Designing Gabor windows using convex optimization. (English) Zbl 1427.42037
Summary: Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal \(\ell{l}^2 \)-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found.
MSC:
| 42C15 | General harmonic expansions, frames |
| 90C25 | Convex programming |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Keywords:
short time Fourier transform; dual window design; tight window design; convex optimization; Gabor systemReferences:
| [1] | Vetterli, M., Filter banks allowing perfect reconstruction, Signal Process., 10, 219-244 (1986) |
| [2] | Samadi, S.; Ahmad, M. O.; Swamy, M. N.S., Characterization of nonuniform perfect-reconstruction filterbanks using unit-step signal, IEEE Trans. Signal Process., 52, 9, 2490-2499 (2004) · Zbl 1369.94272 |
| [3] | Bölcskei, H.; Hlawatsch, F., Oversampled cosine modulated filter banks with perfect reconstruction, IEEE Trans. Circuits Syst. II, 45, 8, 1057-1071 (1998) · Zbl 1001.94502 |
| [4] | Gröchenig, K., Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis (2001), Birkhäuser Boston: Birkhäuser Boston Boston, MA · Zbl 0966.42020 |
| [5] | Feichtinger, H. G.; Strohmer (Eds.), T., Gabor Analysis and Algorithms, ((1998), Birkhäuser: Birkhäuser Boston) · Zbl 0890.42004 |
| [6] | Malvar, H. S., Signal Processing with Lapped Transforms (1992), Artech House Publishers · Zbl 0948.94505 |
| [7] | Debbal, S.; Bereksi-Reguig, F., Time-frequency analysis of the first and the second heartbeat sounds, Appl. Math. Comput., 184, 2, 1041-1052 (2007) · Zbl 1111.92037 |
| [8] | Roach, B. J.; Mathalon, D. H., Event-related eeg time-frequency analysis: an overview of measures and an analysis of early gamma band phase locking in schizophrenia, Schizophr. Bull., 34, 5, 907-926 (2008) |
| [9] | Fujita, K.; Takei, Y.; Morimoto, A.; Ashino, R., Mathematical view of a blind source separation on a time frequency space, Appl. Math. Comput., 187, 1, 153-162 (2007) · Zbl 1118.62007 |
| [10] | Adler, A.; Emiya, V.; Jafari, M.; Elad, M.; Gribonval, R.; Plumbley, M., Audio inpainting, IEEE Trans. Audio Speech Lang. Process., 20, 3, 922-932 (2012) |
| [11] | Zeevi, Y. Y.; Zibulski, M.; Porat, M., Multi-window Gabor schemes in signal and image representations, Gabor Analysis and Algorithms, 381-407 (1998), Springer · Zbl 0890.42017 |
| [12] | Li, Y.-Z.; Zhang, Y., Vector-valued Gabor frames associated with periodic subsets of the real line, Appl. Math. Comput., 253, 102-115 (2015) · Zbl 1338.42046 |
| [13] | Balazs, P.; Dörfler, M.; Jaillet, F.; Holighaus, N.; Velasco, G. A., Theory, implementation and applications of nonstationary Gabor frames, J. Comput. Appl. Math., 236, 6, 1481-1496 (2011) · Zbl 1236.94026 |
| [14] | Christensen, O.; Kim, H.; Kim, R. Y., Gabor windows supported on \([- 1, 1]\) and compactly supported dual windows, Appl. Comput. Harmon. Anal., 28, 1, 89-103 (2010) · Zbl 1177.42029 |
| [15] | Christensen, O.; Kim, H.; Kim, R. Y., Gabor windows supported on \([- 1, 1]\) and dual windows with small support, Adv. Comput. Math., 36, 4, 525-545 (2012) · Zbl 1251.42005 |
| [16] | Feichtinger, H. G.; Nowak, K., A First Survey of Gabor Multipliers, Advances in Gabor Analysis, 99-128 (2003), Springer, (Chapter 5) · Zbl 1032.42033 |
| [17] | Balazs, P., Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl., 325, 1, 571-585 (2007) · Zbl 1105.42023 |
| [18] | Stoeva, D. T.; Balazs, P., Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33, 2, 292-299 (2012) · Zbl 1246.42017 |
| [19] | G. Matz, F. Hlawatsch, Application in time-frequency signal processing, in: A. Papandreou-Suppappola, Eds., Linear Time-Frequency Filters: On-line Algorithms and Applications, CRC Press, Boca Raton, FL, pp. 205-271 (Chapter 6).; G. Matz, F. Hlawatsch, Application in time-frequency signal processing, in: A. Papandreou-Suppappola, Eds., Linear Time-Frequency Filters: On-line Algorithms and Applications, CRC Press, Boca Raton, FL, pp. 205-271 (Chapter 6). |
