THE SPHERICAL HARMONICS ROOT-MUSIC

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1 THE SPHERICAL HARMONICS ROOT-MUSIC Laan Kumar, Guoan Bi Schoo of EEE Nanyang Technoogica University, Singapore Emai : {kumar,egbi}@ntu.edu.sg Rajesh M. Hegde Eectrica Engineering Dept. Indian Institute of Technoogy, Kanpur Emai : rhegde@iitk.ac.in ABSTRACT Spherica harmonics root-music (MUtipe SIgna Cassification) technique for source ocaization using spherica microphone array is presented in this paper. Earier work on root-music is imited to inear and panar arrays. Root-MUSIC for panar array utiizes the concept of manifod separation and beamspace transformation. In this paper, the Vandermonde structure of array manifod for a particuar order is proved. Hence, the vaidity of root-music in the spherica harmonics domain is confirmed. The proposed method is evauated by using simuated experiments on source ocaization. Root mean square error anaysis and statistica anaysis are presented. The experimenta measures at various signa to noise ratios (SNRs) show the robustness of the proposed method. The method is aso verified by using experiment on rea signa acquired over spherica microphone array. Index Terms Root-MUSIC, Spherica microphone array, Spherica harmonics, Manifod separation. INTRODUCTION The use of accurate and search free agorithms for estimating direction of arriva (DOA) has been a very active research area in source ocaization. Root-MUSIC (MUtipe SIgna Cassification) [] and Estimation of Signa Parameters using Rotationa Invariance Techniques, ESPRIT [2], fa into this category. The root-music method estimates DOAs as the roots of MUSIC [3] poynomia owing to Vandermonde structure of array manifod. Such a structure is not observed in array manifod for uniform circuar array (UCA) [4]. Zotowski proposed beamspace transformation based on phase mode excitation to get the Vandermonde structure in array manifod with respect to azimuth ange [5]. Hence, it enabes the appication of root-music to azimuth estimation at a given eevation. The technique was further extended to sparse UCA root-music in order to utiize the modified beamspace transformation [6]. Another approach for extending the ULA root-music to a panar array is presented in [7] using manifod separation. The idea of manifod separation is to write the panar array steering vector (array manifod) as a product of a characteristic matrix of the array and a vector with Vandermonde structure depending on the azimuth ange. The manifod separation which utiizes spherica harmonics (SH) is introduced in [8]. After the introduction of higher order spherica microphone array and associated signa processing in [9] and [], various existing DOA estimation techniques were reformuated in the spherica harmonics domain. The eement space MUSIC was impemented in This work was funded in part by DST project EE/SERB/23277 and in part by BITCOE, IIT Kanpur. terms of spherica harmonics, caed SH-MUSIC, in [] and [2]. The Minimum Variance Distortioness Response (MVDR) spectrum in terms of spherica harmonics, SH-MVDR, was utiized for DOA estimation in []. MUSIC-Group deay [3] was formuated for source ocaization using spherica array in [4] and [5]. Differentia geometry was expored for SH domain source ocaization in [6]. In this work, we have deveoped the theory of root-music in SH domain using manifod separation technique. The theory is vaidated using simuation and rea data experiments. The proposed SH-root-MUSIC (SH-RM) technique provides exact soution without the imitation from the discretization issues associated with the SH-MUSIC and SH-MVDR methods for DOA estimation. 