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Chapter 4: Pitch estimation for music signal processing

Chapter 4: Pitch estimation for music signal processing. KH Wong. Introduction (lecture 4). Pitch estimation is essential to many music signal applications Genre classification Music tutor: detection of playing fault Music style analysis Automatic transcription, audio signal  music score.

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Chapter 4: Pitch estimation for music signal processing

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  1. Chapter 4: Pitch estimation for music signal processing KH Wong Ch4. pitch, v3.c

  2. Introduction (lecture 4) • Pitch estimation is essential to many music signal applications • Genre classification • Music tutor: detection of playing fault • Music style analysis • Automatic transcription, audio signal music score Ch4. pitch, v3.c

  3. Techniques in pitch extraction • Time domain approaches • (1) ACF (Autocorrelation function) and MACF (Modified Autocorrelation function) • (2) Normalized cross correlation function NCCF • (3) AMDF (Average magnitude difference function) • Frequency domain approaches • (4) Cepstrum Pitch Determination (CPD) Ch4. pitch, v3.c

  4. Definition of pitch • What is the pitch (音高) of a tone? • Answer: The perceived frequency of sound. (wiki) Ch4. pitch, v3.c

  5. x n R n Method 1:ACF (Autocorrelation function) • Autocorrelation function (ACF) Symmetrical on both side m Ch4. pitch, v3.c

  6. What is Auto-correlation, R(m)? • E.g. • x=[1 5 7 1 4 ] • N=5, • R(0)=[x(0)*x(0)+x(1)*x(1)+x(2)*x2+x(3)*x(3)+x(4)*x(4)] • R(0)= (1+ 25+49+1+16)=92 • R(1)=[x(0)*x(1)+x(1)*x(2)+x(2)*x(3)+x(3)*x(4)] • x=[1 5 7 1 4 ] • [1 5 7 1 4 ] • (5+ 35+ 7+ 4)=51 • And so on… • R=[92.0000 51.0000 40.0000 21.0000 4.0000] Ch4. pitch, v3.c

  7. Exercise 4.1First, what is auto-correlation? • %matlab code • x=[1 5 7 1 4 8 6 2 4 9 3 ]' • auto_corr_x=xcorr(x) %auto-correlation • figure(1), clf • subplot(2,1,1),plot(x) • grid on, grid(gca,'minor'), hold on • subplot(2,1,2),plot(auto_corr_x) • grid on, grid(gca,'minor') • Exercise: • Show the steps of calculation X[t] t Auto_correlation(x[t]) • We only look at positive n • Gap between two peaks is 4, so period of X is around 4 Ans: ?? Ch4. pitch, v3.c

  8. Auto correlation R(j) Rthe_max (j1) Rsecond_max (j2) Lag Time j in samples j1=0 j2 autocorrelation • When a segment of a signal is correlated with itself, the distance (-=Lag_time_in_samples) between the positions of the maximum and the second maximum correlation is defined as the fundamental period (pitch) of the signal. Ch4. pitch, v3.c

  9. Then the fundamental frequency can be calculated as: • Then the fundamental frequency can be calculated as: • Usually =0, because is at . Ch4. pitch, v3.c

  10. Modified Auto-Correlation Method:Auto-Correlation Method enhanced by Center clipping • It will give more accurate result because higher frequency signals will not interfere with the result X(n) CL CL Cut(remove) the middle part Typical CL =1/4 peak-to-peak of X n y(n) =clc(x) n Ch4. pitch, v3.c

  11. Finding pitchby center clipping X(n) Y(n)= Center Clipped • In R(m) auto correlation of x(n), it is not easy to pick peaks • In R’(m), auto correlation of clipped signal y(n)=clc{x(n)}, peaks are easy to pick R(m) R’(m) T1 T2 T3 T=mean(T1,T2,T3)= Period=1/(pitch_frequency) Ch4. pitch, v3.c

  12. The MACF (Modified Autocorrelation function) algorithm Ch4. pitch, v3.c

  13. Example • For each frame, find a pitch. • Plot pitch against time (blue), you can see the pitch profile X(n) time Pitch (n) frequency Time n (frame) Ch4. pitch, v3.c

  14. Class exercise 4.2 • x=[1 3 7 2 1 9 3 1 8 ], If Fs= sampling frequency= 1Hz. • (a) Find pitch of this signal x using ACF (Autocorrelation function) . • (b) Repeat above of if Fs = 8KHz Ch4. pitch, v3.c

  15. Method 2:Normalized cross correlation function NCCF method [Verteletskaya 2009 ] Ch4. pitch, v3.c

  16. Method 3:Average Magnitude Difference Function (AMDF) Method [Verteletskaya 2009 ] • An intuitive method, just pick the peaks and find the period Find peaks in D, the estimated period is the average gaps between two neighboring –ve peaks peaks Ch4. pitch, v3.c

