Practical Implementation of Radial Filters for Ambisonic Recordings. Ambisonics

Transcription

1 Practical Implementation of Radial Filters for Ambisonic Recordings Robert Baumgartner, Hannes Pomberger, and Matthias Frank Institut für Elektronische Musik und Akustik, Universität für Musik und darstellende Kunst Graz, Austria Abstract Ambisonics is the representation of an incident sound field by a sum of angular modes (spherical harmonics) up to a given order. Recording and playback of natural sound fields is achieved by using compact spherical microphone arrays and a surrounding spherical loudspeaker array. Typically, the radius of the microphone array is much smaller than the radius of the loudspeaker array for playback. Thus, the recording requires an accurate adaptation to the radius of the playback facility. This is achieved by radial filters compensating for the radial propagation of the modal components. Such ideal radial filters exhibit very high gains at low frequencies, especially for higher order components. However, the feasible dynamic range for filtering is limited, mainly by the signal-to-noise ratio of the microphone array. Usually, the dynamic range of the ideal radial filters is limited by applying high-pass filters with adequate cut-off frequencies. Regarding the low frequency range, this high-pass filtering causes losses in terms of amplitude and power, respectively. We present a new approach that accounts for these losses. Whenever higher order components are omitted, the remaining lower order components are amplified accordingly. This is achieved by an appropriate filter bank, applied in the modal domain. Introduction Ambisonics can be regarded as a 3D surround sound recording and playback technique based on the solutions of the Helmholtz equation in spherical coordinates. The angular parts of these solutions are known as spherical harmonics (SH) and the associated radial parts describe the radial propagation. Using a compact spherical microphone array, the sound pressure distribution measured on the surface of the array is decomposed into its SH components. In practice the number of array microphones is limited and thus the theoretically infinite series of SH is truncated to a maximum order N. This truncation yields a finite angular resolution of the recorded sound field. For reproduction on a surrounding spherical loudspeaker array, the Ambisonic signals are distributed to the individual loudspeakers by a decoder matrix. Furthermore, a compensation of the radial propagation is necessary to adapt the recording to the playback radius. This compensation is accomplished by suitable filtering of the Ambisonic signals. This contribution presents a discrete-time IIR implementation of such radial filters, similar to the filters in [1] and [2]. The compensation gains at low frequencies are very high, especially for higher order components. However, the feasible dynamic range is limited, typically due to the finite signal-to-noise ratio (SNR) of the microphone array. One possible solution is to limit the dynamic range by applying high-pass filters with adequate cutoff frequencies to the radial filters, cf. [3]. In this way, more and more higher order components are omitted with decreasing frequency. This yields not only a lower spatial resolution but also a decrease in amplitude and power of the Ambisonic representation in the low frequency range. We present a new approach that allows to compensate for the decrease in amplitude or power by appropriate amplification of the remaining lower order components. Therefore, a suitable filter bank is designed and evaluated. Ambisonics Within this paper, we define the Cartesian direction vector as θ = [cos(ϕ) sin(ϑ), sin(ϕ) sin(ϑ), cos(ϑ)] T with ϕ and ϑ denoting the azimuth and zenith angle, respectively, cf. Figure 1. y θ ϑ Figure 1: Spherical coordinate system. For any incident sound field, the sound pressure p(kr, θ) can be expressed by p(kr, θ) = n n= m= n z ϕ x b nm j n (kr) Y m n (θ), (1) where k denotes the wave number, b nm the wave spectrum, j n (kr) the spherical Bessel function and Y m n (θ) the SH of order n and degree m. Ambisonic playback can be regarded as a discretization of a continuous amplitude excitation f(θ) on a sphere with radius r L. The SH transform of the amplitude excitation, φ m n (kr L ) = f(θ)y m S 2 n (θ) dθ, yields the Ambisonic signals, as they are usually called. These signals are related to the coefficients in Eq. (1) by, cf. [4], b nm = ikh n (kr L ) φ m n (kr L ), (2) with i 2 = 1 and h n (kr) representing the spherical Hankel function of the second kind. Thus, any incident

