cómo funciona la transformada de fourier
DESCRIPTION
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.TRANSCRIPT
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La Transformada de Fourier
Dr. José Enrique Alvarez Estradahttp://www.software.org.mx/~jalvarez/
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Las ideas deJean Baptiste Fourier
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cualquier señal puedeformarse sumando
funciones senode diferentes frecuencias
a diferentes amplitudes
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Y a la inversa...
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Dada una señal compuesta,
¿cuánto tengo que agregar
de cada señal fundamental para recrearla?
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¿Cuánto sen(3t)?
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¿Cuánto sen(3t)?
¿Cuánto sen(4t)?
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¿Cuánto sen(3t)?
¿Cuánto sen(4t)?
¿Cuánto sen(6t)?
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¿Cuánto sen(3t)?
¿Cuánto sen(4t)?
¿Cuánto sen(6t)?
¿Cuánto sen(12t)?
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Así que la palabra transformarsignifica cambiar el dominio de la señal,
pasando del dominio del tiempo aldominio de la frecuencia.
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Fourier
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tiempo
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Fourier
tiempo frecuencia
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Por simplicidad, trabajaremos conuna versión discreta de la señal
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fenómenoanalógicoa estudiar
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ConversorA/D
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ConversorA/D
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20 Hz
Frecuenciade muestreo (Fs)
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ConversorA/D
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20 Hz
versióndiscreta
(muestras)
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10 muestras
ConversorA/D
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20 Hz
tamaño deventana
“N”
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-2.63 -2.62 -2.63 -4.25 0.00 4.25
10 muestras
2.63 2.62 4.25 0.00
ConversorA/D
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20 Hz ventana “m”
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-2.63 -2.62 -2.63 -4.25 0.00 4.25 2.63 2.62 4.25 0.00
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ConversorD/A
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ConversorD/A
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Hasta aquí, tenemos grabadoun archivo WAV...
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Pero un WAV ocupa mucho espacio,porque guarda todas las muestras.
¿Y si sólo almacenamos algunas de sus características?
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-2.63 -2.62 -2.63 -4.25 0.00 4.25
10 muestras
2.63 2.62 4.25 0.00
ConversorA/D
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
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20 Hz 1/Fs
periodo demuestreo
“T”
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-2.63
n=510Hz
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10 muestras
2.63 2.62 4.25 0.00
ConversorA/D
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20 Hz 1/Fs
“n” de la frecuenciacuyo aporte sequiere conocer
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Fourier-2.63
n=510Hz
-2.62 -2.63 -4.25 0.00 4.25
10 muestras
2.63 2.62 4.25 0.00
ConversorA/D
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
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20 Hz 1/Fs
Transformadade Fourier
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Fourier-2.63
10Hz
-2.62 -2.63 -4.25 0.00 4.25
10 muestras
2.63 2.62 4.25 0.00
ConversorA/D
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20 Hz 1/Fs
AFn
Aporte de la frecuencia que
se quiere conocer
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Hagamos un ejemplo
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# de muestra
Inte
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# de muestra
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Esta señal se muestrearáa una frecuencia Fs = 20Hz
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# de muestra
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0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.00
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¿Qué frecuencias básicasdebemos analizar para
reconstruir la señalen la ventana?
![Page 50: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/50.jpg)
La frecuencia de muestreo
debe ser al menos el doblede la frecuencia máximaque se requiere analizar.
![Page 51: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/51.jpg)
F s≥2 Fmáx
La frecuencia de muestreo
debe ser al menos el doblede la frecuencia máximaque se requiere analizar.
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la frecuencia máxima analizable
no puede ser mayor que la mitadde la frecuencia de muestreo
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Fmáx≤F s
2
la frecuencia máxima analizable
no puede ser mayor que la mitadde la frecuencia de muestreo
![Page 54: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/54.jpg)
Así que, paranuestro ejemplo...
![Page 55: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/55.jpg)
Fmáx⩽20 Hz
2⩽10 Hz
![Page 56: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/56.jpg)
Por tanto, los valores de “n”y las frecuenciasa analizar serán:
![Page 57: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/57.jpg)
n=0⇒F0=010
20=0 Hz
![Page 58: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/58.jpg)
n=0⇒F0=0
1020=0 Hz
n=1⇒F1=1
1020=2 Hz
![Page 59: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/59.jpg)
n=0⇒F0=0
1020=0 Hz
n=1⇒F1=1
1020=2 Hz
n=2⇒F2=2
1020=4 Hz
![Page 60: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/60.jpg)
n=0⇒F0=0
1020=0 Hz
n=1⇒F1=1
1020=2 Hz
n=2⇒F2=2
1020=4 Hz
n=3⇒F3=3
1020=6 Hz
![Page 61: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/61.jpg)
n=0⇒F0=0
1020=0 Hz
n=1⇒F1=110
20=2 Hz
n=2⇒F2=2
1020=4 Hz
n=3⇒F3=310
20=6 Hz
n=4⇒F4=4
1020=8 Hz
![Page 62: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/62.jpg)
n=0⇒F0=010
20=0 Hz
n=1⇒F1=1
1020=2Hz
n=2⇒F2=210
20=4 Hz
n=3⇒F3=3
1020=6 Hz
n=4⇒F4=410
20=8 Hz
n=5⇒F5=510
20=10 Hz
![Page 63: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/63.jpg)
¿Cuánto aporta n=0, sen(0t),para formar la señal?
