En el 2012 preparé esta presentación, basada en el libro "Reconocimiento de Voz y Fonética Acústica" de Bernal Bermúdez Et Al, para explicar cómo funciona la Transformada de Fourier.
104. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
105. 1
Fourier (m , n , N , T )=
N
La Transformada
de Fourier...
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
106. 1
Fourier (m , n , N , T )=
N
-2.63 -2.62 -2.63 -4.25 0.00 4.25 2.63 2.62 4.25 0.00
N −1
∑ m [kT ]e
k=0
...de una ventana “m”
de muestras...
−2 π n k
j
N
107. 1
Fourier (m , n , N , T )=
N
n=1
2 Hz
N −1
∑ m [kT ]e
k=0
...para la “n” de la
frecuencia cuyo aporte
se quiere conocer...
−2 π n k
j
N
108. 1
Fourier (m , n , N , T )=
N
10 muestras
-2.63 -2.62 -2.63 -4.25 0.00 4.25 2.63 2.62 4.25 0.00
N −1
∑ m [kT ]e
−2 π n k
j
N
k=0
...con un tamaño
de ventana de
“N” muestras...
109. 1
Fourier (m , n , N , T )=
N
...y cuyo periodo de
muestreo fue “T”...
1/Fs
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
110. 1
Fourier (m , n , N , T )=
N
...se calcula como...
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
111. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
112. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
113. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
114. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
115. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
116. 1
Fourier (m , n , N , T )=
N
N −1
∑ m [kT ]e
k=0
−2 π n k
j
N
143. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
n=0
N = 10
T = 0.05
(
)
144. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
)
145. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
146. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
147. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
1.0000
-4.25
1.0000
0.00
1.0000
4.25
1.0000
2.63
1.0000
2.62
1.0000
4.25
1.0000
0.00
1.0000
-4.25
1.0000
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
148. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
1.0000
-2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
1.0000
2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
)
149. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=0
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
1.0000
-2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
1.0000
2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
Suma:
0.0000
0.0000
)
150. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
1.0000
-2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
1.0000
2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=0
N = 10
T = 0.05
2
)
151. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
1.0000
-2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
1.0000
2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=0
N = 10
T = 0.05
2
1
. 0=0
10
)
152. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
1.0000
-2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
1.0000
2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=0
N = 10
T = 0.05
)
2
1
. 0=0
10
la señal sen(0t)
aportó 0 de amplitud
a la señal analizada
154. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
)
155. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
-0.5878
-0.9511
-0.9511
-0.5878
0.0000
0.5878
0.9511
0.9511
0.5878
)
156. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
0.8090
0.3090
-0.3090
-0.8090
-1.0000
-0.8090
-0.3090
0.3090
0.8090
sen(-2πnk/N)
0.0000
-0.5878
-0.9511
-0.9511
-0.5878
0.0000
0.5878
0.9511
0.9511
0.5878
)
157. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
0.8090
-4.25
0.3090
0.00
-0.3090
4.25
-0.8090
2.63
-1.0000
2.62
-0.8090
4.25
-0.3090
0.00
0.3090
-4.25
0.8090
sen(-2πnk/N)
0.0000
-0.5878
-0.9511
-0.9511
-0.5878
0.0000
0.5878
0.9511
0.9511
0.5878
)
158. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.8090
-2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
-0.9511
4.0420
0.00
-0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
-0.5878
-2.4981
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
0.5878
1.5400
4.25
-0.3090
-1.3133
0.9511
4.0420
0.00
0.3090
0.0000
0.9511
0.0000
-4.25
0.8090
-3.4383
0.5878
-2.4981
)
159. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=1
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.8090
-2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
-0.9511
4.0420
0.00
-0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
-0.5878
