Objectives and mathématical necessities

 

1. Objectives of this lesson of signal processing

The lesson of signal processing relates of :

To deal and to analyze signals, it is usual to study collectively what happens in the temporal domain and in the frequential domain. The use of the temporal domain seems rather natural and allows to put there evidence some caractéristics of the signal:

The spectre of the signal, that is the representation frequentielle, has for object to make apparaitre the beaches of frequences utilisees by the signal, this notably to make compatible the signal and its canal of transmission.

Example of the amplitude modulation which allows to pass on a signal (voice, music) by Hertzian way, in the case of a radio station.

 



On drawings above, we notice:

The drawing obtained in the fréquential domain is very easy to interpret and to build diagrammatically, it is the reason for which we shall work after with these spectres.

The plan of the lesson is the following one:

 

2. Notions of mathematics to be revised (or to see)

Not so as to lose too much time with notions which must be known, you will find below the required mathematical indispensable notions to be able to follow the sihnal processing module of 1-st Year. Do not hesitate to work those that can put you problem.

 

2.1. Integration by parts

Goal : This technique allows among others to calculate complete of type

Or integrales of type :

That is complete of product between a polynôme and exponential function of type or trigonométrique. Actually, this tool will be notably very useful for the calculation of spectre of triangular signals.

Démonstration : If f and g are dérivable on [a,b] :  (f.g)' = f'.g + f.g'

then :  ,

That is :

Use :

Example 1 : be it so to calculate , we put:

then .

Example 2 : be it so to calculate , we put:

then

           

.

 

2.2. Suites, séries

The used mathematical tool is , this tool allows to write sums in a condensed way.

Example 1 :

Example 2 :

An interesting result is the sum of the geometrical series :

 Where a is called the reason of the geometrical series .

 

2.3. Manipulation of the complex numbers (module, argument, imaginary real, left part).

A complex number c expresses himself by c=a+jb, with (a,b) , a is real part and b imaginary part.

The module of a complex number spells

L'argument du nombre complexe c s'écrit :

It is also possible to write c under the shape : .

Example : c = -1 - j, ,

, we can write :

.

 

2.4. Manipulation of the trigonometry and of exponential complex.

cos(a+b) = cosa.cosb - sina.sinb, cos(a-b) = cosa.cosb + sina.sinb,

sin(a+b) = sina.cosb + sinb.cosa, sin(a-b) = sina.cosb - cosa.sinb,

cosa.cosb = 1/2 [cos(a+b) + cos(a-b)], sina.sinb = 1/2 [cos(a-b) - cos(a+b)], sina.cosb = 1/2 [sin(a+b) + sin(a-b)]

, ,

, ,

, sin2a = 2sina.cosa,

,

For quite complete rational n,

Euler's formulae :; ;

;

The other forms of Euler's formulae : ;

.

 

2.5. Derived and primitive

Primitives

Fonctions

Derived

 

2.6. Cardinal sinus

This signal, very useful in telecommunications, does not put particular problem concerning its study; the only problematic point is 0 which résoud in the following way :

Another definition of cardinal sine which one finds sometimes in the literature and notably in Matlab is the following one :