Publication in the conference proceedings of EUSIPCO, Viena, Austria, 200

Andersen, S. V.

Christensen, Mads Græsbøll

Jakobsson,, A.

Jensen, Søren Holdt

Jakobsson, Andreas

Andersen, S.V.

Christensen, Mads

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

SUBSPACE-BASED FUNDAMENTAL FREQUENCY ESTIMATION

    Aalborg UniversitetSubspace-based Fundamental Frequency EstimationChristensen, Mads Græsbøll; Jensen, Søren Holdt; Andersen, Søren Vang; Jakobsson,, A.Published in:Proceedings of the 12th European Signal ProcessingPublication date:2004Document VersionPeer reviewed versionLink to publication from Aalborg UniversityCitation for published version (APA):Christensen, M. G., Jensen, S. H., Andersen, S. V., & Jakobsson, A. (2004). Subspace-based FundamentalFrequency Estimation. In Proceedings of the 12th European Signal Processing (pp. 637-640). IEEE SignalProcessing Society.General 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.Downloaded from vbn.aau.dk on: April 25, 2017SUBSPACE-BASED FUNDAMENTAL FREQUENCY ESTIMATIONMads Græsbøll Christensen†, Søren Holdt Jensen†, Søren Vang Andersen†, and Andreas Jakobsson‡† Dept. of Communication Technology, Aalborg University, DK-9220 Aalborg, DenmarkPhone: +45 9635 {8672,8654,8627}, email: {mgc,shj,sva}@kom.auc.dk‡ Dept. of Electrical Engineering, Karlstad University, SE-651 88 Karlstad, SwedenPhone: +46 (0)54 700 2330, email: andreas.jakobsson@ieee.orgABSTRACTIn this paper, we present a subspace-based fundamental fre-quency estimator based on an extension of the MUSIC spec-tral estimator. A noise subspace is obtained from the eigen-value decomposition of the estimated sample covariance ma-trix and fundamental frequency candidates are selected as thefrequencies where the harmonic signal subspace is closest tobeing orthogonal to the noise subspace. The performance ofthe proposed method is evaluated and compared to that ofthe non-linear least-squares (NLS) estimator and the corre-sponding Cramér-Rao bound; it is concluded that the pro-posed method has good statistical performance at a lowercomputational cost than the statistically efficient NLS esti-mator.1. INTRODUCTIONThe problem of estimating the fundamental frequency of aperiodic signal is a classical problem in signal processing,and throughout the years many different solutions have beensuggested to solve it. It is encountered in such applicationsas, for instance, coding of speech and audio, automatic musictranscription and determination of rotating targets in radar.The problem of fundamental frequency estimation can bestated as follows; consider a harmonic signal with the fun-damental frequency ω0 that is corrupted by an additive whitecomplex circularly symmetric Gaussian noise, w(n), i.e.,x(n) =L∑l=1Alej(ω0ln+φl) +w(n), n = 0, . . . ,N−1 (1)where Al and φl are the (real-valued) amplitude and the phaseof the l’th harmonic, respectively. The problem considered inthis paper amounts to estimating the fundamental frequencyω0 from a set of N measured samples, x(n). Note that thecomplex-valued signal model in (1) can also be applied toreal-valued signals, when there is little or no spectral con-tents of interest in the frequencies near 0 and π , by the useof the discrete-time “analytical” signal [1]. The classicalfundamental frequency estimators are typically time-domaintechniques based on auto-correlation, cross-correlation, theaverage magnitude difference function (AMDF), or averagesquared difference function (ASDF). For a historical reviewof these methods, we refer to [2, 3], and for examples of morerecent work we refer to [4, 5, 6]. While subspace techniques,such as the MUltiple SIgnal Classification (MUSIC) algo-rithm [7], have a rich history in spectral analysis