2. THE SPHERICAL HARMONICS DATA MODEL A spherica microphone array of order N, radius r and the number of sensors I is considered. A sound fied of L pane-waves is incident on the array with wavenumber k. The th source ocation is denoted by Ψ = (θ,φ ). The eevation ange θ is measured from the positive z axis, whie the azimutha ange φ is measured countercockwise from the positivexaxis. Simiary, thei th sensor ocation is given byφ i = (θ i,φ i). In spatia domain, the sound pressure ati microphones, p(k) = [p (k),p 2(k),...,p I(k)] T, is written as p(k) = V(k)s(k) + n(k) () wherep i(k) p(k,r,θ i,φ i),v(k) is ani L steering matrix,s(k) is al vector of signa ampitudes,n(k) is ani vector of zero mean, uncorreated sensor noise and(.) T denotes the transpose. The steering matrixv(k) is expressed as V(k) = [v (k),v 2(k),...,v L(k)], where (2) v (k) = [e jkt r,e jkt r 2,...,e jkt r I ] T (3) k = (ksinθ cosφ,ksinθ sinφ,kcosθ ) T (4) r i = (rsinθ icosφ i,rsinθ isinφ i,rcosθ i) T (5) where j =. The i th term in (3) refers to the pressure due to th unit ampitude panewave with wavevectork at ocationr i. This may aternativey be written as [7] n e jkt r i = b n(kr)[yn m (Ψ )] Yn m (Φ i) (6) n= m= n where b n(kr) is caed mode strength. The far-fied mode strength, b n(kr), is given by b n(kr) = 4πj n j n(kr), for open sphere (7) = 4πj n( j n(kr) j n(kr) ), for rigid sphere (8) h n(kr)

2 where j n(kr) is the spherica Besse function, h n(kr) is n th order spherica Hanke function of second kind and refers to the first derivative. Figure iustrates mode strength b n as a function of kr and n for an open sphere. For kr =., the zeroth order mode ampitude is22 db, whie the first order has an ampitude of 8 db. It is seen that for an order greater than kr, the mode strength b n decreases significanty. Therefore, the summation in (6) is truncated to a finite vaue of N, which is known as the array order. b n (kr) in db 5 5 n= n= n=2 n=3 n=4 5 kr Fig.. Mode ampitude b n for an open sphere as a function of kr andn The spherica harmonic of order n and degree m, Yn m (Ψ), is given by Yn m (2n+)(n m)! (Ψ) = Pn m (cosθ)e jmφ, 4π(n+m)! n N, m n = ( ) m Y m n (Ψ), n m <, (9) where Yn m is soution to the Hemhotz equation [8] and Pn m is the associated Legendre function. Figure 2 shows the pot of three spherica harmonics. It shoud be noted that Y is isotropic whie Y and Y have directiona characteristics. The spherica harmonics are used for spherica harmonics decomposition of a square integrabe function, simiar to the compex exponentia e jωt used for decomposition of rea periodic functions [9]. Fig. 2. Spherica harmonics pot,y,y, Y Substituting (3) and (6) in (2), the expression of steering matrix becomes V(k) = Y(Φ)B(kr)Y H (Ψ) () where Y(Φ) isi (N +) 2 matrix whose i th row is given as y(φ i) = [Y (Φ i),y (Φ i),y (Φ i),y (Φ i),...,y N N (Φ i)]. () The L (N +) 2 matrix Y(Ψ) can be expanded on simiar ines. The(N +) 2 (N +) 2 matrixb(kr) is given by B(kr) = diag(b (kr),b (kr),b (kr),b (kr),...,b N(kr)). (2) With the introduction of spherica harmonics, the spherica harmonics decomposition of the received pressure, p(k), is given as [2] p nm(k) = = 2π π p(k)[yn m (Φ)] sin(θ)dθdφ I a ip i(k)[y nm(φ i)], (3) i= where p nm(k) is spherica Fourier coefficient. The spatia samping of pressure over a spherica microphone array is captured by using samping weights,a i [2]. Re-writing (3) in a matrix form, we have p nm(k) = Y H (Φ)Γp(k), (4) where p nm(k) = [p,p ( ),p,p,...,p NN] T and Γ = diag(a,a 2,,a I). Aso, under the assumption of (3), we have the orthogonaity property of spherica harmonics as foows Y H (Φ)ΓY(Φ) = I, (5) where I is an (N + ) 2 (N + ) 2 identity matrix. Substituting () in (), then mutipying both sides withy H (Φ)Γ and utiizing the reations in (4) and (5), we have the data mode in spherica harmonics domain as p nm(k) = B(kr)Y H (Ψ)s(k)+n nm(k). (6) For a given array configuration, B(kr) is a constant. Therefore, we get the fina spherica harmonics data mode by mutipying both sides of (6) withb (kr) as where (k) = Y H (Ψ)s(k)+z nm(k), (7) z nm(k) = B (kr)n nm(k). (8) 3. THE SPHERICAL HARMONICS ROOT-MUSIC Root-MUSIC estimates DOAs as roots of the MUSIC poynomia. Hence, we first write