T60m are the desired and measured
T60 values, respectively, using the relation between
T60 and decay rate τ .
Now the frequency-dependent exponential windowing can be carried out through the discrete Fouriertransform in the following way. The computation
can be done for the frequency range of interest, such

For each frequency bin a new exponential windowe−τm t is formed as a vector that is then DFT transformed, and these bin-related transformed vectors
are combined to matrix W of Eq. 7. The windowed
impulse response is achieved now simply by multiplication and inverse DFT as
y = Wx,y(n) = DF T −1 (y(k))(12)

This yields the target impulse response of modal
equalization.
4.4 Equalizer ﬁlter design
In the process above we have obtained the measured
impulse and the target impulse response, x(n) andy(n), respectively. The next problem is to realize adigital ﬁlter that properly approximates the required
modal equalizer transfer function
Hc(z) = Y (z)/X(z),hc (n) = DF T −1 Hc (z) (13)This equation actually deﬁnes the way to obtain afull accuracy FIR version of the desired equalizer directly by inverse DFT of Hc (z). Figure 7 shows theimpulse response obtained in this way for our casestudy. The response consists of an initial impulse
and a slowly decaying ripple. This response cannot
simply be shortened by truncation, otherwise the effect of temporal equalization is lost after the lengthof the response used, as can be seen in Fig. 8, where
the impulse response achieved by the equalizer of

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TEMPORAL SHAPING . . .

2

15
10
level [dB]amplitude

1
0
-1
-2

5
0
-5
-10
-15

0

0.2

0.4

0.6
time [s]0.8

1Fig. 8: Equalized response when only part (500 ms)of the equalizer impulse response is used. Notice theecho-like increase in level after 0.5 s.
Fig. 7 is shortened to 500 ms length. Such sudden
increase in response is perceived as a disturbing echo
eﬀect and must be avoided2.
An important question from the practical point ofview is that the frame size for DFT-based windowing
has to be long enough in order to avoid truncationand folding eﬀects. The frequency bin related exponential windows as well as the measured impulse
response become truncated, which introduces error,
and the circularity of DFT may bring problems unless properly dealt with3 .
Another problem with the equalizer response Hc (z),as obtained from Eq. 13, is that it may introduce
boosting of the magnitude response at some frequencies. This can be seen in Fig. 9, which plots
the magnitude response of the equalizer shown in
Fig. 7. Narrow-band peaks higher than a few dB
can be problematic by driving the loudspeaker andampliﬁer to overloading. A simple way to avoid theboosting is to limit the magnitude level before inverse DFT in Eq. 13. This is problematic, however,
because it degrades the Hilbert relation of magnitude vs. phase in minimum-phase systems. A betterway to control this detail is thus needed.
2 Since the temporal shape of the window used in themethod is free, a properly adjusted window with a temporaldip could be used to counteract an echo in a given impulse
response. Furthermore, this windowing technique to generatea new impulse response from a given impulse response canbe used also for audio eﬀects, whereby the peculiarities of a
response may be a useful feature.
3 In fact, the windowed and inverse transformed target response y(n) typically contains a small imaginary component
that has to be removed to get a real-valued signal. However,
the eﬀect of this is typically negligible.

0

50

100
150
frequency [Hz]200

Fig. 9: Magnitude response of the modal equalizer
designed for the case of the study.
The question in techical focus is that of ﬁnding an efﬁcient yet accurate realization for the derived equalization response. The direct solution of FIR ﬁltering with hc (n) of Eq. 13 is of prohibitively high orderfor practical real-time processing at full sample rate,unless some type of fast convolution is used instead,which has a tendency to introduce latency.
Fortunately the need of ﬁltering is here for the lowfrequency range only, which means that by a properly decimated multirate implementation the computational load remains light. For example in ourcase study an FIR of order 512 at a sample rate of
440 Hz can do the frequency range of interest up
to 200 Hz with a ﬁlter response length of 1.16 seconds. Using of most eﬃcient ﬁlter structures means,however, a demanding design and additional latency
compared to more direct computation at a higher
sampling rate.
A more detailed comparison of diﬀerent DSP realizations of the modal equalizer is left for future study.
It is good to notice that for best results FIR typesolutions are necessary, since the main task of theequalizer is to counteract modal pole pairs, and thiscan be done properly only by using enough zeros(zero pairs).
4.5 Analysis of equalization results
The result of applying the designed modal equalizer to the measured response is illustrated in thewaterfall plot of Fig. 10 and the decay rate proﬁleof Fig. 11. Figure 10 shows clearly an accelerated
modal decay at most frequencies, when compared tothe original response in Fig. 5. Only some of thelowest frequences remain long-ringing. The decay

