1. Chapter 4: Pitch estimation for music signal processing
KH Wong
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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
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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)
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4. Definition of pitch What is the pitch (音高) of a tone?
Answer: The perceived frequency of sound. (wiki)
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5. Method 1: ACF (Autocorrelation function)
Autocorrelation function (ACF) R x n
Symmetrical on both side m
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6. What is Auto-correlation, R(m)?
E.g.
x=[ ]
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)= ( )=92
R(1)=[x(0)*x(1)+x(1)*x(2)+x(2)*x(3)+x(3)*x(4)]
[ ]
( )=51
And so on…
R=[ ]
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7. Exercise 4.1 First, what is auto-correlation?
%matlab code
fs=1
x=[ ]'
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')
[pks,locs] = findpeaks(auto_corr_x)
[mm,peak1_ind]=max(pks)
'peak value1 at location'
pks(peak1_ind) %peak
locs (peak1_ind) %location
'peak value2 at location'
pks(peak1_ind+1)%peask next to the top peak
locs (peak1_ind+1) %location
period=locs(peak1_ind+1)-locs(peak1_ind)
pitch_Hz=fs/period %display pitch in Hz
%peaks at t=11,15, dt=15-11=4
Exercise:
Show the steps of calculation
Exercise 4.1 First, what is auto-correlation?
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
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Ans: ??

8. 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 (1/pitch_frequency) of the signal.
Lag Time j
in samples
Auto correlation R(j)
Rthe_max (j1)
Rsecond_max (j2)
j1=0 j2
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9. Then the fundamental frequency can be calculated as:
Usually =0, because is at .
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10. Testing a real sound A5_flute 880Hz, (sampling at fs=44100Hz)
%testing a real sound , matlab code
%x=[ ],
[xx,fs,nbits]=wavread('c:\sounds\A5_flute.wav');
sound(x,fs)%fs=44100Hz,
fs %sampling freuqncy
start=10000; %pitch a fram around t=10000
length=512;
x=xx(start:start+length);
auto_corr_x=xcorr(x); %auto-correlation
figure(1), clf
subplot(2,1,1),plot(x)
title(' one frame of the sound A5-flute=880Hz')
grid on, grid(gca,'minor'), hold on
subplot(2,1,2),plot(auto_corr_x)
title('cross correlation result')
grid on, grid(gca,'minor')
[pks,locs] = findpeaks(auto_corr_x)
[mm,peak1_ind]=max(pks)
'peak value1 at location'
pks(peak1_ind) %peak
locs (peak1_ind) %location
'peak value2 at location'
pks(peak1_ind+1)%peask next to the top peak
locs (peak1_ind+1) %location
period=locs(peak1_ind+1)-locs(peak1_ind)
pitch_Hz=fs/period %display pitch in Hz
Testing a real sound A5_flute 880Hz, (sampling at fs=44100Hz)
(x[t])
Auto_correlation(x[t])
2 peaks at t=513, 563
Use sort( ) in matlab to find the two peaks,
The gap between 2 peaks is dt= =50, hence frequency is fs/dt=44100/50=882 Hz.
Note: Pitch of a flute sound played by a human may not be too stable.
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11. 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
clc(x)=Cut(remove) the middle part
Typical CL =1/4 peak-to-peak of X n
y(n) =clc(x) n
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12. Finding pitch by 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)
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13. The MACF (Modified Autocorrelation function) algorithm
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14. 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)
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15. Class exercise 4.2
x=[ ], If Fs= sampling frequency= 1Hz.
(a) Find pitch of this signal x using ACF (Autocorrelation function) .
(b) Repeat above of if Fs = 8KHz
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16. Method 2: Normalized cross correlation function NCCF method [Verteletskaya 2009 ]
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17. An intuitive method, just pick the peaks and find the period
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
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18. 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.
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19. 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.
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20. 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
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Spectrum of x

21. 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
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22. Recover Glottal excitation and vocal track spectrum
Spectrum of glottal excitation
Cepstrum of glottal excitation
C_high
For
Glottal
excitation
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 :
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23. Measure pitch of musical instruments Example: Find pitch of Oboe A4 sound
A4_Oboe
Spectrogram
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24. Example: Find pitch of Oboe A4 sound http://www. cse. cuhk. edu
The first peak of the cepstrum (in Quefrency)
time= (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= (1/time)=F2=
200Hz
1/200=5x10^-3
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25. Summary Methods of pitch extraction have been studied.
Cepstrum and its use for pitch extraction is discussed.
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26. References
[Naotoshi Seo 2007] Project: Pitch Detection, ]
[Verteletskaya 2009 ] E. Verteletskaya, B. Šimák,” Performance Evaluation of Pitch Detection Algorithms”,
[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): , Oct 1976
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27. Appendix
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28. Music Frequency table http://wc. pima
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29. Music frequency table % source : http://www. angelfire
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30. 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:
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31. Anwer4.1: Exercise 4.1 First, what is auto-correlation?
%matlab code
x=[ ]'
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: [ ]
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32. Answer 4.2 for exercise 4.2 It is using MACF, you can use ACF, and the result for the pitch found is the same for this example.
Question: x=[ ], sampling at 1Hz.Find pitch of this signal x using MACF (Modified Autocorrelation function) .
%%%%%%%%%%%%%%Answer: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
orginal_x =
x =centered_wave =orginal_x-mean_x =
cl=center clipped range= 2
y =center clipped signal=
(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..
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33. Answer 4.2: Class exercise 4.2
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
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34. Matlab Ch4. pitch, v4b %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=[ ]
orginal_x=[ ]
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
cl=dd/4000
cl=dd/4 %center clippped "cl" length is 1/4 of total peak-to_peak span
pause
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