Spatiotemporal Gradient Analysis of Differential Microphone Arrays*

Transcription

1 Satiotemoral Gradient Analysis of Differential Microhone Arrays* MIHAILO KOLUNDŽIJA, CHRISTOF FALLER, AND MARTIN VETTERLI** School of Comuter and Communications Sciences, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland The literature on gradient and differential microhone arrays makes a distinction between the two tyes, but it nevertheless shows how both tyes can be used to obtain the same directional resonses. A more theoretically sound rationale for using delays in differential microhone arrays has not yet been given. A gradient analysis of the sound field viewed as a satiotemoral henomenon is resented, giving a theoretical interretation of the working rinciles of gradient and differential microhone arrays. It is shown that both tyes of microhone arrays can be viewed as devices for aroximately measuring satiotemoral derivatives of the sound field. Furthermore the design of high-order differential microhone arrays using the aforementioned satiotemoral gradient analysis is discussed. 0 INTRODUCTION The concet of gradient microhones dates back to the middle of the last century and the works of Olson [1], [2], who described gradient microhones as arrays of ressure-sensing elements whose signals are combined in a similar way as gradients are aroximated with finite differences. Gradient microhones have been used in a variety of alications, such as surround and sot recording (for examle, see [3], [4]), and as article velocity and sound intensity measurements called - robes [5] [8]. The works of Olson [1], [2] also showed how the combination of resonses from gradient microhones of different order can be equivalently obtained by combining signals from multile ressure microhones with aroriately chosen delays. This more flexible family of microhone arrays found alications in hands-free communication [9] and hearing-aid devices [10], [11]. Later microhone arrays of the latter tye, which use delay elements, have been termed differential microhone arrays, and they were more extensively analyzed in the works of Elko at the start of the last decade [12], [13]. However, to the best of our knowledge the relation between these two microhone array tyes, and the rationale behind the idea of combining the delayed *Presented at the 126th Convention of the Audio Engineering Society, Munich, Germany, 2009 May 7 10; revised 2010 November 2. **Also with the Deartment of Electrical Engineering and Comuter Sciences, University of California at Berkeley, Berkeley, CA. microhone signals in differential microhone arrays, has not been resented yet. This aer resents a slightly different analysis of the sound ressure field, which is viewed as both a satial and a temoral henomenon, that is, as a multivariate function of satial location and time. This analysis then exoses the oerations of taking gradients and directional derivatives of the sound ressure field as combinations of its satial and temoral derivatives. This new ersective gives a clear interretation of gradient and differential microhones and microhone arrays the former as devices used for aroximately measuring only satial derivatives, and the latter as devices used for aroximately measuring satiotemoral derivatives of the sound ressure field. In other words, it shows their equivalence. This aer is organized as follows. Section 1 gives a theoretical analysis of satiotemoral derivatives of the sound ressure field when the latter is viewed as both a satial and a temoral henomenon, that is, as a multivariate function of sace and time. Section 2 shows a number of ractical differential microhone arrays which follow from the theoretical analysis in Section 1. Conclusions are given in Section 3. 1 THEORETICAL ANALYSIS OF SPATIOTEMPORAL DERIVATIVES OF THE SOUND FIELD 1.1 Satial Derivatives of Sound Pressure Field The sound ressure at a osition defined by vector r of a lane wave roagating with wave vector k is given 20 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February

