Modal Equalization of Loudspeaker Room Responses at Low Frequencies *

Transcription

1 Modal Equalization of Loudspeaker Room Responses at Low Frequencies * AKI MÄKIVIRTA, 1 AES Member, POJU ANTSALO, 2 MATTI KARJALAINEN, 2 AES Fellow, AND VESA VÄLIMÄKI, 2 AES Member 1 Genelec Oy, FIN lisalmi, Finland 2 Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, FIN HUT, Espoo, Finland The control of excessively long decays in a listening room with strong low-frequency modes is problematic, expensive, and sometimes impossible with conventional passive means. A systematic methodology is presented to design active modal equalization able to selectively reduce the mode decay rate of a loudspeaker room system at low frequencies in the vicinity of a sound engineer s listening location. Modal equalization is able to increase the rate of initial sound decay at mode frequencies, and can be used with conventional magnitude equalization to optimize the reproduced sound quality. Two methods of implementing active modal equalization are proposed. The first modifies the primary sound such that the mode decay rates are controlled. The second uses separate secondary radiators and controls the mode decays with additional sound fed into the secondary radiators. Case studies are presented of implementing active modal control according to the first method. 0 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 in absorbers. The resulting energy decay is controlled to a desired level by introducing the necessary amount of absorbance in the acoustical space. This is generally feasible as long as the wavelength of sound is small compared to the dimensions of the space. As we move toward low frequencies, passive means of controlling the speed of reverberant decay become more difficult to use because the physical size of the necessary absorbers increases and may become prohibitively large compared to the volume of the listening space, or * Manuscript received 2002 July 15; revised 2003 February 14. Parts of this paper were presented at the 111th Convention of the Audio Engineering Society, New York, 2001 November 30 December 3. absorbers have to be made narrow-band. Consequently the cost of passive control of reverberant decay greatly increases at low frequencies. Methods for optimizing the response at a listening position by finding suitable locations for loudspeakers have been proposed [1] but cannot fully solve the problem of controlling modal decay. Because of these reasons, and because active control becomes technically feasible when the wavelength of sound becomes long relative to the room size, resulting in a less diffuse sound field in the room [2] [6], there has been an increasing interest in methods of active soundfield control at low frequencies. Mode resonances in a room can be audible because they modify the magnitude response of the primary sound or, when the primary sound ends, because they are no longer masked by the primary sound [7], [8]. The detection of a mode resonance appears to be very dependent on the signal content. Olive et al. report detection thresholds for resonances for both continuous broad-band sound and transient discontinuous sound, showing that low-q resonances are more readily audible with continuous signals containing a broad frequency spectrum whereas high-q resonances become more audible with transient discontinuous 324 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

2 signals [8]. The antiresonances (notches) are as audible as the resonances for low Q values. The audibility of antiresonances reduces dramatically for wide-band continuous signals when the Q value becomes high [8]. The detectability of resonances reduces approximately 3 db for each doubling of the Q value [7], [8] and low-q resonances are more readily heard with zero or minimal time delay relative to the direct sound [7]. The duration of the reverberant decay in itself appears an unreliable indicator of the audibility of the resonance [7], as audibility seems to be determined more by the frequency-domain characteristics of the resonance. Traditional magnitude response equalization cannot guarantee the control of the modal decay rate because such equalizers typically have a system transfer function of significantly lower order than the loudspeaker room system. A system having a unity transfer function (no change in magnitude or phase of the transmitted sound) does not modify the transmitted signal. By using a DSP filter it is in theory possible to design a filter that equalizes a loudspeaker room system to approximate a unity transfer function at a single location in the listening space within a limited frequency range. Such filtering may shorten the modal decay times too aggressively and may also lead to a system overload by boosting notched frequency areas, and is therefore often impractical for equalizing real rooms. Also, at higher frequencies the unity transfer function target can lead to a highly local correction in the room space with little practical value. High-order equalizers have often been found to actually decrease frequency response flatness for points other than the design target location. Contrary to the unity transfer function equalization target, the principle of modal equalization is rather to balance the decay rate of low-frequency modes to correspond to the reverberation time at mid and high frequencies in order to minimize the audibility of modal resonances at low frequencies. We also assume that once modal equalization has been performed, a conventional magnitude response equalizer (for example, an equalizer designed on one-third-octave smoothed magnitude response data) is applied to improve the subjective magnitude response balance at low frequencies and between the low-frequency area and higher frequencies. Mode equalization therefore becomes an additional tool to improve the audio quality in rooms beyond what is attainable using conventional magnitude equalization alone. In this paper we present methods to actively control lowfrequency reverberation. We will first present the concept and two principal methods of applying modal equalization. A target for mode decay time versus frequency will be discussed based on existing recommendations for high-quality audio monitoring rooms. Methods to identify and parameterize modes in an impulse response are introduced. The mode equalizer design for an individual mode is discussed with examples. Several case studies of both synthetic modes and modes of real rooms are presented. Finally, aspects about the synthesis of infinite impulse response (IIR) mode equalizer filters are discussed. MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES 1 THE CONCEPT OF MODAL EQUALIZATION To be useful, modal equalization must affect a change in the room decay characteristics, at least within an area sufficiently large to allow binaural monitoring without radical changes in the perceived reverberation or coloration of sound as a result of small movements of head location. The present work is restricted to frequencies below 200 Hz and environments where the sound wavelength relative to the room dimensions is not very small, that is, small rooms such as monitoring rooms instead of halls. We are not aiming at global control of the sound field in a room, but we try to introduce a change at the primary listening position, typically in the vicinity of a sound engineer s listening location. These limitations lead to a problem formulation where the modal behavior of the listening space can be modeled by a distinct number of modes such that they can be individually controlled. Each mode is modeled by an exponential decay function, τ m t m m m m h ^ th A e sin_ ω t φ i. (1) Here A m is the initial envelope amplitude of the decaying sinusoid, τ m is a coefficient that denotes the decay rate, ω m is the angular frequency of the mode, and φ m is the initial phase of oscillation. We define modal equalization as a process that can modify the rate of modal decay. It can be viewed as a special case of parametric equalization, operating individually on selected modes in a room. Modal equalization can be applied in a room by modifying the primary sound using a filter or by introducing in the room one or more secondary sound sources emitting a correcting sound. A mode resonance is represented in the z-domain transfer function as a pole pair with pole radius r and pole angle θ, H m 1 ^zh 1 r j ` e θ z j`1 re 1 jθ 1 z. j 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 resonance Q value must be reduced by moving poles toward the origin. Therefore modal equalization can also be viewed as a process of moving the locations of room transfer function poles. Mode decay time modification can be implemented in several ways either the sound going into a room through the primary radiator is modified or additional sound is introduced in the room with one or more secondary radiators to interact with the primary sound. The first method has the advantage that the transfer function from a sound source to a listening position does not affect mode equalization. In the second case differing locations of primary and secondary radiators lead to different transfer functions from the sound source to the listening position, and this must be considered when calculating a corrective filter. We will now discuss these two cases in more detail, draw- (2) J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 325

