Gil-Cacho, Pepe

Jensen, Søren Holdt

Moonen, Marc

van Waterschoot, Toon

English

Strong acoustic resonances create long room impulse responses (RIRs) which may harm the speech transmission in an acoustic space and hence reduce speech intelligibility. Equalization is performed by cancelling the main acoustic resonances common to multiple room transfer functions (RTFs), i.e., common-acoustical-poles, in the room. This paper discusses the utilization of different norms (i.e., 2-norm and 1-norm) and models (i.e., all-pole and pole-zero) for RTF modelling and then equalization. Acoustic resonances may be modelled by means of the poles of the RTF. In the literature, however, it is not clear what model (i.e., all-pole or pole-zero) generates pole estimates that perform better in RTF equalization. Furthermore, least squares error (i.e., 2-norm) minimization is typically employed for the estimation of the poles. In this paper a least absolute error (i.e., 1-norm) minimization is further proposed for pole estimation. A comparative evaluation of these different norms and models in terms of their residual RTF and residual RIR (i.e., the residuals after equalization) is provided

    Aalborg UniversitetEstimation of acoustic resonances for room transfer function equalizationGil-Cacho, Pepe; van Waterschoot, Toon; Moonen, Marc; Jensen, Søren HoldtPublished in:Proceedings of the International Workshop in Acoustic Echo and Noise Control (IWAENC)Publication date:2010Document VersionEarly version, also known as pre-printLink to publication from Aalborg UniversityCitation for published version (APA):Gil-Cacho, P., van Waterschoot, T., Moonen, M., & Jensen, S. H. (2010). Estimation of acoustic resonances forroom transfer function equalization. Proceedings of the International Workshop in Acoustic Echo and NoiseControl (IWAENC). ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/pepegilcacho/reports/10-127.pdfGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: November 29, 2020Estimation of acoustic resonances for room transfer functionequalizationPepe Gil-Cacho,Toon van Waterschootand Marc Moonen.Katholieke Univeristeit LeuvenESAT-SCD, Kasteelpark Arenberg 10B-3001 Leuven, Belgium.Søren Holdt Jensen.Aalborg UniversityDept. Electronic SystemsNiels Jernes Vej 12, DK-9220.Aalborg, DenmarkAbstract—Strong acoustic resonances create long room impulseresponses (RIRs) which may harm the speech transmission in anacoustic space and hence reduce speech intelligibility. Equalizationis performed by cancelling the main acoustic resonances commonto multiple room transfer functions (RTFs), i.e., common-acoustical-poles, in the room. This paper discusses the utilization of differentnorms (i.e., 2-norm and 1-norm) and models (i.e., all-pole and pole-zero) for RTF modelling and then equalization. Acoustic resonancesmay be modelled by means of the poles of the RTF. In the literature,however, it is not clear what model (i.e., all-pole or pole-zero)generates pole estimates that perform better in RTF equalization.Furthermore, least squares error (i.e., 2-norm) minimization istypically employed for the estimation of the poles. In this paper a leastabsolute error (i.e., 1-norm) minimization is further proposed for poleestimation. A comparative evaluation of these different norms andmodels in terms of their residual RTF and residual RIR (i.e., theresiduals after equalization) is provided.I. INTRODUCTIONThe sound transmission characteristics between a loudspeaker anda microphone are described in the frequency domain by the roomtransfer function (RTF) or in the time domain by the room impulseresponse (RIR) which depends on the loudspeaker-microphone po-sition. In some audio applications (e.g., sound reproduction systemsin train stations or other large spaces) where speech intelligibilityis an issue, an equalization filter is commonly used to compensatefor the frequency response of the room. Such a filter may remove,by inverse filtering, acoustical artefacts present in spaces with longRIR originating from strong acoustic resonances. The equalizationperformance then depends on the model from which the inverse filteris derived. The most common model for the RTF is an all-zero model[1]. It represents the physics of room acoustics where a microphonesignal is a weighted sum of discrete reflections of the loudspeakersignal. Its drawback is that any change in the loudspeaker-microphoneor obstacle position inside the room will change all the coefficientsof the model. The equalization filter is then a single point inversefilter which is only valid for a