Fundamental Frequency Estimation

The f0est function requires findpeaks listed in §G.2.1. It also uses a trivial utility freqlabs defined at the end of the listing below. The blackman function (for the Blackman window) comes standard with Octave and requires the Signal Processing Toolbox in Matlab.

function [f0,fc] = f0est(sig,fs,framesize,npartials,minlevel,debug);
%F0EST  Estimate fundamental frequency F0 from spectral peaks
%
% [f0,fc] = f0est(sig,fs,framesize,npartials,minlevel);
%
% Computes fundamental frequency estimate f0 (Hz) and spectral cut-off
% frequency fc (Hz) from spectral peaks for npartials partials in
% signal sig.  The frame size used in the fft is specified in
% framesize which is rounded up to the next odd number if necessary.
% The sampling rate is passed in fs (default = 1).  Peaks are rejected
% minlevel dB below the highest peak.  fc is returned as the frequency
% (Hz) of the highest partial.  A Blackman window is used.

len = length(sig);

if (nargin<2),
 fs = 1;
end;

if (nargin<3),
 framesize = 512;
end;

if (nargin<5),
 minlevel = -60; % dB
end;

if (nargin<6),
 debug = 0;
end;

if (len < framesize),
 disp(sprintf('*** f0est: signal too short  for fft (%d samples)',len));
 framesize = len;
end;

if mod(framesize,2)==0,
 framesize = framesize - 1;
 disp(sprintf('f0est: framesize -> next odd number = %d',framesize));
end;

minzpadfactor = 5;  % min zero-padding factor (for accurate freqs)
nfft = pwr2(framesize*minzpadfactor);
nspec = nfft/2 + 1;
[fkHz fxlab fylab] = freqlabs(nspec,fs);
nzpad = nfft - framesize;
padding = zeros(1,nzpad);
zpadfactor = nfft/framesize;
locscl = (fs/2) / nfft;		% Converts bin to freq

if (nargin<4),
 npartials = framesize/2;	% i.e., no upper limit
end;

window = blackman(framesize)';

% Parameters established.
% Compute FFT.

minpeakwidth = zpadfactor*5; % 5 for generalized Blackman family
maxpeakwidth = minpeakwidth*2; % reject too much modulation/interference

forigin = 1;
frame = [window .* sig(forigin:forigin+framesize-1), padding];
if debug>1, plot(frame); title(sprintf('Signal Frame')); end;
spec = fft(frame);
specdb = db(spec(1:nspec));
minval = max(specdb) + minlevel;
if debug, plot(fkHz,specdb);
  title('Singal Frame Spectrum');
  xlabel(fxlab); ylabel(fylab);
end;

[peakamps peaklocs peakwidths] = ...
  findpeaks(specdb,npartials,minpeakwidth,maxpeakwidth,minval,debug);

[peaklocs,sorter] = sort(peaklocs);
amps = zeros(size(peakamps));
widths = zeros(size(peakamps));
npeaks = length(peaklocs);
for col=1:npeaks,
	amps(:,col) = peakamps(:,sorter(col));
	widths(:,col) = peakwidths(:,sorter(col));
end;

freqs = (peaklocs - ones(size(peaklocs))) * fs / nfft;
if npeaks>1,
  fspacings = diff(freqs);
  medianfspacing = median(fspacings);
  avgfspacing = mean(fspacings);
else
  fspacings = freqs(1);
  medianfspacing = freqs(1);
  avgfspacing = freqs(1);
end;

f0a = medianfspacing;
harmonic_numbers = round(freqs/f0a);

subharms = find(harmonic_numbers < 0.75);
msh = max(subharms);
if msh > 0 % delete subharmonics, dc peak, etc.
  disp(sprintf(...
    '*** Tossing out the following %d peaks below F0',msh));
  format bank;
  disp('	freqs	amps	widths:');
  [freqs(1:msh)', amps(1:msh)', widths(1:msh)']
  disp('*** PAUSING (RETURN to continue) ***');
  pause;
  freqs = freqs(msh+1:npeaks);
  amps = amps(msh+1:npeaks);
  widths = widths(msh+1:npeaks);
  harmonic_numbers = harmonic_numbers(msh+1:npeaks);
  npeaks = npeaks - msh;
end

if debug,
	format bank;

	disp('	freqs	amps	widths:'); [freqs', amps', widths']
	disp('partial frequency intervals:'); fspacings

	bar(freqs(1:end-1),fspacings-f0a);
	title('Partial Frequency Spacings minus fundamental');
	ylabel('Frequency Spacing (Hz)');
	xlabel('Frequency (Hz)');
        disp('PAUSING - RETURN to continue'); pause;

	disp(sprintf('Median partial frequency interval = %f',medianfspacing));
	disp(sprintf('Average partial frequency interval = %f',avgfspacing));
	disp(sprintf('Fundamental frequency estimate = %f',f0a));
	disp(sprintf('Harmonics versus peaks:'));
		[f0a*harmonic_numbers',freqs']
	disp(sprintf('Relative deviation:'));
	disp(sprintf('Relative deviation (%%):'));
	f0a_error = 100*(freqs - f0a*harmonic_numbers)./freqs

	bar(freqs,f0a_error*1200);
	title('Relative Partial Frequency Deviations (measured - harmonic)');
	ylabel('Cents');
	xlabel('Frequency');
        disp('PAUSING - RETURN to continue'); pause;

	disp(sprintf('Saved output format:'));
	nfreqs = length(freqs);
        [sprintf('partial_freqs = {%f',freqs(1)), ...
		sprintf(', %f',freqs(2:nfreqs)),sprintf('};')]
        [sprintf('partial_numbers = {%d',harmonic_numbers(1)),...
		sprintf(', %d',harmonic_numbers(2:nfreqs)),sprintf('};')]
end;

%
% Optimize fit of f0 * harmonic_numbers to measured peak frequencies.
%
R = harmonic_numbers * harmonic_numbers';
P = harmonic_numbers * freqs';
f0 = R\P;
if debug > 1,
	disp('Refined F0 estimate via harmonic regression:'); f0
	disp('refined harmonics versus peaks:'); [f0*harmonic_numbers',freqs']
	disp('relative deviation (%%):');
        f0_error=100*(f0*harmonic_numbers-freqs)./freqs
	stem(f0_error);
	title('Relative Harmonic Deviation: Harmonic Minus Peak Over Peak');
end;

fc = max(freqs);

specdbc = max(specdb,minval);
if debug,
	ttl = 'Signal Frame Spectrum with F0 and Cut-Off Marked';
        clf;
        hold on;
        f = fkHz * 1000;
	plot(f,specdbc,'-k');
        markline = [min(specdbc),max(specdbc)];
	plot([f0 f0], markline);
	plot([fc fc], markline);
        title(ttl);
        xlabel(fxlab);
        ylabel(fylab);
        disp('PAUSING (CUTOFFS) - RETURN to continue'); pause;
        hold off;
end;

% ----------------------------------------------------------------

function [f,fxlab,fylab] = freqlabs(nspec,fs);
%FREQLABS returns freq axis f (kHz), x label, and y label, given
%         max FFT bin number spec and sampling rate fs (in Hz)
fxlab = 'Frequency (kHz)';
fylab = 'Amplitude (dB)';
f = (fs/2000)*[0:nspec-1]/nspec; % frequency axis for spectral plots

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Subsections

• Test Program for F0 Estimation

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Next: Test Program for F0 Estimation

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.