Pedersen, Christian Fischer

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    Aalborg UniversitetLeja ordering LSFs for accurate estimation of predictor coefficientsPedersen, Christian FischerPublished in:Proceedings of the International Conference on Spoken Language ProcessingPublication date:2011Document VersionEarly version, also known as pre-printLink to publication from Aalborg UniversityCitation for published version (APA):Pedersen, C. F. (2011). Leja ordering LSFs for accurate estimation of predictor coefficients. Proceedings of theInternational Conference on Spoken Language Processing. http://www.isca-speech.org/archive/interspeech_2011/i11_2545.htmlGeneral 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: November 29, 2020Leja ordering LSFs for accurate estimation of predictor coefficientsC.F. PedersenDepartment of Electronic Systems, Aalborg University, Denmarkcfp@es.aau.dkAbstractLinear prediction (LP) is the most prevalent method forspectral modelling of speech, and line spectrum pair (LSP) de-composition is the standard method to robustly represent thecoefficients of LP models. Specifically, the angles of LSP poly-nomial roots, i.e. line spectrum frequencies (LSFs), encode ex-actly the same information as LP coefficients. The conversionof LP coefficients to LSFs and back, has received considerableattention since mid 1970s when LSFs were introduced.The present paper demonstrates how Leja ordering LSFs re-duce amplification of rounding errors when converting LSFs toLP coefficients. The theory behind Leja ordering and the LSFsto LP coefficients conversion is presented. To supplement the-ory, numerical experiments illustrate the accuracy gain achievedby Leja ordering LSFs prior to conversion. Accuracy is mea-sured as the root mean square deviation between estimated co-efficient vectors with and without prior Leja ordering.Index Terms: Line spectral frequencies, linear prediction coef-ficients, Leja order1. IntroductionLinear prediction (LP) is the premier method for spectral mod-elling of speech. The coefficients representing the LP modelare, however, sensitive to quantization errors, i.e. small errorsmay lead to detrimental distortions in the spectral domain. Inthe mid 1970s, a method for robust representation of LP coef-ficients was introduced, cf. [1]. The method, now known asline spectrum pair (LSP) decomposition, decomposes the LPmodel’s denominator polynomial into LSP polynomials withuseful inherent properties; see, e.g. [2, 3, 4] for elaborate pre-sentations of LSP polynomial properties. A notable propertyis that LSP polynomials’ roots are all unit-modulus; hence, theycan unambiguously be represented by their arguments (frequen-cies). Unit-modulus roots appear as vertical lines in the spectraldomain, thus the term line spectral frequencies (LSFs). Com-pared to LP coefficients, LSFs quantize well and encode exactlythe same information. Therefore, LSFs are predominant whenparameterizing analysis and synthesis filters in linear predictivecoding of, e.g. speech and audio.By exploiting properties of LSP polynomials’ roots, e.g.they occur in complex conjugate pairs and interlace on the unitcircle, LSF estimation is done in R1 and very efficient tech-niques exist. See, e.g. [5] for an effective root estimator basedon a Chebyshev series formulation of the LSP polynomials.Matlab c© incorporate functionality to convert LP coefficients toLSFs and back; the functions are poly2lsf and lsf2polyrespectively. Intrinsic Matlab c© functions are, in this paper, setin typewriter font. Both of these methods are based on workpresented in [5].However, even in well conditioned cases, significant per-tubations may occur when computing polynomial coefficientsfrom roots, e.g. LP coefficients from LSFs. This may comeas a surprise, as the procedure appears simple and straightfor-ward, but rounding errors tend to accumulate. The present pa-per demonstrates how the accumulation of rounding errors canbe suppressed by ordering the LSP polynomial roots prior tocomputing the LP coefficients. The ordering scheme is knownas Leja ordering due to the Polish mathematician FranciszekLeja, cf. [6]. Leja ordering, which is in time O(n2) [7],is not conducted as part of lsf2poly. Inspired by [8], aMatlab c© implementation of Leja ordering has been publishedin [9]; the implementation is utilized in the present paper. Thepapers [10, 11] also contain interesting insights into suppres-sion of rounding error accumulation by Leja ordering and Lejasequences.The results of the current paper