3 MÄKIVIRTA ET AL. ing some conclusions on necessary conditions for control in both cases. 1.1 Type I Modal Equalization Type I implementation (Fig. 1) modifies the audio signal fed into the primary loudspeaker to compensate for room modes. The total transfer function from the primary radiator to the listening position represented in the z domain is H^zh G^zh H ^zh (3) m where G(z) is the transfer function of the primary radiator from the electric input to the acoustic output and H m (z) B(z)/A(z) is the transfer function of the path from the primary radiator to the listening position. The primary radiator has essentially a flat magnitude response and a small delay in our frequency band of interest, and can therefore be neglected in the following discussion, G^zh 1. (4) We now design a pole zero filter H c (z) having a zero pair at each identified pole location of the mode resonances in H m (z). This cancels out the existing room response pole pairs in A(z), replacing them with new pole pairs A (z) set to locations producing the desired decay time in the modified transfer function H m (z), A^zh B^zh Hlm ^zh Hc^zh Hm^zh Al^zh A^zh This leads to a correcting filter, B^zh. Al^zh A^zh Hc ^zh. (6) Al^zh 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 mode resonance poles appear to have been moved toward the origin, and the Q value of the mode has been decreased. The coefficient sensitivity of this approach will be discussed later with example designs. 1.2 Type II Modal Equalization The type II method uses a secondary loudspeaker at an appropriate position in the room to radiate sound that interacts with the sound field produced by the primary loudspeakers. The loudspeakers are assumed to be identical in the treatment that follows but this does not need to be the case in (5) PAPERS practical implementations. Loudspeakers can be rendered similar by equalization for the purposes of modal equalization. The transfer function for the primary radiator is H m (z), and for the secondary radiator it is H 1 (z). The acoustical summation in the room (Fig. 2) produces a modified frequency response H m (z) with the desired decay characteristics, B^zh Hlm ^zh Hm^zh HcH1^zh. (7) Al^zh We can solve for a correcting filter H c (z), where H m (z) and H m (z) differ by modified pole radii, Hlm ^zh Hm ^zh Hc ^zh H ^zh H 1 ^zh A1 ^zh B^zh B ^zh A^zh 1 B1^zh A ^zh 1. 1 A^zh Al^zh Al^zh Note that if the primary and secondary radiators are the same source, Eq. (8) reduces into a parallel formulation of a cascaded correction filter, equivalent to the type I method, (8) (9) Hlm ^zh Hm^zh 81 Hc^zh B. (10) A necessary but not sufficient condition for a solution to exist is that the secondary radiator can produce sound levels at the listening location in frequencies where the primary radiator can, within the frequency band of interest, H1_ fi! 0, for Hm _ fi! 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 such that at some frequency it does not couple well into the room. At such frequencies the condition of Eq. (11) may not be fulfilled, and a secondary radiator placed in such a location will not be able to affect modal equalization. Because of this it may be advantageous to have multiple secondary radiators in the room. In the case of multiple secondary radiators, Eq. (7) is modified to the form Hl ^zh H ^zh H ^zh H ^zh (12) m N m! c, n 1, n n 1 primary source H c G H m H m listening position room primary source listening position H c H 1 secondary source room Fig. 1. Type I modal equalization using the primary sound source. Fig. 2. Type II modal equalization using a secondary radiator. 326 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

4 where N is the number of secondary radiators. After the decay times of individual modes have been equalized in this way, the magnitude response of the resulting system may be corrected to achieve a subjectively flat overall response. This correction can be implemented with several well-known magnitude response equalization methods, and is not addressed further in this study. We will now discuss the identification and parameterization of modes, and review some case examples of applying modal equalization to various synthetic and real rooms, using the type I mode equalization method. The type II method using one or more secondary radiators will be left for future study. 2 TARGET OF MODAL EQUALIZATION After the in-situ impulse response at the primary listening position has been measured using one of the standard techniques for room response measurements capable of producing an impulse response estimate, the process of modal equalization starts by defining a target for equalization. This target describes the desired maximum decay time at the frequencies of interest. There are several rational ways to set this target. Current recommendations for rooms intended for critical listening [9] [11] propose a requirement for the average reverberation time T m in seconds for midfrequencies (200 Hz to 4 khz) that depends on the volume V of the room, 13 / J V N Tm 025. K. V O (13) L 0 P The reference room volume V 0 of 100 m 3 yields a reverberation time of 0.25 s. An increase in reverberation time at low frequencies is typical, particularly in rooms where the passive control of the reverberation time by absorption is compromised. These rooms are likely to have isolated modes with long decay times, particularly at low frequencies. Recommendations [10] and [11] allow the reverberation time to increase linearly below 200 Hz by 0.3 s as the frequency decreases to 63 Hz, with a maximum relative change of 25% between adjacent one-third-octave bands. Below 63 Hz there is no requirement. These requirements are motivated by the goal to achieve a natural sounding environment for monitoring [11]. Along these lines of reasoning, one way to define the decay-time target is to define it relative to the midfrequency decay time (Fig. 3). To comply with [10], [11] we can define the target decay time relative to the mean T 60 in midfrequencies (500 Hz to 2 khz), increasing linearly (on a log frequency scale) by 0.2 s as the frequency decreases from 300 Hz down to 50 Hz. There are also other approaches to setting the decaytime target. Because the aim of modal equalization is to minimize the audibility of mode resonances, it is rational to aim at reducing the mode decay time down to the general decay time within the frequency range of interest. One way of doing so is to look at the statistical distribution of the decay time in the frequency band of interest. A percentile value of the decay-time distribution can be chosen MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES as the target decay time. This may be a suitable choice to reduce the audibility of individual mode frequencies in rooms where it is difficult to achieve the low decay times proposed in recommendations. 3 MODE IDENTIFICATION AND PARAMETER ESTIMATION The next task is to find and identify the modes that need to be equalized. They can be identified in the frequency response by assuming that a mode produces a level increase in the magnitude response at a mode resonance, or by inspecting directly the measured short-term Fourier transform as frequencies of prolonged decay time. We will discuss both methods of identifying modes. One identification method is to search for magnitude response level increases produced by modes. A method of doing this is to note within the frequency range of interest (in our case f < 200 Hz) where the magnitude response exceeds an average midfrequency magnitude response level (500 Hz to 2 khz). Then local maxima above this midfrequency reference level are noted and considered as potential mode frequencies. AR and ARMA modeling methods can be used to identify resonance frequencies in order to find the resonances with the largest radii instead of directly inspecting the magnitude response data [12]. Because the selection of a proper model order for this task can be problematic, a detection function has been proposed based on the fact that at a modal resonance there is concurrently a high value of gain H(Ω) and a rapid change in the phase [12], G^Ωh H^Ωh max' 0, D9arg`H^ΩhjC1 (14) where H(Ω) is the Fourier transform of the measured response, Ω is the normalized angular frequency, and D is a frequency-domain differentiation operator. A positive peak in this detection function may indicate a mode that needs equalization. One identification method is to inspect short-term Fourier transform data decay profiles directly. A decay profile P L (ω) describes at frequency ω the time t 0 from the impulse response maximum level after which the shortterm Fourier transform bin level has decreased permanently below a profile detection level L, P ^ωh t : H^ω, th < L, t > t, ω ω. (15) L Reverberation time measured Frequency target Fig. 3. Principle of setting reverberation time target relative to measured midfrequency octave-band reverberation time. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 327

