Audio Engineering Society
Convention Paper Presented at the 134th Convention 2013 May 4–7 Rome, Italy This paper was peer-reviewed as a complete manuscript for presentation at this Convention. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society.
Optimal Directional Pattern Design Utilizing Arbitrary Microphone Arrays: A Continuous-Wave Approach Symeon Delikaris-Manias1 , Constantinos A. Valagiannopoulos2 , and Ville Pulkki1 1
Department of Signal Processing and Acoustics, Aalto University, 02150 Espoo, Finland
2
Department of Radio Science and Engineering, Aalto University, 02150 Espoo, Finland
Correspondence should be addressed to Symeon Delikaris-Manias (
[email protected]) ABSTRACT A frequency-domain method is proposed for designing directional patterns from arbitrary microphone arrays. This theoretical approach employs the complex Fourier series to decompose audio signals from microphone arrays into the complex Fourier coefficients. A target directional pattern is defined and an optimal set of sensor weights is determined in a least-squares sense. The analyzed method is presented in the cylindrical coordinate system adopting a continuous-wave approach. It is based on discrete measurements with high spatial sampling ratio, which mitigates the potential aliasing affect. A set of criteria is defined for employing the optimal number of Fourier coefficients and microphones in the least-squares solution. Furthermore, the low-frequency robustness is increased by smoothing the target patterns at those bands. The performance of the algorithm is assessed by calculating the directivity index and the sensitivity. Applications, such as synthesizing virtual microphones, beamforming, binaural and loudspeaker rendering are presented.
1.
INTRODUCTION
A common research question is how to combine signals captured with the sensors of a microphone array in a way that signals with arbitrary directional patterns can be formed [1, 2, 3, 4, 5]. Typical application of such microphone array processing are found in multichannel surround sound, teleconference and
noise control applications. Uniform microphone arrays, in which the sensors are placed uniformly on a sphere for three-dimensional arrays and on a circle for circular arrays, sample spatially the sound field in a similar way to time-domain sampling. The accuracy of the sampling schemes depends on the number of sensors and their geometrical arrangement. One common approach is the anal-
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ysis of the sound field, sampled on the surface of the sphere or a plane, into a set of signals with directional patterns following harmonic functions [6], [7]. Essentially, that decomposition provides a transformation from the time-space or frequency space domain to the spatial Fourier coefficients domain. The accuracy of the sampling scheme depends on how the harmonic components can be computed from the spatial samples with a minimum error. The desired directional pattern can be then expressed as a weighted sum of harmonic functions. In practice, the sensors of a microphone array are matrixed by using coefficients derived from the cylindrical or spherical harmonics. These are then processed either by using the theoretical approach, meaning to calculate the equalization functions according to the array geometry and sensors specifications [6], or by using inverse filtering and a regularization parameter to control the power output [8]. The least-squares approach has been employed previously to solve the problem of weight calculation. Specifically, an algorithm for designing virtual microphone directional patterns design has been proposed in which the constrains of the analysis of the sound-field into a set of harmonic functions is released [9]. A target pattern is defined analytically from which a set of frequency-dependent gains is calculated by solving a least-squares problem. The advantage of this method is the lack of intermediate processing the input signal, to analyze them into a set of harmonic functions. Synthesizing a directional pattern from a microphone array is performed by applying a set of weights that matrix and equalize the sensors of the microphone array. In [10] a leastsquares approach is used to calculate the harmonic coefficients and control the inversion by regularization means. Thus, it is possible to calculate the weights for each sensor in a microphone array with existing methods, in order to obtain a desired directional pattern. However, these methods do not take into consideration the exact geometrical position and the spectral variability of each separate sensor. Sensor misalignment or mismatch affect the overall quality of the system. The aim of this work is to provide an algorithm capable of deriving arbitrary directional patterns, employing in principle arbitrary microphone arrays. In the proposed method, a set of
