The document discusses the Fast Fourier Transform (FFT) algorithm. It begins with an introduction to the Discrete Fourier Transform (DFT) and compares the computational complexity of directly calculating the DFT versus using the FFT algorithm. The FFT algorithm reduces the computation time of the DFT from O(N2) to O(NlogN) operations. The document then derives the radix-2 FFT algorithm, showing how it decomposes the DFT calculation into interleaved even and odd numbered terms. This allows computation of the DFT using primarily additions and multiplications by complex roots of unity.

2. PROCESAMIENTO DIGITAL DE
SEÑALES
SEÑALES Y SISTEMAS EN TIEMPO DISCRETO
Marcelo Fernando Valdiviezo Condolo
Quinto ‘A’
Carrera de Telecomunicaciones

3. FFT
JULIO - 2020

4. INTRODUCCIÓN
Existen varios algoritmos para el
cálculo eficiente de la DFT
Función matemática
o
Transformación
Familia específica de
algoritmos para el
cálculo de la DFT
DFT FFT

5. INTRODUCCIÓN
OPERACIONES PARA LA
EVALUACIÓN DIRECTA
OPERACIONES MEDIANTE EL
ALGORITMO DE FFT
( )2
o N ( )2logo N N

6. RELACIÓN DE LA FFT CON LA DFT
La transformada rápida de Fourier (Fast Fourier Transform -FFT) es un
algoritmo que reduce el tiempo de cálculo de la DFT.
3
( ) sin(2 1000 ) 0.5sin 2 2000
4
s sx n nt nt

 
 
=   +   + 
 
1
2 /
0
( ) ( )
N
j nk N
n
X k x n e 
−
−
=
= 
( )
1
0
Re ( ) ( )Cos 2 /
N
n
X k x n nk N
−
=
=  ( )
1
0
Im ( ) ( )Sin 2 /
N
n
X k x n nk N
−
=
= −

7. CÁLCULO DE LA DFT
( )
( )
( )
( )
( )
( )
(1) 0.3535 1.0 0.3535 0.0 0
0.3535 0.707 0.3535 0.707 1
0.6464 0.0 0.6464 1.00 2
1.0607 0.707 1.0607 0.707 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.707 1.0607 0.707 5
1.3535 0.0 1
X j n
j n
j n
j n
j n
j n
j
=  −   =
+  −   =
+  −   =
+ − − −  =
+ − −   =
− − − − −  =
−  − −( )
( )
.3535 1.0 6
0.3535 0.707 0.3535 0.707 7
n
j n
−  =
−  − − −  =
( )
( )
( )
( )
( )
( )
( )
(2) 0.3535 1.0 0.3535 0.0 0
0.3535 0.0 0.3535 1.0 1
0.6464 1.0 0.6464 0.0 2
1.0607 0.0 1.0607 1.0 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.0 1.0607 1.0 5
1.3535 1.0 1.3535 0.0 6
0
X j n
j n
j n
j n
j n
j n
j n
=  −   =
+  −   =
+ − −   =
+  − −  =
+  −   =
−  − −   =
− − − −   =
− ( ).3535 0.0 0.3535 1.0 7j n − − −  =
( )
( )
( )
( )
( )
( )
(3) 0.3535 1.0 0.3535 0.0 0
0.3535 0.707 0.3535 0.707 1
0.6464 0.0 0.6464 1.0 2
1.0607 0.707 1.0607 0.707 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.707 1.0607 0.707 5
1.3535 0.0 1.
X j n
j n
j n
j n
j n
j n
j
=  −   =
+ − −   =
+ − − −  =
+  −   =
+ − −   =
−  − − −  =
−  − −( )
( )
3535 1.0 6
0.3535 0.707 0.3535 0.707 7
n
j n
  =
− − − − −  =

