DETECTION OF ROOM REFLECTIONS FROM A BINAURAL ROOM IMPULSE RESPONSE

Transcription

1 Sampo Vea and Tapio Lokki Detection of room reflection from a binaural room impule repone. In: Proceeding of the 9th International Conference on Digital Audio Effect (DAFx 2006). Montreal, Canada September 2006, page by author

2 DETECTION OF ROOM REFLECTIONS FROM A BINAURAL ROOM IMPULSE RESPONSE Sampo Vea, Tapio Lokki Telecommunication Software and Multimedia Laboratory Helinki Univerity of Technology ampo.vea@tml.hut.fi ABSTRACT A novel analyi method for binaural room impule repone (BRIR) i preented. It i baed on the analyi of ear canal ignal with continuou wavelet tranform (CWT). Then, the crowavelet tranform (XWT) i ued for detection of the direct ound and individual reflection from a BRIR. The new method eem to time-localize the reflection quite accurately. In addition, the propoed analyi method enable detailed tudy of the frequency content of the early reflection. The algorithm i teted with both meaured and modeled impule repone. A comparion with an FFT-baed cro-pectrogram i made. The reult how that XWT ha potential in audio ignal analyi. 1. INTRODUCTION In many cae detailed time cale analyi of binaural impule repone i needed. For example, a dene pattern of early reflection i uually aociated with good concert hall acoutic and it would be great to be able to tudy thee reflection individually from meaured repone. One practical application in which detail of early reflection i needed i the auralization in low motion uing meaured binaural impule repone [1]. When uing imulated (binaural) impule repone the auralization in low motion i imple: the peed of ound i lowed down by a certain factor in the auralization. Thi allow perception of the time and direction of arrival of individual reflection. If meaured binaural repone are to be lowed down, the ituation become more complicated, ince individual reflection hould be very accurately localized in time in order to iolate them from the original repone. Thi call for a method that can time-localize the individual reflection a accurately a poible. The problem tudied in thi paper i time-localization of early reflection from a binaural impule repone. It i aumed that if a reflection occur there i correlation in hort time window between left and right ear canal ignal. Auditorily motivated approache typically employ filter bank and calculation of cro-correlation between channel. Here, a more ignal proceing oriented approach i taken. An obviou method would be the frame-wie calculation of Fourier cro-pectrum between the left and right channel. A wavelet-baed approach wa hypotheized to be a better alternative, becaue of better time reolution compared to framebaed approache. Therefore, both frame-baed and wavelet-baed approache are evaluated in the tudy. An introductory paper on wavelet analyi with application to time erie analyi [2] provided inpiration for uing wavelet method for localizing the reflection. Wavelet analyi, in the form of filter bank decompoition, ha been ued for approximating room impule repone in imulation [3]. The continuou wavelet tranform (CWT) ha alo been applied to audio ignal proceing previouly for noie reduction and ignal compreion [4], intermodulation effect analyi [5], ound ynthei [6] and ound ignal modeling [7]. The CWT ha alo been ued for decompoition of room and loudpeaker impule repone [8]. In the current work, the continuou, non-orthogonal and complex cro-wavelet tranform (XWT) i ued, becaue the interet i in the correlation between two time erie, i.e., the left and right ear canal ignal of a binaural impule repone. A far a we know, the cro-wavelet tranform eem not to be applied to audio ignal proceing previouly. 