5 MÄKIVIRTA ET AL. A mode detection function C(ω) is constructed as a weighted sum of k decay profiles in the Fourier transform data detected at several profile levels L(k), N! C^ωh a * ap ^ωh min 8P ^ωhbk 14 k 1 k Lk () Lk () p (16) where the possible weight for each decay profile is denoted by a k. Decay-time differences can be enhanced by setting the exponent p > 1. This mode detection function accumulates value at frequencies with long decay times, typically having the maximum at the longest decay time. A sample decay profile and mode detection function will be depicted later in the case examples. 3.1 Mode Parameters After finding the frequency of a possible mode with any of the methods described, we must parameterize the mode in order to design a mode equalizer filter. The parameters that describe a mode are the decay rate coefficient τ m, angular frequency ω m, and initial phase φ m. Assuming the wavelength of sound within the frequency range of interest to be large relative to the room size, the initial phase is typically close to zero, and is assumed to have the value zero henceforth. Mode parameters are estimated using short-term Fourier transforms of the transfer function impulse response. When certain time windowing is used, this is also called the cumulative spectral decay [13]. A plot of the short-term Fourier transform data is frequently referred to as a waterfall plot. To determine the exact center frequency of a mode, we exploit a special property of windowed Fourier transforms. It can be shown that the spectral peak of a Gaussian-windowed stationary sinusoid calculated using a Fourier transform has the form of a parabolic function [14]. Therefore the precise center frequency of a mode can be calculated by fitting second-order parabolic function, 2 G_ fi af bf c (17) into the three Fourier transform bin values around a local magnitude maximum indicated by the mode detection methods described earlier. The frequency where the secondorder function derivative assumes the value zero is taken as the center frequency of the mode, PAPERS 2 G_ fi b 0 & f. (18) 2f 2a When time windowing is used, this fitting technique allows us to determine the mode frequency more precisely than the frequency bin spacing of a Fourier transform. A 20-fold improvement has been demonstrated in comparison to raw fast Fourier transform bin spacing by using the Hamming window [15]. After determining the mode frequency, the pole radius of the mode resonance model must be determined. This estimate can be based on one of two parameters, the Q value of the steady-state resonance or the actual measurement of the decay time T 60. While the Q value can be estimated for isolated modes, it may be difficult or impossible to define it for modes closely spaced in frequency. Because of this we use the decay time measurable directly in short-term Fourier transform data to determine the mode pole radius. The method employed in this work [16] provides a direct estimate of the decay rate coefficient τ. This then enables the calculation of T 60 to obtain a representation of the decay time in a form readily related to the concept of reverberation time, T ln ` j.. (19) τ τ The decay time for each detected potential room mode is calculated by fitting an exponential decay noise model into time-series data formed by a particular frequency bin of consecutive short-term Fourier transforms. The mode decay is modeled by an exponentially decaying sinusoid [Eq. (1) is reproduced here for convenience], τ m t m m m m h ^ th A e sin_ ω t φ i (20) where A m is the initial envelope amplitude of the decaying sinusoid, τ m is a coefficient defining the decay rate, ω m is the angular frequency of the mode, and φ m is the initial phase of mode oscillation. We assume that this decay is corrupted in practical measurements by an amount of noise n b (t), n ^th A n^th (21) b n and that this noise is uncorrelated with the decay. Statistically the decay envelope of this signal is 2 2τ t 2 m n a^h t A e m A. (22) The values of A m, τ m, and A n are found by least-squares fitting this model to the measured time series of short-term Fourier transform bin values. For further details of this method, the reader is referred to [16]. A sufficient dynamic measurement range is required to allow reliable detection of the room mode parameters, although the least-squares fitting method used has been shown to be resilient to rather high noise levels. Noise level estimates by using the least-squares fitting method across the frequency range provide a measurement of the frequencydependent noise level A( f ), and this information can be used later to check data validity. An alternative method employing ARMA modeling has been proposed in [12], enabling the direct determination of the pole model coefficients instead of going through the intermediate stage of determining first a center frequency and a decay rate or resonance Q value. In practice some further error checking of the identified modes can be useful before moving onward to design mode equalizers in order to discard obvious measurement artifacts and not consider them as modes to be equalized. A candidate mode may have to be rejected if the noise level estimated at that mode frequency is too high, implying an insufficient signal-to-noise ratio for reliable measurement. Also, candidate modes that show unrealistically 328 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

6 slow decay or no decay at all can be rejected if they represent technical problems in the measurement, such as mains hum, ventilation noise in the room, or other unrelated stationary error, and not any true mode decay. These rejections can be confirmed by inspecting the short-term Fourier transform data and with additional measurements. 3.2 Discrete-Time Representation of a Mode To obtain a discrete-time representation of a mode, consider now a second-order all-pole transfer function having pole radius r and pole angle θ, 1 H^zh 1 r j ` e z j`1 re 1 j r cos θ z r z z j (23) Taking the inverse z transform yields the impulse response of this system as n r sin8θ^n 1hB h^nh u^nh (24) sinθ where u(n) is a unit step function. The envelope of this sequence is determined by the term r n. To obtain a decay rate equal to T 60,we require that the decay of 60 db be accomplished in N 60 steps given the sample rate f s, N 20 log`r 60j 60, N60 T60 fs. (25) We can now solve for the pole radius r of the discrete-time representation, 3 r 10. (26) T60 fs We have now obtained a discrete-time model of a room mode described by two parameters, the mode frequency ω yielding the pole phase angle θ ωt s (where T s denotes the sampling interval) and the pole radius r. We can also determine the desired location of a pole having the target resonance decay time by using the same approach, selecting the same mode frequency but a modified decay time T 60. This will be needed later when constructing a mode equalizer filter. MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES 4 MODE EQUALIZER DESIGN For the sake of simplicity only the design of the type I modal equalizer is presented here. The type I equalizer is based on the concept of modifying the primary sound such that target modes decay faster. The mode resonance is modeled in discrete time by a pole pair z F(r, θ) determined in short-term Fourier data using the method presented. The desired decay time is produced by a modified pole pair z c F(r c, θ). The correction filter of Eq. (6) for a single mode becomes A^zh Hc ^zh Al^zh `1 re z j`1 re 1 r ej c z ` j`1 r e c j 1 j 1 z j. c z j 1 j 1 c (27) To give an example of the equalizer function, consider a system defined by a pole pair (at radius r 0.95 and angular frequency ω 0.l8π) and a zero pair (at r 1.9 and ω 0.09π) In order to reduce the decay time of the poles, we want to shift the location of the poles to the radius r 0.8. The resulting type I equalizer has a notchtype magnitude response (Fig. 4) because the numerator gain of the correction filter is larger than the denominator gain. Cascaded with the system response, poles at radius r 0.95 are canceled and new poles created at the desired radius (Fig. 5). Impulse responses of the two systems (Fig. 6) demonstrate a reduction in the mode resonance Q value. The decay envelope of the impulse response (Fig. 7) now shows a rapid decay. The precision of a mode pole location estimate determines the success of mode equalization. Errors in the estimated mode frequency and decay rate will displace the zero compensating an actual room mode pole and reduce the accuracy of control. For example, an estimation error of 1% in the mode pole radius (Fig. 7) or angle (Fig. 8) greatly reduces control. Despite the error we still see rapid initial decay, but then the decay follows the original decay rate of the mode. This demonstrates that accurate modeling of the pole locations is paramount to the success of modal equalization. Fig. 4. Effect of modal equalization on example system and magnitude response of modal equalizer. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 329