![Page 64: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/64.jpg)
sen(0t)0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.0000
10 0.000011 0.000012 0.000013 0.000014 0.000015 0.000016 0.000017 0.000018 0.000019 0.000020 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
El valor de sen(0t)en los 21 instantes
de muestreo
![Page 65: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/65.jpg)
t sen(0t) m[t]0 0.0000 0.001 0.0000 4.252 0.0000 2.633 0.0000 2.624 0.0000 4.255 0.0000 0.006 0.0000 -4.257 0.0000 -2.638 0.0000 -2.629 0.0000 -4.25
10 0.0000 0.0011 0.0000 4.2512 0.0000 2.6313 0.0000 2.6214 0.0000 4.2515 0.0000 0.0016 0.0000 -4.2517 0.0000 -2.6318 0.0000 -2.6219 0.0000 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
El valor de las21 muestras
![Page 66: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/66.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
¿Cómo podemoscomparar una señal
con la otra?
t sen(0t) m[t]0 0.0000 0.001 0.0000 4.252 0.0000 2.633 0.0000 2.624 0.0000 4.255 0.0000 0.006 0.0000 -4.257 0.0000 -2.638 0.0000 -2.629 0.0000 -4.25
10 0.0000 0.0011 0.0000 4.2512 0.0000 2.6313 0.0000 2.6214 0.0000 4.2515 0.0000 0.0016 0.0000 -4.2517 0.0000 -2.6318 0.0000 -2.6219 0.0000 -4.2520 0.0000 0.00
![Page 67: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/67.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
¿Y si multiplicamos ambas?El resultado nos dará
el área de un rectángulopor cada muestra
sen(0t) Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
t m[t]
![Page 68: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/68.jpg)
sen(0t) Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
t m[t]
Si ambas se parecen,las áreas de los
rectángulos serán grandes.
Si no se parecen,serán pequeñas.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
![Page 69: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/69.jpg)
sen(0t) Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
t m[t]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Como las señalesno se parecen, las
áreas de los rectángulosson nulas.
![Page 70: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/70.jpg)
t sen(0t) m[t] Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
La sumatoria de lasáreas de los
rectángulos equivalea 0 unidades.
![Page 71: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/71.jpg)
t sen(0t) m[t] Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Luego sen(0t) no aporta nadaa la señal original.
![Page 72: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/72.jpg)
¿Cuánto aporta n=1, sen(2t),para formar la señal?
![Page 73: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/73.jpg)
sen(2t)0 0.00001 0.58782 0.95113 0.95114 0.58785 0.00006 -0.58787 -0.95118 -0.95119 -0.5878
10 0.000011 0.587812 0.951113 0.951114 0.587815 0.000016 -0.587817 -0.951118 -0.951119 -0.587820 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 74: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/74.jpg)
t sen(2t) m[t]0 0.0000 0.001 0.5878 4.252 0.9511 2.633 0.9511 2.624 0.5878 4.255 0.0000 0.006 -0.5878 -4.257 -0.9511 -2.638 -0.9511 -2.629 -0.5878 -4.25
10 0.0000 0.0011 0.5878 4.2512 0.9511 2.6313 0.9511 2.6214 0.5878 4.2515 0.0000 0.0016 -0.5878 -4.2517 -0.9511 -2.6318 -0.9511 -2.6219 -0.5878 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 75: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/75.jpg)
sen(2t) Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 0.9511 2.63 2.5013 0.9511 2.62 2.4924 0.5878 4.25 2.4985 0.0000 0.00 0.0006 -0.5878 -4.25 2.4987 -0.9511 -2.63 2.5018 -0.9511 -2.62 2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 0.9511 2.63 2.50113 0.9511 2.62 2.49214 0.5878 4.25 2.49815 0.0000 0.00 0.00016 -0.5878 -4.25 2.49817 -0.9511 -2.63 2.50118 -0.9511 -2.62 2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
t m[t]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 76: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/76.jpg)
t sen(2t) m[t] Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 0.9511 2.63 2.5013 0.9511 2.62 2.4924 0.5878 4.25 2.4985 0.0000 0.00 0.0006 -0.5878 -4.25 2.4987 -0.9511 -2.63 2.5018 -0.9511 -2.62 2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 0.9511 2.63 2.50113 0.9511 2.62 2.49214 0.5878 4.25 2.49815 0.0000 0.00 0.00016 -0.5878 -4.25 2.49817 -0.9511 -2.63 2.50118 -0.9511 -2.62 2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
Sumatoria: 39.957Normalizada: 1.998
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
La sumatoria de las áreas delos rectángulos equivale a
casi 40 unidades. Normalizada representa casi 2.
![Page 77: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/77.jpg)
t sen(2t) m[t] Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 0.9511 2.63 2.5013 0.9511 2.62 2.4924 0.5878 4.25 2.4985 0.0000 0.00 0.0006 -0.5878 -4.25 2.4987 -0.9511 -2.63 2.5018 -0.9511 -2.62 2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 0.9511 2.63 2.50113 0.9511 2.62 2.49214 0.5878 4.25 2.49815 0.0000 0.00 0.00016 -0.5878 -4.25 2.49817 -0.9511 -2.63 2.50118 -0.9511 -2.62 2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
Sumatoria: 39.957Normalizada: 1.998
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
Luego sen(2t) aporta 2a la señal original.