measured transfer functions provides both geometrical and spectral information and by analyzing these transfer functions with the Fourier series, a more detailed information is acquired for each sensor of the microphone array. By using that information in a least-squares approach, an accurate and realistic inversion can be achieved. Section 2 presents the proposed method, including the criteria on setting the optimal parameters and how to modify the target pattern to increase robustness. A set of common performance criteria is mentioned in Section 3.1. A real array has been assembled in Section 3 to demonstrate the method by reconstructing directional pattern from analytical formulas. In Section 4 example applications such as beamforming, binaural and loudspeaker array rendering are shown. Section 5 summarizes the advantages of the method and Section 6 concludes the paper. 2. METHOD FOR DERIVING DIRECTIONAL PATTERNS The coordinate system is the cylindrical coordinate system with φ ∈ [0, 2π) and r the radius, shown in Fig. 1. The cartesian coordinates are given as x
= r cos φ,
y
= r sin φ, p x2 + y 2 . =
r
(1)
2.1. Overview The proposed algorithm employs a target directional pattern G(φ, f ) for continuous azimuthal angle φ and frequency f . That pattern is approximated by using a set of functions gn (φ, f ), n = 1, .., N , the auxiliary components in the two-dimensional space. More specific, the complex Fourier coefficients of the target pattern G are approximated by taking a frequency-dependent weighted sum of the complex Fourier coefficient of the function g for each component n. In a practical application, a continuous function gn (φ, f ) is characterized by a set of discrete transfer functions between the nth sensor of a microphone array and an incoming plane wave. The resulting responses include the information of each sensor behavior in term of directionality and spectral properties. That information shows how its response
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functions gn (φ, f ), ∀φ ∈ [0, 2π)
z
G(φ, f ) =
N X
wn (f )gn (φ, f ),
(2)
n=1
where gn (φ, f ) is a function representing the response of the nth microphone at angle φ of an arbitrary microphone array for n = 1, . . . , N , with N being the total number of microphones, and wn (f ) the frequency dependent gains. The set of gains w is applied to each sensor to approximate the directional pattern G.
y r
x
Fig. 1: The cylindrical coordinate system.
According to Fourier series, the function G(φ, f ) can be decomposed into +∞ X
G(φ, f ) = varies across the frequencies due to its placement in the microphone array. A set of criteria is introduced to optimize the leastsquares solutions: • The number of complex Fourier coefficients is limited when decomposing the input functions at lower frequencies, where directional patterns are smoother and of lower order, a feature that require less harmonics. The number of coefficients increases as the frequency, and hence complexity of the measured transfer functions, increases.
du (f )eiuφ ,
(3)
u=−∞
where du are the complex Fourier coefficients and u = −∞, . . . , +∞ indicates the number of corresponding harmonics. The complex Fourier coefficients of the function G(φ, f ) are defined as du (f ) =
1 2π
2π
Z
G(φ, f )eiuφ dφ.
(4)
0
Similarly, the microphone array responses gn (φ, f ) can be decomposed into +∞ X
gn (φ, f ) =
cun (f )eiuφ ,
(5)
u=+∞
1
• The number of microphones used in the leastsquares solution is adjusted for each frequency. Employing less and the suitable sensors at low frequencies results to less noise amplification when applying the calculated weights. • The target directional pattern is modified at lower frequency. It is essentially smoothed towards an omnidirectional pattern to minimize low-frequency instability errors due to lowfrequency sensor noise.
and the complex Fourier coefficients cun (f ) are 1 cun (f ) = 2π
Z
2π
gn (φ, f )eiuφ dφ.
(6)
0
By substituting Eq. (3) and Eq. (5) to Eq. (2), a relation between the complex Fourier coefficients of the target directional pattern and the sensors is obtained N X
cun (f )wn (f )
= du eiuφ .
(7)
n=1
2.2. Theory The starting point in the proposed method is that a directional pattern G(φ, f ) varying with angle φ and frequency f can be expressed as a weighted sum of
Eq. (7) holds ∀φ ∈ [0, 2π) and ∀u = −∞, . . . , +∞. In practice u is limited to a number of finite complex coefficients U . By confining u to u = −U, . . . , U a linear system of finite equation is derived from Eq.
