Christensen, Mads Græsbøll

So, Hing-Cheung

Zhou, Zhenhua

English

In this paper, the problem of fundamental frequency and direction-of-arrival (DOA) estimation for multi-channel harmonic sinusoidal signal is addressed. The estimation procedure consists of two stages. Firstly, by making use of the subspace technique and Markov-based eigenanalysis, a multi- channel optimally weighted harmonic multiple signal classification (MCOW-HMUSIC) estimator is devised for the estimation of fundamental frequencies. Secondly, the spatio- temporal multiple signal classification (ST-MUSIC) estimator is proposed for the estimation of DOA with the estimated frequencies. Statistical evaluation with synthetic signals shows the high accuracy of the proposed methods compared with their non-weighting versions

    Aalborg UniversitetTwo stage DOA and Fundamental Frequency Estimation based on SubspaceTechniquesZhou, Zhenhua; Christensen, Mads Græsbøll; So, Hing-CheungPublished in:International Conference on Signal Processing. ProceedingsDOI (link to publication from Publisher):10.1109/ICoSP.2012.6491637Publication date:2012Document VersionEarly version, also known as pre-printLink to publication from Aalborg UniversityCitation for published version (APA):Zhou, Z., Christensen, M. G., & So, H-C. (2012). Two stage DOA and Fundamental Frequency Estimation basedon Subspace Techniques. International Conference on Signal Processing. Proceedings, 1, 210-214.https://doi.org/10.1109/ICoSP.2012.6491637General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            ? You may not further distribute the material or use it for any profit-making activity or commercial gain            ? You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Two Stage DOA and Fundamental Frequency Estimation Based on Subspace Techniques Zhenhua Zhou†, M. G. Christensen‡, H. C. So† †Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China ‡Department of Architecture, Design and Media Technology, Aalborg University, Aalborg, DenmarkE-mail: sjtu.zzh2010@gmail.com, mgc@create.aau.dk, hcso@ee.cityu.edu.hkAbstract—In this paper, the problem of fundamental frequency and direction-of-arrival (DOA) estimation for multi-channel harmonic sinusoidal signal is addressed. The estimation procedure consists of two stages. Firstly, by making use of thesubspace technique and Markov-based eigenanalysis, a multi-channel optimally weighted harmonic multiple signal classification (MCOW-HMUSIC) estimator is devised for the estimation of fundamental frequencies. Secondly, the spatio-temporal multiple signal classification (ST-MUSIC) estimator is proposed for the estimation of DOA with the estimated frequencies. Statistical evaluation with synthetic signals shows the high accuracy of the proposed methods compared with theirnon-weighting versions.Keywords-fundamental frequency estimation; DOA estimation;subspace method; harmonic sinusoidal signal; Markov optimum weighting I. INTRODUCTIONThe problem of fundamental frequency or pitch estimationhas been of interest to the signal processing community formany years, finding applications in a wide range of areas such as audio and speech coding, classification of music, and speech analysis. Kinds of algorithms have been proposed for both the single-pitch and multi-pitch scenarios. The nonlinear leastsquares (NLS) estimator [1] is statistically efficient for single pitch scenario, but performs poorly in the multi-pitch scenariodue to the decoupling difficulty [2]. In [2]-[3], the harmonic multiple signal classification (HMUSIC) algorithm is proposedfor single-pitch and multi-pitch estimation. However, the HMUSIC estimation is not statistically efficient, and itsaccuracy cannot attain Cramér-Rao lower bound (CRLB) [4]. Recently, multi-channel approaches have attractedconsiderable attention in both single-pitch and multi-pitch scenarios [5]-[7], where both direction-of-arrival (DOA)estimation and fundamental frequency estimation are ofimportance. In [8], the multi-channel multi-pitch harmonic multiple signal classification (MC-HMUSIC) is presented for solving the problem of joint fundamental frequency and DOA estimation. The simulations show its increased accuracy in DOA and fundamental frequency estimation compared with other state-of-the-art method. Nevertheless, the two-dimensional search is utilized for the estimates of fundamentalfrequency and DOA, which makes the computation complexityhigh. Still, its fundamental frequency estimation is notstatistically efficient. To overcome the disadvantages of the MC-HMUSIC estimator, we propose to estimate the fundamental frequencyand DOA in two stages. At first, the multi-channel optimally weighted harmonic multiple signal classification (MCOW-HMUSIC) estimator is devised, and the estimation offundamental frequency is conducted. The MCOW-HMUSICestimator is the extension of the HMUSIC estimator to the multi-channel