the MUSIC spectrum in spherica harmonics domain. Comparing the spatia data mode in () with spherica harmonics data mode in (7),[Y H (Ψ)] (N+) 2 L is the steering matrix in spherica harmonics domain. Hence, the SH-MUSIC spectrum is written as P SH MUSIC(Ψ) = y(ψ)s NS [S NS ] H y H (Ψ), (9) wherey H (Ψ) is a steering vector defined in (). S NS is the noise subspace obtained from eigenvaue decomposition of autocorreation matrix, S anm = E[(k)(k) H ]. Frequency smoothing and whitening of noise shoud be appied as in []. The SH-MUSIC spectrum is shown in Figure 3(a) for two sources at (2,4 ) and (2,8 ). The two peaks in the figure correspond to the two sources. The SH-MUSIC spectrum in (9) resuts in a peak which corresponds to a source owing to orthogonaity between noise eigenvector and steering vector. A comprehensive search agorithm is needed to estimate the DOA of the desired source. The resoution is aso imited by the resoution of discretization at which the spectrum is evauated. The SH-root-MUSIC overcomes these imitations in estimating the DOAs. We first iustrate the Vandermonde structure in the steering vector using manifod separation technique. Utiizing

3 Actua poe Noisy poe Imaginary Part.5.5 (a).5.5 Rea Part (b) Fig. 3. (a) SH-MUSIC spectrum (b) SH-root-MUSIC iustrating the actua DOA estimates (red poes) and noisy DOA estimates (bue poes) (order of the spherica array, N = 4, sources at (2,4 ), (2,8 ) and SNR=5dB). (9) and (), the steering vector for co-eevation θ can be written in a more compact form as y H (Ψ) = y H (θ,φ) where, f nm = = [f, f ( ) e jφ,f,f e jφ,,f NNe jnφ ] T (2) (2n+)(n m )! P n m (cosθ ). (2) 4π(n+ m )! Then (2) is rewritten in a matrix form as y H (θ,φ) = F(θ )d(φ) (22) where, F(θ ) = diag(f, f ( ),f,f,,f NN) (23) d(φ) = [,e jφ,,e jφ,,e jnφ ] T. (24) 4. PERFORMANCE EVALUATION Simuation experiments based on source ocaization were carried out to evauate the proposed SH-root-MUSIC method. Additionay, experiments were performed on rea signa acquired over spherica microphone array to verify the agorithm. The experiments utiized an Eigenmike R system [22] which is shown in Figure 4. It consists of 32 microphones, embedded in rigid sphere of radius 4.2cm. The order of the microphone array was taken to be 4. Root mean square error (RMSE) and probabiity of resoution vaues were used to evauate the source ocaization performance of the proposed method. The performance of the proposed method is compared to SH-MUSIC and SH-MVDR. The matrix d(φ) consists of ony the exponent terms containing the azimuth ange and, each submatrix corresponding to a particuar order foows the Vandermonde structure with common ratio as e jφ. From (9) and (22), the SH-MUSIC cost function can be written as P SHM(φ) = d H (φ)f H (θ )S NS [S NS ] H F(θ )d(φ) = d H (φ)f H (θ )CF(θ )d(φ) (25) where C = S NS [S NS ] H. By defining z = e jφ, the SH-MUSIC cost function now assumes a poynomia form of degree 4N, given by P SHM(φ) = 2N u= 2N C uz u (26) where the coefficient C u is obtained mathematicay. The poynomia has 4N roots. If z is a root of the poynomia then wi z aso be the root. Hence, 2N roots are within the unit circe whie the other2n roots are outside the unit circe. Of the2n roots within the unit circe, L roots cose to the unit circe correspond to the DOAs. This is iustrated in Figure 3(b) for a fourth order spherica microphone array. The roots are potted for two sources with co-eevation ange 2 and azimuth ange (4,8 ) at SNR 5dB. A the roots within and near the unit circe are shown in the figure. The DOA can be estimated from the roots by using the reation, φ = I(n(z)), where I() is the imaginary part of (). Fig. 4. The Eigenmike R setup in an anechoic chamber at IIT Kanpur for acquiring a far-fied source. 4.. Simuation Experiments on DOA Estimation The RMSE anaysis and statistica anaysis are presented here for 5 independent Monte Caro trias. The additive noise is assumed to be zero mean Gaussian distributed. Two sources with co-eevation 2 are considered RMSE Anaysis The experiments on source ocaization are presented as cumuative RMSE (CRMSE) computed by CRMSE = t= = [(φ ) 2 ], (27) where t is the tria index whie denotes the source index. The CRMSE vaues are potted in Figure 5(a) with various SNR vaues