![Page 78: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/78.jpg)
¿Cuánto aporta n=2, sen(4t),para formar la señal?
![Page 79: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/79.jpg)
sen(4t)0 0.00001 0.95112 0.58783 -0.58784 -0.95115 0.00006 0.95117 0.58788 -0.58789 -0.9511
10 0.000011 0.951112 0.587813 -0.587814 -0.951115 0.000016 0.951117 0.587818 -0.587819 -0.951120 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 80: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/80.jpg)
t sen(4t) m[t]0 0.0000 0.001 0.9511 4.252 0.5878 2.633 -0.5878 2.624 -0.9511 4.255 0.0000 0.006 0.9511 -4.257 0.5878 -2.638 -0.5878 -2.629 -0.9511 -4.25
10 0.0000 0.0011 0.9511 4.2512 0.5878 2.6313 -0.5878 2.6214 -0.9511 4.2515 0.0000 0.0016 0.9511 -4.2517 0.5878 -2.6318 -0.5878 -2.6219 -0.9511 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 81: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/81.jpg)
t sen(4t) m[t] Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 0.5878 2.63 1.5463 -0.5878 2.62 -1.5404 -0.9511 4.25 -4.0425 0.0000 0.00 0.0006 0.9511 -4.25 -4.0427 0.5878 -2.63 -1.5468 -0.5878 -2.62 1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 0.5878 2.63 1.54613 -0.5878 2.62 -1.54014 -0.9511 4.25 -4.04215 0.0000 0.00 0.00016 0.9511 -4.25 -4.04217 0.5878 -2.63 -1.54618 -0.5878 -2.62 1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 82: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/82.jpg)
t sen(4t) m[t] Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 0.5878 2.63 1.5463 -0.5878 2.62 -1.5404 -0.9511 4.25 -4.0425 0.0000 0.00 0.0006 0.9511 -4.25 -4.0427 0.5878 -2.63 -1.5468 -0.5878 -2.62 1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 0.5878 2.63 1.54613 -0.5878 2.62 -1.54014 -0.9511 4.25 -4.04215 0.0000 0.00 0.00016 0.9511 -4.25 -4.04217 0.5878 -2.63 -1.54618 -0.5878 -2.62 1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
La sumatoria de las áreas delos rectángulos equivale a
0 unidades.
![Page 83: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/83.jpg)
t sen(4t) m[t] Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 0.5878 2.63 1.5463 -0.5878 2.62 -1.5404 -0.9511 4.25 -4.0425 0.0000 0.00 0.0006 0.9511 -4.25 -4.0427 0.5878 -2.63 -1.5468 -0.5878 -2.62 1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 0.5878 2.63 1.54613 -0.5878 2.62 -1.54014 -0.9511 4.25 -4.04215 0.0000 0.00 0.00016 0.9511 -4.25 -4.04217 0.5878 -2.63 -1.54618 -0.5878 -2.62 1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
Luego sen(4t) no aporta nadaa la señal original.
![Page 84: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/84.jpg)
¿Cuánto aporta n=3, sen(6t),para formar la señal?
![Page 85: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/85.jpg)
sen(6t)0 0.00001 0.95112 -0.58783 -0.58784 0.95115 0.00006 -0.95117 0.58788 0.58789 -0.9511
10 0.000011 0.951112 -0.587813 -0.587814 0.951115 0.000016 -0.951117 0.587818 0.587819 -0.951120 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 86: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/86.jpg)
t sen(6t) m[t]0 0.0000 0.001 0.9511 4.252 -0.5878 2.633 -0.5878 2.624 0.9511 4.255 0.0000 0.006 -0.9511 -4.257 0.5878 -2.638 0.5878 -2.629 -0.9511 -4.25
10 0.0000 0.0011 0.9511 4.2512 -0.5878 2.6313 -0.5878 2.6214 0.9511 4.2515 0.0000 0.0016 -0.9511 -4.2517 0.5878 -2.6318 0.5878 -2.6219 -0.9511 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 87: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/87.jpg)
sen(6t) Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 -0.5878 2.63 -1.5463 -0.5878 2.62 -1.5404 0.9511 4.25 4.0425 0.0000 0.00 0.0006 -0.9511 -4.25 4.0427 0.5878 -2.63 -1.5468 0.5878 -2.62 -1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 -0.5878 2.63 -1.54613 -0.5878 2.62 -1.54014 0.9511 4.25 4.04215 0.0000 0.00 0.00016 -0.9511 -4.25 4.04217 0.5878 -2.63 -1.54618 0.5878 -2.62 -1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
t m[t]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 88: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/88.jpg)
t sen(6t) m[t] Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 -0.5878 2.63 -1.5463 -0.5878 2.62 -1.5404 0.9511 4.25 4.0425 0.0000 0.00 0.0006 -0.9511 -4.25 4.0427 0.5878 -2.63 -1.5468 0.5878 -2.62 -1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 -0.5878 2.63 -1.54613 -0.5878 2.62 -1.54014 0.9511 4.25 4.04215 0.0000 0.00 0.00016 -0.9511 -4.25 4.04217 0.5878 -2.63 -1.54618 0.5878 -2.62 -1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
Sumatoria: 19.992Normalizada: 1.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
La sumatoria de las áreas delos rectángulos equivale a
casi 20 unidades.Normalizada representa casi 1.