in general,they have only rarely been used in fundamental frequencyThis work was conducted within the ARDOR project, EU grant no.IST–2001–34095.estimation. In [6, 8], MUSIC is used for finding the individ-ual harmonics independently, and in [9], a noise estimate isobtained from MUSIC and used in a cepstral pitch estimator.In this paper, we propose an extension of the classical MU-SIC algorithm by imposing the assumed harmonic structurein (1) on the MUSIC criterion. The paper is organized as fol-lows. In Section 2, the covariance matrix model of the signalmodel (1) is presented along with some definitions. Then, inSection 3, we present the proposed fundamental frequencyestimator termed the harmonically constrained MUSIC esti-mator. In Section 4, some numerical results are presentedand, finally, Section 5 concludes on the work.2. COVARIANCE MATRIX MODELIn this section, we present the covariance matrix model andintroduce some useful vector and matrix definitions beforewe proceed to discuss the proposed extension. By assumingthat the phases of the harmonics are independent and uni-formly distributed in the interval [−π,π], the covariance ma-trix R ∈ CM×M can be written as [10]R = E{x˜(n)x˜H(n)}= A(ω0)PAH(ω0)+σ2wI, (2)where E{·} denotes the statistical expectation, (·)H the con-jugate transpose, and x˜(n) is a signal vector containing Msamples of the observed signal, i.e.,x˜(n) = [ x(n) x(n−1) · · · x(n−M+1) ]T , (3)with (·)T denoting the transpose. Further,P= diag([A21 · · · A2L]) (4)and the full rank Vandermonde matrix A(ω0) ∈ CM×L is de-fined asA(ω0) = [ a(ω0) · · · a(ω0L) ] , (5)wherea(ω) =[1 e− jω · · · e− jω(M−1) ]T . (6)Also, σ2w denotes the variance of the additive noise, w(n), andI is the M×M identity matrix. We note thatrank(A(ω0)PAH(ω0))= L, (7)and that the number of harmonics in A(ω0) is bounded byL =⌊ωmaxω0⌋, (8)whereωmax may go up to π for real signals, although it is typ-ically well below this. This is, for example, the case for audiosampled at 44.1 kHz or speech signals sampled at 16 kHz.Here, the constant M ≥ L+ 1 is a user parameter that deter-mines the accuracy of the resulting MUSIC frequency esti-mator, with larger M yielding increasing resolution. Thus,M should be selected as large as possible while still allowingfor a reliable estimate of the covariance matrix [10].3. THE HARMONIC MUSIC ALGORITHMThe MUSIC algorithm [7, 11] (see also [12]) is based on theeigenvalue decomposition (EVD) of the covariance matrixR, exploiting the structure in (2). LetR=UΛUH (9)where U is formed from the M orthonormal eigenvectors ofR, i.e.,U= [ u1 · · · uM ] , (10)and Λ is a diagonal matrix with the eigenvalues, λk, on thediagonal. The following decomposition requires a prioriknowledge of the number of harmonic components, L. Here,we will instead determine L using (8), and as a result thisL will be frequency dependent; hereafter, we will use thenotation L(ω0) to stress this dependence. Now, let G(ω0)be formed from the M− L(ω0) eigenvectors correspondingto the M − L(ω0) least significant eigenvalues (G(ω0) is afunction of ω0 through L(ω0)), i.e.,G(ω0) =[uL(ω0)+1· · · uM]. (11)Then, it can be shown that the noise subspace spanned byG(ω0) will be orthogonal to the Vandermonde matrixA(ω0)spanned by the L harmonic sinusoids [10], i.e.,AH(ω0)G(ω0) = 0. (12)We stress that whereA is a function of the set of frequencies{ωl}Ll=1 in MUSIC, it is here only a function of the funda-mental frequency ω0 as the frequencies of the harmonics aregiven by ωl = ω0l. As R is typically unknown, one needs toform an estimate of it; here, we estimate the sample covari-ance matrix asR̂=1NN∑n=Mx˜(n)x˜H(n). (13)and note that the orthogonality in (12) will only hold approx-imately for the eigenvectors found