Time-Variant Least Squares Harmonic Modeling

Transcription

1 Tme-Varant Least Squares Harmonc odelng Qn L and Les Atlas Department of Electrcal Engneerng, Unversty of Washngton Seattle, WA , USA ABSTRACT An algorthm for harmonc decomposton of tme-varant sgnals s derved from a least squares harmonc (LSH) technque. The estmates of harmonc ampltudes and phases are formulated as the soluton of a set of lnear equatons whch mnmzng mean square error; the sgnal frequency s modeled by a lnear or quadratc polynomal and obtaned va a local search over polynomal coeffcents. An ntal estmate of sgnal frequency s necessary to reduce computaton tme. Ths method s capable of producng accurate and robust harmonc estmaton n low SR stuatons. We show applcablty to hgh accuracy speech ptch and heart sound beat epoch estmaton.. ITRODUCTIO Harmonc modelng, whch s also nown as a snusodal representaton, has been wdely appled n a number of areas such as speech codng, compresson, enhancement, synthess, and ptch estmaton. Bascally, harmonc modelng can be descrbed as a fnte combnaton of snusodal components, and was frst ntroduced by caulay and Quater [4]. Usually speech sgnals can be decomposed nto two parts: a quas-perodc (harmonc component) and a non-perodc part (nose component). A crucal step for harmonc modelng s to fnd the parameters of harmonc components, e.g. ther ampltudes, frequences, and phases. A varety of technques have been proposed for ths decomposton of harmonc sgnals, such as the hgh-resoluton analyss of the short-tme Fourer Transform (STFT) [4], the multband exctaton model (BE) [3], and the least squares harmonc model (LSH) []. The frst two technques have been successfully used for low bt-rate speech codng; however ther performance degrades at low SR. The LSH model s capable of producng more accurate and robust harmonc analyss, even at very low SR; however, as wll be shown, ts performance degrades sgnfcantly wth rapd changes n sgnal frequency. In ths paper, we propose a harmonc analyss method for tme-varant sgnals, whch s a substantally modfed verson of the LSH approach developed by Abu-Shhah and Derche []. The ey dfference from LSH [] s that the fundamental frequency of sgnal s allowed to vary wth tme wthn the data segment. A lnear or quadratc polynomal s used to ft the frequency varaton n the data segment. The best-ft harmonc estmaton s then obtaned va mnmzng mean square error (SE). As we wll show, our extended LSH (ELSH) model has been successfully used to detect heartbeats from acoustc sgnals recorded from body-worn sensors wth presence of very strong body-moton nterference and ambent nose [7]. We also apply ths technque to estmate ptch epochs from lnear predcton (LP) resduals of speech sgnals. The quanttatve modelng accuracy of harmonc estmaton s also avalable as a relable ndcator to classfy voced versus unvoced sounds.. EXTEDED LEAST SQUARES HAROIC ODEL Suppose we have a harmonc sgnal that conssts of a set of snusods, whch s gven by s ( ) = C cos( ( ) + φ), () where s() s a segment of a harmonc sgnal wth length. =,,, - s the dscrete tme ndex, s the total number of harmonc components, C and φ are the ampltude and phase angle for each harmonc component,,,, and () s the normalzed fundamental frequency. For ELSH, our ey dfference from LSH [], where was constant, s that () s allowed to vary wth tme. The LSH model, wth our tme-varyng extenson, assumes that the sgnal s the sum of a harmonc sgnal and a nose, so that the sgnal s gven by s( ) = h( ) + n( ) = Ccos( ( ) + φ) + n( ). () The harmonc component h() n equatons () can be rewrtten as: h ( ) = C cos( ( ) + φ) (3) = C cos( ( ) )cos( φ ) sn( ( ) )sn( φ ) = [ ] [ A B ] = cos( ( ) ) sn( ( ) ), where A = Ccos( φ), B = C sn( φ ), and tan B, f A A φ =. B tan A + π, f A < The weghted mean square error (SE) between s() and h() s then: E = { s( ) h( ) } W( ) (5) = = s( ) [ Acos( ( ) ) Bsn( ( ) ) ] W( ), = where W() s weght of the th sample pont for SE calculaton. (4)