  17. Method 4:Cepstrum Pitch Determination (CPD)[Verteletskaya 2009 ] Peak at Q’, Pitch =1/0.006= 166Hz. Q’ The problem : For human voice, the peak may be the result of glottal excitation. Ch4. pitch, v3.c

  18. For human voice pitch detection (or recognition ) • We must study its structure of the vocal system and find out how to get the accurate answer. • vocal system has 2 elements • Glottal excitation (no use for pitch measurement) • Vocal tract filter • Use liftering to remove glottal excitation before we use the spectrum of the vocal tract filter for pitch extraction. Ch4. pitch, v3.c

  19. Cepstrum of speech • A new word by reversing the first 4 letters of spectrum  cepstrum. • It is the spectrum of a spectrum of a signal • Why we need this? • Answer: remove the ripples • of the spectrum caused by • glottal excitation. Too many ripples in the spectrum caused by vocal cord vibrations. But we are more interested in the speech envelope for recognition and reproduction Fourier Transform Speech signal x Spectrum of x Ch4. pitch, v3.c http://isdl.ee.washington.edu/people/stevenschimmel/sphsc503/files/notes10.pdf

  20. Liftering method: Select the higher and lower samples • Signal X(n) • Cepstrum= • C(n)=fft|(log|fft(x(n))|)| • Select high time liftering, select C_high (lower frequency):glottal excitation • Select low time liftering, • Select C_low (higher frequency) :Vocal tract filter response Quefrency is in time domain (in second) So Higher Quefrency lower frequency Ch4. pitch, v3.c

  21. Recover Glottal excitation and vocal track spectrum Spectrum of glottal excitation Cepstrum of glottal excitation C_high For Glottal excitation C_high For Vocal track Frequency Spectrum of vocal track filter Cepstrum of vocal track Frequency quefrency (sample index) This peak may be the pitch period: This smoothed vocal track spectrum can be used to find pitch For more information see : http://isdl.ee.washington.edu/people/stevenschimmel/sphsc503/files/notes10.pdf Ch4. pitch, v3.c

  22. Measure pitch of musical instrumentsExample: Find pitch of Oboe A4 sound http://www.cse.cuhk.edu.hk/%7Ekhwong/www2/cmsc5707/A4_oboe.wav • A4_Oboe • Spectrogram Ch4. pitch, v3.c

  23. Example: Find pitch of Oboe A4 sound http://www.cse.cuhk.edu.hk/%7Ekhwong/www2/cmsc5707/A4_oboe.wavhttp://www.cse.cuhk.edu.hk/%7Ekhwong/www2/cmsc5707/demo_ceps_note_v3.zip The first peak of the cepstrum (in Quefrency) time=0.002268(1/time)=F1=440.91Hz is the pitch, it has the strongest energy Input: Oboe A4 X(n) Fourier Transform X(w)=fft(x) Cepstrum C(n)=fft|(log|fft(x(n))|)| From range 200 To 900 Hz Cepstrum C(n) All range, around From 30 to  Hz Found two Harmonics 440, 220Hz This axis is in x10^-3  Hz 900Hz 1/900=1.11x10^-3 The second peak: time=0.004535(1/time)=F2=220.507 200Hz 1/200=5x10^-3 Ch4. pitch, v3.c

  24. Summary • Methods of pitch extraction have been studied. • Cepstrum and its use for pitch extraction is discussed. Ch4. pitch, v3.c

  25. References • [Naotoshi Seo 2007] Project: Pitch Detection, ]http://note.sonots.com/SciSoftware/Pitch.html#ke283f3a • [Verteletskaya 2009 ] E. Verteletskaya, B. Šimák,” Performance Evaluation of Pitch Detection Algorithms”, http://access.feld.cvut.cz/view.php?cisloclanku=2009060001 • [Rabiner1976] Rabiner, L.; Cheng, M.; Rosenberg, A.; McGonegal, C." A comparative performance study of several pitch detection algorithms",IEEE Transactions on Acoustics, Speech and Signal Processing, Volume: 24, Issue:5 page(s): 399 - 418, Oct 1976 Ch4. pitch, v3.c

  26. Appendix Ch4. pitch, v3.c

  27. Music Frequency tablehttp://wc.pima.edu/~manelson/MUS%20102/MIDI%20tunings%20per%20note.jpg Ch4. pitch, v3.c

  28. Music frequency table% source : http://www.angelfire.com/in2/yala/t4scales.htm Ch4. pitch, v3.c

  29. Autocorrelation • In signal processing, given a signal f(t), the continuous autocorrelation is the continuous cross-correlation of f(t) with itself, at lag τ, and is defined as: • In discrete system, autocorrelation R at lag j for signal is defined as: Ch4. pitch, v3.c