2 by 1 ðr; tþ ¼Ae jðxtþkrþ ð1þ where x is the temoral frequency of the lane wave. The satial derivative of the sound ressure field along the direction defined by vector u quantifies the rate of change of the sound ressure in that direction. It is given by the rojection of the satial gradient of the sound ressure field onto the vector u, D u ðr; tþ ¼rðr; tþu ¼ jðr; tþðk uþ ¼ jk cos hðr; tþ ð2þ where h is the angle between vectors k and u. Generalizing to the nth-order satial derivative along the direction defined by vector u gives D n u ðr; tþ ¼ðjkÞn ðcos hþ n ðr; tþ: ð3þ The nth-order satial derivative of the sound ressure field comosed of a lane wave has a bidirectional characteristic whose shae has the form d n ðhþ ¼AðxÞðcos hþ n ð4þ where A(x) ¼ (jk) n is a comlex frequency-deendent gain, and h is the angle between the direction along which the satial derivative is taken and the direction of roagation of a lane wave. The directional characteristics of a lane wave s satial derivatives of different orders n along the ositive direction x are shown in Fig Also the lane wave s nth-order satial derivative has at all angles a high-ass magnitude frequency characteristic roortional to ( jk) n or, equivalently, ( jx/c) n, as shown in Fig Satiotemoral Derivatives of Sound Pressure Field Without loss of generality, the analysis will be given in a two-dimensional lane, such that the ressure field can be written as a function (x, y, t) of two satial coordinates x and y, and one temoral coordinate t. Since such a ressure field is a function of three indeendent coordinates, its gradient is given by rðx; y; tþ ¼ ] ] ] T : ð5þ ]x ]y ]t The gradient of the sound ressure field, as defined in Eq. (5), shows how it and subsequently all its rojections can be measured in a oint. In articular one needs to measure the satial derivatives of the sound field along directions x and y, 3 and its temoral derivative in the given oint. This is the rincile of - sound 1 In this aer the wave vector k is chosen to oint to the direction from which waves emanate, as oosed to the direction of wave roagation used in standard texts, such as [14]. 2 The directional characteristics are lotted in the lane z ¼ 0. 3 In three dimensions, one needs to add the z direction. GRADIENT ANALYSIS OF DIFFERENTIAL MICROPHONES intensity robes (see, for examle, [5], [6], [8]), which can measure the sound intensity vector in two and three dimensions, and also some sound field microhones [7]. The following analysis for a sound field comosed of a lane wave with temoral frequency x and wave vector k ¼ [k cos h k sin h] T is given by the exression jðxtþkx cos h ky sin hþ ðx; y; tþ ¼Ae ð6þ where the angle h defines the direction of roagation of a lane wave First-Order Satiotemoral Derivatives of Sound Pressure Field The gradient of the sound ressure field given in Eq. (6) has the form rðx; y; tþ ¼jkðx; y; tþ½cos h sin h cš T ð7þ where c is the seed of sound roagation (k ¼ x/c). Let the unit vector u, onto which the ressure gradient is rojected, be defined as u ¼½q u cos / u q u sin / u u t Š T ð8þ where q u (q u 2 [0,1]) and / u (/ u 2 [0,2]) define the satial coordinates, and u t (u t 2 [0,1]) is the temoral coordinate of the vector u. Note that since the vector u has a unit norm, q 2 u þ u2 t ¼ 1, the ratio q u /u t gives the relation between its satial art and its temoral art. The derivative of the sound ressure field along the satiotemoral direction defined by the vector u is given by D u ðx; y; tþ ¼rðx; y; tþu ¼ jkðx; y; tþ½q u cosðh / u Þþcu t Š: ð9þ The directional characteristic of a satiotemoral derivative of a lane wave with wave vector k and temoral frequency x is a combination of a first-order (bidirectional) directional characteristic of its satial gradient, given by the term q n cos (h / u ), and a zeroorder (omnidirectional) directional characteristic of a temoral differentiator, given by the term cu t. The relative contributions of the two derivatives satial and temoral given by the ratio q u /u t