-2.4981
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
0.5878
1.5400
4.25
-0.3090
-1.3133
0.9511
4.0420
0.00
0.3090
0.0000
0.9511
0.0000
-4.25
0.8090
-3.4383
0.5878
-2.4981
Suma:
-19.0025
6.1678
)
160. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.8090
-2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
-0.9511
4.0420
0.00
-0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
-0.5878
-2.4981
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
0.5878
1.5400
4.25
-0.3090
-1.3133
0.9511
4.0420
0.00
0.3090
0.0000
0.9511
0.0000
-4.25
0.8090
-3.4383
0.5878
-2.4981
Suma:
-19.0025
6.1678
√−19.0025 + 6.1678 ≈20
2
n=1
N = 10
T = 0.05
2
)
161. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.8090
-2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
-0.9511
4.0420
0.00
-0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
-0.5878
-2.4981
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
0.5878
1.5400
4.25
-0.3090
-1.3133
0.9511
4.0420
0.00
0.3090
0.0000
0.9511
0.0000
-4.25
0.8090
-3.4383
0.5878
-2.4981
Suma:
-19.0025
6.1678
√−19.0025 + 6.1678 ≈20
2
n=1
N = 10
T = 0.05
1
. 20=2
10
2
)
162. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.8090
-2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
-0.9511
4.0420
0.00
-0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
-0.5878
-2.4981
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
0.5878
1.5400
4.25
-0.3090
-1.3133
0.9511
4.0420
0.00
0.3090
0.0000
0.9511
0.0000
-4.25
0.8090
-3.4383
0.5878
-2.4981
Suma:
-19.0025
6.1678
√−19.0025 + 6.1678 ≈20
2
n=1
N = 10
T = 0.05
)
1
. 20=2
10
2
la señal sen(2t)
aportó 2 de amplitud
a la señal analizada
164. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
)
165. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
-0.9511
-0.5878
0.5878
0.9511
0.0000
-0.9511
-0.5878
0.5878
0.9511
)
166. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
0.3090
-0.8090
-0.8090
0.3090
1.0000
0.3090
-0.8090
-0.8090
0.3090
sen(-2πnk/N)
0.0000
-0.9511
-0.5878
0.5878
0.9511
0.0000
-0.9511
-0.5878
0.5878
0.9511
)
167. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
0.3090
-4.25
-0.8090
0.00
-0.8090
4.25
0.3090
2.63
1.0000
2.62
0.3090
4.25
-0.8090
0.00
-0.8090
-4.25
0.3090
sen(-2πnk/N)
0.0000
-0.9511
-0.5878
0.5878
0.9511
0.0000
-0.9511
-0.5878
0.5878
0.9511
)
168. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.3090
-0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
-0.5878
2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
0.9511
4.0420
2.63
1.0000
2.6300
0.0000
0.0000
2.62
0.3090
0.8096
-0.9511
-2.4918
4.25
-0.8090
-3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
-4.25
0.3090
-1.3133
0.9511
-4.0420
)
169. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=2
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.3090
-0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
-0.5878
2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
0.9511
4.0420
2.63
1.0000
2.6300
0.0000
0.0000
2.62
0.3090
0.8096
-0.9511
-2.4918
4.25
-0.8090
-3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
-4.25
0.3090
-1.3133
0.9511
-4.0420
Suma:
0.0000
0.0000
)
170. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.3090
-0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
-0.5878
2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
0.9511
4.0420
2.63
1.0000
2.6300
0.0000
0.0000
2.62
0.3090
0.8096
-0.9511
-2.4918
4.25
-0.8090
-3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
-4.25
0.3090
-1.3133
0.9511
-4.0420
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=2
N = 10
T = 0.05
2
)
171. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.3090
-0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
-0.5878
2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
0.9511
4.0420
2.63
1.0000
2.6300
0.0000
0.0000
2.62
0.3090
0.8096
-0.9511
-2.4918
4.25
-0.8090
-3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
-4.25
0.3090
-1.3133
0.9511
-4.0420
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=2
N = 10
T = 0.05
2
1
. 0=0
10
)
172. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
0.3090
-0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
-0.5878
2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
0.9511
4.0420
2.63
1.0000
2.6300
0.0000
0.0000
2.62
0.3090
0.8096
-0.9511
-2.4918
4.25
-0.8090
-3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
0.5878