2. THE CONTINUOUS WAVELET TRANSFORM AND THE CROSS-WAVELET TRANSFORM The CWT i a method that can be ued for time erie analyi. It give a highly redundant time-frequency repreentation of a time erie, being very different from the dicrete wavelet tranform (DWT), which give a compact repreentation of the ignal and i thu better uited for ignal proceing [2]. For time erie analyi, the CWT i much more uitable. The DWT alo ha the diadvantage of an aperiodic hift in the time erie giving a different wavelet pectrum [2]. The continuou wavelet tranform for a ignal x(t) i defined a [9, 2, 8]: W x(t, ) = 1 Z «x(t )ψ t t dt (1) where the aterik ( ) indicate complex conjugation, t i time (tranlation), i cale (dilation) and ψ(t) i the wavelet function. For a dicrete equence, the continuou wavelet tranform i defined a a convolution um [2]: W x(n, ) = 1 N 1 X n =0 «x(n )ψ n n For more efficient computation, the Fourier tranform of Eq. (2) i ued. Two complex wavelet are ued in thi tudy, the firt one being the Morlet wavelet, defined a [2]: ψ(t) =π 1/4 e jω0t e t2 /2 (3) where t i a dimenionle time parameter and ω 0 i a nondimenional frequency. Since the CWT can alo be een a a filter bank, ω 0 i alo called the center or ocillating frequency of the wavelet. A wavelet with a better time reolution and poorer frequency reolution, compared to the Morlet wavelet, i the Paul wavelet [2]: (2) DAFX-215

3 ψ(t) = 2m j m m! p π(2m)! (1 jt) (m+1) (4) where m i the order of the Paul wavelet. Becaue the interet in audio ignal analyi i on frequency, the cale (non-dimenional) hould be converted into frequency f (in Hz), uing the following equation [8]: f = ff0 (5) where f i the ampling frequency (in Hz) and f 0 = ω0 i the 2π non-dimenional wavelet center frequency. It hould be noted that Eq. (5) only hold for the Morlet wavelet. When generalizing to all poible analyzing wavelet, f 0 can be een a a proportionality contant which depend on the particular choice of wavelet bae and order/center frequency. The relationhip between cale and frequency for different wavelet can be found in [2]. The CWT enable alo to tudy imilaritie of two ignal in the ame way a FFT-baed cro-pectrum. The equivalent CWTbaed tool i the cro-wavelet tranform (XWT) [10], aloknown a cro-wavelet pectrum: W xy(t, ) =W x(t, )Wy (t, ) (6) Becaue the interet in thi paper i on time-localizing reflection, we are only concentrated on the power of the XWT, which i ued for all plot. The phae of the XWT could poible be ued for calculating the azimuth angle of each individual reflection, but uch a tudy i left for future work. 3. CROSS-WAVELET ANALYSIS OF BRIRS The cro-wavelet tranform give information on the dependence between two ignal a a function of time, imilar to cro-correlogram (or the cro-pectrogram, i.e., the hort-time cro-pectrum preented a a function of time). In our cae thee two ignal are the left and right ear canal ignal of a binaural impule repone. Therefore, the XWT hould be ueful in localizing individual reflection of a binaural room impule repone, which manifet themelve a correlation between left and right ear ignal, the time lag of the correlation maximum being proportional to the azimuth angle of the reflection. Fig. 1 compare the Fourier cro-pectrogram and crowavelet tranform calculated from a BRIR, meaured in a mall room (the litening room of the Laboratory of Acoutic and Audio Signal Proceing at TKK, ee Fig. 2). The Fourier pectrogram wa calculated uing a time-domain window length of 64 ample and a very mall hop ize of 2 ample. The time-domain window were zero-padded to yield a 512-point FFT per frame. The cro-wavelet tranform were calculated uing the Morlet and Paul wavelet, the cale ranging logarithmically (bae 2) from 2 to 512 with 24 cale per octave, reulting in the total number of cale being 193. The frequency range covered were Hz for the Morlet wavelet and Hz for the Paul wavelet. A can be een in Fig. 1, the maller detail are given more emphai in the cro-wavelet pectrogram. The difference in time and frequency reolution of the Morlet and the Paul wavelet i alo very evident. The logarithmic frequency reolution of the wavelet tranform i alo clear in the figure. It i alo clear how the time reolution of the wavelet tranform i much wore at low than high frequencie. Frequency [khz] Frequency [khz] Frequency [khz] Time [m] Figure 1: Comparion between the Fourier cro-pectrogram (top panel) and the cro-wavelet pectrogram calculated uing the Morlet (middle panel) and the Paul (bottom panel) wavelet. 