![Page 89: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/89.jpg)
t sen(6t) m[t] Producto0 0.0000 0.00 0.0001 0.9511 4.25 4.0422 -0.5878 2.63 -1.5463 -0.5878 2.62 -1.5404 0.9511 4.25 4.0425 0.0000 0.00 0.0006 -0.9511 -4.25 4.0427 0.5878 -2.63 -1.5468 0.5878 -2.62 -1.5409 -0.9511 -4.25 4.042
10 0.0000 0.00 0.00011 0.9511 4.25 4.04212 -0.5878 2.63 -1.54613 -0.5878 2.62 -1.54014 0.9511 4.25 4.04215 0.0000 0.00 0.00016 -0.9511 -4.25 4.04217 0.5878 -2.63 -1.54618 0.5878 -2.62 -1.54019 -0.9511 -4.25 4.04220 0.0000 0.00 0.000
Sumatoria: 19.992Normalizada: 1.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
Luego sen(6t) aporta 1a la señal original.
![Page 90: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/90.jpg)
¿Cuánto aporta n=4, sen(8t),para formar la señal?
![Page 91: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/91.jpg)
sen(8t)0 0.00001 0.58782 -0.95113 0.95114 -0.58785 0.00006 0.58787 -0.95118 0.95119 -0.5878
10 0.000011 0.587812 -0.951113 0.951114 -0.587815 0.000016 0.587817 -0.951118 0.951119 -0.587820 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 92: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/92.jpg)
t sen(8t) m[t]0 0.0000 0.001 0.5878 4.252 -0.9511 2.633 0.9511 2.624 -0.5878 4.255 0.0000 0.006 0.5878 -4.257 -0.9511 -2.638 0.9511 -2.629 -0.5878 -4.25
10 0.0000 0.0011 0.5878 4.2512 -0.9511 2.6313 0.9511 2.6214 -0.5878 4.2515 0.0000 0.0016 0.5878 -4.2517 -0.9511 -2.6318 0.9511 -2.6219 -0.5878 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 93: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/93.jpg)
sen(8t) Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 -0.9511 2.63 -2.5013 0.9511 2.62 2.4924 -0.5878 4.25 -2.4985 0.0000 0.00 0.0006 0.5878 -4.25 -2.4987 -0.9511 -2.63 2.5018 0.9511 -2.62 -2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 -0.9511 2.63 -2.50113 0.9511 2.62 2.49214 -0.5878 4.25 -2.49815 0.0000 0.00 0.00016 0.5878 -4.25 -2.49817 -0.9511 -2.63 2.50118 0.9511 -2.62 -2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
t m[t]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 94: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/94.jpg)
t sen(8t) m[t] Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 -0.9511 2.63 -2.5013 0.9511 2.62 2.4924 -0.5878 4.25 -2.4985 0.0000 0.00 0.0006 0.5878 -4.25 -2.4987 -0.9511 -2.63 2.5018 0.9511 -2.62 -2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 -0.9511 2.63 -2.50113 0.9511 2.62 2.49214 -0.5878 4.25 -2.49815 0.0000 0.00 0.00016 0.5878 -4.25 -2.49817 -0.9511 -2.63 2.50118 0.9511 -2.62 -2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
La sumatoria de las áreas delos rectángulos equivale a
0 unidades.
![Page 95: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/95.jpg)
t sen(8t) m[t] Producto0 0.0000 0.00 0.0001 0.5878 4.25 2.4982 -0.9511 2.63 -2.5013 0.9511 2.62 2.4924 -0.5878 4.25 -2.4985 0.0000 0.00 0.0006 0.5878 -4.25 -2.4987 -0.9511 -2.63 2.5018 0.9511 -2.62 -2.4929 -0.5878 -4.25 2.498
10 0.0000 0.00 0.00011 0.5878 4.25 2.49812 -0.9511 2.63 -2.50113 0.9511 2.62 2.49214 -0.5878 4.25 -2.49815 0.0000 0.00 0.00016 0.5878 -4.25 -2.49817 -0.9511 -2.63 2.50118 0.9511 -2.62 -2.49219 -0.5878 -4.25 2.49820 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
Luego sen(8t)no aporta nadaa la señal original.
![Page 96: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/96.jpg)
¿Cuánto aporta n=5, sen(10t),para formar la señal?
![Page 97: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/97.jpg)
sen(10t)0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.0000
10 0.000011 0.000012 0.000013 0.000014 0.000015 0.000016 0.000017 0.000018 0.000019 0.000020 0.0000
t
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 98: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/98.jpg)
t sen(10t) m[t]0 0.0000 0.001 0.0000 4.252 0.0000 2.633 0.0000 2.624 0.0000 4.255 0.0000 0.006 0.0000 -4.257 0.0000 -2.638 0.0000 -2.629 0.0000 -4.25
10 0.0000 0.0011 0.0000 4.2512 0.0000 2.6313 0.0000 2.6214 0.0000 4.2515 0.0000 0.0016 0.0000 -4.2517 0.0000 -2.6318 0.0000 -2.6219 0.0000 -4.2520 0.0000 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 99: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/99.jpg)
sen(10t) Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
t m[t]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5
![Page 100: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/100.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5t sen(10t) m[t] Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
La sumatoria de las áreas delos rectángulos equivale a
0 unidades.
