Analogical signals (or continuous)

1. Introduction

A physical greatness, translated by a sensor under the shape of an electric signal, depends continuously on time.

The signal will be then a continuous function of time : s : t => s(t)

T is a continuous real variable representing time. S( t) is a real number (sometimes complex, S * (t) represents then the combined).

s(t), the bearer of information is also a bearer of energy;

power is proportional in
s²(t) = [s(t) s*(t)] And energy on an interval [t1 , t2] will be proportional in :

        Generally, one will consider that signals are generated from certain moment taken as origin of times. One will suppose consequently that these signals do not exist for t < 0, that is that they are invalid for t < 0. These signals are called causal.

2. Classic signals

2.1. The harmonious or sinusoidal signal

        The sinusoidal, very used signal, is the periodic signal par excellence. It is an eternal sinusoïde (not causal clause). One can so incite a system by a sinusoidal signal - test and investigate the answer varying frequency, to observe for example echos. One uses also the mathematical signal ejwt which is easier to manipulate

(Reminder : ejwt = cos(wt) + jsin(wt)).

2.2. The unitarian banister r(t), causale

r(t) = (Causality)

In physical appearance(physics), the hillside is rarely 1.

She(it) expresses the speed of variation of the considered greatness.

q(t) = ar(t) ; a = 0.2 Degree by second expresses for example the linear growth of the temperature of an oven.

2.3. Heaviside's unitarian rung G(t) or existence function

G(t) = (causality)

G(t) By-product is (cease originally) of r(t) ; G(t) is not defined for t = 0 ;

G(0+) = 1 et G(0-) = 0.

        It is possible to make analogy with an electric system. Let us apply a continuous tension to a system. Tension is known well before, then after the lock in t = 0 of the switch. If the amplitude is Eo tension will be a not unitarian rung EoG ( t ). One can be tried to express duration e finished with lock of the switch and the continuance of the evolution of the tension, here besides linéarisée, by G1(t).

                                                 

2.4. Dirac's unitarian impulse d(t)

       Let us divert G1, we obtain d1 ( t ) which is worth 1/e in the interval ] - e /2 , + e /2 [.One observes that the area is 1 which that or e so :
                                                    "e

        If e decrease to 0, d1 as no limit in the direction of functions, but in the direction of distributions because d1(t) is not dérivable the two points of break. This limit is d(t), which is called the distribution of Dirac.

                                                     

The intégrale constitute a sort of definition of d(t).

        Dirac's impulse d(t) is called unitarian because its measure, or its weight, or its area is worth 1. In practice, we shall consider that a brief impulse of some shape can be approached, the point of view of the effects, by Dirac's impulse of measure A, A being the area of the brief impulse

Properties of Dirac's impulse :
          1)
2)
3) Change of variable :
Particular case :

2.5. The delayed causal signal

                                                     

                                                          g(t) = f(t - t) ; g(t) is null for t < t.

        The sum causal signals and delayed signals allows to create signals of some forms.

So 3G(t) - 3G(t - To) is a causal crenel of height 3 and duration To.