single point in the room and thereforea recalculation of every coefficient will be needed at each and everyother point in the room [6].One of the alternatives is to use a pole-zero model for the RTF[1]. This model represents the physics of room acoustics by alsoincluding the modelling of the acoustic resonances by means ofthe poles of the transfer function. Poles can represent long impulseresponses caused by resonances, while zeros represent time delaysand antiresonances [7]. Yet another alternative model is the all-pole model, which represents only the acoustic resonances in aneffort to model the RTF spectral envelope [1]. This provides anappropriate strategy to cancel only the main resonances discardingthe cancellation of the zeros. Moreover, the concept of commonacoustical poles, first introduced in [7], can be applied to these twoalternative models. The underlying idea is that the acoustic resonancesin a room depend on the dimensions and shape of the enclosure andnot on the loudspeaker-microphone position. Each RTF in the roommay then be expressed using a common set of poles and differentzeros. Hence an inverse filter that cancels the common acousticalpoles can be created that equalizes the physically common mainresonances in multiple points in the room [6]. Finding the polesof the transfer function may be seen as an optimization problemin which the set of poles that minimizes the error between themodel and the measurements (i.e., the actual impulse response) issought for. Different models and error criteria will obviously renderdifferent pole estimates and consequently different equalization filtersand residual RTFs. Typically a least squares error criterion (2-normminimization) is employed. In the literature, however, it is not clearwhich of these models generate pole estimates that perform better inthe RTF equalization contest.In this paper the use of a least absolute error criterion (1-normminimization) for RTF pole estimation is proposed. In [2] sparselinear prediction of speech signals is used to obtain sparse residualswith minimum number on non-zeros elements. With this idea in mind,1-norm minimization is proposed here to calculate the poles of anall-pole model, so that the inverse filtering will render a residualimpulse response having discrete separated reflection rather than adense impulse response. On the other hand, it is found that the polescalculated from a pole-zero model using 1-norm minimization givesresults similar to the 2-norm pole-zero approach and therefore thiswill not be considered any further. The aim of this paper is hence tomake a comparative evaluation involving 2-norm minimization usingan all-pole and a pole-zero model and 1-norm minimization using anall-pole model. The comparative evaluation will be presented both intime and frequency domain. Several questions on how the choice of amodel and norm affects the residual RTF and RIR will be addressed.The paper is organized as follows. In section 2, the mathematicalformulation is presented for each model. In section 3, equalizationresults using the presented techniques on real measured room impulseresponses are presented. Finally, section 4 concludes the paper.II. POLE ESTIMATION USING DIFFERENT NORMSAlthough the RTFs are different for each loudspeaker-microphoneposition, all RTFs in a room share the same resonance frequencies.These resonance frequencies may be visible as spectral peaks in theRTFs [1]. If only the zeros cause RTF variation then the RTFs canbe expressed using a common denominator for all and a differentnumerator for each of them. This can be represented by eithercommon poles, p(k), and distinct zeros, zi(k, t), or in polynomialform using common autoregressive (AR), a(k), and distinct movingaverage (MA), bi(k, t), coefficients [7],Hi(q, t) =∏Qk=1(1− zi(k, t)q−1)∏Pk=1(1− p(k)q−1)=∑Qk=1bi(k, t)q−k1−∑Pk=1a(k)q−k(1)where Q and P are the order of numerator and denominator respec-tively, i = 1, ...,M the number of RTFs and where q denotes thetime shift operator, i.e., q−ku(t) = u(t− k).Mathematically the class of problems considered in this papercan be cast into one general optimization problem associated withfinding the filter coefficient vector x from a set of measured impulseresponses cast in v and W, so that the error e = v−Wx is minimizedminx‖v−Wx‖pp (2)where ‖·‖p is the p-norm defined for p > 1 as‖x‖p =(∑Nt=1|x(t)|p)1/p. Matrix W and vector v involvedin the minimization problem (2) depend on whether the modelfor pole estimation is all-pole or pole-zero and whether commonacoustical poles are considered. In the pole-zero model with commonacoustical poles case, matrix W and vector v are formed asv = [h1, h2, ..., hM ]Thi = [hi(0), hi(1), ..., hi(N − 1), 0, 0, 0]Tx = [a, b1, ..., bM ]Ta = [a(1), a(2), ..., a(P )]Tbi = [bi(0), bi(1), ..., bi(Q)]TW =W1 D 0 0 0W2 0 D 0 0... · · · 0. . . 