| [20] | Balazs, P.; Dörfler, M.; Kowalski, M.; Torrésani, B., Adapted and adaptive linear time-frequency representations: a synthesis point of view, IEEE Signal Process. Mag., 30, 6, 20-31 (2013) |
| [21] | Christensen, O., Frames and Bases. An Introductory Course, Applied and Numerical Harmonic Analysis (2008), Basel Birkhäuser · Zbl 1152.42001 |
| [22] | Bölcskei, H., A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported, J. Fourier Anal. Appl., 5, 5, 409-419 (1999) · Zbl 0933.42029 |
| [23] | Gröchenig, K.; Stöckler, J., Gabor frames and totally positive functions, Duke Math. J., 162, 6, 1003-1031 (2013) · Zbl 1277.42037 |
| [24] | Christensen, O.; Kim, H. O.; Kim, R. Y., On entire functions restricted to intervals, partition of unities, and dual Gabor frames, Appl. Comput. Harmon. Anal., 38, 1, 72-86 (2015) · Zbl 1302.42048 |
| [25] | Christensen, O., Pairs of dual Gabor frame generators with compact support and desired frequency localization, Appl. Comput. Harmon. Anal., 20, 3, 403-410 (2006) · Zbl 1106.42030 |
| [26] | Christensen, O.; Kim, R., On dual Gabor frame pairs generated by polynomials, J. Fourier Anal. Appl., 16, 1, 1-16 (2010) · Zbl 1210.42048 |
| [27] | Christensen, O.; Goh, S., Pairs of dual periodic frames, Appl. Comput. Harmon. Anal., 33, 3, 315-329 (2012) · Zbl 1266.42073 |
| [28] | Laugesen, R. S., Gabor dual spline windows, Appl. Comput. Harmon. Anal., 27, 2, 180-194 (2009) · Zbl 1174.42038 |
| [29] | Gröchenig, K.; Lyubarskii, Y., Gabor frames with hermite functions, C. R. Math., 344, 3, 157-162 (2007) · Zbl 1160.42013 |
| [30] | Abreu, L. D.; Gröchenig, K., Banach Gabor frames with hermite functions: polyanalytic spaces from the Heisenberg group, Appl. Anal., 91, 11, 1981-1997 (2012) · Zbl 1266.46020 |
| [31] | Seip, K.; Wallstén, R., Density theorems for sampling and interpolation in the Bargmann-Fock space. II, J. Reine Angew. Math, 429, 107-113 (1992) · Zbl 0745.46033 |
| [32] | Harris, F., On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, 66, 1, 51-83 (1978) |
| [33] | Nuttall, A., Some windows with very good sidelobe behavior, IEEE Trans. Acoust. Speech Signal Process., 29, 1, 84-91 (1981) |
| [34] | Romero, J. L., Surgery of spline-type and molecular frames, J. Fourier Anal. Appl., 17, 1, 135-174 (2011) · Zbl 1218.42014 |
| [35] | Romero, J. L., Characterization of coorbit spaces with phase-space covers, J. Funct. Anal., 262, 1, 59-93 (2012) · Zbl 1237.42015 |
| [36] | Strohmer, T., Numerical algorithms for discrete Gabor expansions, Gabor Analysis and Algorithms, 267-294 (1998), Springer · Zbl 0890.42013 |
| [37] | Wexler, J.; Raz, S., Discrete Gabor expansions, Signal Process., 21, 3, 207-220 (1990) |
| [38] | Daubechies, I.; Landau, H. J.; Landau, Z., Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl., 1, 4, 437-478 (1994) · Zbl 0888.47018 |
| [39] | Li, S.; Liu, Y.; Mi, T., Sparse dual frames and dual Gabor functions of minimal time and frequency supports, J. Fourier Anal. Appl., 19, 1, 48-76 (2013) · Zbl 1304.42075 |
| [40] | Krahmer, F.; Kutyniok, G.; Lemvig, J., Sparsity and spectral properties of dual frames, Linear Algebra Appl., 439, 4, 982-998 (2013) · Zbl 1280.42026 |
| [41] | Combettes, P.; Pesquet, J., Proximal splitting methods in signal processing, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, 185-212 (2011), Springer · Zbl 1242.90160 |
| [42] | Combettes, P.; Pesquet, J., A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery, IEEE J. Sel. Top. Signal Process., 1, 4, 564-574 (2007) |
| [43] | Combettes, P.; Wajs, V., Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4, 4, 1168-1200 (2005) · Zbl 1179.94031 |
| [44] | Adler, A.; Emiya, V.; Jafari, M.; Elad, M.; Gribonval, R.; Plumbley, M., A constrained matching pursuit approach to audio declipping, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 329-332 (2011), IEEE |