show that by introducing theLeja ordering, the root mean square deviation (RMSD) betweentrue and estimated LP coefficient vectors is in the neighborhoodof machine epsilon, ε, up to the tested maximum coefficientvector length of 160. In this paper, ε ≈ 2.22 · 10−16, i.e. IEEE754 double precision. These results are significantly better thanwhat is obtained when estimating the coefficients without priorLeja ordering with lsf2poly. Especially without Leja order-ing, the RMSD increases as a function of LP coefficient vectorlength, and for lengths beyond 50, the rounding errors accumu-late to such an extent as to dominate LP coefficient estimations.The remainder of this paper is organized as follows. Sec-tion 2 introduces the preliminaries of the study, i.e. LSP poly-nomials, LSP decomposition, LSF, and Leja ordering. Section3 presents the proposed method, and in section 4 the methodis tested with regard to LP coefficient estimation accuracy. Sec-tion 5 presents the results of the tests conducted in the numericalexperiment, and in the closing section, section 6, the results arediscussed along with future perspectives.2. PreliminariesThis section introduces the preliminaries of the current paper,i.e. LSP polynomials, LSP decomposition, LSF and Leja order-ing.2.1. Line spectrum pair polynomialsDecomposing the denominator polynomial of a LP model intoLSP polynomials is often referred to as LSP decomposition. Tointroduce the decomposition, the following definition is useful.Definition 1 Palindromic and anti palindromic polynomial.A real polynomial, a(x) =∑Nn=0anxn, is palindromic iffan = aN−n and anti-palindromic iff an = −aN−n.Note the following properties of (anti-) palindromic polynomi-als; cf. [2, 3, 4] for elaborate presentations of LSP polynomialproperties.Property 1 (Anti-) palindromic polynomials.1: Every real polynomial that has all of its roots on the unitcircle is either palindromic or anti-palindromic.2: Conversely, not every palindromic or anti-palindromicpolynomial has all its roots on the unit circle.In LSP decomposition, the idea is to define (anti-) palindromicpolynomials with all roots on the unit circle, cf. property 1,2.Definition 2 LSP decomposition.Any real polynomial, a(x), of orderN can be stated as the sumof a palindromic polynomial, p(x), and an anti-palindromicpolynomial, q(x):a(x) = 12(p(x)+q(x)) wherep(x) = a(x) + xN+1a(x−1)q(x) = a(x)− xN+1a(x−1)The LSP decomposition is bijective and the polynomials p(x)and q(x) are referred to as LSP polynomials. Notable propertiesof a(x), p(x), and q(x), proved in [3], are:Property 2 LSP polynomials.1: If all the roots of a(x) are inside the unit circle, then allthe roots of p(x) and q(x) are interlaced on the unit circle.2: Conversely, if the roots of two real polynomials of thesame order, one palindromic and one anti-palindromic, e.g.p(x) and q(x), are interlaced on the unit circle, then their sum,e.g. a(x), always has all its roots inside the unit circle.Polynomial p(x) has a real root at -1, and q(x) has a real rootat 1; all other roots occur in complex conjugate pairs. A rootvector, e.g. of a(x), is denoted byλ = [λ1, λ2, ..., λN ]T ∈ CN2.2. Line spectrum frequencyAs LSP polynomials’ roots lie on the unit circle they can un-ambiguously be expressed by their arguments, i.e. frequencies.This leads to the following definition:Definition 3 Line spectrum frequencies.LSFs are the arguments (frequencies) of LSP polynomials’roots.Since LSP polynomials’ roots occur in complex conjugate pairs,except for the two real roots at ±1, it suffices to determine theLSFs on the upper half unit circle, i.e. in the interval ]0;π[ . See[5] for an effective root estimator based on a Chebyshev seriesformulation of the LSP polynomials. A LSF vector, e.g. for theN ’th order polynomial a(x), that leaves room for the two realroots’ arguments can be denoted byθ = [θ1, θ2, ..., θN+2]T ∈ RN+2+The roots of a polynomial define the polynomial’s coefficientsup to scaling. Hence, the LSFs define the LSP polynomials,p(x) and q(x), which in turn define the LP model’s denomina-tor polynomial, a(x), cf. definition 2. Further, stability of theestimated LP model is ensured as a(x) will have all roots in-side the unit circle, cf. property 2. This illustrates the path ofconversion from LSFs to LP coefficients.Roots on the unit circle can unambiguously be representedby their arguments (frequencies) and appear in the spectral do-main as vertical lines; hence, the name line spectral frequencies.2.3. Leja orderingLeja ordering proves useful when the coefficients