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(7). This system can be expressed now in matrix form as follows C(f )w(f ) = d(f ). The matrices C(f ) and d(f ) are c−U 1 (f ) ... ... c−U N (f ) ... ... ... ... c (f ) ... ... c C(f ) = 0N (f ) 01 ... ... ... ... cU 1 (f ) ... ... cU N (f ) d−U (f ) ... , d (f ) d(f ) = 0 ... dU (f )
(8)
,
(9)
(10)
cannot be represented by lower-order harmonics and thus U cannot be very small. Many constraints need to be satisfied with much less degrees of freedom. On the other hand, when 2U + 1 is much larger than N , meaning that one would use higher-order components to reconstruct low-order harmonics, the Moore-Penrose optimal solution of the linear system is rarely acceptable. Such an optimal selection of U is possible for a single frequency when the target and reconstructed patterns are fixed. As the pattern G(φ, f ) varies per frequency, a frequency dependent U (f ) is necessary. In order to capture that complexity the following criteria are adopted for the choice of U for both G(φ, f ) and gn (φ, f ) PUG
where C(f ) is a (2U + 1) × N matrix, w(f ) a N × 1 vector and d(f ) a (2U + 1) × 1 vector. The solutions of the linear system in Eq. (8) depends on how the number of coefficients U relates to the number of microphones N . In particular the well known behavior of a linear system is if 2U + 1 < N
→ no solutions,
(11)
if 2U + 1 = N
→ unique solutions,
(12)
if 2U + 1 > N
→ infinite solutions.
(13)
The case of infinite solutions, where there are more equations that unknowns, is addressed with the Moore-Penrose inverse. In the next section a set of criteria is proposed to optimize the general solution of the linear system in Eq. (8). 2.3. Criteria for optimal parameters Assuming a large set of transfer function measurements, enough so that aliasing can be neglected, the complexity of a directional pattern increases as the frequency increases. Hence, the number of harmonics U is an increasing function of f . While measuring a transfer function, the same distance between a receiver and a sound source at different angles, can be seen differently by an acoustic wave at different frequencies, due to its wavelength. That means that in order to reconstruct a directional pattern varying in frequency, more harmonics U are needed for higher frequencies. The selection of the number of harmonics U is crucial. In the case where 2U + 1 < N , the function G(φ, f ) contains a higher-order component that
2 u=−UG |du (f )| R 2π 1 2 2π 0 |G(z, f )| dz
PUg
u=−Ug
1 2π
R 2π 0
|cun0 (f )|2
|gn0 (z, f )|2 dz
> 1 − α,
> 1 − α,
(14)
(15)
where UG is the total number of coefficient for G(φ, f ), Ug for gn0 (φ, f ) and α the corresponding tolerance level. In other words a single sensor n0 is chosen, assuming that all sensors have similar behavior, n = n0 , and it is required that most of its pattern’s spectral energy (1 − α) is contained into the first U (f ) harmonics of its pattern. That means that α provides a control on the number of complex Fourier coefficients that the input functions are decomposed into. The lower the α values the higher the number of coefficients U . In the case where the microphone array consists of sensors with varying behavior, the criteria are modified accordingly. Depending on which of the directional patterns G(φ, f ) and the behavior of a single sensor gn0 (φ, f ) is more complex, the optimal U (f ) at each frequency f , is defined as U (f ) = max(UG (f ), Ug (f )).
(16)
The number of sensors is also adjusted to optimize the solution of the linear system in Eq. (8). The choice of N , in the rare case that the linear system has infinite solutions (2U (f ) + 1 > N ) can be adjusted to M (f ) < N . This is feasible by nullifying the corresponding weights wn of those N −M (f ) microphones that are not necessary. In particular the
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total number of sensor for each frequency is ( N, if 2U (f ) + 1 > N M (f ) = 2U (f ) + 1, if 2U (f ) + 1 ≤ N.
