Fundamental Frequency Estimation

The f0est function requires findpeaks listed in §F.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.

sig = sig(:).'; % ensure row-vector
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 = 2^nextpow2(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/interf.

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('Signal 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

Test Program for F0 Estimation

The following test program generates a harmonic signal and uses f0est to determine the fundamental frequency, first with all harmonics present, and second with the fundamental suppressed by a notch filter.

% Tested using Octave 3.0.0, Aug 2008

clear all;

f0 = 440;    % true fundamental frequency
n = 1:200;   % time in samples
fs = 10000;  % sampling rate

N = length(n);
fund = sin(2*pi*f0*n/fs); % sine at fundamental frequency
npartials = 7;
sig = zeros(1,N);
for k=1:npartials
  ampk = 1 / k^2; % give a natural roll-off
  sig = sig + ampk * sin(2*pi*k*f0*n/fs);
end
%rmsnoise = 0.01;
rmsnoise = 0.0;
sig = sig + rmsnoise * randn(1,N); % add some noise for realism

framesize = N;
minlevel = -60; % Lowest relative partial amplitude to accept
                % (-40 good with Hamming window family)
                % (-60 good with Blackman window family)
debug = 0;
f0 = f0est(sig,fs,framesize,npartials,minlevel,debug)
% Trick to leave all internal f0est variables defined
% (comment-out first line of f0est.m declaring the function):
% nargin = 6; f0est;

% Notch out fundamental and repeat:
b1 = -2*cos(2*pi*f0/fs);
sig = filter([1 b1 1],1,sig);
f0 = f0est(sig,fs,framesize,npartials,minlevel,debug)

% Example output:
% f0 =  439.99
% f0 =  437.50


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Daniel Bernoulli's Modal Decomposition
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Non-Parametric Frequency Warping