of the LPmodel’s denominator polynomial, a(x), are to be determinedaccurately from the LSFs. In theory, the mapping between LPcoefficients and LSFs is bijective up to order and scaling, but innumerical computations, accumulation of rounding errors canbecome detrimental. In the present paper, Leja ordering is con-sidered as a remedy to alleviate rounding error accumulation inthe LSFs to LP coefficients conversion.Definition 4 Weighted Leja ordering [8]|λn|n−1∏i=1|λn − λi| = maxn≤l≤N|λl|n−1∏i=1|λl − λi|for n = 1, 2, ..., N .For the first root, n = 1, the equation reduces to|λ1| = max1≤l≤N|λl|For the second root, n = 2, the equation is|λ2| · |λ2 − λ1| = max2≤l≤N|λl| · |λl − λ1|Example 1 illustrates that Leja ordering is not unique as themaximization may yield more candidates.Example 1 Leja ordering inR1λ in = [1, 2, 3, 4, 5]λ1 = 5, λ2 = 2 ∨ 3, λ3 = 4 ∨ 1,λ4 = 1 ∨ 4, λ5 = 3 ∨ 2λ out = [5, 2, 4, 1, 3]λ out = [5, 3, 1, 4, 2]Already, implementations of the Leja ordering scheme exist,e.g. a Matlab c© implementation - inspired by [8] - is publishedin [9]. The ordering is in time O(n2) [7].3. Proposed methodTable 1 outlines the proposed method and algorithm for estimat-ing LP coefficients from Leja ordered LSFs. The real coefficientvectors for a(x), p(x) and q(x) are denoted by a, p and q re-spectively. Root vectors are denoted by λ and LSF vectors byθ. The LP polynomial, a(x), is of order N .1 Form unit-modulus roots λ ∈ C2N from θ ∈ RN .Complex conjugates included.2 De-interlace λ and form λp,λq ∈ CN , cf. prop. 2.3 Leja order λp and λq , cf. def. 4.4 Compute coefficients ap, aq ∈ RN+1 by expandingN∏n=1(x− λp,n) andN∏n=1(x− λq,n)5 Convolve real roots ±1 into aq and ap respectively.6 Compute a = 12(ap + aq) ∈ RN+1, cf. def. 2.Table 1: Outline of the proposed method and algorithm.The significant difference between lsf2poly and the pro-posed method is the Leja ordering, i.e. step 3 in table 1. Theordering is in time O(n2) [7].4. Numerical experiment4.1. Experiment setupIn this numerical experiment, the proposed method, cf. table1, is compared to lsf2poly. As Leja ordering is applied toreduce rounding error accumulation, the objective of the exper-iment is to measure potential improvements in accuracy. Thisis done by evaluating the root mean square deviation (RMSD)between true LP coefficient vectors and vectors estimated withlsf2poly and the proposed method.4.2. Data materialThe data material is generated by converting LP coefficient vec-tors into LSF vectors. The coefficient vectors are randomizedand range in length. Theorem 1 is employed in coefficient vec-tor generation to ensure that all roots of the LP model’s denom-inator polynomial, a(x), lie inside the unit circle. Now, the es-timations can be compared with the true LP coefficient vectors.Theorem 1 Eneström-Kakeya [12]If a(x) =N∑n=0anxnwith a0 ≥ a1 ≥ ... ≥ aN > 0,then all the roots of a(x) lie outside the open unit disk. Con-versely, if aN ≥ aN−1 ≥ ... ≥ a0 > 0, then all the roots ofa(x) lie in the closed unit disc.Minimum phase sequences are generated, i.e. aN ≥ aN−1 ≥... ≥ a0 > 0 all in R, by reversing the coefficient ordering ofaN = 1, aN−1−i = aN−i + r, i ∈ [0;N − 1], and makingthe polynomial monic. The uniform distribution is denoted byU and r ∼ U [0, 1]. Figure 1 exemplifies a coefficient vectorand the pertaining LSFs. In the upper panel, the coefficientsare ordered in descending powers, i.e. how Matlab c© orderspolynomial coefficients.0 10 20 30 40 50 60 70 8000.20.40.60.81Coefficient index [1] a  [1]−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 100.20.40.60.81RealImaginaryFigure 1: Upper: Coefficient vector, a, from the data set, N =80. Lower: LSFs computed from the example vector above.5. ResultsIn figure 2, a typical example of error between the true and theestimated coefficient vectors is illustrated. The dataset is a sin-gle minimum phase sequence of length 80.0 10 20 30 40 50 60 70 80−1000−50005001000Coefficient index [1] a− u  [1]0 10 20 30 40 50 60 70 80−4−202x 10−14Coefficient index [1] a− o  [1]Figure 2: Typical instance of error between true, a, and esti-mated coefficient vectors. Upper: Without Leja ordering, u.Lower: With Leja ordering, o.Figure 3 illustrates the RMSD between true and estimated co-efficient vectors. The dataset consists of 31 minimum phasesequences that range in length 50-80. The range has been cho-sen to illustrate the abrupt accumulation of inaccuracies whenLSP polynomial roots are not Leja ordered.50 55 60 65 70 75 80050100150200Sequence length [1]RMSD [1]50 55 60 65 70 