![Page 101: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/101.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-5
-4
-3
-2
-1
0
1
2
3
4
5t sen(10t) m[t] Producto0 0.0000 0.00 0.0001 0.0000 4.25 0.0002 0.0000 2.63 0.0003 0.0000 2.62 0.0004 0.0000 4.25 0.0005 0.0000 0.00 0.0006 0.0000 -4.25 0.0007 0.0000 -2.63 0.0008 0.0000 -2.62 0.0009 0.0000 -4.25 0.000
10 0.0000 0.00 0.00011 0.0000 4.25 0.00012 0.0000 2.63 0.00013 0.0000 2.62 0.00014 0.0000 4.25 0.00015 0.0000 0.00 0.00016 0.0000 -4.25 0.00017 0.0000 -2.63 0.00018 0.0000 -2.62 0.00019 0.0000 -4.25 0.00020 0.0000 0.00 0.000
Sumatoria: 0.000Normalizada: 0.000
Luego sen(10t) no aporta nadaa la señal original.
![Page 102: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/102.jpg)
¡Son demasiados cálculos!
![Page 103: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/103.jpg)
Se necesita crear un método abreviado
que requiera menos esfuerzo.
![Page 104: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/104.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 105: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/105.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
La Transformadade Fourier...
![Page 106: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/106.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
-2.63 -2.62 -2.63 -4.25 0.00 4.25 2.63 2.62 4.25 0.00
...de una ventana “m”de muestras...
![Page 107: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/107.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
n=12 Hz ...para la “n” de la
frecuencia cuyo aporte se quiere conocer...
![Page 108: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/108.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
10 muestras ...con un tamañode ventana de
“N” muestras...-2.63 -2.62 -2.63 -4.25 0.00 4.25 2.63 2.62 4.25 0.00
![Page 109: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/109.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
1/Fs...y cuyo periodo de
muestreo fue “T”...
![Page 110: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/110.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
...se calcula como...
![Page 111: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/111.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 112: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/112.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 113: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/113.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 114: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/114.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 115: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/115.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 116: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/116.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
![Page 117: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/117.jpg)
¡Difícil de entender!
![Page 118: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/118.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
?
![Page 119: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/119.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj
?
![Page 120: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/120.jpg)
Para facilitar el cálculo...
![Page 121: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/121.jpg)
aplicando micélebre fórmula...
![Page 122: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/122.jpg)
exj=cos(x)+ sen(x) j
![Page 123: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/123.jpg)
exj=cos(x)+ sen(x) j
![Page 124: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/124.jpg)
exj=cos(x)+ sen(x) j
![Page 125: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/125.jpg)
mi Transformadaqueda como...
exj=cos(x)+ sen(x) j
![Page 126: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/126.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 127: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/127.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 128: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/128.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 129: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/129.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 130: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/130.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 131: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/131.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 132: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/132.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 133: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/133.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
En vez de operar sobretodo el conjunto de datos,elegiremos una ventana “m” de tamaño “N”
![Page 134: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/134.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
Ventanade análisis
N = 10 muestrasm[0..9]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 135: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/135.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
la ventanase recorre a
m[1..10]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 136: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/136.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
m[2..11]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 137: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/137.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
m[3..12]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 138: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/138.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
m[4..13]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 139: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/139.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
m[5..14]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 140: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/140.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
m[6..15]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 141: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/141.jpg)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5
-4
-3
-2
-1
0
1
2
3
4
5
# de muestra
Inte
nsi
da
d
usaremos éstacomo ejemplo:
m[7..16]
Muestra Valor0 0.001 4.252 2.633 2.624 4.255 0.006 -4.257 -2.638 -2.629 -4.25
10 0.0011 4.2512 2.6313 2.6214 4.2515 0.0016 -4.2517 -2.63
![Page 142: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/142.jpg)
¿Cuánto aporta n=0, sen(0t),para formar la señal en la ventana?
![Page 143: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/143.jpg)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
n = 0N = 10T = 0.05
![Page 144: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/144.jpg)
n = 0N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 145: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/145.jpg)
n = 0N = 10T = 0.05
0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.0000
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 146: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/146.jpg)
n = 0N = 10T = 0.05
0 1.0000 0.00001 1.0000 0.00002 1.0000 0.00003 1.0000 0.00004 1.0000 0.00005 1.0000 0.00006 1.0000 0.00007 1.0000 0.00008 1.0000 0.00009 1.0000 0.0000
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 147: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/147.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 1.0000 0.00002 -4.25 1.0000 0.00003 0.00 1.0000 0.00004 4.25 1.0000 0.00005 2.63 1.0000 0.00006 2.62 1.0000 0.00007 4.25 1.0000 0.00008 0.00 1.0000 0.00009 -4.25 1.0000 0.0000
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 148: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/148.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 1.0000 -2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 1.0000 2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 1.0000 4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 1.0000 -4.2500 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 149: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/149.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 1.0000 -2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 1.0000 2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 1.0000 4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 1.0000 -4.2500 0.0000 0.0000
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 150: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/150.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 1.0000 -2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 1.0000 2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 1.0000 4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 1.0000 -4.2500 0.0000 0.0000
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 151: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/151.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 1.0000 -2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 1.0000 2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 1.0000 4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 1.0000 -4.2500 0.0000 0.0000
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 152: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/152.jpg)
n = 0N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 1.0000 -2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 1.0000 2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 1.0000 4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 1.0000 -4.2500 0.0000 0.0000
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0la señal sen(0t)
aportó 0 de amplituda la señal analizada
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 153: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/153.jpg)
¿Cuánto aporta n=1, sen(2t),para formar la señal en la ventana?