8. CÁLCULO DE LA DFT
( )
( )
( )
( )
( )
( )
( )
(4) 0.3535 1.0 0.3535 0.0 0
0.3535 1.0 0.3535 0.0 1
0.6464 1.0 0.6464 0.0 2
1.0607 1.0 1.0607 0.0 3
0.3535 1.0 0.3535 0.0 4
1.0607 1.0 1.0607 0.0 5
1.3535 1.0 1.3535 0.0 6
0
X j n
j n
j n
j n
j n
j n
j n
=  −   =
+ − −   =
+  −   =
+ − −   =
+  −   =
− − − −   =
−  − −   =
− ( ).3535 1.0 0.3535 0.0 7j n− − −   =
( )
( )
( )
( )
( )
( )
(5) 0.3535 1.0 0.3535 0.0 0
0.3535 0.707 0.3535 0.707 1
0.6464 0.0 0.6464 1.0 2
1.0607 0.707 1.0607 0.707 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.707 1.0607 0.707 5
1.3535 0.0 1.3
X j n
j n
j n
j n
j n
j n
j
=  −   =
+ − − −  =
+  −   =
+  − −  =
+ − −   =
−  − −   =
−  − −( )
( )
535 1.0 6
0.3535 0.707 0.3535 0.707 7
n
j n
−  =
− − − −   =
( )
( )
( )
( )
( )
( )
( )
(6) 0.3535 1.0 0.3535 0.0 0
0.3535 0.0 0.3535 1.0 1
0.6464 1.0 0.6464 0.0 2
1.0607 0.0 1.0607 1.0 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.0 1.0607 1.0 5
1.3535 1.0 1.3535 0.0 6
X j n
j n
j n
j n
j n
j n
j n
=  −   =
+  − −  =
+ − −   =
+  −   =
+  −   =
−  − − −  =
− − − −   =
− ( )0.3535 0.0 0.3535 1.0 7j n − −   =

9. CÁLCULO DE LA DFT
( )
( )
( )
( )
( )
( )
(7) 0.3535 1.0 0.3535 0.0 0
0.3535 0.707 0.3535 0.707 1
0.6464 0.0 0.6464 1.0 2
1.0607 0.707 1.0607 0.707 3
0.3535 1.0 0.3535 0.0 4
1.0607 0.707 1.0607 0.707 5
1.3535 0.0 1
X j n
j n
j n
j n
j n
j n
j
=  −   =
+  − −  =
+  − −  =
+ − − −  =
+ − −   =
− − − −   =
−  − −( )
( )
.3535 1.0 6
0.3535 0.707 0.3535 0.707 7
n
j n
  =
−  − −   =
( ) ( )
1 1
0 0
(0) ( ) Cos 0 Sin 0 ( )
N N
n n
X x n j x n
− −
= =
= − =   

10. CÁLCULO DE LA DFT
OPERACIONES PARA LA
EVALUACIÓN DIRECTA
OPERACIONES MEDIANTE EL
ALGORITMO DE FFT
( )2
o N ( )2logo N N
2
N Multiplicaciones→
( )1N N Sumas− →
2
64N →

11. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
1
2 /
0
( ) ( )
N
j nk N
n
X k x n e 
−
−
=
= 
( )
( )
( )
( )2 1 2 1
2 2 / 2 2 1 /
0 0
( ) (2 ) (2 1)
N N
j n k N j n k N
n n
X k x n e x n e 
− −
− − +
= =
= + + 
( )
( )
( )
( )2 1 2 1
2 2 / 2 2 /2 /
0 0
( ) (2 ) (2 1)
N N
j n k N j n k Nj k N
n n
X k x n e e x n e 
− −
− −−
= =
= + + 

12. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
2 /j N
NW e −
=
( ) ( )2 1 2 1
2 2
0 0
( ) (2 ) (2 1)
N N
nk k nk
N N N
n n
X k x n W W x n W
− −
= =
= + + 
FACTOR DE ÁNGULO DE
FASE COMPLEJO
2
2 2 2/ 2
N
j
j N
NW e e