2 sound field can be generated by choosing an appropriate amplitude excitation f(θ). In practice, a surrounding spherical loudspeaker array accomplishes a discretized excitation ˆf(θ) = L l=1 δ (θ θ l) g l, where θ l and g l are the positions and gains of the L individual loudspeakers, respectively. The SH transform of the discretized excitation yields ˆφ m n (kr L ) = L l=1 Y n m (θ l ) g l, or expressed in matrix-vector notation with ˆφ = Y L g, (3) ˆφ := [ ˆφ (kr L ),, ˆφ m n (kr L ),, ˆφ ] T N N(kr L ), g := [g 1,... g L ] T, Y L := [y (θ 1 ),..., y (θ L )], y(θ l ) := [ Y (θ l ),, Y m n (θ l ),, Y N N (θ l ) ] T. Given the Ambisonic signals φ, the signals g driving the loudspeakers are determined by a carefully designed decoder matrix D as g = D φ. (4) Ideally, D is left inverse to Y L, so that ˆφ is identical to φ. However, this is usually infeasible. The design of a suitable decoder is a non-trivial task and crucial for the performance of an Ambisonic playback system. For recent works on decoder design strategies see e.g. [5, 6]. The continuous sound pressure distribution on the surface of an array with radius r M can be expressed as a weighted sum of SH like p(kr M, θ) = n n= m= n ψm n (kr M ) Yn m (θ). Ambisonic recordings are accomplished with spherical microphone arrays, measuring a discretized version of the pressure distribution on M microphone positions. Expressed in matrix-vector notation this yields p = Y M ψ. (5) The coefficient vector ψ is computed from the microphone signals p by matrix inversion, ψ = Y M p, (6) where Y M is the pseudo inverse1 solving Eq. (5) in a least squares sense. The SH coefficients of the sound pressure distribution on a rigid sphere with radius r M are related to the coefficients of an incident field by, cf. [7, 8], ψ m n (kr M ) = b nm i(kr M ) 2 h n(kr M ). (7) Combining this relation with Eq. (2), the Ambisonic signals for a playback facility with radius r L are determined by φ m n (kr L ) = r 2 Mk h n(kr M ) h n (kr L ) ψm n (kr M ). (8) 1 An exact inverse is typically not feasible, as Y M is usually not a square matrix. The term rm 2 k h n (krm) h n(kr L) represents a filter which relates the SH coefficients measured with a compact spherical microphone array of radius r M to Ambisonic signals for playback on a loudspeaker array of radius r L. The following section describes how these radial filters can be efficiently implemented as discrete-time filters. Implementation of Radial Filters With the substitution k = i s c, where c denotes the speed of sound, the radial filter can be expressed as a Laplace-domain transfer function as G n (s) = r2 M c s ih n(s rm c ) h n (s rl c ). (9) Omitting the constant term r2 M c, the above equation can be interpreted as product of a differentiator D (s) = s and a transfer function H n (s) = ih n (srm/c) h. n(sr L/c) The function H n (s) appears in the context of radial filtering for radiation pattern synthesis with compact spherical loudspeaker arrays. It has been shown that H n (s) can be efficiently implemented as a discrete-time IIR filter H n (z), cf.[2]. This is achieved by decomposing H n (s) into a product of second-order sections. Each section is then transformed into the z-domain individually by the corrected impulse invariance technique. The discrete time filter is easy to adjust to different values of r M or r L, as its coefficients can be directly computed from a precomputed set of normalized Laplace-domain coefficients, cf.[2]. The differentiator has no discrete-time counterpart. However, an approximate discrete-time differentiator can be found by designing a discrete-time integrator and inverting its transfer function, cf. [9], D(z) = 1 z (1) 7z 1 This differentiator approximates the magnitude of an ideal derivative up to.78 of the half sampling frequency within a 2% error. A discrete-time approximation of the radial filter for Ambisonic recording can be described as G n (z) = D(z) H n (z). (11) In fact, H n (z) possesses always a pole at z = 1 which cancels the zero of D(z). Thus, the discretetime radial filters for Ambisonic recording and radiation pattern synthesis are very similar and differ only in one pole. Figure 2 shows the frequency response of the ideal continuous-time radial filters and their discretetime approximations as described above. It can be seen that the magnitude response is approximated very well and the phase response differs only for frequencies above 2 khz. Apparently, higher order components need to be amplified enormously in the low frequency range for adjusting the radius of the recording device to the radius of the playback facility. The following section presents the