determine the shae of the directional characteristic of a satiotemoral derivative of a lane wave. In the two extreme cases when q u ¼ 0 and q u ¼ 1, the directional resonses are omnidirectional and bidirectional, resectively. When q u ¼ cu t, the directional resonse has a well-known cardioid olar attern, and when the value of q u is smaller or larger than cu t, the directional resonse is a variation of a subcardioid or a tailed cardioid. Some well-known first-order olar atterns, resulting from different q u /u t ratios, are given in Table 1 and shown in Fig Higher Order Satiotemoral Derivatives of Sound Pressure Field The exression for a general nth-order satiotemoral derivative of the sound ressure field along a single J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February 21

3 KOLUNDŽIJA ET AL. direction given by the vector u is obtained by iterating the oerations of taking its gradient and rojecting it along the vector u. For the sound field of a lane wave given by Eq. (6) the nth-order satiotemoral derivative along the direction given by Eq. (8) has the form D n u ðx; y; tþ ¼ðjkÞn ½q u cosðh / u Þþcu t Š n ðx; y; tþ: ð10þ Also, instead of along a single direction, higher order satiotemoral derivatives can be taken along multile directions. Given an n-tule of vectors U ¼ (u 1,..., u n ), with each vector of the form u i ¼½q ui cos / ui q ui sin / ui u ti Š T ; a mixed derivative of the sound ressure field along directions given by U has the form D n U ðx; y; tþ ¼ðjkÞn ðx; y; tþ Yn i¼1 PAPERS q ui cosðh / ui Þþcu ti : ð11þ As with the satial derivative, the satiotemoral derivative of the sound ressure field comosed of a single lane wave has a high-ass frequency characteristic at all angles which is roortional to (jk) n, as shown in Fig. 2. The directional characteristic is, however, roortional to a linear combination of satial gradients of different orders, resulting from exanding the roduct P n i¼1 ½q u i cos ðh / ui Þþcu ti Š in Eq. (11), or the term [q u cos (h / u ) þ cu t ] n in Eq. (10), which is a secial case of Eq. (11). Fig. 1. Directional characteristics of lane-wave satial derivatives for different derivative orders n. Fig. 2. Magnitude resonses of lane-wave satial derivatives of different orders n. 22 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February

4 Table 1. Some well-known first-order olar atterns exressed through the ratio q u /u t of the satiotemoral derivative. Resonse Tye Cardioid q u /u t Subcardioid (0, c) Hyercardioid Suercardioid c 3c 3 ffiffiffi 3 ffiffiffi c 3 1 GRADIENT ANALYSIS OF DIFFERENTIAL MICROPHONES As with the first order, the shae of the directional characteristic of a higher order satiotemoral derivative of a lane-wave sound field is determined by the choice of the vectors u i, that is, the arameters q ui ;/ ui ; and u ti : Some well-known second-order olar atterns, resulting from different choices of ratios q u1 =u t1 and q u2 =u t2 and angle differences D/ ¼ / u1 / u2 ; are given in Table 2, and shown in Fig PRACTICAL SPATIOTEMPORAL DIFFERENTIAL MICROPHONE ARRAYS Section 1 resented a theoretical analysis of the satiotemoral derivatives of a lane-wave sound field, which serves as a basis for designing gradient and differential microhone arrays with desired directional resonses. Fig. 3. Directional characteristics of lane-wave first-order satiotemoral derivatives for different ratios q u /u t as given in Table 1. Table 2. Some well-known second-order olar atterns exressed through the ratios q u /u t and angle differences D/ of the satiotemoral gradient. Resonse Tye q u1 =u t1 q u2 =u t2 D/ ¼ / u1 / u2 Cardioid c c 0 ffiffiffi ffiffi Hyercardioid ð 6 1Þc ð 6 þ 1Þc 4 ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 þ ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Suercardioid ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 8 3 c 7 0 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February 23

