Computational Models of Music Similarity and their ... - OFAI

22 2 Audio-based Similarity Measures
1
DCT Matrix
Mel Frequency Cepstral Coefficients Reconstructed
80
mfcc
M_dB_rec
60
40
20
1 36
0
1 20
20
0
1 10 20 30 36
Figure 2.7: DCT matrix, MFCCs, and reconstructed Mel power spectrum (dB) for
the audio signal used in Figure 2.3. High values in the DCT matrix are visualized
as black. The dimensions of the DCT matrix are DCT coefficients on the y-axis
and Mel frequency bands on the x-axis.
2.2.2.4 DCT
The Discrete Cosine Transform is applied to compress the Mel power spectrum.
In particular, the num_filt (e.g. 36) frequency bands are represented
by num_coeffs (e.g. 20) coefficients. Alternatives include, e.g., the Principial
Component Analysis. A side effect of the compression is that the spectrum
is smoothed along the frequency axis which can be interpreted as a simple
approximation of the spectral masking in the human auditory system.
The DCT matrix can be computed as follows. 11
01 num_coeffs = 20; (2.8)
02
03 DCT = 1/sqrt(num_filt/2) * ...
04 cos((0:num_coeffs-1)’*(0.5:num_filt)*pi/num_filt);
05 DCT(1,:) = DCT(1,:)*sqrt(2)/2;
Thus the DCT matrix has num_coeffs rows and num_filt columns. Each of
the rows corresponds to an eigenvector, starting with the most important one
(highest eigenvalue) in the first row. The first eigenvector describes the mean
of the spectrum. The second describes a spectral pattern with high energy
in the lower half of the frequencies and low energy in the upper half. The
eigenvectors are orthogonal. The DCT is applied to the Mel power spectrum
(in Decibel) as follows:
mfcc = DCT * M_dB; (2.9)
The effects of the DCT matrix on the Mel power spectrum are shown in
Figure 2.7. The resulting MFCCs are a compressed representation of the
11 This code is based on Malcolm Slaney’s Auditory Toolbox.