3 Gn(f) in db ideal n = n = 1 n = 2 n = 3 n = 4 bands of the filter bank need to be equalized in order to enable smooth crossovers between the bands. For this purpose, a phase-matched all-pass for each bandpass is necessary. Equiphase filter paths are achieved by cascading each band-pass with the phase-matched all-passes of all the other band-passes. Figure 3 shows the resulting structure for the example of a filter bank consisting of five bands. As shown below, the filter bank is derived from a single analog low-pass prototype and a phase matched all-pass prototype by appropriate transformations. Gn(f) in degrees 4 2 f in Hz Figure 2: Magnitude and phase response of radial filters for different orders n (r M = 4.2 cm, r L = 5 m); solid: ideal curves, dashed/dotted: approximated version by modification of existing radial steering filter H n(f). design of a filter bank that limits the dynamic range of the radial filters by omitting higher order components at low frequencies. By appropriate amplification of the remaining components the resulting decrease in amplitude or power is compensated. Filter Bank Design As visible in Figure 2, different SH orders exceed a fixed maximum gain limit, e.g. 5 db, one after another, starting with the highest SH order. Therefore, a filter bank is designed to divide the frequency range into bands. The cut-off frequencies of these bands are chosen according to those frequencies where the radial filters exceed a level threshold. The filter bank accomplishes both the omission of the exceeding channels as well as the level adjustment of the remaining ones. The omission of signals above the threshold causes a decrease of amplitude and power. This decrease is compensated by proper amplification of the remaining signals individually. However, either the amplitude or the power loss can be compensated. It is not clear yet, whether compensating for amplitude or power has more perceptual relevance. Later on, we will present a method to design gains for amplitude compensation which can be easily adapted to power compensation or even to something in between. The filter bank implementation is based on discrete-time IIR band-passes. The phase relations between the sub- φ n BP 1 BP 2 BP 3 BP 4 BP 5 ν 1 AP ν ν 2 AP ν ν 3 AP ν ν 4 AP ν ν 5 AP ν g 1 (n) g 2 (n) g 3 (n) g 4 (n) g 5 (n) Figure 3: Block diagram of Butterworth filter bank with allpass based phase compensation and gain factors g i depending on SH order n 4; products symbolize all-pass cascades. Low-pass and phase-matched all-pass prototype. To obtain maximal flatness in the passband, Butterworth characteristics are used for the filter design. For the choice of filter order two aspects must be considered. First, it depends on the maximum occurring SH order N, because the slope of the magnitude response increases with the order n of the radial filters, cf. Figure 2. More precisely, this increase amounts 2 db per decade and SH order. This behaviour is quite similar to Butterworth characteristics which show a slope of 2 db per decade and filter order as well. Thus, the filter order of the lowpass prototype has to be chosen greater or equal to the maximum SH order N. Secondly, the filter order has an impact on how exactly the phase response of the lowpass can be matched by an all-pass. Regarding only the positive frequency range, an all-pass rotates the phase about 36 for each complex conjugated pole pair it consists of, cf. [11]. Since a Butterworth low-pass rotates the phase about 9 per order, cf. [1], only the use of multiple orders of four enables a good phase match. To conclude, the prototype order should be chosen as the next multiple of four which is greater or equal to N. Thus, assuming N = 4 a low-pass prototype of order four should be taken. Its transfer function is, cf. [1], P (s) = with p 1 = p 3 = , 1 p 2 = , and a cut-off frequency of ω p = 1 rad/s. For this low-pass an all-pass prototype with a matching phase response is φ n 1 s 4 + p 3 s 3 + p 2 s 2 + p 1 s + 1, (12)

