A dedicated algorithm for sparse spectral representation of music sound is presented. The goal is to enable the representation of a piece of music signal as a linear superposition of as few spectral components as possible, without affecting the quality of the reproduction. A representation of this nature is said to be sparse. In the present context sparsity is accomplished by greedy selection of the spectral components, from an overcomplete set called a dictionary. The proposed algorithm is tailored to be applied with trigonometric dictionaries. Its distinctive feature being that it avoids the need for the actual construction of the whole dictionary, by implementing the required operations via the fast Fourier transform. The achieved sparsity is theoretically equivalent to that rendered by the orthogonal matching pursuit (OMP) method. The contribution of the proposed dedicated implementation is to extend the applicability of the standard OMP algorithm, by reducing its storage and computational demands. The suitability of the approach for producing sparse spectral representation is illustrated by comparison with the traditional method, in the line of the short time Fourier transform, involving only the corresponding orthonormal trigonometric basis.

1.
X.
Serra
and
J.
Smith
 III
, “
Spectral modeling synthesis: A sound analysis/synthesis based on a deterministic plus stochastic decomposition
,”
Comput. Music J.
14
,
12
24
(
1990
).
2.
N.
Fletcher
and
T.
Rossing
,
The Physics of Musical Instruments
(
Springer
,
Berlin, Germany
,
1998
), p.
131
.
3.
M.
Davy
and
S. J.
Godsill
, “
Bayesian harmonic models for musical signal analysis
,” in
Bayesian Statistics 7
(
Oxford University Press
,
New York
,
2002
), pp.
105
124
.
4.
J.
Wolfe
,
J.
Smith
,
J.
Tann
, and
N. H.
Fletcher
, “
Acoustic impedance spectra of classical and modern flutes
,”
J. Sound Vib.
243
,
127
144
(
2001
).
5.
J. F.
Alm
and
J. S.
Walker
, “
Time-frequency analysis of musical instruments
,”
SIAM Rev.
44
,
457
476
(
2002
).
6.
J. O.
Smith
 III
,
Spectral Audio Signal Processing
(
W3K Publishing
,
2011
), pp.
231
253
.
7.
S. G.
Mallat
and
Z.
Zhang
, “
Matching pursuits with time-frequency dictionaries
,”
IEEE Trans. Signal Process.
41
,
3397
3415
(
1993
).
8.
R.
Gribonval
and
E.
Bacry
, “
Harmonic decomposition of audion signals with matching pursuit
,”
IEEE Trans. Signal Processing
51
,
101
111
(
2003
).
9.
R.
Baraniuk
, “
Compressive sensing
,”
IEEE Signal Processing Mag.
24
,
118
121
(
2007
).
10.
R.
Baraniuk
, “
More is less: Signal processing and the data deluge
,”
Science
331
,
717
719
(
2011
).
11.
D. L.
Donoho
, “
Compressed sensing
,”
IEEE Trans. Inf. Theory
52
,
1289
1306
(
2006
).
12.
J.
Candès
,
J.
Romberg
, and
T.
Tao
, “
Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
,”
IEEE Trans. Inf. Theory
52
,
489
509
(
2006
).
13.
E.
Candès
and
M.
Wakin
, “
An introduction to compressive sampling
,”
IEEE Signal Processing Mag.
25
,
21
30
(
2008
).
14.
L.
Rebollo-Neira
, “
Cooperative greedy pursuit strategies for sparse signal representation by partitioning
,”
Signal Processing
125
,
365
375
(
2016
).
15.
J. H.
Friedman
and
W.
Stuetzle
, “
Projection pursuit regression
,”
J. Am. Stat. Assoc.
76
,
817
823
(
1981
).
16.
L. K.
Jones
, “
On a conjecute of Huber concerning the convergence of projection pursuit regression
,”
Ann. Stat.
15
,
880
882
(
1987
).
17.
Y. C.
Pati
,
R.
Rezaiifar
, and
P. S.
Krishnaprasad
, “
Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition
,” in
Proceedings of the 27th ACSSC
(
1993
), Vol.
1
, pp.
40
44
.
18.
L.
Rebollo-Neira
, “
Trigonometric dictionary based codec for music compression with high quality recovery
,” http://arxiv.org/abs/1512.04243 (
2016
).
19.
R.
Young
,
An Introduction to Nonharmonic Fourier Series
(
Academic Press
,
San Diego, CA
,
1980
), pp.
154
169
.
20.
I.
Daubechies
, “
Ten lectures on wavelets
,” SIAM,
55
103
(
1992
).
21.
L.
Rebollo-Neira
, “
Constructive updating/downdating of oblique projectors: A generalization of the Gram-Schmidt process
,”
J. Phys. A
40
,
6381
6394
(
2007
).
22.
L.
Rebollo-Neira
and
J.
Bowley
, “
Sparse representation of astronomical images
,”
J. Opt. Soc. Am. A
30
,
758
768
(
2013
).
23.
B. K.
Natarajan
, “
Sparse approximate solutions to linear systems
,”
SIAM J. Comput.
24
,
227
234
(
1995
).
24.
J. R.
Partington
, “
Interpolation, identification, and sampling
,” in
London Mathematical Society Monographs New Series
(
Oxford University Press
,
New York
,
1997
), Vol.
17
, pp.
140
150
.
25.
L.
Rebollo-Neira
and
D.
Lowe
, “
Optimized orthogonal matching pursuit approach
,”
IEEE Signal Process. Lett.
9
,
137
140
(
2002
).
26.
Highly nonlinear approximations for sparse signal representation
,” http://www.nonlinear-approx.info/examples/node02.html (Last viewed September 30,
2016
).
You do not currently have access to this content.