4. FINE-SCALE TIME-LOCALIZATION OF REFLECTIONS Our hypothei i that each reflection manifet itelf a local maxima in the XWT on different cale at the time of the reflection, reulting in vertical tripe in the XWT plot (ee Fig. 1). In order to ue the XWT for time-localizing room reflection, we need to extract information on the time location of the tripe from the XWT. An obviou way to do thi i to integrate out the cale axi, i.e., to um along the cale in the dicrete cae, yielding a one-dimenional ignal, which i a function of time. Location of the local maxima of thi ignal hould correpond to the individual reflection. The um acro cale j 1 to j 2 i calculated from the cro-wavelet tranform W xy(n, ) a: j 2 X W um(n) = W xy(n, j) 2 (7) j=j 1 A an alternative, the maximum acro cale could be ued: W max(n) = max j [j 1,j 2] Wxy(n, j) 2 Since it i expected that um and/or maximum acro cale could have ome minor local maxima that do not correpond to reflection, Savitzky-Golay moothing [11] wa applied to them (uing the function golayfilt in MATLAB). The filter order wa 15 and the frame length wa 23 ample. It turned out, that the maximum acro cale wa mooth enough a uch and thu moothing wa only ued for the um acro cale Chooing the et of cale In contrat to orthogonal wavelet analyi, in non-orthogonal analyi the et of ued cale can be choen arbitrarily. The cale can be pecified a fractional power of two [2]: (8) i = 02 iδ i,i=0, 1,...,I (9) DAFX-216

4 0.7 m left wall 5.6 m front wall 3.8 m back wall 6.3 m 4.0 m 3.0 m 2.0 m 1.0 m 0.5 m 0 m right wall Figure 2: Poition of the binaural manikin in the room where the BRIR were meaured. The room height i 3.0 m and the litener and the ource height wa 1.5 m. The binaural room impule repone were meaured at ource-to-receiver ditance of 0.5, 1, 2, 3 and 4 meter. where 0 i the mallet cale, I i the number of cale and δ i control the cale reolution (D =1/δ i give the number of cale per octave). 5. EXPERIMENTS A an example of the XWT analyi, both meaured and modeled binaural impule repone are analyzed and tudied. Firt, the XWT i calculated for the entire meaured binaural impule repone. Thi yield a matrix ized the length of the ignal (in ample) time the number of cale the XWT wa calculated at. The cale configuration i a decribed in Section 3. Since the larget cale convey little information about the time location of the reflection, the XWT were truncated in cale to the range [2, 64], which i Hz for the Paul wavelet and Hz for the Morlet wavelet. The range wa choen by hand o that the um and maximum acro cale eem to give a reaonable amount of detail, i.e., reaonable amount of local maxima. Including the frequencie below 500 Hz reult in the um and maximum acro cale being overly mooth, and thu ome reflection will not be detected. For comparion, a tandard FFT-baed cro-pectrogram (calculated a decribed in Section 3) i analyzed in the Hz band, which i cloe to the band covered by the Paul wavelet. After calculating the XWT or FFT-baed cro-pectrogram, the um and maximum acro cale/frequency i evaluated uing Eq. (7) and (8). The curve repreenting the um acro cale i then moothed a decribed previouly. Finally, all of the local maxima of the reulting curve are located in time. For a quantitative validation of the time-localization method, the exact location of the room reflection hould be known. For modeled repone, the exact delay for each reflection are known preciely. The ituation i more complicated with meaured repone, when there i no exact information on the timing of the reflection. However, ince the room dimenion are known in thi cae (ee Fig. 2), the image-ource method can be ued to calcu- Figure 3: Top panel: the XWT of a binaural impule repone meaured in a tandard litening room (ource-to-receiver ditance 0.5 m, azimuth angle 0 ). The Paul wavelet wa ued in the analyi. Upper middle panel: maximum of the XWT acro cale, veru time (dahed line) and um of the XWT acro cale, veru time (olid line), preented on a logarithmic cale (bae 2). Lower middle panel: the binaural impule repone, with the localized reflection marked by dotted vertical line (time-localization baed on local maxima of the maximum acro cale) and the firt and econd order reflection calculated from a hoebox model marked by olid vertical line. Bottom panel: ame a the lower middle panel, but with time-localization baed on local maxima of the um acro cale. late the early reflection approximately. In thi work, only