| [45] | Kowalski, M.; Siedenburg, K.; Dörfler, M., Social sparsity! Neighborhood systems enrich structured shrinkage operators, IEEE Trans. Signal Process., 61, 10, 2498-2511 (2013) · Zbl 1393.94307 |
| [46] | Werther, T.; Matusiak, E.; Eldar, Y. C.; Subbana, N. K., A unified approach to dual Gabor windows, IEEE Trans. Signal Process., 55, 5, 1758-1768 (2007) · Zbl 1390.42047 |
| [47] | Bölcskei, H.; Hlawatsch, F.; Feichtinger, H. G., Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process., 46, 12, 3256-3268 (1998) |
| [48] | Vetterli, M.; Kovacevic, J., Wavelets and Subband Coding (1995), Prentice Hall · Zbl 0885.94002 |
| [49] | Balazs, P.; Holighaus, N.; Necciari, T.; Stoeva, D., Frame theory for signal processing in psychoacoustics, (Balan, R.; Benedetto, J. J.; Czaja, W.; Okoudjou, K., Excursions in Harmonic Analysis Vol. 5 (2016), Springer) |
| [50] | Wang, D.; Brown, G. J., Computational Auditory Scene Analysis: Principles, Algorithms, and Applications (2006), Wiley-IEEE Press |
| [51] | Quatieri, T., Discrete-Time Speech Signal Processing: Principles and Practice (2001), Prentice Hall Press: Prentice Hall Press Upper Saddle River, NJ, USA |
| [52] | Portnoff, M., Implementation of the digital phase vocoder using the fast Fourier transform, IEEE Trans. Acoust. Speech Signal Process., 24, 3, 243-248 (1976) |
| [53] | Oppenheim, A. V.; Schafer, R., Discrete-Time Signal Processing (1999), Oldenbourg · Zbl 0994.94005 |
| [54] | Majdak, P.; Balazs, P.; Kreuzer, W.; Dörfler, M., A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps, Proceedings of the 36th International Conference on Acoustics, Speech and Signal Processing, ICASSP 2011, Prag (2011) |
| [55] | Hall, J. W.; Haggard, M.; Fernandes, M. A., Detection in noise by spectro-temporal pattern analysis, J. Acoust. Soc. Am., 76, 50-56 (1984) |
| [56] | Zölzer (Ed.), U., DAFX: Digital Audio Effects, ((2002), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York, NY, USA) |
| [57] | Liuni, M., Phase vocoder and beyond, Musica/Tecnologia, 7, 73-120 (2013) |
| [58] | Driedger, J.; Müller, M., A review of time-scale modification of music signals, Appl. Sci., 6, 2, 57 (2016) |
| [59] | Průša, Z.; Holighaus, N., Phase vocoder done right, Signal Processing Conference (EUSIPCO), 2017 25th European, 976-980 (2017), IEEE |
| [60] | Griffin, D.; Lim, J., Signal estimation from modified short-time Fourier transform, IEEE Trans. Acoust. Speech Signal Process., 32, 2, 236-243 (1984) |
| [61] | Perraudin, N.; Balazs, P.; Soendergaard, P., A fast Griffin-Lim algorithm, Proceedings of 2013 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, USA, 2013 (2013) |
| [62] | Necciari, T.; Balazs, P.; Kronland-Martinet, R.; Ystad, S.; Laback, B.; Savel, S.; Meunier, S., Auditory time-frequency masking: psychoacoustical data and application to audio representations, Speech, Sound and Music Processing: Embracing Research in India, 146-171 (2012), Springer |
| [63] | Balazs, P.; Laback, B.; Eckel, G.; Deutsch, W. A., Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, IEEE Trans. Audio Speech Lang. Process., 18, 1, 34-49 (2010) |
| [64] | Narayanan, A.; Wang, D., The role of binary mask patterns in automatic speech recognition in background noise, J. Acoust. Soc. Am., 133, 5, 3083-3093 (2013) |
| [65] | Perraudin, N.; Holighaus, N.; Soendergaard, P.; Balazs, P., Gabor dual windows using convex optimization, Proceedings of the 10th International Conference on Sampling theory and Applications (SAMPTA 2013) (2013) |
| [66] | Hager, W. W.; Zhang, H., A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2, 1, 35-58 (2006) · Zbl 1117.90048 |
| [67] | Rockafellar, R., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 5, 877-898 (1976) · Zbl 0358.90053 |