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TEMPORAL SHAPING . . .
0
-5
-10
-15

-40

level [dB]-50
Level [dB]

-60
-70

-20
-25

-80

-30

-90

-35

-100
0

0
0.2

0.8

150
1

Time [s]

-45

100

0.6

200

1.2
1.4

-50

Frequency [Hz]250

Fig. 10: Waterfall plot of the modal equalized response of the system under study.
1.5

decay time [s]

-40

50

0.4

0

0.1

0.2

0.3
time [s]

0.4

0.5

0.6

Fig. 12: Decay envelopes (Schroeder-integrated) for
the original impulse response (upper solid line), for
the AMK equalized (dotted line), for the ARMA
equalized (dashed line), and for the windowing mode
equalized response (lower solid line).

1

reduced. However, the overall accuracy achieved isgood from a practical point of view.

0.5

0

50

100
150
frequency [Hz]200

Fig. 11: Decay time as function of frequency fororiginal response (dash-dot line), the target curve(dashed line), and for the equalized response (solidline) using the windowing technique.

rate proﬁle in Fig. 11 gives more information on theperformance of the equalization and nonidealities init.
From Fig. 11 we can ﬁnd that the proposed windowing technique has shortened the (early) decay times
almost as desired. At some frequencies it still exceeds the allowed maximum but only slightly. Thisnonideality is due to several reasons. First, the decay time estimation, applied here both to the original and to the equalized response, is not ideal. Secondly the FIR equalizer was only 0.7 seconds longinstead of 1.3 seconds of the original response. One
undesirable feature is that the decay time is shortened also around 200 Hz although it should not be

From Fig. 10 it can be learnt that the decay rate
remains slow at 50 Hz after about 0.5 seconds andat about 25 Hz already much earlier. This is no
problem in practice, because the loudspeaker usedhardly radiates at 25 Hz, and the reduced sensitivityof hearing at low frequencies relaxes the tolerance forlong decay time.
4.6 Comparison to prior methods of modal
equalization
We applied the AMK and ARMA methods of modal
equalization to the case of study to compare the performance of the windowing method to them. Whenno hand-tuning of design parameters was done, both
the AMK and the ARMA method showed only little if any improvement in this demanding modal
equalization case. The reason to this is that bothof these metods try to correct the response modeby mode. When a prominent mode has been detected and equalized, the magnitude spectrum is also
reduced around this peak remarkably, whereby theneighbouring modes gain in relative prominence. If
there is a high density of problematic modes, thesetechniques are limited in their ability to detect andcorrect all the modes.

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KARJALAINEN ET AL.

TEMPORAL SHAPING . . .

Figure 12 shows very clearly how the windowing
method is able to control the total decay envelopeof the low-frequency part (below 200 Hz) in ourcase, while the AMK and the ARMA methods without hand-tuning of their design parameters show
hardly any improvement. The curves are achieved
by backwards Schroeder integration of their impulse
responses. By careful adjustment the AMK and
ARMA methods may achieve about 5–10 dB of fast
initial decay and then the decay rate follows the original one [3, 4]. Thus the new windowing method isfound more capable and robust particularly in cases
where a densely distributed set of problematicallylong ringing modes are to be corrected.

