% mfcc - Mel frequency cepstrum coefficient analysis.
% [ceps,freqresp,fb,fbrecon,freqrecon] = ...
% mfcc(input, samplingRate, [frameRate])
% Find the cepstral coefficients (ceps) corresponding to the
% input. Four other quantities are optionally returned that
% represent:
% the detailed fft magnitude (freqresp) used in MFCC calculation,
% the mel-scale filter bank output (fb)
% the filter bank output by inverting the cepstrals with a cosine
% transform (fbrecon),
% the smooth frequency response by interpolating the fb reconstruction
% (freqrecon)
% -- Malcolm Slaney, August 1993
% Modified a bit to make testing an algorithm easier... 4/15/94
% Fixed Cosine Transform (indices of cos() were swapped) - 5/26/95
% Added optional frameRate argument - 6/8/95
% Added proper filterbank reconstruction using inverse DCT - 10/27/95
% Added filterbank inversion to reconstruct spectrum - 11/1/95
% (c) 1998 Interval Research Corporation
function [ceps,freqresp,fb,fbrecon,freqrecon] = mfcc(input, samplingRate, frameRate)
global mfccDCTMatrix mfccFilterWeights
[r c] = size(input);
if (r > c)
input=input';
end
% Filter bank parameters
lowestFrequency = 133.3333;
linearFilters = 13;
linearSpacing = 66.66666666;
logFilters = 27;
logSpacing = 1.0711703;
fftSize = 512;
cepstralCoefficients = 13;
windowSize = 400;
windowSize = 256; % Standard says 400, but 256 makes more sense
% Really should be a function of the sample
% rate (and the lowestFrequency) and the
% frame rate.
if (length(input) < windowSize)
error('Input too short; length of input < windowSize');
end
if (nargin < 2) samplingRate = 16000; end;
if (nargin < 3) frameRate = 100; end;
% Keep this around for later....
totalFilters = linearFilters + logFilters;
% Now figure the band edges. Interesting frequencies are spaced
% by linearSpacing for a while, then go logarithmic. First figure
% all the interesting frequencies. Lower, center, and upper band
% edges are all consequtive interesting frequencies.
freqs = lowestFrequency + (0:linearFilters-1)*linearSpacing;
freqs(linearFilters+1:totalFilters+2) = ...
freqs(linearFilters) * logSpacing.^(1:logFilters+2);
lower = freqs(1:totalFilters);
center = freqs(2:totalFilters+1);
upper = freqs(3:totalFilters+2);
% We now want to combine FFT bins so that each filter has unit
% weight, assuming a triangular weighting function. First figure
% out the height of the triangle, then we can figure out each
% frequencies contribution
mfccFilterWeights = zeros(totalFilters,fftSize);
triangleHeight = 2./(upper-lower);
fftFreqs = (0:fftSize-1)/fftSize*samplingRate;
for chan=1:totalFilters
mfccFilterWeights(chan,:) = (fftFreqs > lower(chan) & fftFreqs <= center(chan)).* triangleHeight(chan).*(fftFreqs-lower(chan))/(center(chan)-lower(chan)) + (fftFreqs > center(chan) & fftFreqs < upper(chan)).* triangleHeight(chan).*(upper(chan)-fftFreqs)/(upper(chan)-center(chan));
end
%semilogx(fftFreqs,mfccFilterWeights')
%axis([lower(1) upper(totalFilters) 0 max(max(mfccFilterWeights))])
hamWindow = 0.54 - 0.46*cos(2*pi*(0:windowSize-1)/windowSize);
if 0 % Window it like ComplexSpectrum
windowStep = samplingRate/frameRate;
a = .54;
b = -.46;
wr = sqrt(windowStep/windowSize);
phi = pi/windowSize;
hamWindow = 2*wr/sqrt(4*a*a+2*b*b)*(a + b*cos(2*pi*(0:windowSize-1)/windowSize + phi));
end
% Figure out Discrete Cosine Transform. We want a matrix
% dct(i,j) which is totalFilters x cepstralCoefficients in size.
% The i,j component is given by
% cos( i * (j+0.5)/totalFilters pi )
% where we have assumed that i and j start at 0.
mfccDCTMatrix = 1/sqrt(totalFilters/2)*cos((0:(cepstralCoefficients-1))' * (2*(0:(totalFilters-1))+1) * pi/2/totalFilters);
mfccDCTMatrix(1,:) = mfccDCTMatrix(1,:) * sqrt(2)/2;
%imagesc(mfccDCTMatrix);
% Filter the input with the preemphasis filter. Also figure how
% many columns of data we will end up with.
if 1
preEmphasized = filter([1 -.97], 1, input);
else
preEmphasized = input;
end
windowStep = samplingRate/frameRate;
cols = fix((length(input)-windowSize)/windowStep);
% Allocate all the space we need for the output arrays.
ceps = zeros(cepstralCoefficients, cols);
if (nargout > 1) freqresp = zeros(fftSize/2, cols); end;
if (nargout > 2) fb = zeros(totalFilters, cols); end;
% Invert the filter bank center frequencies. For each FFT bin
% we want to know the exact position in the filter bank to find
% the original frequency response. The next block of code finds the
% integer and fractional sampling positions.
if (nargout > 4)
fr = (0:(fftSize/2-1))'/(fftSize/2)*samplingRate/2;
j = 1;
for i=1:(fftSize/2)
if fr(i) > center(j+1)
j = j + 1;
end
if j > totalFilters-1
j = totalFilters-1;
end
fr(i) = min(totalFilters-.0001, max(1,j + (fr(i)-center(j))/(center(j+1)-center(j))));
end
fri = fix(fr);
frac = fr - fri;
freqrecon = zeros(fftSize/2, cols);
end
% Ok, now let's do the processing. For each chunk of data:
% * Window the data with a hamming window,
% * Shift it into FFT order,
% * Find the magnitude of the fft,
% * Convert the fft data into filter bank outputs,
% * Find the log base 10,
% * Find the cosine transform to reduce dimensionality.
for start=0:cols
first = floor(start*windowStep) + 1;
last = first + windowSize-1;
fftData = zeros(1,fftSize);
fftData(1:windowSize) = preEmphasized(first:last).*hamWindow;
fftMag = abs(fft(fftData));
t=mfccFilterWeights * fftMag';
for i=1:size(t,1)
for j=1:size(t,2)
if (t(i,j) == 0)
t(i,j) = 0.001;
end
end
end
earMag = log10(t);
ceps(:,start+1) = mfccDCTMatrix * earMag;
if (nargout > 1) freqresp(:,start+1) = fftMag(1:fftSize/2)'; end;
if (nargout > 2) fb(:,start+1) = earMag; end
if (nargout > 3)
fbrecon(:,start+1) = mfccDCTMatrix(1:cepstralCoefficients,:)' * ceps(:,start+1);
end
if (nargout > 4)
f10 = 10.^fbrecon(:,start+1);
freqrecon(:,start+1) = samplingRate/fftSize * (f10(fri).*(1-frac) + f10(fri+1).*frac);
end
end
% OK, just to check things, let's also reconstruct the original FB
% output. We do this by multiplying the cepstral data by the transpose
% of the original DCT matrix. This all works because we were careful to
% scale the DCT matrix so it was orthonormal.
if 1 & (nargout > 3)
fbrecon = mfccDCTMatrix(1:cepstralCoefficients,:)' * ceps;
% imagesc(mt(:,1:cepstralCoefficients)*mfccDCTMatrix);
end;