| [68] | Martinet, B., Détermination approchée d’un point fixe d’une application pseudo-contractante. cas de l’application prox, C. R. Acad. Sci. Paris Ser. AB, 274, A163-A165 (1972) · Zbl 0226.47032 |
| [69] | Søndergaard, P. L., Gabor frames by Sampling and Periodization, Adv. Comput. Math., 27, 4, 355-373 (2007) · Zbl 1123.42006 |
| [70] | Balazs, P.; Feichtinger, H. G.; Hampejs, M.; Kracher, G., Double preconditioning for Gabor frames, IEEE Trans. Signal Process., 54, 12, 4597-4610 (2006) · Zbl 1203.65069 |
| [71] | Janssen, A. J.E. M.; Søndergaard, P. L., Iterative algorithms to approximate canonical Gabor windows: computational aspects, J. Fourier Anal. Appl., 13, 2, 211-241 (2007) · Zbl 1133.42044 |
| [72] | Søndergaard, P. L.; Torrésani, B.; Balazs, P., The linear time frequency analysis toolbox, Int. J. Wavelets Multiresolut. Inf. Process., 10, 4, 419-442 (2012) |
| [73] | N. Perraudin, V. Kalofolias, D. Shuman, P. Vandergheynst, UNLocBoX: A MATLAB convex optimization toolbox for proximal-splitting methods, (2014) arXiv:1402.0779; N. Perraudin, V. Kalofolias, D. Shuman, P. Vandergheynst, UNLocBoX: A MATLAB convex optimization toolbox for proximal-splitting methods, (2014) arXiv:1402.0779 |
| [74] | Chen, S. S.; Donoho, D. L.; Saunders, M. A., Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20, 1, 33-61 (1998) · Zbl 0919.94002 |
| [75] | Feichtinger, H. G., On a new Segal algebra, Mon. Math., 92, 269-289 (1981) · Zbl 0461.43003 |
| [76] | Lieb, E. H., Integral bounds for radar ambiguity functions and Wigner distributions, Inequalities, 625-630 (2002), Springer |
| [77] | Feichtinger, H. G.; Onchis-Moaca, D.; Ricaud, B.; Torrésani, B.; Wiesmeyr, C., A method for optimizing the ambiguity function concentration, Proceedings of the 20th European Signal Processing Conference (EUSIPCO), 2012, 804-808 (2012), IEEE |
| [78] | N. Perraudin, B. Ricaud, D. Shuman, P. Vandergheynst, Global and local uncertainty principles for signals on graphs, (2016) arXiv:1603.03030; N. Perraudin, B. Ricaud, D. Shuman, P. Vandergheynst, Global and local uncertainty principles for signals on graphs, (2016) arXiv:1603.03030 · Zbl 1403.94031 |
| [79] | Boyd, S.; Parikh, N.; Chu, E.; Peleato, B.; Eckstein, J., Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends® Mach. Learn., 3, 1, 1-122 (2011) · Zbl 1229.90122 |
| [80] | Combettes, P. L., The foundations of set theoretic estimation, Proc. IEEE, 81, 2, 182-208 (1993) |
| [81] | Raguet, H.; Fadili, J.; Peyré, G., A generalized forward-backward splitting, SIAM J. Imaging Sci., 6, 3, 1199-1226 (2013) · Zbl 1296.47109 |
| [82] | Ricaud, B.; Torrésani, B., A survey of uncertainty principles and some signal processing applications, Adv. Comput. Math., 40, 3, 629-650 (2014) · Zbl 1330.94023 |
| [83] | Afonso, M.; Bioucas-Dias, J.; Figueiredo, M. A.T., Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19, 9, 2345-2356 (2010) · Zbl 1371.94018 |
| [84] | Labate, D.; Wilson, E., Connectivity in the set of Gabor frames, Appl. Comput. Harmon. Anal., 18, 1, 123-136 (2005) · Zbl 1057.42023 |
| [85] | Janssen, A. J.E. M.; Strohmer, T., Characterization and computation of canonical tight windows for Gabor frames, J. Fourier Anal. Appl., 8, 1, 1-28 (2002) · Zbl 1001.42023 |
| [86] | Attouch, H.; Bolte, J.; Redont, P.; Soubeyran, A., Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Lojasiewicz inequality, Math. Oper. Res., 35, 2, 438-457 (2010) · Zbl 1214.65036 |
| [87] | Wiesmeyr, C.; Holighaus, N.; Sondergaard, P., Efficient algorithms for discrete Gabor transforms on a nonseparable lattice, IEEE Trans. Signal Process., 61, 20, 5131-5142 (2013) · Zbl 1393.94492 |
| [88] | Balazs, P.; Dörfler, M.; Holighaus, N.; Jaillet, F.; Velasco, G., Theory, implementation and applications of nonstationary Gabor frames, J. Comput. Appl. Math., 236, 6, 1481-1496 (2011) · Zbl 1236.94026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