0.0000
-4.25
0.3090
-1.3133
0.9511
-4.0420
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=2
N = 10
T = 0.05
)
2
1
. 0=0
10
la señal sen(4t)
aportó 0 de amplitud
a la señal analizada
174. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
)
175. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
-0.9511
0.5878
0.5878
-0.9511
0.0000
0.9511
-0.5878
-0.5878
0.9511
)
176. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
-0.3090
-0.8090
0.8090
0.3090
-1.0000
0.3090
0.8090
-0.8090
-0.3090
sen(-2πnk/N)
0.0000
-0.9511
0.5878
0.5878
-0.9511
0.0000
0.9511
-0.5878
-0.5878
0.9511
)
177. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
-0.3090
-4.25
-0.8090
0.00
0.8090
4.25
0.3090
2.63
-1.0000
2.62
0.3090
4.25
0.8090
0.00
-0.8090
-4.25
-0.3090
sen(-2πnk/N)
0.0000
-0.9511
0.5878
0.5878
-0.9511
0.0000
0.9511
-0.5878
-0.5878
0.9511
)
178. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.3090
0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
0.5878
-2.4981
0.00
0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
-0.9511
-4.0420
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
0.3090
0.8096
0.9511
2.4918
4.25
0.8090
3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
-0.5878
0.0000
-4.25
-0.3090
1.3133
0.9511
-4.0420
)
179. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=3
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.3090
0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
0.5878
-2.4981
0.00
0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
-0.9511
-4.0420
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
0.3090
0.8096
0.9511
2.4918
4.25
0.8090
3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
-0.5878
0.0000
-4.25
-0.3090
1.3133
0.9511
-4.0420
Suma:
5.8625
-8.0966
)
180. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.3090
0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
0.5878
-2.4981
0.00
0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
-0.9511
-4.0420
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
0.3090
0.8096
0.9511
2.4918
4.25
0.8090
3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
-0.5878
0.0000
-4.25
-0.3090
1.3133
0.9511
-4.0420
Suma:
5.8625
-8.0966
√ 5.8625 + (−8.0966) ≈10
2
n=3
N = 10
T = 0.05
2
)
181. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.3090
0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
0.5878
-2.4981
0.00
0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
-0.9511
-4.0420
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
0.3090
0.8096
0.9511
2.4918
4.25
0.8090
3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
-0.5878
0.0000
-4.25
-0.3090
1.3133
0.9511
-4.0420
Suma:
5.8625
-8.0966
√ 5.8625 + (−8.0966) ≈10
2
n=3
N = 10
T = 0.05
1
.10=1
10
2
)
182. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.3090
0.8096
-0.9511
2.4918
-4.25
-0.8090
3.4383
0.5878
-2.4981
0.00
0.8090
0.0000
0.5878
0.0000
4.25
0.3090
1.3133
-0.9511
-4.0420
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
0.3090
0.8096
0.9511
2.4918
4.25
0.8090
3.4383
-0.5878
-2.4981
0.00
-0.8090
0.0000
-0.5878
0.0000
-4.25
-0.3090
1.3133
0.9511
-4.0420
Suma:
5.8625
-8.0966
√ 5.8625 + (−8.0966) ≈10
2
n=3
N = 10
T = 0.05
)
1
.10=1
10
2
la señal sen(6t)
aportó 1 de amplitud
a la señal analizada
184. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
)
185. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
-0.5878
0.9511
-0.9511
0.5878
0.0000
-0.5878
0.9511
-0.9511
0.5878
)
186. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
-0.8090
0.3090
0.3090
-0.8090
1.0000
-0.8090
0.3090
0.3090
-0.8090
sen(-2πnk/N)
0.0000
-0.5878
0.9511
-0.9511
0.5878
0.0000
-0.5878
0.9511
-0.9511
0.5878
)
187. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
-0.8090
-4.25
0.3090
0.00
0.3090
4.25
-0.8090
2.63
1.0000
2.62
-0.8090
4.25
0.3090
0.00
0.3090
-4.25
-0.8090
sen(-2πnk/N)
0.0000
-0.5878
0.9511
-0.9511
0.5878
0.0000
-0.5878
0.9511
-0.9511
0.5878
)
188. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.8090
2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
0.9511
-4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
0.5878
2.4981
2.63
1.0000
2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
-0.5878
-1.5400
4.25
0.3090
1.3133
0.9511
4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
-4.25
-0.8090
3.4383
0.5878
-2.4981
)
189. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=4