7 MÄKIVIRTA ET AL. PAPERS Fig. 5. Poles ( ) and zeros (&) of mode-equalized example system. Equalization of example system mode is produced by canceling original system poles creating the mode resonance and replacing them with a pole pair with less gain, located at the same frequency but closer to origin. Sample Fig. 6. Impulse responses of original and mode-equalized example systems. Fig. 7. Effect of mode pole radius modeling error on decay envelope control by modal equalizer in example system. Fig. 8. Effect of mode pole frequency modeling error on decay envelope control by modal equalizer in example system. 330 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

8 5 CASE STUDIES In this section we present case studies to enlighten the modal equalization process. In the first two examples we use artificially added modes with known center frequency and decay time to explain and demonstrate the principles of modal equalizer design. In the third example a real room measurement is used and the robustness of modal equalization for offset locations is demonstrated. The short-term Fourier transforms used and plotted in these case examples are calculated using a tapered time window, or an apodized rectangular window, as Bunton and Small called it originally [13]. Prior to this calculation the 850-ms impulse response is decimated to a sample frequency of 1179 Hz, resulting in 1003 sample values. The impulse response is zero-padded for short-term Fourier transform calculation. The time window is constructed by attaching a half Hanning time window as the initial taper Fig. 9. Anechoic waterfall plot of two-way loudspeaker response used in cases I and II. MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES (length of taper is 4.24 ms) at the beginning of a time window extending to the end of the data. This one-sided time window is slid over the impulse response data while calculating Fourier transforms (4096 bins) at 30-ms time intervals. Plots of short-term Fourier data (typically called waterfall plots) are limited to 40 db. This floor is useful as it enhances the visibility of the initial decay rate increase produced by modal equalization, and it makes it easier to roughly judge the initial decay rate. The ensemble average level decay within the frequency band of interest is visualized by backward integrating the bandpass-filtered impulse response magnitude [17]. 5.1 Cases with Artificial Modes Case I demonstrates the mode equalizer design. It is based on a free-field response of a compact two-way loudspeaker measured in an anechoic room (Fig. 9). Some slow low-level decay can be observed in the waterfall. This is a combination of the decay of loudspeaker resonances and the fact that the sound absorption in the anechoic room is no longer perfect at very low frequencies, below 50 Hz. An artificial mode with a decay time T s is added to the data at the frequency f 100 Hz (Fig. 10). The target frequency band for mode equalization is Hz. An equalizer is designed to limit the mode decay time to the 70th percentile of decay time distribution measured within the target frequency band. After equalization we obtain a waterfall response very similar to the original (Fig. 11). This trivial example highlights the basic steps of mode equalizer design: 1) The short-term Fourier transform is calculated and scaled to set the maximum value within the target frequency range to ) Using the short-term Fourier transform, decay profiles are detected. We used several detection levels ( 10, 20, 25, 30, 35, and 40 db). Some resulting Fig. 10. Case I. Free-field response of compact two-way loudspeaker with added artificial room mode at f 100 Hz. Fig. 11. Case I. Mode-equalized free-field response of compact two-way loudspeaker with added artificial room mode at f 100 Hz. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 331

9 MÄKIVIRTA ET AL. decay profiles for case I are plotted in Fig. 12 3) The mode equalization decay time target is set using the decay profiles. We use the decay profile P 20 detected at level 20 db. The 70th percentile of the P 20 value distribution is extrapolated to 60 db and set as equalized target decay time T 60. An alternative target can be set based on the reverberation time of the center frequencies of 500 Hz to 2 khz. 4) The mode detection function C is calculated as a weighted sum of the decay profiles. The detection function for case I is plotted in Fig. 13 5) The mode detection function is used to determine the number of modes to equalize. To determine this number, a limit is set in the statistical distribution of the mode detection function values. In the case studies we used the 65th percentile value. The count of peaks above this level in the mode detection function data is taken as the number of modes to equalize. 6) Recording the frequency of the mode detection function maximum produces the first estimate of the mode frequency. Note that the mode detection function value only exists at Fourier transform bin frequencies. We use nearestneighbor fitting with a second-order function [see Eq. (18)] to find an accurate frequency of the maximum interpolated between bin frequencies. This is the estimate of the mode frequency based on the decay profile. 7) The second estimate for the mode frequency is obtained using the maximum magnitude of the Fourier transform. This mode frequency estimate is made more accurate by fitting a second-order function to the magnitude values of the bin where the maximum was found PAPERS and its two neighboring bins [see Eq. (18)]. This is the estimate of the mode frequency based on the maximum magnitude. 8) Now a choice must be made as to which mode frequency estimate to use in subsequent modal equalizer design. It is also possible to combine these two mode frequency estimates, or to take actions based on the similarity of their values. We made the choice prior to the design, based on the type of problem. Decay-profile-based mode frequency estimates may be more reliable in real rooms with complex closely spaced modes because in these rooms magnitude maxima do not necessarily correctly indicate the frequency of mode resonance. The case I design presented in this paper uses maximum-magnitudebased mode frequency estimation, but both maximummagnitude and decay-profile-based estimates yielded very similar designs for case I. 9) The impulse response is bandpass filtered to exclude possible other modes. The corner frequencies of the bandpass filter are chosen as {0.8ω m, 1.2ω m }, where ω m is the previously estimated mode frequency. 10) The decay time is estimated using the nonlinear fitting approach [see Eq. (22) and [16]. 11) The radii of the mode compensating filter pole and zero are determined based on the estimated center frequency and decay time and the decay-time target set earlier. 12) A second-order mode equalizer of Eq. (27) has now been determined. After this, the room impulse response is filtered with the equalizer to equalize this mode. Steps 1 11 are repeated a predetermined number of times to equal- Fig. 12. Case I. Initial decay profiles detected at l0-db, 20- db, 25-dB, and 35-dB levels (bottom to top). Fig. 13. Case I. Initial value of mode detection function. 332 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

10 MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES ize all modes, excluding steps 3, 5, and 8, which are performed only the first time the algorithm is run. The resulting equalizer for case I has the magnitude response of a simple notch at the mode frequency (Fig. 14) and equalization speeds up to the initial decay (Fig. 15). The reduction of the decay speed at time t 0.2 to 0.4 s in Fig. 15 was shown to be a result of noise in the measurement file showing up as an increase in the backwardintegrated level, and does not indicate a reduction in decay speed. This is also evident in the waterfall plots of Figs Case II uses the same anechoic two-way loudspeaker measurement. Five artificial modes with slightly differing decay times are added at mode frequencies (decay times in parentheses) of 50 Hz (1.4 s), 55 Hz (0.8 s), 100 Hz (1.0 s), 130 Hz (0.8 s), and 180 Hz (0.7 s). After modal equal- Fig. 14. Case I. Modal-equalizer magnitude response. (a) (b) (c) Fig. 15. Case I. Impulse responses. (a) Original system with artificial mode at f 100 Hz. (b) Mode-equalized system. (c) Decay envelopes for original ( ) and equalized ( ) systems within frequency band of interest ( Hz). Apparent reduction in decay speed at t s is caused by noise in anechoic measurement. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 333