(17)
Essentially the sensors that are kept are those, whose weights possesses a minimum sum of magnitudes and the remaining are nullified. This criterion results to a minimum energy configuration at those frequencies. 2.4. Target Directional Pattern Smoothing At low frequencies, when the acoustic wavelength is longer than the distance between the sensors of the microphone array, only slight variations occur in phase from sensor to sensor. In that case the sensors are indistinguishable by an incoming acoustic wave. That results to high valued gains, when solving the linear system in Eq. (8), that amplify the reconstructed signal at those frequencies but also the internal sensor noise. An algorithm for varying the directional target pattern is defined. Low order directivities can be used at low frequencies and higher order at high frequencies. Assume a desired target directional pattern G(φ, f ). For f lower than a specific frequency, the corresponding wavenumber λ becomes larger than the distance between the sensors of the microphone array. The target pattern is modified at all frequency bands for which D < t0 , (18) λ where D is the maximum distance between two sensors and t0 a threshold. The target pattern is redefined as Z 1 h φ+φ0 /2 ∗ ˆ G(φ, f, φ0 ) = G (f ) + G(z, f )dz − φ0 φ−φ0 /2 Z φ∗ +φ0 /2 i G(z, f )dz , (19) φ∗ −φ0 /2 ∗
∗
where φ is the angle at which G = max(G). Essentially, the proposed rule smoothens the spatial variation of the target pattern to all directions, in order to increase robustness of the output, with minimum amplification errors at low frequencies. The angular extend of the smoothing is defined by the angle φ0 . If the value G∗ (f ) is one, the existing directional pattern is smoothed towards an omnidirectional pattern at very low bands.
2.5. Complete System Overview The block diagram of the process is shown in Fig. 1. At first the microphone array signals are transformed into the frequency domain to obtain the function gn (φ, f ). A desired target pattern G(φ, f ) is defined and modified at low frequencies according to ˆ f, φ0 ). Both the desired specification to obtain G(φ, functions are decomposed into complex Fourier coefficients utilizing the complex Fourier series. The Directivity Pattern Design algorithm (DPD) receives as inputs the complex Fourier coefficients for both g ˆ the optimal number of the complex Fourier and G, coefficients U and the number of microphones M for each frequency f . The weights wn (f ) are then being calculated. 3.
EXPERIMENTAL VALIDATION
In this section two common performance criteria are defined to evaluate the performance of the proposed method. The method is applied to a real microphone array to synthesize directional patterns of different orders and its performance is assessed. 3.1. Common Performance Measures One measure is the directivity factor. It is the ratio between the power of the model G(φ, f ) in the look direction γ to the spatial integration over all directions [11], [12]. The directivity index is the directivity factor expressed in dB DI(γ, f ) = 10 log10
1 4π
|G(γ, f )|2 . R 2π |G(z, f )|2 dz 0
(20)
The numerator represents the power due to a signal arriving from direction γ and the denominator represents the noise power at the array output due to diffuse noise. The second measure is the sensitivity which is a measure of robustness to gain errors. It is proportional to the square of the weight’s w 2-norm S(f ) = ||w(f )||2 .