4 CRMSE SH MVDR SH MUSIC SH RM SNR (db) (a) CRMSE 5 SH MVDR SH MUSIC SH RM Azimuth Separation( ) (b) Fig. 5. Cumuative RMSE for two sources with co-eevation 2, (a) azimuth(4,8 ) at various SNRs. (b) azimuth of one source is fixed at4 and that of other source is varying in steps of. SNR= 2dB. for two sources at (2,4 ) and (2,8 ). The CRMSEs vaues are aso potted in Figure 5(b) for the case where azimuth of one source is fixed as 4, whie that of the other source varies at a step size of. The SNR in this case is fixed at 2dB. It shoud be noted that the proposed SH-root-MUSIC method performs reasonaby better than SH-MUSIC and SH-MVDR Statistica Anaysis Statistica anaysis of the proposed method is presented in terms of probabiity of resoution for various SNRs. Two sources with DOAs (2,4 ) and (2,8 ) are considered. The confidence interva of ζ = 5 was used for cacuating the probabiity over 5 independent trias. The probabiity of resoution is given by P r = = t= = t= = [Pr( φ [sgn(ζ φ ζ)] )], (28) where Pr(.) denotes the probabiity of an event, and sgn(x) is defined as { ifx sgn(x) = (29) ifx <. The resut is presented in Tabe in which zero probabiity indicates inabiity of the methods to resove sources in the given confidence interva. It is noted that the proposed method has higher probabiity of resoution when compared to other methods at a SNR vaues. It can aso be concuded that a higher SNR is required for SH-MVDR to resove co-eevated sources. Tabe. Probabiity of resoution performance of various methods. Method SNR SNR SNR SNR SNR (5dB) (db) (5dB) (2dB) (25dB) SH-RM SH-MUSIC SH-MVDR Rea Data Experiments The proposed agorithm is aso verified by using rea signa acquired over spherica microphone array. The experimenta set-up for acquiring a source using Eigenmike R system is shown in Figure 4. A smartphone speaker is utiized as an acoustic source. The source is fixed at ocation (9,9 ) in far-fied region. A narrowband signa with frequency of 25Hz is payed. The eevation of the source is assumed to be known and the azimuth is estimated using the proposed SH-root-MUSIC method. A the2n(= 8) roots within the unit circe are potted in Figure 6. The root with argument cose to9 corresponds to the source and is represented by red star. It is noted that noisy roots are aso competing in magnitude. The DOA estimation mismatch and mutipe competing roots are due to the refection of sound from the tripods, non-point sound source and microphone-source physica pacement errors. Imaginary Part.5.5 Actua poe Noisy poe Rea Part Fig. 6. Azimuth estimation of a source at(9,9 ) using SH-root- MUSIC. A roots within unit circe are shown for N = 4. The star denotes the actua estimate. 5. CONCLUSION In this paper, theory of root-music is estabished in spherica harmonics domain. The theory is vaidated using simuation and rea data experiments. The SH-root-MUSIC method does not require any search to estimate the DOAs. It provides DOA estimates as direct roots of SH-MUSIC poynomia. The Vandermonde structure of array manifod in spherica harmonics domain is shown using manifod separation technique. The robustness of the method is iustrated by using source ocaization experiments for various SNRs and anguar separations. The RMSE and probabiity of resoution vaues indicate the reevance of the proposed method. Additionay, the method is verified with rea signa acquired over spherica microphone array. Owing to its robustness and high resoution, a rea time impementation for voiced-based camera steering in a meeting room can be expored.

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