scenario with optimum weighting. As mentioned,the HMUSIC estimator is not statistically efficient, and there is a gap between its mean square error and CRLB. Therefore, a kind of Markov optimum weighting is utilized to improve itsaccuracy. To facilitate the statistical analysis of orthogonality error, we modify the HMUSIC formulation. Then we analyze the perturbation of the orthogonality error, and derive the weighting matrix. The fundamental frequency estimates are calculated in an iterative manner. Simulation results show thatthe MCOW-HMUSIC estimator performs better than its non-weighting version. Especially in the single-pitch scenario, its performance can attain the CRLB. Secondly, we make use of the spatio-temporal multiple signal classification (ST-MUSIC) estimator to estimate the DOA with the estimated fundamental frequency. Since we estimate the frequency and DOA at two stages, the computationally intensive two-dimensional search in [8] is avoided. The rest of this paper is organized as follows. The proposed estimators for fundamental frequency and DOA,MCOW-HMUSIC and ST-MUSIC, are developed in Section II.The statistical properties of the orthogonality error is analyzed, and the optimum weighting matrix is derived. In Section III,simulation results are included to show the performance of the proposed estimators by comparing with their non-weightingversions as well as CRLB. Finally, conclusions are drawn inSection IV. II. ALGORITHM DEVELOPMENTA. Spatio-Temporal Signal ModelWithout multi-path propagation of sources, the multi-channel signal model is given as follows. The signalreceived by the microphone element (channel) arranged in a uniform linear array (ULA) configuration, , isgiven by [8],                           (1) ,        (2) for , with , ,  and  denoting unknown fundamental frequency, scaled DOA, amplitudes and initial phases of the k-th source. The number of sources and the number of harmonics of each source , areassumed to be known, or found in some way such as [9]. The  ( ) are assumed to be uncorrelated whiteGaussian noise with variance . The objective is to estimatethe nonlinear parameters  and . Once they are estimated, the remaining linear parameters can be estimated as a linear least squares (LLS) solution. B. Fundamental Frequency EstimationFirstly, we focus on the single-pitch case, that is, . Construct the data matrix for each channel  [10]: ,                                (3) where  is the data matrix with elements, . The matrices  and are the noise-free and noise components of , respectively. It is straightforward that  can be factorized as: ,                                  (4) where ,                       (5) ,                        (6) ,                        (7) ,                        (8) ,                       (9) ,                    (10) for , and . On the other hand,can be decomposed using singular value decomposition(SVD) as: ,     (11)with  and spanning the signal and noise subspace, respectively. Comparing (4) and (11), the HMUSIC estimate of from channel , denoted by , is obtained by minimizing the orthogonality error between  and : ,          (12) with the variable for , and denoting the Frobeniusnorm. As demonstrated in [2]-[3], this kind of estimator is notstatistically efficient. To improve its accuracy, we introduce Markov optimum weighting. Due to the non-uniqueness of the columns of , that is any linear combination of these vectors still spans the noise space, it is impossible to analyze theirstatistical properties. To overcome such difficulty, we reformulate the HMUSIC estimator to a different but equivalent form as follows:,                       (13) where is the orthogonality error vector of the form  with being the vectorizationoperator. According to the results of [11], and after some extra manipulations, the first-order approximation of  originating from  is derived as ,   (14) where stands for the Kronecker product,  is the perturbation of , and ,  are defined as ,                   (15) ,                             (16) ,              (17), for . Based on the aboveperturbation analysis, the Markov optimum weighting matrixfor the channel , is determined as [12],                  (18) where stands for pseudoinverese. Then for each channel , is estimated as .         (19) When dealing with the multi-channel signal, we stack the error vectors  as , and the fundamental frequency estimate  is computed as ,                         (20) where  dueto the uncorrelatedness among the channels. Then the fundamental frequency estimate  is expressed as .                        (21) As are functions of the unknown , the following iterative procedure is employed to solve for : � Step 1. Set which is the identity matrix. � Step 2. Find  by searching for the minimum of (21).