6 ACKGNOWLEDGMENTS

5 SUMMARY AND CONCLUSIONS[2] EBU Document Tech. 3276-1998 (second ed.),“Listening Condition for the Assessment ofSound Programme Material: Monophonic and
Two-Channel Stereophonic,” (1998).

In this paper we have proposed a new method torealize modal equalization of loudspeaker-room responses, i.e., to control by DSP the temporal decay
of system response in sound reproduction for rooms
with too low level of absorption at low frequencies.
The method is based on shaping the time-frequency
properties of a given impulse response by applying
a frequency-dependent windowing technique.
The method is able to implement temporal response
shaping for complex impulse responses, also for nonexponential decay envelopes, although in this paper we have only applied exponential windowing toshorten excessive modal decay times. The windowing method is found more accurate and robust thanthe AMK and ARMA methods proposed before.
The critical phases in the realization of the method
are: ﬁrst the measurement of system impulse response with good enough signal-to-noise ratio at low
frequencies is needed, then the decay time proﬁlemust be estimated reliably, and ﬁnally an accurate
yet eﬃcient implementation for the equalization ﬁlter is to be found. Tradeoﬀ between processing latency and computational eﬃciency must be done inﬁlter design, since eﬃcient multirate or frame-based
structures introduce processing delay, while directﬁlter implementation may be beyond present processor capabilities. A remaining research task is also
to gain deeper understanding of perceptual aspects
related to room mode behavior.

This study has been 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 Academyof Finland (project number 201050).
7 REFERENCES[1] AES Technical Committee on Multichannel
and Binaural Audio Technology (TC-MBAT),“Multichannel Surround Sound Systems andOperations,” Technical Document, version 1.5(2001).

[3] A. Mäkivirta, P. Antsalo, M. Karjalainen, andV. Välimäki, “Low-Frequency Modal Equalization of Loudspeaker-Room Responses,”
Preprint 5480, AES 111th Convention, New
York, USA, 2001 Nov/Dec.
[4] M. Karjalainen, P. A. A. Esquef, P. Antsalo,
A. Mäkivirta, and V. Välimäki, ”FrequencyZooming ARMA Modeling of Resonant and Reverberant Systems,” J. Audio Eng. Soc., vol. 50,
no. 12, pp. 1012-1029, 2002 Dec.
[5] M. Karjalainen and T. Paatero, “FrequencyDependent Signal Windowing,” in Proc. Workshop Applications of Signal Processing to Audio and Acoustics (WASPAA’2001), New Paltz,
NY., USA, pp. 35–38, 2001 Oct.
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[7] 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
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[8] P. Antsalo, M. Karjalainen, A. Mäkivirta, andV. Välimäki, “Comparison of Modal Equalizer
Design Methods,” Accepted to AES 114th Convention, Amsterdam, 2003 March.

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KARJALAINEN ET AL.
[9] M. Karjalainen, P. Antsalo, A. Mäkivirta,
T. Peltonen, and V. Välimäki, “Estimation of
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[10] T. Saramäki, “Finite impulse response ﬁlter design,” Ch. 4 in Handbook for Digital Signal Processing, ed. S. K. Mitra and J. F. Kaiser, Wiley
1993.
[11] A. Härmä, M. Karjalainen, L. Savioja,
V. Välimäki, U. K. Laine, and J. Huopaniemi,”Frequency-warped signal processing for audio
applications,” J. Audio Eng. Soc., vol. 48, no.
11, pp. 1011-1031, Nov. 2000.

TEMPORAL SHAPING . . .
[12] J. O. Smith, Techniques for digital ﬁlter designand system identiﬁcation with application to theviolin, Ph.D. thesis, Elec. Eng. Dept., Stanford
University, 1983.
[13] Churchill R. V., Complex Variables and Applications. McGraw-Hill, New York, 1960.
[14] T. Paatero and M. Karjalainen, “Kautz Filters
and Generalized Frequency Resolution: Theory
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[15] A. Farina, “Simultaneous Measurement of Impulse Response and Distortion with a SweptSine Technique,” Preprint 5093, AES 108th
Convention, Paris, France, Feb. 2000.

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