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On the chirp decomposition of Weierstrass-Mandelbrot functions, and their time-frequency interpretation. (English) Zbl 1054.94002
The authors provide a time-frequency interpretation of Weierstrass-Mandelbrot functions which puts emphasis on their possible decomposition on chirps, i.e. amplitude and frequency modulated signals of the form $a(t)\exp\{i\psi(t)\}$ with $\psi(t)=2\pi f\log t$, as an alternative to their standard Fourier-based representation. The authors consider examples of so-defined deterministic functions and their randomised variants and the relevant estimation problems.

MSC:
94A12Signal theory (characterization, reconstruction, filtering, etc.)
28A80Fractals
94A14Modulation and demodulation
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
Software:
Matlab
WorldCat.org
Full Text: DOI
References:
[1] Auger, F.; Flandrin, P.: Improving the readability of time--frequency and time-scale representations by the reassignment method. IEEE trans. Signal proc. 43(5), 1068-1089 (1995)
[2] F. Auger, P. Flandrin, P. Gonçalvès, O. Lemoine, Time--frequency toolbox for Matlab, user’s guide and reference guide. Freeware available at: http://iut-saint-nazaire.univ-nantes.fr/ auger/tftb.html
[3] Berry, M. V.; Lewis, Z. V.: On the Weierstrass--Mandelbrot fractal function. Proc. roy. Soc. London A 370, 459-484 (1980) · Zbl 0435.28008
[4] Bertrand, J.; Bertrand, P.; Ovarlez, J. Ph.: The Mellin transform. The transforms and applications handbook (1990)
[5] Borgnat, P.; Flandrin, P.; Amblard, P. -O.: Stochastic discrete scale invariance. IEEE signal process. Lett. 9, No. 6, 181-184 (2002)
[6] P. Borgnat, Modèles et outils pour l’invariance d’échelle brisée: Variations sur la transformation de Lamperti et contributions aux modèles statistiques de vortex en turbulence, PhD thesis (in French), École Normale Supérieure de Lyon, France, 2002
[7] Borgnat, P.; Amblard, P. -O.; Flandrin, P.: Lamperti transformation for finite size scale invariance. Proc. internat. Conf. on physics in signal and image proc. PSIP-03, Grenoble, 177-180 (2003)
[8] Falconer, K.: Fractal geometry. (1990) · Zbl 0689.28003
[9] Flandrin, P.: Time--frequency/time-scale analysis. (1999) · Zbl 0954.94003
[10] Flandrin, P.: Time--frequency and chirps, wavelet applications VIII, SPIE, vol. 4391. Proc. of aerosense’01 (2001)
[11] P. Flandrin, P. Borgnat, P.-O. Amblard, From stationarity to self-similarity, and back: Variations on the Lamperti transformation, in: G. Raganjaran, M. Ding (Eds.), Processes with Long-Range Correlations, Springer, 2003, in press
[12] Gluzman, S.; Sornette, D.: Log-periodic route to fractal functions. Phys. rev. E 6503, No. 3 (2002)
[13] Hardy, G. H.: Weierstrass’s non-differentiable function. Trans. amer. Math. soc. 17, 301-325 (1916) · Zbl 46.0401.03
[14] Lamperti, J.: Semi-stable stochastic processes. Trans. amer. Math. soc. 104, 62-78 (1962) · Zbl 0286.60017
[15] Mandelbrot, B. B.; Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications. SIAM rev. 10, 422-437 (1968) · Zbl 0179.47801
[16] Mandelbrot, B. B.: Fractals: forms, chance and dimension. (1977) · Zbl 0376.28020
[17] Mandelbrot, B. B.: Gaussian self-affinity and fractals. (2002) · Zbl 1007.01020
[18] Pipiras, V.; Taqqu, M. S.: Convergence of the Weierstrass--Mandelbrot process to fractional Brownian motion. Fractals 8, 369-384 (2000) · Zbl 0976.60042
[19] Samorodnitsky, G.; Taqqu, M. S.: Stable non-Gaussian random processes: stochastic models with infinite variance. (1994) · Zbl 0925.60027
[20] Sornette, D.: Discrete scale invariance and complex dimensions. Phys. rep. 297, 239-270 (1998)
[21] Tricot, C.: Courbes et dimension fractale. (1993) · Zbl 0783.28004
[22] Weierstrass, K.: Mathematische werke II. (1967)