**6. References**


226 Fourier Transform Applications

A peak around the frequency 1 / 4 nearby at position 2500 corresponds to a satellite (Fig. 16 subfigure a). This frequency derives from repetitions of certain dinucleotides in the area. The spectrogram reveals the presence of a satellite with multiple frequencies; this is manifested clearly in the 3D graph in the form of horizontally aligned peaks colored in red,

In This chapter, we investigate the contribution of each coding technique: the linear, the two-dimensional and the structural one in the enhancement of the peaks related to the C. elegans genome periodicities. For this purpose, we use a mean values of smoothed Discrete Fourier Transform applied on sliding window along the DNA sequence to follow the peak evolution for specific frequency points around the frequencies. We detect periodicities around 3, 6, 9 and 10 and found periodicities 3 and 10 related respectively to genes and the positions of the nucleosomes. First we evaluate the frequencies spread through the chromosomes with a 1-D spectrum. Second, we consider the 2-D and 3-D DNA spectrograms to visually detect the specific parts of chromosomes related with protein

The time frequency analysis made it possible to follow the periodicities' evolution. We studied the contribution of a range of binary indicators for the raising of exons' peak frequency. We also studied the localization of the areas being able to form nucleosomes. Thanks to the spectrogram with two dimensions, we visualized the localization of the areas corresponding to periodicity 10 in the limits and not in the center of the helix. The threedimensional spectrogram showed that the raised peaks do not correspond to the periodicity 10 but we see clearly in certain sequences and for some indicators two lines of peaks of variable powers around this periodicity. This result can explain the variation between 10 and 10.7 of the periodicities associated with the nucleosomes presented in the literature. It is also observable that these peaks are alternated around two periodicities; this result could be

Almeida, J.S., Carrico, J.A., Maretzek, A., Noble P.A. & Fletcher M. (2001) "*Analysis of* 

Anastassiou D. (2001), "*Genomic Signal processing*", IEEE Signal Processing Magazine, 18 (4),

Berger J. A., Mitra S. K.& Astola J. (2003), "*Power spectrum analysis for DNA sequences*", Proc.

Cénac, P. (2006) "Étude statistique de séquences biologiques et convergence de martingales",

Cénac P., Fayolle G., Lasgouttes J.M., (2004) " *Dynamical Systems in the Analysis of Biological* 

Cohanim, A.B., Kashi Y. & Trifinov E.N. (2005), "*Yeast Nucleosome DNA Pattern: Deconvolution* 

ISSN 0739-1102, volume 22, Issue Number 6, Adenine Press, pp: 687-693.

*genomic sequences by Chaos GameRepresentation*", Bioinformatics Vol. 17, n°5, pp 429–

PhD thesis on Applied Mathematics, Paul Sabatier University, Toulouse III, pp

*from Genome Squences of S. cerevisiae*, Journal of Biomolecular Structure & Dynamics

coding regions, nucleosomes positioning regions, and other particular regions.

associated with the phenomena of chromatin compaction.

of ISSPA 2003, pp 29-32, France, 1-4 July.

*Sequences*", research report n° 5351,pp 3–50.

the higher frequencies.

**5. Conclusion** 

**6. References** 

437.

pp: 8-20.

17−25.


**Part 3** 

**Fourier and Helbert Transform Applications** 