5 KOLUNDŽIJA ET AL. Practical differential microhone arrays are based on the rincile of the finite-difference aroximation of the satiotemoral derivatives of a sound ressure field. They combine values of the sound ressure field in multile closely saced oints in sace and time, 4 either acoustically (ressure at two faces of a diahragm, and differentlength acoustic aths to the two faces of a diahragm) or electronically (ressure at different microhones of a microhone array combined with delay elements). This section will resent a few ractical differential microhone array realizations based on the analysis from the revious section. 2.1 First-Order Differential Microhone Arrays: Cardioid, Hyercardioid, and Suercardioid The first-order directional resonses of the satiotemoral derivatives of a sound field, resented in Section 1 and shown in Fig. 3 can be obtained by a finite-difference aroximation of the soatiotemoral derivative of a sound field, which involves taking differences of the sound ressure, both in sace and in time. Two closely saced microhones, saced at a distance d, together with a delay element, as shown in Fig. 5, can be used for a ractical realization of any first-order differential microhone array. The resonse of a ractical first-order differential microhone array, as shown in Fig. 5, in a sound field 4 Points are saced at a distance much shorter than the wavelength, and at a time much shorter than the eriod. of a lane wave, exressed in Eq. (1), is given by d ðtþ ¼2j sin k 2 ðd cos h þ ct dþ r; t t d 2 PAPERS ð12þ where k is the wavenumber, d the intermicrohone distance, t d the delay used, and r the osition of the microhone array center (midoint between the two microhones). At low frequencies, Eq. (12) can be aroximated by d ðtþ jkðd cos h þ ct d Þ r; t t d ð13þ 2 from which it can be seen that the ratio d/t d determines the directional resonse of a ractical differential microhone array in the same way the ratio q u /u t determines the directional resonse of the satiotemoral derivative in Eq. (9) of the lane-wave sound field. Fig. 6 shows directional resonses of the ractical cardioid, suercardioid, and hyercardioid microhones realized with the microhone combination shown in Fig. 5, with d ¼ 20 mm. From Fig. 6 it can be seen that the shae of the directional resonses of ractical first-order microhone arrays is frequency deendent, and that it corresonds to the desired resonses, shown in Fig. 3, only at low frequencies. Above the aliasing frequency 5 the directional characteristics 5 The aliasing frequency of a first-order gradient microhone array is deendent on the intermicrohone distance d and the delay t d used. Fig. 4. Directional characteristics of lane-wave second-order satiotemoral derivatives for different ratios q u /u t and angle differences D/ as given in Table J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February

6 deviate from the desired ones, as can be observed in Fig. 6 for frequency f ¼ 7000 Hz. 2.2 Second-Order Differential Microhone Arrays In this art it is shown how clover-leaf directional resonses sin 2h and cos 2h can be realized in two different ways based on the analysis from Section Clover-Leaf Resonse sin 2h: Quadruole Microhone Array The clover-leaf directional resonse sin 2h can be reresented as the roduct of the directional resonses of GRADIENT ANALYSIS OF DIFFERENTIAL MICROPHONES two satial derivatives of a lane-wave sound field, the satial derivative along the axis x, which has a directional resonse cos h, and the satial derivative along the axis y, which has a directional resonse sin h [or cos (h /2)]. As such the directional resonse sin 2h can be realized as a cascade of two satial derivative aroximations first along the axis x and then along the axis y, or vice versa. Fig. 7 illustrates a configuration of four ressure microhones used as an aroximation of the reviously described cascade of satial derivatives of the sound field. Fig. 8 shows the directional resonses at various frequencies of the quadruole microhone array shown in Fig. 7 when the intermicrohone distance d ¼ 20 mm is used Clover-Leaf Resonse cos 2h: Three-Microhone Line