(21)
3.2. Real Microphone Array A cylindrical array is employed, consisting of 8 sensors in a circular arrangement, baffled by a rigid
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-
1 2
.. . N criteria for optimal parameters
reconstructed pattern
Fig. 1: Block diagram of the proposed algorithm in the cylindrical coordinate system including the Low Frequency Modification (LFM) scheme and Directional Pattern Design process (DPD). cylinder. The radius of the array is 0.013m. Measurements are performed every 5o to obtain impulse responses in an anechoic environment employing the swept-sine method [13]. The resulting measurements are transformed into the frequency domain with a block length of 1024 points. The sampling rate is set at 48kHz. This set of measurements assembles the function gn (φ, f ). As a target directivity a 1st , 2nd and 3rd cardioid pattern is defined as G(φ, f ) = (0.5 + 0.5 cos φ)m for m = 1, 2, 3. By taking the set of responses gn (φ, f ), a reconstruction ˆ f, φ0 ) is performed according to of G(φ, f ) and G(φ, Eq. (8) by first calculating the Fourier coefficients according to Eq. (4) and Eq. (6). The choice of optimal U (f ) and M (f ) is performed according to Eq. (16) and Eq. (17) and the low frequency modification according to Eq. (19). The reconstructed directivities are shown for each separate order in Fig. 2. The directivity index is shown in Fig. 3 for all target patterns and it is calculated ∀φ0 ∈ [0, 2π). It provides an estimate of the array performance in one specific direction, in terms of Signal to Noise Ratio
(SNR) improvement, in a diffuse field as it calculates the power of the array in one direction over all directions. A constant value of 4dB is achieved for the 1st order cardioid without the Low Frequency Modification (LFM), which drops at high frequencies for both arrays due to the spatial aliasing [14]. The 2nd and 3rd order cardioids result to a directivity index of approximately 6dB and 7dB. The LFM process for all target patterns results in a reduction of the directivity index below 150Hz as the target pattern is modified to converge into an omni-directional shape at those frequencies. It should be noted that these values are very close to the theoretical values of the corresponding cardioid patterns: 4.6dB for 1st order, 7dB for 2nd order and 8.45dB for 3rd order, derived from Eq. (20). The sensitivity function is shown in Fig. 4. For the reconstructed patterns without the LFM high sensitivity values at low frequencies are due to the wavelength, which is longer than to the inter-element spacing. That results in high-valued positive weights wn (f ) when calculating the Fourier coefficients. The values of sensitivity are lower at higher frequencies
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Fig. 2: Target and reconstructed directivities for the first three orders of a cardioid directivity employing the eight sensor real cylindrical microphone array.
60
8
1st order cardioid with LFM 50
7
2nd order cardioid with LFM
40
5
30 SENS (dB)
DI (dB)
3rd order cardioid with LFM 6
4 1st order cardioid with LFM 3rd order cardioid with LFM 2
0
2nd order cardioid
−10
3rd order cardioid 2
10
3
10 frequency (Hz)
3rd order cardioid
20
0
1st order cardioid
1
2nd order cardioid
10
2nd order cardioid with LFM
3
1st order cardioid
−20
4
Fig. 3: Directivity Index (DI) for different order of reconstructed cardioid microphones with and without the Low Frequency Modification (LFM) employing the eight sensor real cylindrical microphone array.
3
10 frequency (Hz)
4
10
Fig. 4: Sensitivity function for different order of reconstructed cardioid microphones with and without the Low Frequency Modification (LFM) employing the eight sensor real cylindrical microphone array.
4.
as the effective inter-element distance is shorter than the wavelength and thus the amplitude of the calculated weights oscillates between positive and negative values. By enabling the LFM, it is clear that a control over the sensitivity of the array is possible, resulting in a wider directional performance at those frequencies.
2
10
10
APPLICATIONS
In this section, three specific applications are chosen to demonstrate the ability of the system to synthesize directional patterns. In particular the applications under consideration are beamforming, binaural rendering and loudspeaker reproduction. A simulated microphone array is used consisting of eight sensors in a equidistant cylindrical arrangement of
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2cm radius. 4.1. Virtual Microphones - Beamforming Beamforming applications usually require directional patterns which are constant for most frequency bands. The same principle applies when generating virtual microphones. An example of pattern reconstruction is shown here. The target directional pattern is a dual-beam towards 0o and 150o composed as the product of a cardioid and a dipole. It can be expressed as G(φ, f ) = G(φ) = [0.5+0.5 sin(φ)] cos(φ). The pattern is reconstructed with a tolerance α = 10−4 . The resulted directivities of the target and reconstructed pattern are shown in Fig. 5. There is a good match between the two directivities except frequencies above 10kHz where spatial aliasing commences.