2 Report Documentaton Page Form Approved OB o Publc reportng burden for the collecton of nformaton s estmated to average hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 5 Jefferson Davs Hghway, Sute 4, Arlngton VA -43. Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OB control number.. REPORT DATE 3. REPORT TYPE 3. DATES COVERED - 4. TITLE AD SUBTITLE Tme-Varant Least Squares Harmonc odelng 5a. COTRACT UBER 5b. GRAT UBER 5c. PROGRA ELEET UBER 6. AUTHOR(S) 5d. PROJECT UBER 5e. TASK UBER 5f. WORK UIT UBER 7. PERFORIG ORGAIZATIO AE(S) AD ADDRESS(ES) Army Research Laboratory,Adelph,D, PERFORIG ORGAIZATIO REPORT UBER 9. SPOSORIG/OITORIG AGECY AE(S) AD ADDRESS(ES). SPOSOR/OITOR S ACROY(S). DISTRIBUTIO/AVAILABILITY STATEET Approved for publc release; dstrbuton unlmted 3. SUPPLEETARY OTES. SPOSOR/OITOR S REPORT UBER(S) 4. ABSTRACT An algorthm for harmonc decomposton of tme-varant sgnals s derved from a least squares harmonc (LSH) technque. The estmates of harmonc ampltudes and phases are formulated as the soluton of a set of lnear equatons whch mnmzng mean square error; the sgnal frequency s modeled by a lnear or quadratc polynomal and obtaned va a local search over polynomal coeffcents. An ntal estmate of sgnal frequency s necessary to reduce computaton tme. Ths method s capable of producng accurate and robust harmonc estmaton n low SR stuatons. We show applcablty to hgh accuracy speech ptch and heart sound beat epoch estmaton. 5. SUBJECT TERS 6. SECURITY CLASSIFICATIO OF: 7. LIITATIO OF ABSTRACT a. REPORT b. ABSTRACT c. THIS PAGE 8. UBER OF PAGES 4 9a. AE OF RESPOSIBLE PERSO Standard Form 98 (Rev. 8-98) Prescrbed by ASI Std Z39-8

3 For a gven sequence of fundamental frequency (), the mnmum SE s found by E E =, =, for j=,,,. (6) Aj B j The above equaton ends up wth a lnear equaton Y Q R A =, (7) Y S T B where A and B are unnown vectors to be determned, and other matrces are defned as: Q( j, ) = cos( ) W( )cos( j ), = = = R( j, ) = sn( ) W( )cos( j ), S( j, ) = cos( ) W( )sn( j ), T( j, ) = sn( ) W( )sn( j ), = ( j) = s( ) W( )cos( j), = ( j) = s( ) W( ) sn( j), Y Y = where, j =,,,, and =,, -. Solvng equaton set (7) for a gven () results n A Q R Y =, (9) B S T Y and then C and φ can be obtaned from equaton (4). Solvng equaton (7) only yelds ampltudes and phases gven a specfed fundamental frequency (). There s no closedform analytcal soluton for the unnown nput sgnal frequency. In order to estmate the unnown sgnal frequency, we need to repettvely solve equaton (7) for a range of dscrete () s, and select the () that gves the mnmum SE wth correspondng C and φ as the fnal results. Snce the result of a Fourer decomposton s unque, the true sgnal fundamental frequency () always yelds the mnmum SE among all possble () s. 3. FUDAETAL FREQUECY ESTIATE If we allow () to vary ndependently for each =,,,- and then search for all possble combnatons of (), the number of computatons would be mpractcal. So here we choose a polynomal to model () s evoluton n (dscrete tme). The polynomal could be zeroth order (constant frequency, one free parameter), frst order (lnear chrp, two free parameters), second order (quadratc frequency, three parameters), or hgher order. Snce the computaton tme ncreases exponentally wth polynomal order, the computaton tme for hgh order polynomals s consderable. An accurate ntal estmaton of frequency s necessary to narrower down the search range and lower computaton tme. In order to lower computaton tme, we mplemented the polymonal models va a step-up recurson. We start wth a constant frequency model and search over all canddates of a (8) range of dscrete. Ths step s same as the algorthm descrbe n []. Then we step up wth a lnear frequency model. We formulate the nstantaneous frequency of the sgnal as nst ( ) = a+ b. () Snce the nstantaneous frequency s the dervatve of the whole phase term ( ( ) + φ ) wth respect to tme, () n equaton () s therefore wrtten as ( ) = a+ b. () For convenence, we set,..., -,,,..., f s even = (),..., -,,,..., f s odd wthout losng loss of generalzaton, so that the nstantaneous frequency nst () has mean of a and slope of b. We use calculated from the constant frequency model as the ntal estmate of a. Then we search all combnatons of a and b to fnd the one wth mnmum SE. If necessary, we can contnue to step up to a quadratc tmevaryng frequency model, where we model nstantaneous frequency as nst ( ) = a+ b + c( ), (3) and the correspondng phase term () s wrtten as ( ) = a+ b + c( ). (4) 3 Addng a constant term convenently maes a the mean and b the slope of the nstantaneous frequency. Therefore we can drectly use a and b calculated from the lnear model as the ntal estmate. We can ostensbly contnue to step up to obtan hgh order frequency model va a smlar formulaton. However hgher order frequency models requre substantally more computaton tme and tend to ft nose. Usng overlapped data wndows also helps to obtan a better ft to frequency varaton. In practce, the polynomal order should be determned from data qualty, the nature of the unnown sgnal, and computaton effcency. 