Time-frequency Analysis And Pitch Estimation Algorithm

This thesis introduce several method about the time-frequency analysis of music and make a comparison,analysis their strongpoint and disadvantage. Then it introduce several method about the pitch estimation problem and make a comparison. At last,it introduce a sawtooth waveform inspried pithc estimator (SWIPE)for music. SWIPE is shown to outperform existing algorithms on several pbulicly available speech/musical-instruments databases and a disorderd speech database.Given a music passage s(t),according to the method introduced in this thesis, we can make an analysis of the instant information ,that is to transform the data to the time-frequency plane.In this way,we can make the pitch estimation and so on.The earliest analysis method is fourier series,it just can extract the feature in either time domain or frequency domain,can not joint them together.But it also give us many illumination. The expression of the fourier series is:As a conventional and efficient analysis method,the short time fourier transform(STFT) have a broad applycation in music analysis.As said in the thesis,STFT have a uniform resolution in time-frequency plane,this conflicts with the requirement that music signal analysis need a better time resolution at high frequency and better frequency resolution at low frequency. The expression ofSTFT is:Another time-frequency analysis tool,wavelet analysis,can provide a constant-Q frequency resolution that almost matches the requirement of music signal analysis. The expression of wavelet transform is: The expression of Wigner-Distribution is:The expression of the impulse response in time domain of the Gammatone filte is:The basic steps that most PEAs perform to track the pitch of a signal are as follows: First,the signal is split into windows;then,for each window the following steps are performed: (i)the spectrum is estimated using STFT,(ii)a score is computed for each pitch candidate within a predefind range by computing an integral tranform(IT) over the spectrum,and (iii)the candidate with the highest score is selected as the estimated pitch.In the forth part of the chapter three,I briefly introduced several common pitch estimation algorithms: HPS/ SHS/ HS/ AC /AMDF,ASDF/ CEP.Harmonid Procuct Spectrum (HPS) estimates the pitch as the frequency that maximaizes the product of the spectrum at harmonics of that frequency,i.e. aswhere X is the estimated spectrum of the signal,n is the number of harmonics to be used,and p is the estimated pitch.A pitfall of this algorithms is that if any of the harmonics is missing, the product will be zero for the candidate corresponding to the pitch,and therefore the pitch will not be recoginzed.Sub-Harmonic Summation(SHS) solves the missing harmonics problem.It estimates the pitch as(?), (7.8)A pitfall of this algorithms is that since it gives the same weight to all theharmonics, subharmonics ot the pitch may have the same score as the pitch,and therefore they are valid candidates for being recoginized as the pitch.Suvhaarmonic to Harmonic Ratio algorithm(SHR) can be written as(?) (7.9)It uses the logarithm of the spectrum,and therefore has the problem previously discussed for HPS.Also,it gives the same weight to all the harmonics and therefor it suffers from the subharmonics problem.Harmonics Sieve(HS) is similar to SHS,but has two key differences:instead of using pulses it uses rectangles,and instead of computing the inner product between the spectrum and the rectangles,it counts the number of rectangles that contain at least one component.It can be espressed methematically as (?) (7.10)A pitfall of HS is that,when a component is close to an edge of a rectangle,a slight change in its frequency could put it in or out of the rectangle,possibly changing the estimated pitch drastically.The autocorelation-based pitch estimation algorithm(AC)estimates the pitch as the frequency whose inverse maximizes the autocorrelation function of the signal,i.e.,as(?) (7.11)Like all the algorithrms presented ,except SHS,AC eshibits the subharmonics problem caused by the equal weight given to all the harmonics.The cepstrum-based pitch estimation algorithm(CEP) is similar to AC.It estimates the pitch as the frequency whose inverse maximizes the cepstrum of the signal,i.e.,as(?) (7.12) The most common problems found in these algorithrms were the inability to deal with missing harmonics(HPS,SHR,CEP)and inharmonics signal(HPS,SHS,SHR) ,and the tendency to produce high scores for subharmonics of the pitch(all the algorithms).Aiming to improve upon the algorithrms ,a Sawtooth Waveform Inspired Pitch Estimator was developed. The seed of SWIPE is the implicit idea of the algorithrms: to find the frequency that maximizes the average peak-to-valley distance at harmonics of that frequency. However,this idea will be implemented trying to avoid the problem-causing features found in those algorithrms. This will be achieved by avoiding the use of the logarithrm of the spectrum,applying a monotonically dacaying weight to the harmonics,observing the spectrum in the neighborhood of the harmonics and middle points between harmonics,and using smooth weighting functions.After these modifications,SWIPE estimates the pitch as the fundamental frequency of the sawtooth waveform whose spectrum best matchs the spectrum of the input signal.The SWIPE estimate of the pitch at time t can be formulated aswithotherwise, Its variation,SWIPE',uses only the first and prime harmonics of the signal,producing a large reduction in subharmonic errors by reducing significantly the scores of subharmonics of the pitch.SWIPE and SWIPE' were tested using several published speech and musical instruments databases and their performance was compared other algorithms.SWIPE' was shown to outperform all the databases.SWIPE was ranked second in the normal speech and musical instruments databases,and was ranked third in the disordered speech database.The time-frequency analysis of music is an establishment in the multidisciplinary foundation comprehensive technology.Along with many new theories,the new method,the new technology unceasing appearances, the content in this field will grow richer and achieve new development.