* University of Piteşti, Electronics, Communications and Computers Department, Târgul din Vale Str., No. 1, Postal Code 110040, Piteşti - Argeş, Romania, E-Mail: firstname.lastname@example.org
Dumitru ŞCHEIANU * and Ion TUTĂNESCU **
Department of Importance, University of Piteşti, Electronics, Communications and Computers Department, Târgul din Vale Str., No. 1,Postal Code 110040, Piteşti - Argeş, Romania, E-Mail: email@example.com
Abstract - "Fractal" term - which in Latin language defines something fragmented anomalous - was introduced in mathematics by B. B. Mandelbrot in 1975. He avoided to define it rigorously and used it to designate some "rugged" and "self-similar" geometrical forms. Fractals were involved in the theory of chaotic dynamicsystems and used to designate certain specific sets; finally, they were “captured” by geometry and remarked in tackling of the boundary problems. It proved that the fractals can be of interest even in the signal’s theory. I. INTRODUCTION In the category of fractals there are included also the images whose description by conventionally ways of mathematics is, in principle, impossible. If to a 2Dimage (x, y) is added a third dimension (t), we have a bi-dimensional fractal signal. In such a signal, the fractal nature is manifested in (x, y) plan, having no connection with temporal dimension. In order to consider a scalar signal as a fractal signal, the scalar signal has to fulfill three conditions related to the time domain: - signal’s chart extension on time domain has to be endless, -scalar signal has to be continuous everywhere and - signal has not to have a differential form on time domain. The structure of fractal signal proposed in this paper is only one from an infinity of structures to be imagined. II. THE CONSTRUCTION OF A FRACTAL SIGNAL The construction of a fractal signal is realized on the base of some periodic pulses, as shown in Figure 1. Let’s consider a period oftime T0 = 4τ0 for the first alternation (half-period) T+ (considered centered by the axis of the time). The first component, x0(t), is a continuous signal and its amplitude is equal with the unit. The following components, x1(t), x2(t), a.s.o. are bipolar pulses derived by division to 3 of the alternation (half-period). 250
The components xi(t) are described by the following mathematicalequations: 0 1 2 ; t ∈ T+ = [− τ 0 , τ 0 ] (1) x 0 (t ) = 0 ; t ∉ T+
1 21 ; t ∈ T1 = [− (1 3)τ 0 , (1 3)τ 0 ] x 1 (t ) = − 1 21 ; t ∈ T+ − T1 0 ; t ∉ T + 1 2 2 ; t ∈ T2 = [− (7 9)τ 0 ,−(5 9 )τ 0 ] ∪ [− (1 9)τ 0 , (1 9)τ 0 ] ∪ [(5 9)τ 0 , (7 9 )τ 0 ] x 2 (t ) = − 1 2 2 ; t ∈ T+ − T2 0 ; t ∉ T +
x i (t ) = . . .
x0 1 0 t 0,5 x1 2τ0/3 2τ0
t x2 0,25 2τ0/32 t x3 T+T0 2τ0/33 Tt
Figure 1 - Fractal signal’s components.
If, for the first period, the summation of the signal’s components is done according to the following relations
x + (t ) = ∑ x i (t )
i =0 ∞ ∞
; t ∈ T+
x − (t ) = − ∑ x i (t − 2τ 0 ) ; t ∈ T−
and the result is divided into periods, we obtain the fractal signal:
s f (t ) = x + (t ) + x − (t )
The evolution of signal’s aspect during its making-up is shown in Figure 2. The superscripts "+" and "-" assigned to function x(t),
x+(t), respective x-(t), say that these are defined on the half-periods T+ and T-, presented in the Figure 1. The described fractal signal does not belong to the class of the functions defined in this paper as signals. Its
s0 1 2τ0/31 s1 0,5 2τ0/30continuity and non-differentiability are thoroughly remarked from its making-up process; also, it is remarked that it integrates a infinity of first rang discontinuity points.
t s2 2τ0/32
0,25 t s3 2τ0/33
x-(t) 0,12 5 t x+(t)
Figure 2 - Fractal signal’s making-up process.
Every time this signal’s amplitude belongs to the interval [-2, 2]. Clearly, its contour length comprises...