11 MÄKIVIRTA ET AL. ization the magnitudes of the mode resonances have been decreased (Figs. 16 and 17). There is an initial fast decay (Fig. 18). The same anechoic measurement as in case I PAPERS was used for generating the synthetic case II. Again, the reduction of the decay speed at time t 0.2 to 0.4 s in Fig. 18 was shown by inspection to be a result of noise in the Fig. 16. Case II. Five artificial modes added to impulse response of compact two-way loudspeaker anechoic response. Modes are added at frequencies (decay times in parentheses) of 50 Hz (1.4 s), 55 Hz (0.8 s), 100 Hz (1.0 s), 130 Hz (0.8 s), and 180 Hz (0.7 s). Fig. 17. Case II. Waterfall of mode-equalized system. (a) (b) (c) Fig. 18. Case II. Impulse responses. (a) Original system with five artificial modes. (b) Mode-equalized system. (c) Decay envelopes for original ( ) and equalized ( ) systems within frequency band of interest ( Hz). Apparent reduction in decay speed at t s is caused by noise in anechoic measurement. 334 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

12 measurement file, and is not a true indication of a reduced speed of decay. We tried two mode frequency identification methods. Both the magnitude-maximum and the decay-rate-based mode frequency identification methods produced similar results. 5.2 Case with a Real Room Response Case III is a measurement in a hard-walled approximately rectangular room with 40.9 m 2 floor area (length 7.18 m by width 5.69 m by height 2.5 m). The room is constructed of concrete except for the front half of the floor, which is a 30-mm chipboard construction on 200-by 50-mm studs having 0.6-m spacing, covering a 0.6-mdeep concrete floor cavity in the front half of the room to form a flat floor surface. The impulse response measurement for the filter design is taken in the center of the room ( design point ) at the height of 1.2 m. The sound source is a three-way active studio monitor loudspeaker (Genelec 1037B) placed on a (a) MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES stand of 0.7-m height in the right front corner of the room. The vertical middle point of the front baffle is located 0.5 m from both walls in the corner of the room, with the loudspeaker acoustical axis aimed to 1 m forward from the design point. Additional measurements were taken 1 m to the left and right of the reference point, and also 1 m forward toward the front of the room, and 1 m to the left and right from that point, totaling six measurements, including the measurement of the design point. The floor area covered by the measurements is 2 m 2. The loudspeaker microphone arrangement has front back symmetry, but not left right symmetry, and therefore the six measurements will give an idea about how local the effect of modal equalization is in this room over a floor area of 4 m 2. The listening location of the recording engineer is always in some way variable, and the measurements outside the design point give a chance to get an idea of the spatial robustness of modal equalization. To remove infrasound frequencies prior to processing, the original impulse responses are filtered by a fourthorder Butterworth high-pass filter with a corner frequency at 20 Hz. The reverberation time of the room is 0.37 s in the midfrequency band of 500 Hz to 2 khz, and 1.2 s in the target frequency band (30 to 200 Hz) measured in the initial 30 db of level decay. The low-frequency waterfall responses (Fig. 19) reveal strong distinct modes, with the longest decay time occurring at the second-order axial resonance along the length of the room at the frequency of 47 Hz. The first- and second-order eigenfrequencies for a rectangular cavity with the dimensions of the listening room can be used as an estimate of the expected modal frequencies (Table 1). It is typical that not all of the theoretically existing eigenmodes are measurable in a real room. The equalizer was designed with the algorithm presented in case I, but incorporating additional optimization steps to be detailed. The decay-profile-based method was used in estimating mode frequencies. Designs where mode frequencies were identified using the magnitude- (b) Fig. 19. Case III. Waterfall plots in real listening room. (a) 1.0 m forward and 1.0 m left of design point. (b) 1.0 m forward from design point. (c) 1.0 m right of design point. (d) 1.0 m left of design point. (e) Design point (center of room). (f ) 1.0 m right of design point. Measurements taken with omnidirectional microphone. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 335 (c)

13 MÄKIVIRTA ET AL. (d) PAPERS maximum identification were not successful. The case III room response has some closely spaced modes, and is an example of a real room where a single magnitude maximum is in fact constituted of several closely spaced mode resonances. Some of these resonances decay faster, causing the apparent frequency of maximum magnitude to shift in frequency, rendering the use of maximum-magnitudebased identification problematic. Additional optimization steps were incorporated in the case III design to allow the interaction of closely spaced correction filter notches to be taken into account during filter design. The equalizer was designed with the iterative approach presented in case I, but incorporating the following additional optimizing steps: a) For every algorithm cycle, at step 12 of the basic iterative design algorithm presented for case I, calculate the distance ε from the newly found mode center frequency ω to the center frequencies ω k of all N 1 previously determined mode equalizers, ε ωk ω ω k < ε0, for k 1, f, N 1. (28) If the distance ε is smaller than ε 0 (the value 0.01 was used), consider this to be the same filter as the closest previous filter. Discard the newly defined filter and change the pole radius of the closest equalizer filter to move the filter pole toward the origin by a design step to increase its notch depth. The design step is taken by modifying the pole radius r by an exponent a (the value 1.19 was used; note that r < 1.0), a r^nh r ^n 1h. (29) Table 1. First- and second-order eigenfrequencies in a rectangular cavity with case III listening room dimensions. (e) (f) Fig. 19. Continued Frequency p q r [Hz] Type of Mode Axial Axial Tangential Axial Tangential Axial Tangential Axial Tangential Tangential Tangential Oblique Tangential Oblique Tangential Oblique Oblique Axial Tangential Tangential Oblique Tangential Oblique Tangential Oblique Oblique Note: p, q, r order of eigenfrequency along length, width, and height of room. 336 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

14 The motivation of this optimization step is to handle the case where in fact the algorithm picks up the same mode again because the mode equalizer notch filter attenuation is not sufficiently large, and this optimization step allows the notch attenuation to be adapted to a sufficient level. b) During the design process, after every M cycles (the value of 10 cycles was used), a simulated annealing step [18] is performed. This moves all previously designed equalizer poles outward, away from the origin, decreasing the depth of their respective notch filters by an exponent derived from an annealing noise generator N having a logarithmic positive value distribution, TN () n r^nh r ^n 1h, T, N^nh > 0. (30) The magnitude scaling of the log noise variable N is determined by a positive annealing temperature coefficient T, reduced after each annealing step. The purpose of the simulated annealing step is to allow a readjustment of gains in MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES all mode filters during the design to accommodate the interaction of closely spaced filters. This is the mechanism in the design process that reduces the notch depth of a mode equalizer, and it can also eventually eliminate some mode equalizer filters by reducing their notch depth to zero if they are no longer needed to achieve equalization. Note that when the annealing step reduces the filter notch depth, the basic algorithm reconsiders these mode frequencies if necessary, and will tend to return their notch depth to a sufficient level. c) Repeat the basic filter design loop with these two optimization steps until the mode detection function C(ω) assumes a value less than a preset design target. This terminates the design process. When the optimization steps described here are used in the design process, the termination condition based on a predetermined number of modes (step 5 described in case I) cannot be used. The equalized waterfall response (Fig. 20) shows that decays are now well controlled within decay profile cal- (a) (c) (b) Fig. 20. Case III. Waterfall plots in mode-equalized real listening room. (a) 1.0 m forward and 1.0 m left of design point. (b) 1.0 m forward from design point. (c) 1.0 m right of design point. (d) 1.0 m left of design point. (e) Design point (center of room). (f ) 1.0 m right of design point. Mode equalizer design based on omnidirectional microphone measurement at design point (e). J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 337 (d)