from this matrix. Exploit-ing the harmonic structure in (1), the estimated fundamentalfrequency can be found asargminω0‖AH(ω0)G(ω0)‖F , (14)where ‖ · ‖F denotes the Frobenius norm. By the Cauchy-Schwarz inequality, we have that‖AH(ω0)G(ω0)‖F ≤ ‖AH(ω0)‖F‖G(ω0)‖F . (15)As the M−L(ω0) columns of G(ω0) are orthonormal, andall the L(ω0) columns of A(ω0) have norm√M, we get‖AH(ω0)G(ω0)‖F ≤√L(ω0)M√M−L(ω0) (16)and thus‖AH(ω0)G(ω0)‖F√L(ω0)M(M−L(ω0))≤ 1. (17)We now define the harmonic pseudo-spectrum asP(ω0) =L(ω0)M(M−L(ω0))‖AH(ω0)G(ω0)‖2F, (18)and find the estimated fundamental frequency asωˆ0 = arg maxω0∈Ω0P(ω0). (19)Thus, the fundamental frequency candidates can be foundfrom (18) by sweeping ω0 over a finite set of frequencies Ω0and then project the harmonic subspace onto the noise sub-space. In the rest of this paper, we refer to this estimator asthe harmonically constrained MUSIC or HMUSIC in short.The algorithm can be summarized as the following steps:1. Estimate R̂ using (13).2. Perform an EVD of R̂.3. For each ω0 ∈Ω0,(a) Determine L(ω0) from (8).(b) Construct A(ω0) using (5), and G(ω0) using (11).(c) Compute P(ω0) using (18).4. Find fundamental frequency candidates as the maxima ofP(ω0).We note that it is possible to use a noise subspace with a fixeddimension by estimating an upper bound on L. For example,in speech the fundamental frequency is typically limited tothe range 60 Hz - 400 Hz, which would result in an upperbound of the dimension of the signal subspaceL =⌊ωmax fs2π60⌋, (20)with fs being the sampling frequency. In our experience, thepeaks of the harmonic pseudo-spectra computed using a fixedL are often more distinct and thus appear less noisy comparedto the variable dimension approach, but the latter seems togive a better response at low frequencies.A classical problem in fundamental frequency estimationis erroneous estimates at k or 1/k times the true fundamentalfrequency for k = 2,3, . . ., commonly referred to as doublingsand halvings. These problems also exist in HMUSIC. Espe-cially, doublings of the fundamental occur, because A(kω0),for k = 2,3, . . ., is spanned by the column space ofA(ω0) andthese columns are thus also orthogonal to the noise subspacewhen this is kept fixed. This is less of a problem for variabledimension noise subspace. Halvings of the fundamental fre-quency also occur, but these are generally much weaker thanthe doublings as only a subset of the harmonics will be or-thogonal to the noise subspace. In order to build a practicalfundamental frequency estimator from HMUSIC, we need tolimit the search space Ω0 of ω0 to some interval that does notinclude doublings and halvings. This can, for example, beachieved by pitch tracking, some coarse initial estimate, orby some post-processing of the harmonic pseudo-spectrum.In this paper, we defer from any further discussion of thisand instead concentrate on the statistical performance of theestimator.4. EXPERIMENTAL RESULTS4.1 Reference MethodsFor reference, we use a non-linear least-squares (NLS) fun-damental frequency estimator similar to that of [6]. This isa particularly simple version of the NLS frequency estima-tor (see, e.g., [10]) because of the harmonic relation betweenthe sinusoidal components. As is well known, the NLS fre-quency estimator is statistically efficient under white noiseconditions; furthermore, it can be shown that the NLS esti-mator is asymptotically efficient also for the coloured noisecase [13]. We note that the NLS fundamental frequency es-timator can be stated as the minimizer of the squared errorbetween the signal and the harmonic sinusoidal model, andbe found by sweeping over a finite set Ω0 of frequencies.Here, we use the same grid as in HMUSIC. We refer to thismethod