4 designed. This is achieved by applying a damped Gauss- Newton method, as described in [12]. The resulting allpass has the transfer function A(s) = s2 + b 1 s + 1 s 2 + a 1 s + 1, (13) with a 1 = b 1 = Band-passes and phase-matched all-pass pairs. The low-pass prototype is scaled to a desired 3 db-cutoff frequency ω c by the transform, cf. [1], s ω p ω c s. (14) Applying the low-pass to high-pass transform ω p ω c s ω c ω p s (15) yields a high-pass with the cut-off frequency ω c. These low- and high-pass filters are combined to build the desired band-pass filters. Since the phase responses of all bands will be matched, the crossover frequencies should be located at the frequency with 6 db attenuation. The transformations described above are preformed similarly with the all-pass prototype. Afterwards, the resulting band-pass and all-pass filters are separated into secondorder sections and undergo bilinear transformation to obtain discrete-time IIR filters. Amplitude compensation. Assuming an incident field with distinct but arbitrary directivity, all SH orders are needed similarly to decompose the field. In other 2n+1 words, each SH order contains a portion of (N+1) of 2 the entire surround sound. By omitting higher order components the representable maximal order N < N shrinks and consequently the quantity of effective components decreases to (N + 1) 2. Since the filter bank is used to fade out the modal orders step by step, the index ν of each band can be related to N by ν = N + 1. Hence, the order dependent gain factor of each band is defined by g ν (n) = { (N+1) 2 for n < ν ν 2 for n ν. (16) Here, it should be noted that this approach enforces constant amplitude for a specific source direction, which is wanted especially in free field conditions or beam forming. However, in diffuse sound fields this might amplify sounds in the low frequency range since the amplitude compensation widens the directivity. Thus, for diffuse conditions equalization of the power might be favored which uses the square root of the compensation gains above. For amplitude equalization the sum over all responses in the SH domain is a representative estimate of the overall frequency response. Since each SH order n consists of 2n + 1 components, the response estimate B N can be defined as B N (f) = N n= F n (f) 2n + 1 4π, (17) with F n (f) denoting the order dependent frequency response of the filter bank. Since this estimate is based on the assumption of representing a highly directed source distribution B N also represents the beam amplitude of an N th order directional hyper-cardioid microphone. Design example. We assume the following scenario. The playback radius is r L = 5 m, according to the Ambisonics setup of the IEM CUBE. The mhacoustics Eigenmike EM32 is used for recording. This compact spherical microphone array consists of M = 32 sensor elements and thus enables a SH decomposition up to the order N = M 1 = 4. The microphones are located on a sphere with radius r M = 4.2 cm and are distributed according to a truncated icosahedron. The SNR of the recording device is assumed to be 8 db. Leaving some headroom, a maximum gain limit of 5 db for the radial filters seems to be appropriate. First, the gain factors of the filter bank have to be fixed to be able to extract suitable cut-off frequencies. Table 1 includes the linear gain factors g ν (n) for all SH orders n N and bands 1 ν N + 1 computed according to Eq. (16). g ν (n) n = n = 1 n = 2 n = 3 n = 4 g 1 (n) 25 g 2 (n) g 3 (n) g 4 (n) g 5 (n) Table 1: Values of linear gain factors g ν(n) depending on filter step ν and SH order n to prevent amplitude losses. Considering these additional gain factors the absolute limit λ n of each modal order yields λ n = 5 db 2 log 1 (g n+1 ()). (18) In the magnitude response of the radial filters, the points of intersection between each filter curve and its absolute limit λ n lead to the lower cut-off frequencies f n. Furthermore, the first and last band-pass are bounded by the beginning and the end of the audible range, namely at 2 Hz and 2 khz. Finally, before building the band-pass filters by applying frequency scaling and the low-pass to high-pass transform according to Eq. (14) and Eq. (15), the cut-off frequencies f n should be corrected by a factor of 1 3/8 to receive the commonly used 3 db cut-off frequencies. The resulting magnitude response F n (f) of the order dependent filter bank is shown at the top of Figure 4. Since the band-pass filters are just combinations of highand low-passes, narrow bands like the lowest one, which is only passed by SH order n =, exhibit a rounded shape in the pass band. But apart from that, this leads to