15 MÄKIVIRTA ET AL. culation levels (20 40-dB decay levels) at the design point [Fig. 20(e)]. Decay below the target frequency range ( f < 30 Hz) is not included in the equalizer filter design, and therefore is not controlled. Moving away from the design position, the modal decay remains controlled, although the amount of modal equalization reduces slightly. This demonstrates that for the low frequencies considered in this work, the modal equalizer is able to control modes in an area large enough for practical sound engineering work, and can therefore potentially bring an improvement to the sound quality within a limited area in a room. Note in this case study that the modal equalizer is not able to control a mode at about 90 Hz for positions outside the design point. This is because this mode is not measurable at the design point [Fig. 19(e)], so the filter design is blind to this particular mode. This emphasizes the importance of the point in space chosen as the design point, and the significance of sufficient study of the modal structure in the room in the vicinity of the intended design point in order to choose a response that is sufficiently representative for successful modal equalizer design. If a mode happens to have a null at the chosen design point, as in case III, this mode is not going to be controlled. This mode may then become disturbingly audible when other modes PAPERS are controlled at points close to the design point, where the level of this uncontrolled mode increases rapidly. The reverberation time within the target frequency range after equalization is 0.7 s measured in the initial 30 db of level decay (original reverberation time was 1.2 s measured in the same way). The resulting total equalizer (Fig. 21) is a collection of notch filters, and has a very small positive gain outside the target frequency band. The equalized system magnitude response (Fig. 22) shows smaller magnitude deviation, and equalization has reduced the overall level close to mode frequencies. Shortening of the decay is also visible in the bandpass-filtered ( Hz) impulse response [Fig. 23(a) and (b)], and the energy decay shows fast initial decay and then follows the decay speed of the original system [Fig. 23(c)]. The resulting equalizer filter contains nine second-order sections. It is somewhat complex, but given that we are working at low frequencies, multirate techniques allow an efficient and cost-effective implementation. A total of nine optimization cycles were needed in the filter design. For this design, a simulated annealing step was not used because the design converged so rapidly, but an equalizer gain optimization step was used. The notch frequencies of the modal equalizer (Table 2) coincide fairly well with the eigenmodes calculated for a rectangular cavity with the (e) Fig. 20. Continued (f) Fig. 21. Case III. Modal equalizer magnitude response. 338 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

16 dimensions of the room (Table 1). Naturally, only a part of the modes are measurable at the design point, therefore becoming subject to equalization. MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES 6 MODAL EQUALIZER IMPLEMENTATION 6.1 Type I Filter Implementation To correct N modes with a type I mode equalizer, we need an order-2n IIR transfer function. The most immediate method is to design a second-order filter defined by Eq. (27) for each mode. The final order-2n filter is then formed as a cascade of second-order subfilters, N % c, k k 1 H ^zh H ^zh. (31) c Another formulation allowing the design for individual modes is served by the formulation in Eq. (10), a parallel formulation of a cascaded correction filter. This leads to a Fig. 22. Case III. Magnitude response of design point in listening room. Unequalized ( ) and after modal equalization ( ). (a) (b) (c) Fig. 23. Case III. Impulse responses. (a) Original system. (b) Mode- equalized system. (c) Decay envelopes for original ( ) and equalized ( ) systems within frequency band of interest ( Hz). J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 339

17 MÄKIVIRTA ET AL. parallel structure where the total filter is implemented as N Hc^zh 1! Hlc, k ^zh. (32) k Asymmetry in Type I Equalizers At low angular frequency the maximum gain of a resonant system may no longer coincide with the pole angle [19]. This is also true with modal equalizers, and must be compensated for in the design of an equalizer. A basic type I modal equalizer becomes increasingly unsymmetrical as the frequency approaches ω 0. A case example (Fig. 24) shows a standard design with pole and zero at ω p,z 0.01 rad/s, zero radius r z 0.999, and pole radius r p There is a significant gain change for frequencies below the resonant frequency. This asymmetry may cause a problematic cumulative change in gain. It is possible to avoid asymmetry by decreasing the sampling frequency in order to bring the mode resonances higher on the discrete frequency scale. If sufficient sample rate alteration is not possible, we can symmetrize a mode equalizer by moving the pole slightly downward in frequency. This will symmetrize the response, but it will also shift the notch frequency. These effects must be accounted for when symmetrizing a mode equalizer at low frequencies. This can be handled by an iterative fitting procedure with a target to achieve the desired mode decay time simultaneously with a symmetrical response. Table 2. Case III modal equalizer represented as a cascade of second-order notch filters. f 0 G Zero Pole N [Hz] [db] [rad] Radius Radius Note: f 0 notch center frequency; G notch gain; ω angular frequency of zero and pole. Fig. 24. Modified type I modal equalizer with symmetrical gain having zero radius r at ω 0.01 rad/s and pole radius r at ω rad/s ( ), and standard type I modal equalizer having both a pole and a zero at ω 0.01 rad/s ( ). PAPERS 6.3 Type II Filter Implementation The type II mode equalizer uses one or more secondary radiators in producing additional audio in the room to control the decay time of modes. The type II equalizer requires a solution of Eq. (8) for each secondary radiator. The correcting filter H c (z) can be implemented directly according to Eq. (8) as a difference of two transfer functions convolved by the inverse of the secondary radiator transfer function. Then it is important to recognize the requirement of Eq. (11) that the secondary radiators be able to couple sufficiently to the room at frequencies of interest in order to control the sound field. A more optimized implementation can be found by calculating the correcting filter transfer function H c (z), then designing an FIR or IIR filter approximating this transfer function. This filter can then be used as the correcting filter. There are several standard filter design techniques that can be used in designing such a filter. In the case of more than one secondary radiator the solution becomes more convoluted as the contributions of all secondary radiators must be considered if they have overlapping frequency ranges. For example, the solution of Eq. (12) for the correction filter of the first secondary radiator is H c, 1 N! 2 Hlm ^zh Hm^zh n Hc, n^zhh1, n^zh ^zh. H ^zh 11, (33) All secondary radiators interact to form the correction. Therefore the filter design process becomes a multidimensional optimization task where all correction filters must be optimized together. A suboptimal solution is to design one secondary filter at a time such that the subsequent secondary sources only handle those frequencies not controllable by the previous secondary sources, for instance, because of poor radiator location in the room. 7 DISCUSSION In this paper we have presented two different methods to apply a mode equalizer in a room, type I modifying the sound input into the room using the primary loudspeakers, and type II using separate loudspeakers to output a sound that controls the modal decay. Type I systems are typically minimum-phase systems. Type II systems, because the secondary radiator is separate 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 implementation, type II systems can also be considered minimum-phase systems. An optimization algorithm was presented for modal equalizer design. Several methods for finding and identifying modal resonances were presented. Both the maximum magnitude and the information about decay time were used as methods to find the dominant mode to be equalized in the short-term Fourier transform of the transfer function impulse response. Parameters describing the mode were identified in the short-term Fourier transform 340 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