as harmonically constrained NLS (HNLS). Whilethe HMUSIC gives strong false peaks at integer multiplesof the fundamental frequency, the HNLS estimator is veryprone to halvings (1/k with k = 2,3, . . .) because a funda-mental frequency of 0.5ω0 will capture more signal energythan the true fundamental, especially under noisy conditions.Thus, like the HMUSIC, we need to limit the search rangeΩ0 in order to get the correct result. For each grid point inΩ0, HNLS is computationally more complex than HMUSICas HNLS involves a matrix inversion and matrix productswhereas HMUSIC involves only a matrix product for eachfrequency point. There is, however, some additional compu-tational overhead associated with HMUSIC as it requires thecalculation of the sample covariance matrix and an EVD. Asthe resolution of the grid increases, the relative influence ofthis overhead decreases. As an additional reference, we alsouse spectral MUSIC [7, 11] on the same grid as HMUSICand HNLS to locate the frequency of the first harmonic. Thismethod does not take the harmonic structure of the spectruminto account.4.2 Speech SignalIn this section, we show harmonic pseudo-spectra of a speechsignal (female speaker, sampled at 8 kHz) and illustrate thedifference between using a fixed dimensional noise subspaceand a variable dimensional one. In Figure 1(b), the harmonicpseudo-spectrum of the segment of voiced speech in Figure1(a) is depicted. This pseudo-spectrum has been calculatedusing a fixed noise subspace in the sweep over ω0. It can beseen that the fundamental frequency stands out very clearly atapproximately 159 Hz and that the double is very noticeablypresent at 318 Hz. As a comparison, the harmonic pseudo-spectrum with a variable dimension noise subspace is shownin Figure 1(c), clearly illustrating the reduced risk for a pitch-doubling. It can also be seen that the peaks of Figure 1(b) aremore distinct compared to the noise floor than those of 1(c).4.3 Synthetic SignalsTo investigate the statistical efficiency of HMUSIC, we per-form an evaluation of the fundamental frequency estimatoron a synthetic signal using a technique similar to those of[6, 14]. As a comparison, we also show the asymptoticCramér-Rao bound (CRB) as derived in [14]. First, we inves-tigate the effects of varying SNR for a fixed segment length.The SNR is defined as SNR = 10log10(σ2s /σ2w), with σ2s be-ing the variance of the sinusoidal part of (1) and σ2w being5 10 15 20 25−0.3−0.2−0.100.10.20.3Time [ms]Amplitude(a)100 150 200 250 300 350 400102030405060708090Frequency [Hz]Harmonic Pseudo−Spectrum(b)100 150 200 250 300 350 4001112131415161718Frequency [Hz]Harmonic Pseudo−Spectrum(c)Figure 1: (a) Voiced speech segment, (b) harmonic pseudo-spectrum of speech segment using a fixed noise subspace,and (c) using a variable dimension subspace.the variance of the noise. In Figure 2, the standard devisionof MUSIC, HNLS, HMUSIC and the CRB are shown as afunction of the SNR for a segment length of 256 samples.These were found by 200 Monte Carlo simulations, where in−10 −5 0 5 10 15 20 25 3010−610−410−2100SNR [dB]Standard DeviationMUSICHMUSICHNLSCRBFigure 2: Standard deviation of the estimates ωˆ0 and the CRBfor varying SNR for N = 256.each run the additive noise sequence and the phases of theharmonics have been randomized. A fundamental frequencyof ω0 = 2π0.08 (corresponding to 640 Hz at 8 kHz samplingfrequency) was used in all simulations. Five real harmonics(L = 10) were used, all with an amplitude of 1, and the step-size of the grid searches of MUSIC, HNLS and HMUSICwas set to 0.01 Hz. Further, Ω0 was constrained to be in thevicinity of ω0 by ±10%, and M was set to 128.The effects of varying segment lengths, N, for a fixedSNR have also been