![Page 154: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/154.jpg)
n = 1N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 155: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/155.jpg)
n = 1N = 10T = 0.05
0 0.00001 -0.58782 -0.95113 -0.95114 -0.58785 0.00006 0.58787 0.95118 0.95119 0.5878
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 156: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/156.jpg)
n = 1N = 10T = 0.05
0 1.0000 0.00001 0.8090 -0.58782 0.3090 -0.95113 -0.3090 -0.95114 -0.8090 -0.58785 -1.0000 0.00006 -0.8090 0.58787 -0.3090 0.95118 0.3090 0.95119 0.8090 0.5878
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 157: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/157.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 0.8090 -0.58782 -4.25 0.3090 -0.95113 0.00 -0.3090 -0.95114 4.25 -0.8090 -0.58785 2.63 -1.0000 0.00006 2.62 -0.8090 0.58787 4.25 -0.3090 0.95118 0.00 0.3090 0.95119 -4.25 0.8090 0.5878
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 158: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/158.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.8090 -2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 -0.9511 4.04203 0.00 -0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 -0.5878 -2.49815 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 0.5878 1.54007 4.25 -0.3090 -1.3133 0.9511 4.04208 0.00 0.3090 0.0000 0.9511 0.00009 -4.25 0.8090 -3.4383 0.5878 -2.4981
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 159: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/159.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.8090 -2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 -0.9511 4.04203 0.00 -0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 -0.5878 -2.49815 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 0.5878 1.54007 4.25 -0.3090 -1.3133 0.9511 4.04208 0.00 0.3090 0.0000 0.9511 0.00009 -4.25 0.8090 -3.4383 0.5878 -2.4981
Suma: -19.0025 6.1678
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 160: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/160.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.8090 -2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 -0.9511 4.04203 0.00 -0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 -0.5878 -2.49815 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 0.5878 1.54007 4.25 -0.3090 -1.3133 0.9511 4.04208 0.00 0.3090 0.0000 0.9511 0.00009 -4.25 0.8090 -3.4383 0.5878 -2.4981
Suma: -19.0025 6.1678
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√−19.00252+ 6.16782≈20
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 161: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/161.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.8090 -2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 -0.9511 4.04203 0.00 -0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 -0.5878 -2.49815 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 0.5878 1.54007 4.25 -0.3090 -1.3133 0.9511 4.04208 0.00 0.3090 0.0000 0.9511 0.00009 -4.25 0.8090 -3.4383 0.5878 -2.4981
Suma: -19.0025 6.1678
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√−19.00252+ 6.16782≈20
110
.20=2
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 162: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/162.jpg)
n = 1N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.8090 -2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 -0.9511 4.04203 0.00 -0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 -0.5878 -2.49815 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 0.5878 1.54007 4.25 -0.3090 -1.3133 0.9511 4.04208 0.00 0.3090 0.0000 0.9511 0.00009 -4.25 0.8090 -3.4383 0.5878 -2.4981
Suma: -19.0025 6.1678
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√−19.00252+ 6.16782≈20
110
.20=2la señal sen(2t)
aportó 2 de amplituda la señal analizada
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 163: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/163.jpg)
¿Cuánto aporta n=2, sen(4t),para formar la señal en la ventana?