0WM · · · · · · · · · D⇒ size [M(N + P − 1)× (P +M(Q+ 1))]D =11 0·0 10 0· ·· ·0 · · 0⇒ size [M(N + P − 1)× (P +M(Q+ 1))](3)Wi =0 0 · · · 0hi(0) 0 · · · 0hi(1) hi(0) · · · 0..................... hi(0)hi(N − 1) 0 · · ·...0 hi(N − 1) · · ·......... · · ·...0 0 · · · hi(N − 1)⇒ size [(N + P − 1)× P ](4)where hi is the ith length-N measured impulse response. If an all-pole model is considered, matrix D and vectors bj are just set tozero and vector x will only consist of AR coefficients (i.e., x =[a]). Conversely, when a pole-zero model is considered, vector x willconsist of both AR and MA coefficients (i.e., x = [a, b1, ..., bM ]T ).The set of a coefficients is estimated using either 2-norm or 1-normminimization and forms the filter polynomial A(q) = 1−a(1)q−1−... − a(P )q−P . The residual B̃i(q) is the result of multiplying theactual RTF with the inverse filter Ã(q).Ã(q)Hi(q, t) = Ã(q)Bi(q)A(q)= B̃i(q) (5)where Ã(q) ∼= A(q)−1. The description of B̃i(q) in both time andfrequency domain (i.e., residual RIR and residual RTF respectively)is directly affected by how Ã(q) is calculated. In 2-norm (i.e., leastsquares error) minimization the optimal filter coefficient vector maybe given in closed-form asx =(WT W)−1WT v (6)However in 1-norm (i.e., least absolute error) minimization thereexists no closed-form solution and therefore the filter coefficientvector is calculated as a solution to a convex optimization problem,which can be solved efficiently, e.g. using CVX [8]. It is found, ex-perimentally, that similar ARMA coefficients estimates are extractedfrom a pole-zero model using 1-norm or 2-norm minimization, soonly 2-norm pole-zero model will be considered in the sequel.III. RESULTS FROM MEASURED IMPULSE RESPONSESIn this section results obtained from two measured room impulseresponses are presented. The three methods (i.e., 1-norm all-pole and2-norm all-pole and pole-zero) are compared by inspecting both theresidual RTF and the residual RIR. Matlab computer simulations wereperformed at fs = 16kHz. The impulse responses, shown in Fig.1, (h1 and h2) of length N = 2001 samples were measured in arectangular room of about 5 × 3 × 3 m. The order was chosen asP = 500 and in the ARMA case also Q = 500. This order was foundto be sufficient to accurately model the measured impulse response inthe ARMA case, which served as a quality reference. Two objectivemeasures are employed:• Spectral Flatness (SF )The spectral flatness is calculated by dividing the geometric mean ofthe power spectrum by the arithmetic mean of the power spectrum,i.e.SF =N√∏N−1f=0P (f)∑N−1f=0P (f)N(7)where P (f) represents the magnitude of the f th bin, with N = 512.• Sparseness Degree (SD)The sparseness degree is the number of elements in the residual RIRthat have an absolute value smaller than some threshold close to zero,i.e.SD = {R(n) : |R(n)| < threshold} (8)where R(n) represents the residual RIR and the threshold is set to2 · 10−6.Two different cases for each method are presented: In the first case,the coefficients of the inverse filter are calculated from one impulseresponse h1 (shown in Fig. 1 a). This filter is used to equalize h1so as to clearly observe the residual RTF and residual RIR. In thesecond case, the coefficients of the inverse filter to equalize h1 arecalculated using the set of two measured impulse responses (i.e., withcommon acoustical poles). The frequency response of h1 is shownin Fig. 1 b.• First caseWhen Ã(q) is calculated from the all-pole model using 2-normminimization (Fig. 2(a), 2(d)) the flattest residual RTF is achieved.On average, the magnitude difference between peaks and dips is verysmall; however, it exhibits a long noise-like residual RIR because the2-norm minimization shapes the residual into coefficients that exhibitGaussian like features [2].When Ã(q) is calculated from the pole-zero model using 2-normminimization (Fig. 2(b), 2(e)) a highly coloured residual RTF isachieved, which is a perceptually undesirable characteristic in RTFequalization [5]. However, the short residual RIR may be desirablein other audio applications such as acoustic echo or feedback can-cellation [3], [4].When Ã(q) is calculated from the all-pole model using 1-normminimization (Fig. 2(c), 2(f)), the residual RTF has been flattenedwith respect to the true RTF and, in addition, the main low-frequencyresonances have been cancelled. The residual RIR exhibits a sparsedistribution of non-zero coefficients, which implies that the acousticreflections modelled by the residual RIR are forced to be more spacedin time. The residual RIR from the 1-norm all-pole model showsthe largest number of zero coefficients, i.e., the highest degree ofsparseness. These results are summarized in Table I, where it canbe seen that the 1-norm all-pole model and 2-norm all-pole modelpresent the highest SD and the highest SF respectively.