18 data and used to obtain modal equalizer coefficients. An iterative method of designing a modal equalizer was presented, where the currently identified modes are equalized in the impulse response to remove the effect of these modes, and the mode equalizer order is increased, including optimization of the already determined mode equalizer coefficients, until sufficient mode equalization is achieved. Case examples demonstrated that mode equalization can achieve at least rapid initial decay with type I designs. At low frequencies the control of modal decay was achieved also in a real room (case III) over a listening area reasonably sized for a sound engineer. Case III also demonstrated that the design method is able to deal with closely spaced modes in a real room because of iterative design and optimization of modal filter coefficients. The conventionally applied equalization target is to equalize a system to approximate a unity transfer function, causing no change to the magnitude or phase within a certain bandwidth of interest. An example of a room having a transfer function very close to unity is the anechoic room. This may not be a very practical equalization target for loudspeakers in real rooms as the goal of room equalization is usually not to render a listening room anechoic by eliminating the room resonances entirely, but to improve the perceived quality of sound reproduction in a listening environment. In fact, listeners may even prefer to hear some room response in the form of liveliness creating a spatial impression and some envelopment [20]. The conventional goal of practical listening room equalization is to minimize or eliminate the subjective experience of coloration in audio due to room effects. Magnitude equalization can achieve a subjectively flat magnitude response at the listening location for either the early arriving or the steady-state sound. Both can improve the perceived audio quality for compromised loudspeaker room systems significantly. However, coloration due to the reverberant sound field and slowly decaying sound due to high-q room modes cannot be sufficiently controlled with traditionalmagnitude equalization. Such coloration may deteriorate the sound clarity and definition and become audible as a change in the temporal characteristics of the audio, or it may cause a change in the frequency-domain characteristics, becoming audible as a change in the timbre of audio. Modal equalization is a novel approach specifically addressing problematic modal resonances. By decreasing their Q value, excessively long decay rates in low frequencies can be reduced and set similar to decay rates at higher frequencies, in order to minimize the audibility of energy storage in low-frequency modes. Because modal equalization also reduces the gain of mode resonances, it does produce an amount of magnitude equalization as well, although it does not guarantee magnitude equalization, and conventional magnitude equalization must typically be performed after modal equalization to optimize the audio quality. It is important to note that conventional-magnitude equalization does not achieve modal equalization as a byproduct, nor does a modal equalizer necessarily achieve a system response subjectively flat in frequency. There is no guarantee that zeros in a magnitude equalizer transfer MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES function have correct gain or frequency to achieve control of mode resonance decay time. In fact, this is rather improbable. Modal equalization should be used with conventional magnitude equalization because modal equalization alone does not achieve a subjectively flat magnitude response. There are several reasons why modal equalization is particularly interesting at low frequencies where passive means to control the decay rate by room absorption may be prohibitively expensive, fail because of constructional faults, or simply were not even considered at the time a facility was constructed. At higher frequencies methods to control modes are economical to implement and readily available. Furthermore, modal equalization becomes technically feasible at low frequencies, where the wavelength of sound becomes large relative to the room size and objects in the room, rendering the sound field no longer diffuse, and therefore making (at least) local control of the sound field progressively easier to achieve with decreasing frequency. Modal equalization at low frequencies does not only address the steady-state sound in a room. By modifying the goodness of the modal resonances it actually modifies the transfer function in the room, at least in the vicinity of the design target point, causing related changes to the transient properties of the room as well. At low frequencies, where the sound field is no longer diffuse, it may be misleading to talk about the direct sound or steadystate sound. Modal equalization modifies both the transient sound and the steady-state sound at low frequencies. This can be seen in the case examples in both the waterfalls and the impulse responses. The choice of the mode detection and parameter identification method affects the success of modal equalizer design. The modal equalizer design method suggested in this paper can also model closely spaced and coupled modes with different decays. It tends to place a single or a few equalizers to account for very closely spaced modes, and may not be able to discern every individual mode affecting the decay in that case. A modal equalizer technique based on applying ARMA modeling has recently been suggested [12], and may provide additional robustness in equalizing closely spaced or strongly coupled modes with complex decay characteristics. What should then be the design target for the decay rate at low frequencies? Recommendations [9] [11] suggest that it is psychoacoustically desirable to have approximately equal reverberant decay rates 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. We proposed also another possible target, based on the statistical distribution of decay rates within the frequencies of interest, with a similar psychoacoustical motivation. These targets may serve as a starting point, but further research is needed in order to find the limits to modal decay perception in order to determine a psychoacoustically well-founded design target for modal equalization. Attempts toward this have been made quite recently in the area of psychoacoustic study [21], [22] and modeling [23], [24]. Finally, because it appears clear that conventional- J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 341