the firt and econd order reflection were calculated Qualitative evaluation Fig. 3 and 4 preent the XWT of the firt 30 m of a binaural impule repone meaured in the tandard litening room (refer to Section 3) with T , it maximum and um acro cale, and the localized reflection baed on both the maximum and um acro cale. In Fig. 3, the Paul wavelet wa the analyzing wavelet, while in Fig. 4, the Morlet wavelet wa ued. A localized reflection i et to each local maximum of the maximum or um acro cale, and i indicated by dotted vertical line. The firt and econd order reflection given by the image-ource model [12] are hown by olid vertical line. The better time reolution of the Paul wavelet (Fig. 3) i evident when compared to the Morlet wavelet (Fig. 4). The FFTbaed analyi of a meaured repone (Fig. 5) eem to have a time-localization accuracy comparable to the Paul wavelet. By viual inpection it i hard to tell which peak in the time-domain impule repone are true reflection, but uing the Paul wavelet or the FFT cro-pectrogram eem to locate many detail from the repone. Many of thee detail are likely caued by reflection. Some of the reflection given by the image-ource model alo eem to coincide with the localized reflection. The room model agree well with viual inpection of the time-domain repone in the cae of the almot imultaneou floor and ceiling reflection (at DAFX-217

5 Method Avg. ab. error [m] back wall front wall left wall right wall floor ceiling 1t ord. 1t & 2nd ord. XWT, Paul wavelet (max) XWT, Paul wavelet (um) XWT, Morlet wavelet (max) XWT, Morlet wavelet (um) Fourier cro-pectrogram (max) Fourier cro-pectrogram (um) Table 1: Average abolute error for the detected reflection uing um and maximum acro cale (meaured repone). Method Average abolute error [m] XWT, Paul wavelet (max) 0.10 XWT, Paul wavelet (um) 0.19 XWT, Morlet wavelet (max) 0.28 XWT, Morlet wavelet (um) 0.33 Fourier cro-pectrogram (max) 0.29 Fourier cro-pectrogram (um) 0.35 Table 2: Average abolute error of reflection up to 30 m uing um and maximum acro cale (artificial repone). Figure 4: Same a Fig. 3, but the Morlet wavelet wa ued in the analyi. Figure 5: Same a Fig. 3, but a tandard FFT-baed cropectrogram (with a hop ize of one ample) wa ued. 7.2 and 7.5 m, repectively, according to the model). The analyi method fue the two reflection together, except with the combination of the FFT cro-pectrogram the maximum acro frequency. The back wall reflection (at 4.2 m) i correctly localized by both viual inpection and the analyi method. However, room model may be lightly inaccurate, and therefore the three later reflection coming from the ide and the front wall may be inaccurate. By inpecting the two bottom panel of Fig. 3, it i very hard to tell, which of the bump in the repone are actually due to the three later reflection. Beide the firt and econd order reflection from the wall, higher-order reflection and reflection from object in the room appear in the repone. Fig. 6 preent the XWT analyi for an artificial modeled BRIR, uing the Paul wavelet. The repone i computed with the DIVA oftware [13] from a hoebox-haped room. The reflection are modeled up to fifth order and each reflection i proceed with the appropriate head-related tranfer function to get a binaural repone. A can be een in Fig. 6, the reflection contain energy over the whole frequency region and no background noie exit. Such fact make the artificial repone eaier for the propoed algorithm and the algorithm ha localized mot of the peak correctly. Thi i alo verified by comparing the localized reflection to the ground truth (olid vertical line) in the two bottom panel of Fig. 6. By looking at the top panel of Fig. 3, 4 and 6 it i een that the XWT can alo be ued to analyze the frequency content of the early reflection 1. A an example, the meaured repone (topmot panel of Fig. 3 and 4) contain a clear reflection (a darker area) right before the 5 m time tamp. Thi reflection contain energy 1 Becaue of it linear frequency reolution, the FFT cro-pectrogram (Fig. 5) i not o well uited for detailed viual analyi of the frequency content, at leat at the low frequencie. DAFX-218