investigated. The results are shown inFigure 3 for an SNR of 10 dB. Here a stepsize of 0.1 Hz wasused and the dimensions of the sample covariance matrix wasset to M = N/2. Note that the HMUSIC and MUSIC al-gorithms are sensitive to the choice of M relative to N. Fromthese figures, it can be seen that HMUSIC has very goodstatistical performance approaching the Cramér-Rao bound.From observing the performance of HMUSIC compared toMUSIC, it can also be seen that there is a big gain in takingthe harmonic structure into account in the estimation.5. CONCLUSIONIn this paper, a subspace-based fundamental frequency esti-mator has been proposed. This estimator is based on a har-monic extension of the classical MUSIC estimator, lettingthe dimensionality of the noise signal subspace depend onthe underlying fundamental frequency. The resulting estima-tor is obtained by sweeping over a set of frequencies. Theperformance of the estimator has been evaluated and com-pared to both the non-linear least-squares estimator, the clas-sical MUSIC algorithm, and the Cramér-Rao bound. Fromthe simulations, we conclude that the estimator has good sta-tistical performance at a computational complexity, which islower than the nonlinear least-squares for high resolutions.REFERENCES[1] S. L. Marple, “Computing the discrete-time "analytic"signal via FFT,” in IEEE Trans. Signal Processing,Sept. 1999, vol. 47.[2] W. Hess, Pitch Determination of Speech Signals,Springer-Verlag, Berlin, 1983.[3] W. Hess, “Pitch and voicing determination,” in Ad-vances in Speech Signal Processing, S. Furui and M. M.100 200 300 400 50010−610−410−2100Segment LengthStandard DeviationMUSICHMUSICHNLSCRBFigure 3: Standard deviation of the estimates ωˆ0 and the CRBfor varying segment lengths N with SNR = 10dB.Sohndi, Eds., pp. 3–48. Marcel Dekker, New York,1992.[4] A. de Cheveigné and H. Kawahara, “YIN, a funda-mental frequency estimator for speech and music,” inJournal of ASA, Apr. 2002, vol. 111(4).[5] D. E. Terez, “Robust pitch determination using nonlin-ear state-space embedding,” in Proc. IEEE Int. Conf.Acoust., Speech, Signal Processing, 2002.[6] H. Li, P. Stoica, and J. Li, “Computationally efficientparameter estimation for harmonic sinusoidal signals,”in Signal Processing, 2000, vol. 80, pp. 1937–1944.[7] R. O. Schmidt, “Multiple emitter location and signalparameter estimation,” in IEEE Trans. Antennas andPropagation, Mar. 1986, vol. 34(3).[8] N. Malik and W. H. Holmes, “Pitch estimation anda measure of voicing from pseudo-spectra,” in FifthInternational Symposium on Signal Processing and itsApplications, 1999.[9] M. S. Andrews, J. Picone, and R. D. Degroat, “Robustpitch determination via SVD based cepstral methods,”in Proc. IEEE Int. Conf. Acoust., Speech, Signal Pro-cessing, 1990.[10] P. Stoica and R. Moses, Introduction to Spectral Anal-ysis, Prentice Hall, 1997.[11] G. Bienvenu, “Influence of the spatial coherence of thebackground noise on high resolution passive methods,”in Proc. IEEE Int. Conf. Acoust., Speech, Signal Pro-cessing, 1979.[12] G. H. Händel, “On the History of MUSIC,” in IEEESignal Processing Magazine, March 1999, p. 13.[13] P. Stoica, A. Jakobsson, and J. Li, “Cisiod parame-ter estimation in the coloured noise case: AsymptoticCramér-Rao bound, maximum likelihood, and nonlin-ear least-squares,” in IEEE Trans. Signal Processing,Aug. 1996, vol. 45(8).[14] A. Nehorai and B. Porat, “Adaptive comb filtering forharmonic signal enhancement,” in IEEE Trans. Acoust.,Speech, Signal Processing, Oct. 1986, vol. 34(5).

Subspace-based Fundamental Frequency Estimation

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Subspace-based Fundamental Frequency Estimation

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