  30. Anwer4.1: Exercise 4.1First, what is auto-correlation? • %matlab code • x=[1 5 7 1 4 8 6 2 4 9 3 ]' • auto_corr_x=xcorr(x) %auto-correlation • figure(1), clf • subplot(2,1,1),plot(x) • grid on, grid(gca,'minor'), hold on • subplot(2,1,2),plot(auto_corr_x) • grid on, grid(gca,'minor') • Exercise: • Show the steps of calculation X[t] t Auto_correlation(x[t]) • We only look at positive n • Gap between two peaks is 4, so period of X is around 4 Ans: [302 214 142 183 194 116 65 88 70 24 3 0] Ch4. pitch, v3.c

  31. Answer 4.2 for exercise 4.2It is using MACF, you can use ACF, and the result for the pitch found is the same for this example. • Question: x=[1 3 7 2 1 9 3 1 8 ], sampling at 1Hz.Find pitch of this signal x using MACF (Modified Autocorrelation function) . • %%%%%%%%%%%%%%Answer: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% • orginal_x = 1 3 7 2 1 9 3 1 8 • x =centered_wave =orginal_x-mean_x = • -2.8889 -0.8889 3.1111 -1.8889 -2.8889 5.1111 -0.8889 -2.8889 4.1111 • cl=center clipped range= 2 • y =center clipped signal= • -2.8889 0 3.1111 0 -2.8889 5.1111 0 -2.8889 4.1111 • (a) if the sampling frequency Fs = 1KHz • >> Answer: from the autocorrelation result of y in the figure, we can see that the distance between 2 peaks is 3, so pitch is 1/3 Hz, since the sampling is 1 Hz.. Ch4. pitch, v3.c

  32. Answer 4.2: Class exercise 4.2 • R=[ 24.3333, 9.6667, 8.2222, 16.3333, 6.5556, 4.5556, ,6.8889, 2.7778, 0.8889] • 2nd diagram, R(+ve only) , pick 2 peaks, Period is 3, frequency =1/3 hz • (b) if FS = 8KHz • Answer: If the sampling frequency is Fs=8KHz, sampling period is dt=1/Fs=(1/8)ms , the period of x is 3 units, therefore the actual time is 3*dt= 3*(1/8)ms. The frequency of x is 1/dt=(8/3) KHz Ch4. pitch, v3.c

  33. Matlab %assume the signal x is voltage against time %center clip means set those signals with levels within the clipped regions %center = mean voltage level of the whole signal %positive peak = maxim,um of the signal voltage %negative peak = minimum of the signal voltage %center clip regions are:(i) from center to 1/2 of center_to_positive peak % (ii) from center to -1/2 from center_to_negative peak for t=1:n if x(t)<cl & x(t) > -1*cl %those within center clipped region set to 0 y(t)=0; else y(t)=x(t); end; end ; auto_corr_y=xcorr(y) %auto correlation figure(2) clf subplot(3,1,1),plot(x) ylabel('x=centered wave') subplot(3,1,2),plot(y) ylabel('y=center clipped wave') hold on subplot(3,1,3),plot(auto_corr_y) ylabel('auto correlation of y') xlabel('time ') max_list=max(y) fs 'orginal_x ' , orginal_x 'x =centered_wave =orginal_x-mean_x ' , x 'cl=center clipped range', cl 'y =center clipped signal' , y • %Ver2, MACF (Modified Autocorrelation function)using center clipping • clear • %select one of the followings • %real_data=1 %1 or 0 • real_data=0 • if real_data==1 • %use real sound • %[x,fs]=wavread ('d:\0music\sounds\violin3.wav'); • [orginal_x,fs]=wavread ('violin3.wav'); • x=x(10000:11000); • else • %use test data • %x=[1 2 5 6 7 6 1 0 4 3 4 8 6 7 3 2 4 9 3 ] • orginal_x=[1 3 7 2 1 9 3 1 8 ] • fs=1 %assume frquecy is 1Hz • end • %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% test • x=orginal_x-mean(orginal_x) • n=length(x) • maxx=max(x) • minx=min(x) • dd=maxx-minx • figure(1) • clf • plot(x) • %pause • %center clipping algo for pitch extraction • if real_data==1 • cl=dd/4000 • else • cl=dd/4 %center clippped "cl" length is 1/4 of total peak-to_peak span • pause • end Ch4. pitch, v3.c

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