 
−  −  
= =
2
2
N NW W

13. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( ) ( )2 1 2 1
0 02 2
( ) (2 ) (2 1)
N N
nk k nk
N N N
n n
X k x n W W x n W
− −
= =
= + + 
2
2
N NW W :0 1
2
N
k a −
2
N
k k+ →
( ) ( )2 1 2 1
2 2 2
0 02 2
(2 ) (2 1)
2
N N NN Nn k k n k
N N N
n n
N
X k x n W W x n W
     − −+ + +     
     
= =
 
+ = + + 
 
 

14. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( )
2 2
2 22
2 2 2 2 2 2
1
N N j n Nn k n
nk nk nk nkN
N N N N N NW W W W e W W
 
+  −   = = = = 
 
( ) ( )2 1 2 1
2 2 2
0 02 2
(2 ) (2 1)
2
N N NN Nn k k n k
N N N
n n
N
X k x n W W x n W
     − −+ + +     
     
= =
 
+ = + + 
 
 
( ) ( )2 22 2
1
N Nk
k k j N N k k
N N N N N NW W W W e W W
 
+  − 
= = = − = −
( ) ( )2 1 2 1
0 02 2
(2 ) (2 1)
2
N N
nk k nk
N N N
n n
N
X k x n W W x n W
− −
= =
 
+ = − + 
 
 

15. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( ) ( )2 1 2 1
0 02 2
(2 ) (2 1)
2
N N
nk k nk
N N N
n n
N
X k x n W W x n W
− −
= =
 
+ = − + 
 
 
( ) ( )2 1 2 1
0 02 2
( ) (2 ) (2 1)
N N
nk k nk
N N N
n n
X k x n W W x n W
− −
= =
= + + 

16. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2

17. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( ) ( )2 1 2 1
0 02 2
(2 ) (2 1)
2
N N
nk k nk
N N N
n n
N
X k x n W W x n W
− −
= =
 
+ = − + 
 
 
( ) ( )2 1 2 1
0 02 2
( ) (2 ) (2 1)
N N
nk k nk
N N N
n n
X k x n W W x n W
− −
= =
= + + 
( ) ( ) ( )k
NX k A k W B k= +
( ) ( )
2
k
N
N
X k A k W B k
 
+ = − 
 
( ) ( ) ( ) ( ) ( )
1 1 1
2 4 4
2 12
0 0 02 2 2
2 4 4 2
N N N
n knk nk
N N N
n n n
A k x n W x n W x n W
− − −
+
= = =
= = + +  

18. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
2
2 4
nk nk
N NW W=
( ) ( ) ( )
1 1
4 4
0 04 2 4
4 4 2
N N
nk k nk
N N N
n n
A k x n W W x n W
− −
= =
= + + 
( ) ( )2 1 2 1
0 02 2
(2 ) (2 1)
2
N N
nk k nk
N N N
n n
N
X k x n W W x n W
− −
= =
 
+ = − + 
 
 
SIMILARES

19. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( ) ( ) ( )
1 1
4 4
0 04 2 4
4 1 4 3
N N
nk k nk
N N N
n n
B k x n W W x n W
− −
= =
= + + + 
( ) ( ) ( )
1 1
4 4
0 04 2 4
4 4 2
N N
nk k nk
N N N
n n
A k x n W W x n W
− −
= =
= + + 

20. DERIVACIÓN DEL ALGORITMO FFT
RADIX-2
( ) ( ) ( )
1 1
4 4
0 04 2 4
4 1 4 3
N N
nk k nk
N N N
n n
B k x n W W x n W
− −
= =
= + + + 
( ) ( ) ( )
1 1
4 4
0 04 2 4
4 4 2
N N
nk k nk
N N N
n n
A k x n W W x n W
− −
= =
= + + 
( ) ( ) ( )k
NX k A k W B k= +

22. PREGUNTAS

23. GRACIAS…