4. APPLICATIOS AD PERFORACE Our extended LSH (ELSH) approach has been successfully appled to detect acoustc heartbeat sgnal at very low SR. We also demonstrate ts applcaton to ptch estmaton. 4. Acoustc Heartbeat Detecton For healthy and safety reasons there are needs for, say, an army or fre department to montor solder or frefghter s physologcal ndcators va body-worn acoustc sensors whle they are dong ther mssons [7]. Heart rate s the most mportant physologcal ndcator for human health. Relable algorthms are needed to estmate heart rate from acoustc heartbeat sounds. In such stuatons, the acoustc heartbeat sgnals: () have strong perodcty, but usually bured nto

4 strong ambent nose and body moton nterference; () are typcally corrupted by addtve nose sources that are nonstatonary and dverse n structure; (3) have a rate whch, due to vared actvty, can change rapdly over a short tme perod. In other words, we are dealng wth a harmonc analyss problem wth potentally abrupt and rapd fundamental frequency change and low SR Sgnal Instantaneous Frequency Constant Frequency odel Lnear Frequency odel Quadratc Frequency odel Fgure. Comparson of modelng changes of nstantaneous frequences. Frequency odel Constant Lnear Quadratc ormalzed Waveform SE 54.% 53.4%.% Table. Comparson of modelng errors 4. Ptch estmaton and vocng detecton In recent years, harmonc analyss methods have receved much attenton on speech codng [3, 6] and ptch estmaton [, 5]. We proposed a method to estmate the ptch frequency of speech sgnals usng ELSH. The schematc dagram for ptch estmaton s shown n fgure 3. Frst a lnear predcton (LP) analyss was performed for every ms and a LP resdual are generated. We parttoned the resdual sgnal nto 5 ms segments wth a 5 ms overlap between segments to avod dscontnutes. The ELSH was then performed for each resdual segment to obtan harmonc parameters and harmonc model resynthess. A cos wndowng functon was appled for accurate model reconstructon throughout overlapped regons of the segments. Usually for speech codng or ptch estmaton, a vocedunvoced (U/V) classfer s also needed. One of the advantages of our approach s that the U/V dscrmnaton ndcator can drectly be obtaned from harmonc modelng accuracy, e.g. a harmonc-to-nose rato (HR) h ( ) HR log (5) =, [ s ( ) h ( )] whch s very smlar to HR defned n []. Our HR was calculated over a much shorter wndow ( ms) to qucly follow transtons between voced and unvoced sounds. 6 4 Ampltude - -4 Orgnal data ELSH model Tme (sce) Sgnal nstantaneous frequency Quadratc frequency model Fgure. Results of ELSH approach on low SR (- db) data. We frst gve an example for ELSH appled to a clean heartbeat sgnal (SR = db) wth a rapd heart rate change over a short tme perod. Fgure depcts the modeled nstantaneous frequences for the constant, lnear and quadratc model, respectvely. Table shows the correspondng normalzed mean square error (SE) between the ELSH model and true waveform. From the constant to lnear frequency model, there s a mnor mprovement; yet from lnear to quadratc frequency model, the mprovement s sgnfcant. Another example s gven for presence of colored nose wth low SR of - db. In fgure, the upper panel depcts the orgnal sgnal and the ELSH model; the lower panel depcts the nstantaneous frequency of the sgnal and our results from quadratc frequency model. The results clearly show that our harmonc analyss algorthm s very stable and relable at low SR. Fgure 3. Schematc dagram of ptch estmaton approach. We tested the performance and robustness of the ELSH approach wth polynomal tme-varyng frequency models. Speech test data was obtaned from the Carnege ellon Unversty speech group webste, We hand mared ptch epochs for a comparson reference. ELSH was performed on both orgnal speech and the same speech wth added whte nose. For each case, constant, lnear and quadratc tme-varyng frequency models were used. We also recorded computaton tme to compare computatonal effcency. Typcal results for clean data (no addtve nose) are shown n fgure 4 and fgure 5. In fgure 4, the upper panel depcts comparson of ptch estmaton results and hand mared ptches; the lower panel depcts the HR and the subsequent result for U/V classfcaton. The U/V dscrmnaton threshold was set at 3 db by averagng results of other utterances. In fgure 5, we can clearly see that the HR performs very well as an ndcaton of U/V classfcaton. The comparsons for dfferent nose levels and frequency models are shown n table. We used a prevously-defned relatve accuracy [8] to qualfy the performance of the algorthm, whch s defned as