75 800.511.52x 10−14Sequence length [1]RMSD [1]Figure 3: RMSD between true and estimated coefficient vec-tors. Upper: Without Leja ordering. Lower: With Leja order-ing.Table 2 lists the mean and standard deviation of 14 RMSDpopulations. For each coefficient sequence length, N =40, 60, ..., 160, the RMSDs between 50 true and estimated co-efficient vectors are obtained. Again, estimations are done withand without Leja ordering.Without Leja ordering With Leja orderingFormat: µ; σ Format: µ; σN=40 2.27·10−8; 1.00·10−8 4.90·10−15; 1.71·10−15N=60 2.08·10−3; 9.46·10−4 9.62·10−15; 3.12·10−15N=80 1.73·102 ; 7.04·101 1.63·10−14; 5.03·10−15N=100 1.93·107 ; 6.99·106 2.10·10−14; 6.71·10−15N=120 1.70·1012 ; 7.65·1011 2.56·10−14; 6.92·10−15N=140 1.82·1017 ; 7.97·1016 3.20·10−14; 1.02·10−14N=160 2.03·1022 ; 7.73·1021 4.78·10−14; 1.67·10−14Table 2: Population mean and standard deviation of RMSD. Thepopulation size is 50 for each N .6. DiscussionThe results express differences in rounding error accumulationwith and without Leja ordering the LSP polynomial roots. Itis evident, cf. figure 3 and table 2, that the errors accumulateto such an extent as to dominate the LP coefficient estimationswhen Leja ordering is not applied. Quantization or rounding er-rors are unavoidable whenever a continuous space is discretizedto allow for numerical evaluation. In the present paper, the com-putations have been done in Matlab c© using IEEE 754 doubleprecision floating point numbers, i.e. the machine epsilon isε = 2−52 ≈ 2.22 · 10−16. That is, between 2n and 2n+1the numbers are equispaced with increments of 2n−52; as n in-creases, the spacing increases. The spacing between 1 and 2, i.e.n = 0, yields ε. The rounding procedure is round-to-nearestand round-half-up. Hence, the maximum relative error inducedby rounding the result of a single arithmetic operation is ε/2;broadly, the rounding level is about 16 decimal digits.In speech processing, the order of a linear predictive modelis typically 10-12. To fit 4 resonant peaks, i.e. formants, 8 polesare required; a few extra poles may increase modelling accu-racy. However, the decrease in prediction error as function ofmodel order is not pronounced beyond order 10-12. From thispractical viewpoint, the results in the present paper are mostlyof theoretical interest.7. References[1] F. Itakura, “Line spectrum representation of linear predictor co-efficients of speech signals,” Journal of the Acoustical Society ofAmerica, vol. 57, pp. S35(A), April 1975.[2] Hans W. Schüssler, “A stability theorem for discrete systems,”IEEE Transactions on Acoustics, Speech and Signal Processing,vol. 24, pp. 87–89, Feb. 1976.[3] Frank K. Soong and Biing-Hwang Juang, “Line spectrum pair(LSP) and speech data compression,” in Acoustics, Speech, andSignal Processing. IEEE, 1984, vol. 9.[4] T. Bäckström and C. Magi, “Properties of line spectrum pair poly-nomials - A review,” Signal Processing, vol. 86, no. 11, pp. 3286–3298, February 2006.[5] Peter Kabal and Ravi Prakash Ramachandran, “The computationof line spectral frequencies using Chebyshev polynomials,” IEEETransactions on Acoustics, Speech, and Signal Processing, vol.34, no. 6, pp. 1419–1426, Dec. 1986.[6] F. Leja, “Sur certaines suites liées aux ensembles plan et leurapplication à la représentation conforme,” Ann. Polon. Math., vol.4:8-13, 1957.[7] Stefano De Marchi, “On Leja sequences: Some results and ap-plications,” Elsevier, Applied Mathematics and Computation, vol.152, pp. 621–647, 2004.[8] Noël M. Nachtigal, Lothar Reichel, and Lloyd N. Trefethen,“A hybrid GMRES algorithm for nonsymmetric linear systems,”SIAM, Journal of Matrix Analysis and Applications, vol. 13, no.3, pp. 796–825, July 1992.[9] Markus Lang and Bernhard-Christian Frenzel, “A new and effi-cient program for finding all polynomial roots,” Tech. Rep. TR93-08, Rice University ECE, Houston, TX, USA, 1993.[10] Lothar Reichel, “The application of Leja points to Richardson it-eration and polynomial preconditioning,” Elsevier, Linear algebraand its applications, vol. 154-156, pp. 389–414, Aug-Oct 1991.[11] J. Baglama, D. Calvetti, and L. Reichel, “Fast Leja points,” Elec-tronic Transactions on Numerical Analysis, vol. 7, pp. 124–140,1998.[12] P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequal-ities, Springer, 1995.

Leja ordering LSFs for accurate estimation of predictor coefficients

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Leja ordering LSFs for accurate estimation of predictor coefficients

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