� Step 3. Compute  by (18). � Step 4. Repeat Steps 2 and 3 until a stopping criterion is reached. This estimator is termed as the MCOW-HMUSIC method.We now proceed to focus on the multi-pitch case. Following the idea of the single-pitch case,  can be factorized as: ,                                   (22) where , , , and .  and  can be found similarly. On the other hand, the left singular vectors of the data matrix associated with its ( ) smallest singular values span the noise subspace . Then theorthogonality error vector for each source is given as. Following the similar steps like finding the initial values of fundamental frequencies, constructing weighting matrices of (18) for each of them, and searching for its closest minimum of (21), we can solve theMCOW-HMUSIC estimates of the multi-pitch fundamental frequencies in an iterative way. C. DOA EstimationFirst, the time-domain data samples of the array output are collected to form the data matrix , which has the form of ,                    (23) and can also be expressed as ,                (24)where is the noise part of , , , , , , , and the vector  contains the complex amplitudes ,         (25) with . The multiple sources impinge on the array with different DOAs consisting of various frequencycomponents may, for certain frequency combinations, give the same array steering vector, which causes the matrix  tobe rank deficient. Here, the ambiguities and the rank-deficiency are avoided by introducing temporal smoothness in order to restore the rank of . The temporally smoothed data matrix is obtained by stacking  times temporallyshifted versions of the original data matrix, given as [8] ,      (26) where is the temporally smoothed data matrix, and is its noise part. On one hand,  can befactorized as ,       (27) where , whose submatrix for each individual source  is given by.   (28) On the other hand,  can be decomposed using SVD as:,                          (29) with  and spanning the signal and noise subspace, respectively. Comparing (27) and (29), the DOA andfundamental frequency for each source can be estimated jointly by minimizing the cost function ,                         (30) with  and the variable for  and . Now that we obtain the estimate for the fundamental frequency in the last subsection, the DOA is estimated with the frequency estimate as .                           (31) Note that in the two-stage estimation, only two one-dimensional searches are required and the two-dimensionalsearch in the joint estimation of DOA and fundamental frequency is avoided, which decreases the computation complexity. This DOA estimation method is termed as spatio-temporal multiple signal classification (ST-MUSIC) estimator. III. SIMULATION RESULTSIn this section, we perform Monte Carlo simulations toevaluate the fundamental frequency and DOA estimation performance of the proposed method. The estimation accuracy is evaluated using the mean square error (MSE):  and , with ,  and , being the true fundamental frequency, DOA and their estimates, respectively, and being the number of trials. Weuse the number of iterations as the stopping criterion in theMCOW-HMUSIC algorithm, which is assigned as 3. In the fundamental frequency estimation, the row number of the datamatrix for each channel is set as  withdenoting the smallest integer larger than . In the DOA estimation, the row number of the data matrix is set as. Such row number settings are found to resultin good performance empirically. The microphone array size isset as . All the results are averages of 100 independentruns. First, we provide an example of single-pitch estimation. The harmonic signal consists of  sinusoids withfundamental frequency of and DOA of . The parameter setting is listed in Table 1. Fig.1 shows the MSEs of fundamental frequency estimates by MCOW-HMUSIC and its non-weighting version (which is termed asMC-HMUSIC here) as well as CRLB, with  and , respectively. It is seen that the MCOW-HMUSIC estimateattains the optimum accuracy when . However,there is about 3 dB gap between the MSE of MC-HMUSIC and CRLB. Fig.2 shows the corresponding MSEs of DOAestimates by ST-MUSIC with frequency estimate fromMCOW-HMUSIC and MC-HMUSIC as well as CRLB. Forthe DOA estimation, the results are the same with frequency estimates from any method, and there are about 3 dB and 5 dBgaps from CRLB when  and , respectively.TABLE 1. SIMULATION SETTING OF SINGLE-PITCH ESTIMATIONDOA l Frequency Amplitude Initial Phase1 0.5 2.0 1 2 1.0 1.5 2 3 1.5 2.5 3 0.44 2.0 4.0 4 The next example is about multi-pitch estimation. The harmonic signal consists of pitches, each with tones. The parameter setting is listed in Table 2. Fig.3 and Fig.4 show the MSEs of fundamental frequency and DOA,respectively. From Fig.3, we can still see that the MCOW-HMUSIC estimator is superior to MC-HMUSIC by 5–10 dB,and its MSE can be close to CRLB. In addition, when the datalength is larger, the MCOW-HMUSIC estimator performs withbetter accuracy. For the DOA