(HRTFs). The proposed method is employed to design a system that renders signals acquired from a microphone array into a set of HRTFs for binaural reproduction. A set of horizontal HRTF measurements is used [16] to assemble a target pattern varying in frequency and that pattern is reconstructed. The difference from the previous examples of virtual microphones and/or beamforming is that the target pattern is now frequency dependent. Let hl (φ, f ) and hr (φ, f ) be the set of target responses for sources at the azimuthal plane for the left and right ear with φ ∈ [0, 2π). The binaural rendering utilizing a microphone away with N sensors is: hl (φ, f ) =
N X
wnl (f )gn (φ, f ),
(22)
wnr (f )gn (φ, f ),
(23)
n=1
hr (φ, f ) =
N X n=1
where wnl (f ) wnr (f ) are the weights for the left and right ear, applied to the microphone array measured responses gn (φ, f ). The reconstructed pattern is shown in Fig. 6. Although only the angular frequency responses are shown in Fig. 6, the Interaural Time Difference (ITD) is preserved when applying the weights from Eq. 23 to the sensors of the microphone array. That is due to the fact that the weight calculation is based on a dense angular set of measured transfer functions between the array and a sound source.
Fig. 5: Target and reconstructed dual-beam directional pattern employing an eight sensor simulated microphone array. 4.2. Binaural rendering Binaural rendering using microphone arrays has been studied before [15] for cylindrical arrays and its usefulness is based on the fact that recordings obtained with a microphone array can provide multiple recordings, either mono, binaural or other multichannel recordings, which is not the case when employing a binaural head. An example is shown here to indicate the potential of the system to reconstruct frequency varying patterns with high complexity such as Head Related Transfer Functions
4.3. Loudspeaker array reproduction The last application is similar to the creation of virtual microphones but is intended for multichannel rendering, where the loudspeaker signal directivities are calculated according to VBAP algorithm [17]. In all cases, the desired patterns are modeled with a set of sensor array responses and the method presented in Section 2. An irregular five channel system is used. The microphone pattern are designed according to calculated VBAP gains for azimuthal angles [70o , 120o , 180o , 240o , 290o ] corresponding to left surround, left, center, right and right surround channels. These pattern are then reconstructed by using the assembled microphone array. Fig. 7 shows the reconstructed pattern for one specific frequency. A robust performance can be observed. At lowfrequencies, below 800Hz the directivity patterns appears wider and can result to an higher inter channel
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frequency = 1031.25 5
0
−5
magnitude (dB)
−10
−15
−20 Left Right Center Surround Left Surround Right Model − Left Model − Right Model − Center Model − Surround Left Model − Surround Right
−25
−30
−35
−40
Fig. 6: Target and reconstructed HRTF employing an eight sensor simulated microphone array for the left ear
This study proposes a framework to calculate the appropriate weights for an arbitrary microphone array to obtain a desired directional pattern. It is shown that the method deduces the appropriate coefficients for reconstructing a target directivity. The main advantages of the proposed method can be summarized as follows
• Arbitrary microphone arrays and sensors: The current algorithm is capable of employing any microphone array geometry. That results to an immunity of the algorithm to microphone positioning errors since the spectral properties of the sensors are taken into consideration in the analysis part. Furthermore, there are no constrains on the type of sensors. The microphone array can consists of sensors of different
100
150 200 angle (degrees)
250
300
350
characteristics either in terms of directivity or frequency response. The radiation and the geometrical positioning of each sensor is incorporated in the measurements and analyzed with the Fourier series.
DISCUSSION
• Arbitrary target pattern: The use of the proposed algorithm can approximate not only directional patterns that are constant per frequency, such as the cylindrical harmonics or other predefined patterns, but also frequency varying patterns such as HRTFs.
50
Fig. 7: Angular response of 5 channel reproduction system with loudspeaker directivities reconstructed employing an eight sensor simulated microphone array.
coherence which might degrade the performance of the system. 5.
0
• Low frequency robustness: The robustness at low frequencies is increased by smoothing the target directivity patterns at the frequencies with high sensitivity values. That means that there is a compensation on the width of the directivity for system robustness. 6.