Array The clover-leaf directional resonse of the form cos 2h can be reresented as Fig. 5. First-order differential microhone realization using two ressure microhones and a delay element. cos 2h ¼ 2 cos 2 h 1 or, equivalently, as ffiffi ffiffiffi cos 2h ¼ð 2 cos h 1Þð 2 cos h þ 1Þ ð14þ ð15þ which is a roduct of the directional resonses of two first-order satiotemoral derivatives of a lane-wave Fig. 6. Directional resonses at variousfrequencies ffiffi of first-order ffiffiffi differential microhones realized as shown in Fig. 5, with d ¼ 20 mm and t d ¼ d/c (cardioid), t d ¼ dð 3 1Þ=cð3 3 Þ (suercardioid), and td ¼ d/3c (hyercardioid). J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February 25

7 KOLUNDŽIJA ET AL. sound ressure field. Consequently the resonse cos 2h can be obtained by cascading two satiotemoral derivative oerations: one with q u =u t ¼ ffiffi 2 and the other with q u =u t ¼ ffiffiffi 2 or, equivalently, two satiotemoral finite differences: the first with d=t d ¼ ffiffi 2 c and the second with d=t d ¼ ffiffi 2 c; or vice versa, as shown in Fig. 9. Fig. 10 shows the directional resonses at various frequencies of the microhone array shown in Fig. 9, with the intermicrohone distance d ¼ 20 mm and the delay t d ¼ d= ffiffi 2 c: Like the first-order differential microhone arrays, the second-order differential microhone array has a directional resonse that is frequency deendent. At low frequencies it corresonds well to the desired resonse, and above the aliasing frequency it deviates from the desired resonse. This can be observed in Fig. 10, which shows how the shae of the directional resonse of the microhone array from Fig. 9 deforms at the frequency f ¼ 7000 Hz. PAPERS Note that the directional resonse of the form sin 2h can also be obtained by rotating by 458 the microhone array of Fig CONCLUSIONS This aer resented an analysis of the sound ressure field as a multivariate function of satial location and time, which hels exlaining the working rinciles of gradient microhones, differential microhones, and arrays as devices for aroximately measuring the satiotemoral derivatives of a sound ressure field and shows their equivalence. The resented analysis framework enables not only analyzing the resonse of a given gradient or differential microhone or microhone array, but it can also be used for designing differential microhone arrays. The aroriate adjustment of the microhone array arameters such as array orientation and shae, intermicrohone distances, and microhone signal delays enables meeting the desired resonse requirements of the microhone array. Fig. 7. Quadruole microhone array used to obtain cloverleaf directional resonse sin 2h. Fig. 9. Line array with three microhones used to obtain clover-leaf directional resonse cos 2h. Fig. 8. Directional resonses at various frequencies of quadruole microhone array shown in Fig. 7, with intermicrohone distance d ¼ 20 mm. Fig. 10. Directional resonses at various frequencies of microhone array shown in Fig. 9, with intermicrohone distance d ¼ 20 mm and intermicrohone delay t d ¼ d= ffiffiffi 2 c. 26 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February

8 4 REFERENCES [1] H. F. Olson, Gradient Microhones, J. Acoust. Soc. Am., vol. 17, (1946). [2] H. F. Olson, Directional Microhones, J. Audio Eng. Soc., vol. 15, (1967). [3] A. Blumlein, Imrovements in and Relating to Sound Transmission, Sound Recording and Sound Reroduction Systems, British atent (1931), rerinted in Stereohonic Techniques (Audio Engineering Society, New York, 1986). [4] R. Streicher and W. Dooley, The Bidirectional Microhone: A Forgotten Patriarch, J. Audio Eng. Soc., vol. 51, (2003 Ar.). [5] H. E. de Bree, M. Iwaki, K. Ono, T. Sugimoto, and W. Woszczyk, Anechoic Measurements of Particle- Velocity Probes Comared to Pressure Gradient and Pressure Microhones, Convention of the Audio Engineering Society, (Abstracts) 122ndWraU.df, (2007 May), convention aer [6] R. Raangs, W. F. Druyvesteyn, and H. E. De Bree, A Low-Cost Intensity Probe, J. Audio Eng. Soc., vol. 51, (2003 May). [7] J. Merimaa, Alications of a 3-D Microhone Array, resented at the 112th Convention of the Audio GRADIENT ANALYSIS OF DIFFERENTIAL MICROPHONES Engineering Society, J. Audio Eng. Soc., (Abstracts), vol. 50,. 