6 Figure 6: Same a Fig. 3, but a modeled BRIR wa analyzed. The vertical olid line in the two bottom panel indicate the ground truth location for the reflection. only at middle frequencie while next reflection a little bit later contain only low frequencie. From the artificial repone analyi (Fig. 6) it can be een that modeled reflection are all wideband reflection. Such analyi might be ueful, e.g., in optimizing the loudpeaker placement in the home theater, where ome clear reflection typically occur Quantitative evaluation Quantitative analyi of the accuracy of the time-localization i preented in Table 1 and 2, where the average abolute error of the time difference between the true reflection location and the nearet localized reflection are preented. Thi wa done eparately for the firt and econd order reflection in the cae of the real repone, and for all of the reflection falling in the 30 m time window following the direct ound in the cae of the artificial repone. For the meaured repone (Table 1), the average error i taken a the arithmetic mean over the analyi reult of et of binaural room impule repone meaured at ource-toreceiver ditance of 0.5, 1, 2, 3 and 4 meter (ee Fig. 2). The average error are preented for each of the ix firt order reflection eparately. A total average for each analyi method i alo given for the firt order reflection and all calculated reflection, i.e., both firt and econd order reflection. Only the reflection falling within 30 m of the direct ound are conidered. A ingle artificial repone wa analyzed and the average in Table 2 preent the total average error over all the reflection in the 30 m time window following the direct ound, a calculated for each of the analyi method. By looking at Table 1 it eem that on average, the Paul wavelet and the FFT method perform equally well, while the Morlet wavelet perform lightly wore, on a real meaured ignal. The average abolute error are alway le than 1 m. On average, the floor reflection are clearly located more accurately than the other reflection. Beide the floor reflection being a very trong one (it i actually uperimpoed with the ceiling reflection in thi cae), the theoretically calculated reflection i cloe to the true reflection. With the other firt order reflection lited in Table 1, a well a the econd order reflection, the theoretical time location might not hold exactly due to inaccuracie in the meaured ditance (Fig. 2) ued to calculate the theoretical location. Becaue of thi, the reult preented in Table 1 give only an approximate ene of how the method perform in real pace, and compared to each other. The method alo differ in term of the denity of localized reflection. Some of the localized reflection might not actually correpond to a real reflection, but are jut tatitical fluctuation. Thi alo affect the reult of Table 1 o that a low abolute error may alo be due to a localized reflection being cloe to the theoretical location jut by chance. Table 2 indicate that for the artificial repone, the Paul wavelet perform ignificantly better than the Morlet wavelet or the FFT method. The average error of the Paul wavelet i almot half that of the Morlet wavelet or the FFT. The performance of the wavelet method i alo ignificantly better than that of with meaured repone. With the artificial repone, reflection up to fifth order are conidered, not jut the firt and econd order reflection. The FFT method eem to be unable to localize each of thee reflection individually. However, with the real repone (Fig. 5) it eem that the FFT method give more detected reflection than the wavelet method. Concluion from the uperiority of the any of the method can not be therefore drawn baed on thi. The maximum acro cale/frequency eem to be uperior to the um acro cale/frequency for each method lited in Table 1 and 2. The only exception i with the Morlet wavelet for the meaured repone. The uperiority of the maximum might be explained o that when a reflection i preent in the ignal at a certain time point, there i typically a clear peak in the XWT/cropectrogram at ome frequency at that point. When the maximum acro cale/frequency i evaluated a a function of time, a peak in the curve reult at the very ame time point, and thu the reflection i localized accurately. The um acro cale/frequency average thi peak