![Page 164: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/164.jpg)
n = 2N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 165: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/165.jpg)
n = 2N = 10T = 0.05
0 0.00001 -0.95112 -0.58783 0.58784 0.95115 0.00006 -0.95117 -0.58788 0.58789 0.9511
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 166: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/166.jpg)
n = 2N = 10T = 0.05
0 1.0000 0.00001 0.3090 -0.95112 -0.8090 -0.58783 -0.8090 0.58784 0.3090 0.95115 1.0000 0.00006 0.3090 -0.95117 -0.8090 -0.58788 -0.8090 0.58789 0.3090 0.9511
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 167: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/167.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 0.3090 -0.95112 -4.25 -0.8090 -0.58783 0.00 -0.8090 0.58784 4.25 0.3090 0.95115 2.63 1.0000 0.00006 2.62 0.3090 -0.95117 4.25 -0.8090 -0.58788 0.00 -0.8090 0.58789 -4.25 0.3090 0.9511
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 168: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/168.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.3090 -0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 -0.5878 2.49813 0.00 -0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 0.9511 4.04205 2.63 1.0000 2.6300 0.0000 0.00006 2.62 0.3090 0.8096 -0.9511 -2.49187 4.25 -0.8090 -3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 0.5878 0.00009 -4.25 0.3090 -1.3133 0.9511 -4.0420
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 169: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/169.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.3090 -0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 -0.5878 2.49813 0.00 -0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 0.9511 4.04205 2.63 1.0000 2.6300 0.0000 0.00006 2.62 0.3090 0.8096 -0.9511 -2.49187 4.25 -0.8090 -3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 0.5878 0.00009 -4.25 0.3090 -1.3133 0.9511 -4.0420
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 170: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/170.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.3090 -0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 -0.5878 2.49813 0.00 -0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 0.9511 4.04205 2.63 1.0000 2.6300 0.0000 0.00006 2.62 0.3090 0.8096 -0.9511 -2.49187 4.25 -0.8090 -3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 0.5878 0.00009 -4.25 0.3090 -1.3133 0.9511 -4.0420
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 171: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/171.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.3090 -0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 -0.5878 2.49813 0.00 -0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 0.9511 4.04205 2.63 1.0000 2.6300 0.0000 0.00006 2.62 0.3090 0.8096 -0.9511 -2.49187 4.25 -0.8090 -3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 0.5878 0.00009 -4.25 0.3090 -1.3133 0.9511 -4.0420
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√02+ 02=0
110
.0=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 172: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/172.jpg)
n = 2N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 0.3090 -0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 -0.5878 2.49813 0.00 -0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 0.9511 4.04205 2.63 1.0000 2.6300 0.0000 0.00006 2.62 0.3090 0.8096 -0.9511 -2.49187 4.25 -0.8090 -3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 0.5878 0.00009 -4.25 0.3090 -1.3133 0.9511 -4.0420
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√02+ 02=0
110
.0=0la señal sen(4t)
aportó 0 de amplituda la señal analizada
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 173: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/173.jpg)
¿Cuánto aporta n=3, sen(6t),para formar la señal en la ventana?
![Page 174: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/174.jpg)
n = 3N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 175: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/175.jpg)
n = 3N = 10T = 0.05
0 0.00001 -0.95112 0.58783 0.58784 -0.95115 0.00006 0.95117 -0.58788 -0.58789 0.9511
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 176: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/176.jpg)
n = 3N = 10T = 0.05
0 1.0000 0.00001 -0.3090 -0.95112 -0.8090 0.58783 0.8090 0.58784 0.3090 -0.95115 -1.0000 0.00006 0.3090 0.95117 0.8090 -0.58788 -0.8090 -0.58789 -0.3090 0.9511
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 177: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/177.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 -0.3090 -0.95112 -4.25 -0.8090 0.58783 0.00 0.8090 0.58784 4.25 0.3090 -0.95115 2.63 -1.0000 0.00006 2.62 0.3090 0.95117 4.25 0.8090 -0.58788 0.00 -0.8090 -0.58789 -4.25 -0.3090 0.9511
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 178: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/178.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.3090 0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 0.5878 -2.49813 0.00 0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 -0.9511 -4.04205 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 0.3090 0.8096 0.9511 2.49187 4.25 0.8090 3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 -0.5878 0.00009 -4.25 -0.3090 1.3133 0.9511 -4.0420
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 179: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/179.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.3090 0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 0.5878 -2.49813 0.00 0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 -0.9511 -4.04205 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 0.3090 0.8096 0.9511 2.49187 4.25 0.8090 3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 -0.5878 0.00009 -4.25 -0.3090 1.3133 0.9511 -4.0420
Suma: 5.8625 -8.0966
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 180: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/180.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.3090 0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 0.5878 -2.49813 0.00 0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 -0.9511 -4.04205 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 0.3090 0.8096 0.9511 2.49187 4.25 0.8090 3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 -0.5878 0.00009 -4.25 -0.3090 1.3133 0.9511 -4.0420
Suma: 5.8625 -8.0966
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√5.86252+ (−8.0966)
2≈10
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 181: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/181.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.3090 0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 0.5878 -2.49813 0.00 0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 -0.9511 -4.04205 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 0.3090 0.8096 0.9511 2.49187 4.25 0.8090 3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 -0.5878 0.00009 -4.25 -0.3090 1.3133 0.9511 -4.0420
Suma: 5.8625 -8.0966
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√5.86252+ (−8.0966)
2≈10
110
.10=1
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 182: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/182.jpg)
n = 3N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.3090 0.8096 -0.9511 2.49182 -4.25 -0.8090 3.4383 0.5878 -2.49813 0.00 0.8090 0.0000 0.5878 0.00004 4.25 0.3090 1.3133 -0.9511 -4.04205 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 0.3090 0.8096 0.9511 2.49187 4.25 0.8090 3.4383 -0.5878 -2.49818 0.00 -0.8090 0.0000 -0.5878 0.00009 -4.25 -0.3090 1.3133 0.9511 -4.0420
Suma: 5.8625 -8.0966
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√5.86252+ (−8.0966)
2≈10
110
.10=1la señal sen(6t)
aportó 1 de amplituda la señal analizada
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 183: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/183.jpg)
¿Cuánto aporta n=4, sen(8t),para formar la señal en la ventana?