496 (2002 June), convention aer [8] F. J. Fahy, Measurement of Acoustic Intensity Using the Cross-Sectral Density of Two Microhone Signals, J. Acoust. Soc. Am., vol. 62, (1977). [9] G. W. Elko, Steerable and Variable First-Order Differential Microhone Array, U.S. atent 6,041,127 (2000). [10] R. J. Geluk and L. de Klerk, Microhone Exhibiting Frequency-Deendent Directivity, E.U. atent alication (2001). [11] D. Preves, T. Peterson, and M. Bren, In-the-Ear Hearing Aid with Directional Microhone System, U.S. atent 5,757,933 (1998). [12] G. W. Elko, Microhone Array Systems for Hands-Free Telecommunication, Seech Commun., vol. 20, (1996). [13] G. W. Elko, Suerdirectional Microhone Arrays, in Acoustic Signal Processing for Telecommunication (Kluwer Academic, Boston, MA, 2000), [14] E. G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holograhy (Academic Press, London, 1999). THE AUTHORS M. Kolundžija C. Faller M. Vetterli Mihailo Kolundžija received a Dil. Ing. degree in electrical engineering from the University of Novi Sad, Serbia, in 2004, an M.Sc. degree in communication systems from the Ecole Polytechnique Fédérale (EPFL), Lausanne, Switzerland, in 2007, and is currently a Ph.D. student there. His diloma thesis work, carried out at Micronas GmbH, Freiburg, Germany, involved develoing video uscaling algorithms and uscaling units on FPGAs. His M.Sc. thesis work at EPFL was on techniques for satial sound cature using comact microhone arrays. He is currently working on techniques for satial sound reroduction using loudseaker arrays and sound field analysis using arrays of microhones. His main research interests are satial sound acquisition and reroduction, array signal rocessing, and room acoustics. X Christof Faller received an M.S. (Ing.) degree in electrical engineering from ETH Zurich, Switzerland, in 2000, and a Ph.D. degree for his work on arametric multichannel audio coding from Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland, in From 2000 to 2004 he worked in the Seech and Acoustics Research Deartment at Bell Labs Lucent and its sin-off Agere Systems, where he worked on audio coding for satellite radio, MP3 Surround, and MPEG Surround. He is currently managing director at Illusonic, a comany he founded in 2006, and art-time research associate at EPFL in Lausanne. X Martin Vetterli received an engineering degree from the Eidgenössiche Technische Hochschule (ETHZ), Zurich, J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February 27

9 KOLUNDŽIJA ET AL. Switzerland, an M.S. degree from Stanford University, Stanford, CA, and a Ph.D. degree from the Ecole Polytechnique Fédérale (EPFL), Lausanne, Switzerland. In 1986 he joined Columbia University, New York, first with the Center for Telecommunications Research and then with the Deartment of Electrical Engineering, where he was an associate rofessor of electrical engineering. In 1993 he joined the University of California at Berkeley, where he was a full rofessor until Since 1995 he has been a rofessor at EPFL, where he headed the Communication Systems Division ( ) and now is heading the Audiovisual Communications Laboratory. From 2001 to 2004 he directed the National Cometence Center in Research on mobile information and communication systems. He is also vice resident for international affairs at EPFL. He has held visiting ositions at ETHZ (1990) and Stanford (1998). His research interests are in the areas of alied mathematics, signal rocessing, and communications. Dr. Vetterli is the coauthor of Wavelets and Subband Coding with J. Kovacevic, Signal Processing for Communications with P. Prandoni, and Fourier and Wavelet Signal Processing with J. Kovacevic and V. Goyal. He has ublished about 150 journal aers. PAPERS 28 J. Audio Eng. Soc., Vol. 59, No. 1/2, 2011 January/February