5 f( ) felsh ( ) (6) Relatve Accuracy = %, = f( ) where f() and f ELSH () are the hand mared ptch and estmated ptch from ELSH model at th ptch perod, respectvely. The accuracy values show that the ELSH approach performs extremely well n low SR stuatons. Gong from clean data to nose level of -db, there s only about a.3 percent drop n overall accuracy. It should also be noted that best results were obtaned by usng dfferent wndow sze for dfferent nose levels. HR (db) Quadratc frequency model Hand-mared ptches (a) Threshold = 3 db Fgure 4. Results of ptch estmaton (no addtve nose). (a) shows results of lnear frequency model and hand-mared ptches. (b) shows HR (sold lne) and decsons of U/V classfcaton (dotted lne, hgh level voced; low level unvoced). The threshold s 3 db. LP resdual Harmonc synthess Fgure 5. LP resdual and Harmonc Synthess (no addtve nose). The upper trace s resdual sgnal and lower trace s harmonc synthess. The dotted lne separates estmated voced and unvoced regons. Frequency odel Constant Lnear Quadratc o addtve nose 5 ms data wndow 98.9% 99.% 99.% Whte nose (SR= -5 db) 5 ms data wndow 98.4% 98.6% 98. 7% Whte nose (SR= - db) ms data wndow 98.% 97.8% 97.9% Whte nose (SR= -5 db) ms data wndow 9.7% 9.6% 9.6% Approx. Computaton Table. Relatve accuracy for dfferent nose levels and frequency models. The algorthm s mplemented n ATLAB R3 runnng on AD Athlon.5GHz system wth Wndows. The computaton tme s for 5 second speech sampled at 6 KHz. (b) There are dfferences between frequency models. At low nose level (SR -5dB), a hgher order model generates better results, but also taes more computaton tme; at hgh nose level (SR -db), a hgher order model tends to ft nose and thus generates worse results. In practce, a proper order should be chosen by balancng between accuracy and computaton effcency. Order s also affected by data qualty and wndow sze. In general, a short wndow potentally has less frequency varaton and a low order model may be suffcent; a long wndow potentally has more frequency varaton and thus may need a hgh order model, yet offers hgher tolerance to nose. For the speech tests, nose levels, and wndow szes presented n ths paper, snce the advantages of the quadratc model were slght, we preferred the lnear frequency model at low nose level. 5. COCLUSIO In ths paper, we presented an approach for harmonc analyss, whch s an extenson to LSH. It has been demonstrated that ths extended LSH approach not only s exceptonally robust and accurate at low SR, but also s capable of capturng rapd frequency change. Two applcatons of ths approach were shown n the paper. The applcaton to acoustc heartbeat detecton, where the quadratc tme-varyng model was used, has shown success on dffcult data. The applcaton to ptch estmaton has potental for hgh resoluton ptch estmaton. The dsadvantage of ths method s that the computaton complexty s hgh. It s thus not currently effcent to use ths approach for U/V classfcaton only. Future wor on a large speech data base s needed to confrm the concluson of suffcency of the lnear tme-varyng model. We acnowledge help from techncal dscussons wth Prof. ar Ostendorf. Ths wor was funded by the Army Research Lab. 6. REFERECES []. Abu-Shhah and. Derche, "A robust technque for harmonc analyss of speech," n Proc. IEEE ICASSP', vol., pp , Pscataway, J,. [] Ahn R, Holmes Wh, Derche, oody, and Bennamoun, "Harmonc-plus-nose decomposton and ts applcaton n voced/unvoced classfcaton," n Proc. IEEE TECO '97, vol., pp , Brsbane, Australa, 997. [3] D. W. Grffth and J. S. Lm, "ultband exctaton vocoder," IEEE Transactons on Acoustcs, Speech and Sgnal Processng, vol. 36, pp. 3-35, 988. [4] R. J. caulay and T. F. Quater, "Speech analyss/synthess based on a snusodal representaton," IEEE Transactons on Acoustcs, Speech and Sgnal Processng, vol. 34, pp , 986. [5] R. J. caulay and T. F. Quater, "Ptch estmaton and vocng detecton based on a snusodal speech model," n Proc. IEEE ICASSP'9, vol., pp. 49-5, 99. [6] R. J. caulay and T. F. Quater, "The snusodal transform coder at 4 b/s," n IEEE ILCO '9, vol., pp , 99. [7]. Scanlon, "Acoustc montorng of frst responder's physology for health and performance survellance," n SPIE 6th Annual Internatonal Symposum on Aerospace/Defense Sensng, Smulaton, and Controls, Orlando, Florda, USA,. [8] A. Shah, R. P. Ramachandran, and. A. Lews, "Robust ptch estmaton usng an event based adaptve gaussan dervatve flter," n IEEE ISCAS', vol., pp ,.