5 Fn(f) in db 2 n = n = 1 n = 2 n = 3 n = 4 Gn(f) in db BN(f) in db τ(f) in ms f in Hz Figure 4: Properties of a filter bank designed for a setup with r M = 4.2 cm, r L = 5 m and N = 4; top: magnitude responses of the filter bank for each SH order n, middle: magnitude response of the normalized beam amplitude, bottom: mean group delay of all responses. equivalent slopes of the bands and thus to a quite smooth crossover between them, even though their bandwidths are very different. This profit can be also shown regarding the magnitude response of the corresponding beam amplitude as illustrated in the middle of Figure 4. The overall ripple contains less than 1 db above 1Hz, which causes quite little coloration. Additionally, at the bottom of Figure 4 the group delay τ(ω) defined as τ n (ω) = F n(ω) ω (19) is plotted, whereby τ(ω) τ n (ω) n, since the phase responses are conformed using the cascaded all-pass structure. In this context, only for the low frequency range high group delays occur, but for ascending frequencies the delays decrease quickly. Hence, at 5 Hz the group delay contains 3.7 ms whereas two octaves higher this amount shrinks to.7 ms. According to [13] these values are located nearby the audible threshold for narrow band signals and thus seem to be acceptable. Figure 5 illustrates the resulting frequency response G n (f) of the radial filters cascaded with the designed filter bank. The magnitude response G n (f) is bounded to the fixed limit of 5 db, cf. top of Figure 5. The phase response G n (f), shown in the middle of Figure 5, consists of the phase response of the radial filters plus an additional common phase response caused by the filter bank. The phase response without the common part Gn(f) in degrees Gn(f) in degrees 1, 2, 3, 4 6 n = n = 1 n = 2 n = 3 n = 4 f in Hz Figure 5: Overall filtering results for a setup with r M = 4.2 cm, r L = 5 m and N = 4; top: magnitude responses, middle: phase responses, bottom: differences between the phase responses. G n (f), as depicted at the bottom of Figure 5, is quite similar to the phase response of the ideal radial filter, cf. bottom of Figure 2. Conclusion An efficient implementation of radial filters and a gain limitation using an appropriate filter bank has been presented. This filter bank is used to prevent amplitude or power losses caused by attenuated components. The filter bank design is based on a single analog low-pass prototype from which all band-passes and phase-matched all-passes used for phase conformation are derived. Thus, this design procedure is very flexible and can be easily applied to different requirements. A method to design gains for amplitude compensation has been described which is quite similar for power compensation. Since the whole filter structure is based on second-order sections, it is easy to realize and needs small computational effort. The design process has been described in detail for an application example and it has been shown that the resulting system is practicable in terms of spectral flatness and group delay.

6 Acknowledgment We thank Franz Zotter for fruitful discussions and helpful suggestions. References [1] J. Daniel, Spatial sound encoding including near field effect: Introducing distance coding filters and a viable, new Ambisonic format, in Proc. of the AES 23rd Int. Conf., Copenhagen, 23. [2] H. Pomberger, Angular and radial directivity control for spherical loudspeaker arrays, Master s thesis, IEM Graz, 28. [3] S. Bertet, J. Daniel, and S. Moreau, 3D sound field recording with higher order Ambisonicsobjective measurements and validation of spherical microphone, in Proc. of the 12th AES Conv., Paris, 26. [4] F. Zotter, H. Pomberger, and M. Frank, An alternative Ambisonics formulation: Modal source strength matching and the effect of spatial aliasing, in Proc. of the 126th AES Conv., Munich, 29. [5] F. Zotter, H. Pomberger, and M. Noisternig, Energy preserving Ambisonic decoding, Acta Acustica united with Acustica, accepted for publication Jan [6] F. Zotter, M. Frank, and A. Sontacchi, The Virtual t-design Ambisonics-Rig using VBAP, in Proc. EAA EUROREGIO, Ljubljana, 21. [7] B. Rafaely, Analysis and design of spherical microphone arrays, Speech and Audio Processing, IEEE Transactions on, vol. 13, no. 1, pp , 26. [8] Z. Li and R. Duraiswami, Flexible and optimal design of spherical microphone arrays for beamforming, Audio, Speech, and Language Processing, IEEE Transactions on, vol. 15, no. 2, pp , 27. [9] M. Al-Alaoui, Novel digital integrator and differentiator, Electronics Letters, vol. 29, pp , [1] U. Tietze and C. Schenk, Halbleiter- Schaltungstechnik. Springer, 22. [11] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Zeitdiskrete Signalverarbeitung. Pearson, 24. [12] J. John E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, [13] J. Blauert and P. Laws, Group delay distortions in electroacoustical systems, Journal of the Acoustical Society of America, vol. 63, no. 5, pp , 1978.