signal_15_12_23.py

Created on December 18, 2023
1.56 KB
série de Fourier m(t) :

m(t) = a0 + ∑ (an cos(2πnf0t)
+ bn sin(2πnf0t))

an= 2/T * int(0,T,m(t)*cos(2πnf0t)dt)

bn= 2/T * int(0,T,m(t)*sin(2πnf0t)dt)

a0= 1/T * int(0,T,m(t)dt)

/////////////////////////
transformée de Fourier p(t)=cos(2πf0t)

      e^(2πjfpt)+e^(-2πjfpt)
P(t)= ----------------------
                2

P(t)= 1/2 *(int(-oo,oo,e^(2πjfpt)dt)
+ int(-oo,oo,e^(-2πjfpt)dt))

       δ(f-fp)+δ(f+fp)
P(t)= -----------------
              2
              
/////////////////////////
transformée de Fourier p(t)=sin(2πf0t)

      e^(2πjf0t)-e^(-2πjf0t) 
P(t)= ----------------------
                2
                
P(t)= 1/2 *(int(-oo,oo,e^(2πjf0t)
-e^(-2πjf0t)dt))

       δ(f-f0)-δ(f+f0)
P(t)= -----------------
              2

/////////////////////////
autocorrélation (sign carré) de x

[Rxx, tau] = xcorr(x,x)

Rxx(T) = ∑ x(t).x(t-T)

         rect(t)  rect(t-T)
Rxx(T) = -------.----------
           [T]       [T]

/////////////////////////
autocorrélation (sign sin) de p

[Rpp, tau] = xcorr(p,p)

          x(t) - x(t-T)
Rpp(T) = ---------------
                T

        X(f)-X(f)e^(-2πjft)
Rpp(T) = ------------------
                 T

////////////////////////
transformée de Hilbert

x(t)=A.rect(t)/[T]

xh(t)= x(t) (*) 1/πT

    rect(t)       1
= A.------- (*) ----
      [T]        πT
      
              -T/2
= A/π (ln|t-u|)
               T/2

   A     |t+(T/2)|
= --- ln(---------)
   π     |t-(T/2)|
   
   
tracer :
  
      |     |    /|\
      |     |---/ | \---
------------------------
  --\ | /---|     |
     \|/    |     |