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.8090
2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
0.9511
-4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
0.5878
2.4981
2.63
1.0000
2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
-0.5878
-1.5400
4.25
0.3090
1.3133
0.9511
4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
-4.25
-0.8090
3.4383
0.5878
-2.4981
Suma:
0.0000
0.0000
)
190. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.8090
2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
0.9511
-4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
0.5878
2.4981
2.63
1.0000
2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
-0.5878
-1.5400
4.25
0.3090
1.3133
0.9511
4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
-4.25
-0.8090
3.4383
0.5878
-2.4981
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=4
N = 10
T = 0.05
2
)
191. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.8090
2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
0.9511
-4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
0.5878
2.4981
2.63
1.0000
2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
-0.5878
-1.5400
4.25
0.3090
1.3133
0.9511
4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
-4.25
-0.8090
3.4383
0.5878
-2.4981
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=4
N = 10
T = 0.05
2
1
. 0=0
10
)
192. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-0.8090
2.1196
-0.5878
1.5400
-4.25
0.3090
-1.3133
0.9511
-4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
4.25
-0.8090
-3.4383
0.5878
2.4981
2.63
1.0000
2.6300
0.0000
0.0000
2.62
-0.8090
-2.1196
-0.5878
-1.5400
4.25
0.3090
1.3133
0.9511
4.0420
0.00
0.3090
0.0000
-0.9511
0.0000
-4.25
-0.8090
3.4383
0.5878
-2.4981
Suma:
0.0000
0.0000
√ 0 + 0 =0
2
n=4
N = 10
T = 0.05
)
2
1
. 0=0
10
la señal sen(8t)
aportó 0 de amplitud
a la señal analizada
194. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
)
195. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
196. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
cos(-2πnk/N)
1.0000
-1.0000
1.0000
-1.0000
1.0000
-1.0000
1.0000
-1.0000
1.0000
-1.0000
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
197. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
m[kT] cos(-2πnk/N)
-2.63
1.0000
-2.62
-1.0000
-4.25
1.0000
0.00
-1.0000
4.25
1.0000
2.63
-1.0000
2.62
1.0000
4.25
-1.0000
0.00
1.0000
-4.25
-1.0000
sen(-2πnk/N)
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
)
198. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-1.0000
2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
-1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
-1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
-1.0000
4.2500
0.0000
0.0000
)
199. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
n=5
N = 10
T = 0.05
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-1.0000
2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
-1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
-1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
-1.0000
4.2500
0.0000
0.0000
Suma:
-0.0200
0.0000
)
200. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-1.0000
2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
-1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
-1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
-1.0000
4.2500
0.0000
0.0000
Suma:
-0.0200
0.0000
√−0.0200 + 0.0000 ≈0
2
n=5
N = 10
T = 0.05
2
)
201. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-1.0000
2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
-1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
-1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
-1.0000
4.2500
0.0000
0.0000
Suma:
-0.0200
0.0000
√−0.0200 + 0.0000 ≈0
2
n=5
N = 10
T = 0.05
1
. 0=0
10
2
)
202. 1
N
N −1
k
k
∑ m[kT ] cos(−2 π n N )+ sen(−2 π n N ) j
k =0
(
k
0
1
2
3
4
5
6
7
8
9
m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
-2.63
1.0000
-2.6300
0.0000
0.0000
-2.62
-1.0000
2.6200
0.0000
0.0000
-4.25
1.0000
-4.2500
0.0000
0.0000
0.00
-1.0000
0.0000
0.0000
0.0000
4.25
1.0000
4.2500
0.0000
0.0000
2.63
-1.0000
-2.6300
0.0000
0.0000
2.62
1.0000
2.6200
0.0000
0.0000
4.25
-1.0000
-4.2500
0.0000
0.0000
0.00
1.0000
0.0000
0.0000
0.0000
-4.25
-1.0000
4.2500
0.0000
0.0000
Suma:
-0.0200
0.0000
√−0.0200 + 0.0000 ≈0
2
n=5
N = 10
T = 0.05
)
1
. 0=0
10
2
la señal sen(10t)
aportó 0 de amplitud
a la señal analizada