o that the time accuracy of localization i wore and ome narrow-band reflection are eaily mied completely. On the contrary, the maximum acro cale alo give rie to many more localization, many of which may be jut due to tatitical fluctuation. From Fig. 3, 4 and 6 it i evident that there are more localized reflection when uing the maximum acro cale. 6. CONCLUSIONS The cro-wavelet tranform i applied for time-localizing reflection from binaural room impule repone. The method perform accurately for artificial repone and find almot all early reflection. Location of the firt and econd order reflection were calculated for the room and a comparion to the reflection found by the propoed method wa made. At leat the tronget of the ix firt order reflection were localized quite accurately. However, the performance with real meaured repone i hard to evaluate, ince the exact location of all of the reflection are not known. The propoed XWT with Morlet and Paul wavelet wa alo compared to the conventional FFT cro-pectrum. The Paul wavelet ha the bet time reolution, and thu eem to be the mot accurate one of the teted method, at leat in the cae of the artificially generated room repone. For the meaured repone, the FFT cro-pectrogram method gave equally good reult. Future work hould concentrate on improving the accuracy of the method. The algorithm hould be made robut o that no reflection are mied and no fale detection are made either. The DAFX-219

7 pectral content of the reflection hould alo be taken into account there might be need to differentiate between low and high frequency reflection, a well a narrowband and wideband reflection. More advanced algorithm for detecting the reflection could be developed. Worth of invetigation i alo how the direction, i.e., azimuth angle, of the individual reflection could be calculated. One poibility i the ue of the phae of the XWT for azimuth localization. 7. ACKNOWLEDGMENT A freely available MATLAB toolbox for calculating the crowavelet tranform and wavelet coherence wa ued [14]. Thi work wa partly funded by the HeCSE graduate chool. 8. REFERENCES [1] T. Lokki, Auralization of imulated impule repone in low motion, in 118th Conv. Audio Eng. Soc., Barcelona, Spain, May 2005, paper no [2] C. Torrence and G. P. Compo, A practical guide to wavelet analyi, Bulletin Am. Meteorological Soc., vol. 79, no. 1, pp , [3] M. Schönle, N. Fliege, and U. Zölzer, Parametric approximation of room impule repone baed on wavelet decompoition, in Proc. IEEE Workhop Appl. of Dig. Sig. Proc. to Audio and Acout., New Palz, NY, Oct. 1993, pp [4] P. J. Wolfe and S. J. Godill, Audio ignal proceing uing complex wavelet, in 114th Conv. Audio Eng. Soc., Amterdam, The Netherland, Mar [5] J. R. Beltrán, J. P. de León, and E. Etopiñán, Intermodulation effect analyi uing complex bandpa filterbank, in Proc. Int. Conf. on Digital Audio Effect (DAFx-05), Madrid, Spain, Sept. 2005, pp [6] J. R. Beltrán and F. Beltrán, Additive ynthei baed on the continuou wavelet tranform: a inuoidal plu tranient mode, in Proc. Int. Conf. on Digital Audio Effect (DAFx- 03), London, UK, Sept [7] P. Guillemain and R. Kronland-Martinet, Characterization of acoutic ignal through continuou linear time-frequency repreentation, Proc. IEEE, vol. 84, no. 4, pp , [8] S. J. Loutridi, Decompoition of impule repone uing complex wavelet, J. Audio Eng. Soc., vol. 53, no. 9, pp , [9] A. Cohen and J. Kovacevic, Wavelet: The mathematical background, Proc. IEEE, vol. 84, no. 4, pp , [10] A. Grinted, J. C. Moore, and S. Jevrejeva, Application of the cro wavelet tranform and wavelet coherence to geophyical time erie, Nonlinear Procee in Geophyic, vol. 11, pp , [11] W. H. Pre, S. A. Teukolky, W. T. Vetterling, and B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing. Cambridge Univerity Pre, 1992, ch [12] J. B. Allen and D. A. Berkley, Image method for efficiently imulating mall-room acoutic, J. Acout. Soc. Am., vol. 65, no. 4, pp , [13] T. Lokki, Phyically-baed auralization deign, implementation and evaluation, Ph.D. diertation, Helinki Univerity of Technology, Epoo, Finland, 2002, isbn X. [14] A. Grinted, J. C. Moore, and S. Jevrejeva, Cro wavelet and wavelet coherence MATLAB toolbox, 2006, [Online] DAFX-220