19 MÄKIVIRTA ET AL. magnitude equalization should be applied with a modal equalizer, there is the question of the incremental benefit of applying a modal equalizer once magnitude equalization has been performed. How much does the sound quality increase by also applying modal equalization? This question is a subject of future study and particularly relevant. A recent preliminary listening test by the authors appears to suggest that the incremental improvement of applying modal equalization may be small compared to the initial incremental improvement provided by proper magnitude equalization, in itself not able to equalize the modes in a room [25]. 8 SUMMARY AND CONCLUSIONS In this paper we introduced the principle of modal equalization and formulations for type I and type II modal equalization filters. The type I system implements modal equalization by a filter in cascade with the main sound source, modifying the sound input into the room. The type II system implements modal equalization by one or more secondary sources in the room, requiring a correction filter producing a compensatory sound for each secondary source. Methods for identifying and modeling modes in impulse response measurements were presented. The error sensitivity of modeling and the implementation of system transfer function poles were discussed. Case examples of mode equalizers with demonstrations of the design procedure were given of both simulated and real rooms. The spatial robustness of modal equalization in a real room was demonstrated. Finally, aspects related to the implementation of both type I and type II mode equalizer filters were discussed. Modal equalization is a method to control the reverberation in a room when conventional passive means are not feasible, do not exist, or would present a prohibitively high cost. Modal equalization is an interesting design option, particularly for low-frequency room reverberation control when the mode density is not very high. Conventional-magnitude response equalization and modal equalization supplement each other in optimizing the psychoacoustical quality of a monitoring space. 9 ACKNOWLEDGMENT This study is part of the VÄRE technology program project TAKU (Control of Closed Space Acoustics) funded by the National Technology Agency of Finland (Tekes). Until the fall 2001 the work of Vesa Välimäki was financed with a postdoctoral research grant by the Academy of Finland. Part of this work was conducted when he was with the Pori School of Technology and Economics, Tampere University of Technology, Pori, Finland, during the academic year In 2003 the work by Poju Antsalo was partially funded by the Spatial Audio and Room Acoustics (SARA), Academy of Finland project no REFERENCES [1] A. R. Groh, High-Fidelity Sound System Equalization by Analysis of Standing Waves, J. Audio Eng. PAPERS Soc., vol. 22, pp (1974 Dec.). [2] S. J. Elliott and P. A. Nelson, Multiple-Point Equalization in a Room Using Adaptive Digital Filters, J. Audio Eng. Soc., vol. 37, pp (1989 Nov.). [3] 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, pp (1994 Dec.). [4] J. Mourjopoulos, Digital Equalization of Room Acoustics, presented at the 92th Convention of the Audio Engineering Society, J. Audio Eng. Soc. (Abstracts), vol. 40, p. 443 (1992 May), preprint [5] J. Mourjopoulos and M. A. Paraskevas, Pole and Zero Modelling of Room Transfer Functions, J. Sound Vibr., vol. 146, pp (1991). [6] R. P. Genereux, Adaptive Loudspeaker Systems: Correcting for the Acoustic Environment, in Proc. AES 8th Int. Conf. (Washington, DC, 1990 May), pp [7] F. E. Toole and S. E. Olive, The Modification of Timbre by Resonances: Perception and Measurement, J. Audio Eng. Soc., vol. 36, pp (1988 Mar.). [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, pp (1997 Mar.). [9] ITU-R BS , Methods for the Assessment of Small Impairments in Audio Systems Including Multichannel Sound Systems, International Telecommunications Union, Geneva, Switzerland (1994). [10] AES Tech. Comm. on Multichannel and Binaural Audio Technology (TC-MBAT), Multichannel Surround Sound Systems and Operations, Tech. Doc., version 1.5 (2001). [11] EBU Tech. Doc , Listening Condition for the Assessment of Sound Programme Material: Monophonic and Two-Channel Stereophonic, 2nd ed. (1998). [12] 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, pp (2002 Dec.). [13] J. D. Bunton and R. H. Small, Cumulative Spectra, Tone Bursts, and Apodization, J. Audio Eng. Soc., vol. 30, pp (1982 June). [14] J. 0. Smith and X. Serra, PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation, in Proc. Int. Computer Music Conf. (Urbana, IL, 1987), pp [15] M. Desainte-Catherine and S. March, High- Precision Fourier Analysis of Sounds Using Signal Derivatives, J. Audio Eng. Soc., vol. 48, pp (2000 July/Aug.). [16] 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, pp (2002 Nov.). [17] M. R. Schroeder, New Method of Measuring Reverberation Time, J. Acoust. Soc. Am., vol. 37, pp (1965). [18] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, 342 J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May

20 Optimization by Simulated Annealing, Science, vol. 220, pp (1983). [19] K. Steiglitz, A Note on Constant-Gain Digital Resonators, Computer Music J., vol. 18, no. 4, pp (1994). [20] R. Walker, Equalisation of Room Acoustics and Adaptive Systems in the Equalisation of Small Room Acoustics, presented at the AES 15th Conf. (1998 Oct.), paper [21] T. I. Niaounakis and W. J. Davies, Perception of Reverberation Time in Small Listening Rooms, J. Audio Eng. Soc., vol. 50, pp (2002 May). [22] L. Fielder, Analysis of Traditional and Reverberation- MODAL EQUALIZATION OF LOUDSPEAKER ROOM RESPONSES Reducing Methods of Room Equalization, J. Audio Eng. Soc., vol. 51, pp (2003 Jan/Feb.). [23] T. Paatero and M. Karjalainen, New Digital Filter Techniques for Room Response Modeling, presented at the AES 21st Int. Conf., St. Petersburg, Russia (2002 June 1 3). [24] M. Avis, Q Factor Modification for Low- Frequency Room Modes, presented at the AES 21st Int. Conf., St. Petersburg, Russia (2002 June). [25] P. Antsalo, M. Karjalainen, A. Mäkivirta, and V. Välimäki, Comparison of Modal Equalizer Design Methods, presented at the 114th Convention of the Audio Engineering Society, Amsterdam, The Netherlands (2003 Mar.). THE AUTHORS A. Mäkivirta P. P. Antsalo V. Välimäki Aki Mäkivirta was born 1960 in Jyväskylä, Finland. He received Diploma Engineer, Licentiate in Technology, and Doctor of Science in Technology degrees in electrical engineering from Tampere University of Technology, Tampere, Finland, in 1985, 1989, and 1992, respectively. His doctor s thesis described applications of nonlinear signal processing methods for haemodynamic monitoring in medical intensive care. In 1983, Dr. Mäkivirta joined the Medical Engineering Laboratory of the Research Center of Finland at Tampere, where he worked in various research positions in the field of biomedical signal analysis. In 1990 he joined Nokia Corporation Research Center in Tampere, Finland, where he served as a project manager, and after 1992 was the research manager responsible for building a research group for DSP applications in television audio and high-quality loudspeaker reproduction. In 1995 he joined Genelec Oy, Iisalmi, Finland, where he is a R&D manager responsible for digital system and signal processing applications. Dr. Mäkivirta is a member of the AES and IEEE. He holds 13 patents, and he is the author of more than 40 journal and conference papers. Poju Pietari Antsalo was born in Helsinki, Finland, in He studied electrical and communications engineering at the Helsinki University of Technology, where he obtained a Master's degree in Since then, he has been carrying out research in low-frequency room acoustics at the Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing. antsalo@cc.hut.fi. The biography of Matti Karjalainen was published in the 2003 January/February issue of the Journal. Vesa Välimäki was born in Kuorevesi, Finland, in He received Master of Science in Technology, Licentiate of Science in Technology, and Doctor of Science in Technology degrees in electrical engineering from the Helsinki University of Technology (HUT), Espoo, Finland, in 1992, 1994, and 1995, respectively. Since 1990, Dr. Välimäki has worked mostly at the HUT Laboratory of Acoustics and Audio Signal Processing with the exception of a few periods. In 1996 he spent six months as a postdoctoral research fellow at the University of Westminster, London, UK. During the academic year he was professor of signal processing at the Pori School of Technology and Economics, Tampere University of Technology, Pori, Finland. In August 2002 he returned to HUT, where he is currently professor of audio signal processing. In 2003 he was appointed a docent in signal processing at the Pori School of Technology and Economics. He lectures on digital audio signal processing at the HUT, the Pori School of Technology and Economics, and the Centre for Music and Technology, Sibelius Academy, Helsinki, Finland. His research interests include sound synthesis and digital filter design. Dr. Välimäki is a senior member of the IEEE Signal Processing Society and is a member of the AES, the Acoustical Society of Finland, and the Finnish Musicological Society. He was papers chair of the AES 22nd International Conference on Virtual, Synthetic, and Entertainment Audio, 2002 June, Espoo, Finland. J. Audio Eng. Soc., Vol. 51, No. 5, 2003 May 343