CONCLUSION
A method for synthesizing directional patterns, from arbitrary microphone arrays, is presented in the twodimensional continuous cylindrical coordinate system, approximated by a dense set of measured transfer functions. It is shown that robust performance can be obtained with arbitrary target directional patterns. The potential of this method is to simplify the calculation of weights, that are applied to the sensors of a microphone array in order to obtain a desired directional pattern. That is performed by employing a set of measured responses of a microphone array and without any a priori assumption of
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their spectral behavior. The systematic choice of microphones and number of Fourier coefficients in the least squares problem results to an optimal solution. The low frequency amplification is resolved by a systematic widening of the target directional pattern at those frequencies. The results of both simulated and real arrays show accurate reconstruction of a target directional pattern. ACKNOWLEDGMENT The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [240453]. The Academy of Finland has supported this work. 7.
REFERENCES
[1] T. D. Abhayapala.; B. D. Ward.; , ”Theory and design of high order sound field microphones using spherical microphone array,” Acoustics, Speech, and Signal Processing (ICASSP), 2002 IEEE International Conference on , vol.2, no., pp.II-1949-II1952, 13-17 May 2002 [2] B. Rafaely, “Analysis and Design of Spherical Microphone Arrays”, IEEE Trans Audio, Speech and Language Processing, Vol. 13, No. 1, pp 135-143, January 2005 [3] J. Meyer, G. Elko; , ”A highly scalable spherical microphone array based on an orthonormal decomposition of the soundfield,” Acoustics, Speech, and Signal Processing (ICASSP), 2002 IEEE International Conference on , vol.2, no., pp.II-1781-II-1784, 13-17 May 2002. [4] J Atkins, “Robust Beamforming and Steering of Arbitrary Beam Patterns Using Spherical Arrays”, IEEE Workshop on Applications to Audio and Acoustics. October, 2011. [5] J. Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects”, J. Acoust. Soc. Am., Vol. 109 (1), January 2001. [6] S. Moreau, J. Daniel, S. Bertet, “3D Sound Field Recording with Higher Order Ambisonics - Objective Measurements and Validation of Spherical Microphone”, presented at the AES 120th Convention, Paris, France, 2006, May20–23.
[7] M. A. Poletti, “A Unified Theory of Horizontal Holographic Sound Systems”, J. Audio Eng. Soc., Vol. 48, no. 12 (2000 December). [8] A. Kuntz, “Wave Field Analysis Using Circular Microphone Arrays”, Ph.D. thesis, Erlangen, 2008. [9] A. Farina, A. Capra, L. Chiesi, L Scopece, “A Spherical Microphone Array for Synthesizing Virtual Directive Microphones in Live Broadcasting and in Post Production”, presented at the AES 40th Internation Conference, Tokyo, Japan 2010 October 8–10. [10] A. Laborie, R. Bruno and S. Montoya, “A New Comprehensive Approach to Surround Sound Sound Recording”, Proc. of the 114th Convention of the Audio Engineering Society, Amsterdam, Netherlands, Mar. 22-23, 2003. [11] Michael Brandstein and Darren Ward, “Microphone Arrays: Signal Processing Techniques and Applications”, Springer, 1 edition, June 2001. [12] Harry L. Van Trees, “Optimum Array Processing (Detection, Estimation and Modulation Theory, Part IV)”, Wiley-Interscience; Part IV edition, March 22, 2002 [13] A. Farina, “Simultaneous measurements of impulse response and distortion with a swept-sine technique”, presented at the AES 108th Convention, Paris, France 2000 February 19–22. [14] J. Dmochowski, J. Benesty, and S.Affs ,“On Spatial Aliasing in Microphone Arrays”, IEEE Trans Audio, Speech and Language Processing, Vol. 57, No. 4, April 2009, [15] M.O.J. Hawksfors, “HRTF-enabled Microphone Array for Binaural Synthesis”, presented at the AES 130th Convention, London, UK 2011 May 13– 16. [16] B. J. Gomez, V. Pulkki, “HRIR Database with Measured Actual Source Direction Data”, presented at the AES 133th Convention, San Francisco, USA 2012 October 26–29. [17] V. Pulkki, “Virtual source positioning using vector base amplitude panning”, J. Audio Eng. Soc. 45(6), 1997
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