estimation, the results are thesame with frequency estimates from any method, and there are about 5 dB and 3 dB gaps from CRLB when  and , respectively. TABLE 2. SIMULATION SETTING OF TWO-PITCH ESTIMATIONk DOA lk Frequency Amplitude Initial Phase1 0.3 2.0 1 1 0.42 0.6 1.0 2 1 0.5 2.0 3 2 0.6 2 1.0 1.0 4 IV. CONCLUSIONThe two stage estimation of the fundamental frequency and DOA for the multi-channel harmonic signal is proposed. In the fundamental frequency estimation, the MCOW-HMUSICestimator is developed, where the perturbation of orthogonality error is analyzed, and the Markov optimum weighting isutilized. Then the DOA is estimated with the ST-MUSIC and the estimated frequency. The two-dimensional search in the joint DOA and fundamental frequency estimation is replaced with two one-dimenional searches, which saves thecomputational complexity. Simulation results show that the MCOW-HMUSIC method improves the accuracy of theconventional HMUSIC method, and its performance can attainCRLB for single-pitch estimation. Moreover, the ST-MUSICmethod provides accurate estimation for the DOA. Further works include dealing with more complicated multi-pitchestimation problems, adaptive pitch estimation, and their application to speech and audio signal processing. REFERENCES[1] P. Stoica and A. Nehorai, “Statistical analysis of two non-linear least-squares estimators of sine waves parameters in the colored noise,” in Proc. ICASSP, vol.4, pp.2408-2411, Apr. 11-14, 1988, New York, NewYork, USA.[2] M.G. Christensen, P. Stoica, A. Jakobsson and S.H. Jensen, “Multi-pitch estimation,” Signal Processing, vol.88, no.4, pp.972-983, Apr. 2008.[3] M.G. Christensen, A. Jakobsson and S.H. Jensen, “Joint high-resolutionfundamental frequency and order estimation,” IEEE Transactions onAudio, Speech, and Language Processing, vol.15, no.5, pp.1635-1644,Jul. 2007.[4] S.M. Kay, Fundamentals of Statistical Signal Processing – EstimationTheory, Englewood Cliffs, N.J.: PTR Prentice-Hall, 1993.[5] L. Armani and M. Omologo, “Weighted autocorrelation-based F0 estimation for distant-talking interaction with a distributed microphone network,” in Proc. ICASSP, vol.1, pp.113-116, May 17-21, 2004,Montreal, Quebec, Canada. [6] S.N. Wrigley and G.J. Brown, “Recurrent timing neural networks forjoint F0-localisation based speech separation,” in Proc. ICASSP, vol.1,pp.157-160, Apr. 15-20, 2007, Honolulu, Hawai'i, USA. [7] J.R. Jensen, M.G. Christensen and S.H. Jensen, “Joint DOA and fundamental frequency estimation methods based on 2-D filtering,” in Proc. EUSIPCO, vol.2010, pp.2091-2095, Aug. 23-27, 2010, Aalborg, Denmark. [8] J. X. Zhang, M.G. Christensen, S.H. Jensen and M. Moonen, “JointDOA and multi-pitch estimation based on subspace techniques,”EURASIP Journal on Advances in Signal Processing, vol.2012, no.1,pp.1-11, Jan. 2012.[9] M.G. Christensen and A. Jakobsson, Multi-pitch estimation, [San Rafael, Calif.] : Morgan & Claypool Publishers, 2009.[10] P. Stoica, Introduction to spectral analysis, Upper Saddle River, N.J. : Prentice Hall, 1997.[11] J. Liu, X.-Q. Liu and X.-L. Ma, “First-order perturbation analysis ofsingular value decomposition,” IEEE Transactions on Signal Processing, vol.56, no.7, pp.3044-3049, Jul. 2008.[12] T. Soderstrom and P. Stoica, System Identification, Englewood Cliffs,N.J.: Prentice-Hall International, 1989.0 5 10 15 20 25 30 35 40-110-105-100-95-90-85-80-75-70-65-60SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB0 5 10 15 20 25 30 35 40-90-85-80-75-70-65-60-55-50-45-40SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB0 5 10 15 20 25 30 35 40-90-85-80-75-70-65-60-55-50-45SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB0 5 10 15 20 25 30 35 40-120-115-110-105-100-95-90-85-80-75-70SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB(a) 0 5 10 15 20 25 30 35 40-105-100-95-90-85-80-75-70-65-60-55SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB(b) Figure 3. MSEs of two-pitch fundamental frequency estimation versus SNR for: (a) N = 50 and (b) N = 100.0 5 10 15 20 25 30 35 40-75-70-65-60-55-50-45-40-35-30-25SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB(a) 0 5 10 15 20 25 30 35 40-80-75-70-65-60-55-50-45-40-35SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLB(b) Figure 4. MSEs of two-pitch DOA estimation versus SNR for: (a) N = 50 and (b) N = 100.0 5 10 15 20 25 30 35 40-100-90-80-70-60-50-40-30SNR (dB)MSE (dB)MC-HMUSICMCOW-HMUSICCRLBFigure 2. MSEs of single-pitch DOA estimation versusSNR for: (a) N = 50 and (b) N = 100.(b) (a) Figure 1. MSEs of single-pitch fundamental frequency estimation versus SNR for: (a) N = 50 and (b) N = 100. (b) (a) 

Two stage DOA and Fundamental Frequency Estimation based on Subspace Techniques

https://vbn.aau.dk/ws/files/71779329/zhou2012.pdf

Two stage DOA and Fundamental Frequency Estimation based on Subspace Techniques

Abstract

Similar works

Full text

Available Versions

VBN