• Second caseFig. 3 shows that although in the common-acoustical-poles case theoverall performance has been deteriorated with respect to the singleimpulse response case, the same main features can be observed onthe residuals.TABLE I: SF and SD for the 1-norm all-pole, 2-norm all-pole andpole-zero case2-norm all-pole pole-zero 1-norm all-poleSF 0.7 0.06 0.2SD 5 43 503IV. CONCLUSIONSIn this paper, results from RTF equalization have been presented.Equalization is achieved by cancelling the poles associated withthe main resonances common to multiple RTFs in a room. Poleswere estimated by means of three different methods. Poles estimatedusing 2-norm minimization and an all-pole model offered a residualRTF having the flattest response. Poles estimated using 2-normminimization and a pole-zero model offered a large reduction in themain low-frequency acoustic resonances. However the residual RTFexhibited a highly coloured response, while the residual RIR wasshorter in time. Finally, poles estimated using 1-norm minimizationand an all-pole model offered a flat residual RTF with its mainresonances cancelled and a sparse residual RIR. This means that theresidual RIR will represent sparsely distributed discrete reflections.This feature may be desirable in speech applications although adeeper study taking into account perceptual considerations would beneeded.V. ACKNOWLEDGEMENTSThis research work was carried out at the ESAT Laboratory of Katholieke UniveristeitLeuven, in the frame of the EC FP6 project SIGNAL: ’Core Signal ProcessingTraining Program’ Marie-Curie Fellowship program (http://est-signal.i3s.unice.fr) undercontract No. MEST-CT-2005-021175, the Concerted Research Action GOA-MaNet andthe Belgian Programme on Interuniversity Attraction Poles initiated by the BelgianFederal Science Policy Office IUAP P6/04 (DYSCO, ‘Dynamical systems, control andoptimization’, 2007-2011). The scientific responsibility is assumed by its authors.REFERENCES[1] J. Mourjopoulos and M. A. Paraskevas, “Pole and zero modeling ofroom transfer functions”, J. Sound and Vibration, vol. 146, no. 2, pp.281-302, April 1991.[2] J. A. Cadzow, “Minimum l1, l2, and l∞ Norm Approximate Solutionsto an Overdetermined System of Linear Equations”, Dig. Sig. Proc., vol.12, no. 4, pp. 524-560, Oct. 2002.[3] T. van Waterschoot and M. Moonen, “Adaptive feedback cancellation foraudio applications,” Signal Processing, vol. 89, no. 11, pp. 2185-2201,Nov. 2009.[4] P. S. R. Diniz, Adaptive Filtering: Algorithms and Practical Implemen-tations. Springer, Boston, MA., 2008.[5] D. M. Howard and J. A. S. Angus, Acoustics and Psychoacoustics. FocalPress, Elsevier, 2006.[6] Y. Haneda, S. Makino and Y. Kaneda, “Multiple-Point Equalization ofRoom Transfer Functions by Using Common Acoustical Poles,” IEEETrans. Speech Audio Process., vol. 5, no. 4, pp. 325-333, July 1997.[7] Y. Haneda, S. Makino and Y. Kaneda, “Common Acoustical Pole andZero Modeling of Room Transfer Function,” IEEE Trans. Speech AudioProcess., vol. 2, no. 2, pp. 320-328, Apr. 1994.[8] M. Grant and S. Boyd, CVX: Matlab software for disciplined convexprogramming (web page and software). http://cvxr.com/cvx, April, 2010.(a) Time evolution of h1 and h2 (b) Spectrum of h1 and h2 (RTF1 and RTF2)Fig. 1: Frequency response function and time evolution of h1 and h2(a) 2-norm all-pole model (b) pole-zero model (c) 1-norm all-pole model(d) 2-norm all-pole model (e) pole-zero model (f) 1-norm all-pole modelFig. 2: a,b,c) h1 residual RIRs and d,e,f) h1 residual RTFs, form single impulse response estimation(a) 2-norm all-pole model (b) pole-zero model (c) 1-norm all-pole model(d) 2-norm all-pole model (e) pole-zero model (f) 1-norm all-pole modelFig. 3: a,b,c) h1 residual RIRs and d,e,f) h1 residual RTFs, from common-acoustical-poles calculation

Estimation of acoustic resonances for room transfer function equalization

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Estimation of acoustic resonances for room transfer function equalization

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