![Page 184: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/184.jpg)
n = 4N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 185: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/185.jpg)
n = 4N = 10T = 0.05
0 0.00001 -0.58782 0.95113 -0.95114 0.58785 0.00006 -0.58787 0.95118 -0.95119 0.5878
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 186: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/186.jpg)
n = 4N = 10T = 0.05
0 1.0000 0.00001 -0.8090 -0.58782 0.3090 0.95113 0.3090 -0.95114 -0.8090 0.58785 1.0000 0.00006 -0.8090 -0.58787 0.3090 0.95118 0.3090 -0.95119 -0.8090 0.5878
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 187: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/187.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 -0.8090 -0.58782 -4.25 0.3090 0.95113 0.00 0.3090 -0.95114 4.25 -0.8090 0.58785 2.63 1.0000 0.00006 2.62 -0.8090 -0.58787 4.25 0.3090 0.95118 0.00 0.3090 -0.95119 -4.25 -0.8090 0.5878
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 188: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/188.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.8090 2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 0.9511 -4.04203 0.00 0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 0.5878 2.49815 2.63 1.0000 2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 -0.5878 -1.54007 4.25 0.3090 1.3133 0.9511 4.04208 0.00 0.3090 0.0000 -0.9511 0.00009 -4.25 -0.8090 3.4383 0.5878 -2.4981
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 189: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/189.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.8090 2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 0.9511 -4.04203 0.00 0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 0.5878 2.49815 2.63 1.0000 2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 -0.5878 -1.54007 4.25 0.3090 1.3133 0.9511 4.04208 0.00 0.3090 0.0000 -0.9511 0.00009 -4.25 -0.8090 3.4383 0.5878 -2.4981
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 190: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/190.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.8090 2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 0.9511 -4.04203 0.00 0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 0.5878 2.49815 2.63 1.0000 2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 -0.5878 -1.54007 4.25 0.3090 1.3133 0.9511 4.04208 0.00 0.3090 0.0000 -0.9511 0.00009 -4.25 -0.8090 3.4383 0.5878 -2.4981
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 191: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/191.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.8090 2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 0.9511 -4.04203 0.00 0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 0.5878 2.49815 2.63 1.0000 2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 -0.5878 -1.54007 4.25 0.3090 1.3133 0.9511 4.04208 0.00 0.3090 0.0000 -0.9511 0.00009 -4.25 -0.8090 3.4383 0.5878 -2.4981
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 192: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/192.jpg)
n = 4N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -0.8090 2.1196 -0.5878 1.54002 -4.25 0.3090 -1.3133 0.9511 -4.04203 0.00 0.3090 0.0000 -0.9511 0.00004 4.25 -0.8090 -3.4383 0.5878 2.49815 2.63 1.0000 2.6300 0.0000 0.00006 2.62 -0.8090 -2.1196 -0.5878 -1.54007 4.25 0.3090 1.3133 0.9511 4.04208 0.00 0.3090 0.0000 -0.9511 0.00009 -4.25 -0.8090 3.4383 0.5878 -2.4981
Suma: 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0la señal sen(8t)
aportó 0 de amplituda la señal analizada
√02+ 02=0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 193: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/193.jpg)
¿Cuánto aporta n=5, sen(10t),para formar la señal en la ventana?
![Page 194: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/194.jpg)
n = 5N = 10T = 0.05
0123456789
k
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 195: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/195.jpg)
n = 5N = 10T = 0.05
0 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 0.0000
k sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 196: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/196.jpg)
n = 5N = 10T = 0.05
0 1.0000 0.00001 -1.0000 0.00002 1.0000 0.00003 -1.0000 0.00004 1.0000 0.00005 -1.0000 0.00006 1.0000 0.00007 -1.0000 0.00008 1.0000 0.00009 -1.0000 0.0000
k cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 197: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/197.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 0.00001 -2.62 -1.0000 0.00002 -4.25 1.0000 0.00003 0.00 -1.0000 0.00004 4.25 1.0000 0.00005 2.63 -1.0000 0.00006 2.62 1.0000 0.00007 4.25 -1.0000 0.00008 0.00 1.0000 0.00009 -4.25 -1.0000 0.0000
k m[kT] cos(-2πnk/N) sen(-2πnk/N)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 198: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/198.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -1.0000 2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 -1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 -1.0000 -4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 -1.0000 4.2500 0.0000 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 199: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/199.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -1.0000 2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 -1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 -1.0000 -4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 -1.0000 4.2500 0.0000 0.0000
Suma: -0.0200 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 200: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/200.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -1.0000 2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 -1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 -1.0000 -4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 -1.0000 4.2500 0.0000 0.0000
Suma: -0.0200 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
√−0.02002+ 0.00002≈0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 201: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/201.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -1.0000 2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 -1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 -1.0000 -4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 -1.0000 4.2500 0.0000 0.0000
Suma: -0.0200 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0
√−0.02002+ 0.00002≈0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 202: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/202.jpg)
n = 5N = 10T = 0.05
0 -2.63 1.0000 -2.6300 0.0000 0.00001 -2.62 -1.0000 2.6200 0.0000 0.00002 -4.25 1.0000 -4.2500 0.0000 0.00003 0.00 -1.0000 0.0000 0.0000 0.00004 4.25 1.0000 4.2500 0.0000 0.00005 2.63 -1.0000 -2.6300 0.0000 0.00006 2.62 1.0000 2.6200 0.0000 0.00007 4.25 -1.0000 -4.2500 0.0000 0.00008 0.00 1.0000 0.0000 0.0000 0.00009 -4.25 -1.0000 4.2500 0.0000 0.0000
Suma: -0.0200 0.0000
k m[kT] cos(-2πnk/N) m[kT]*cos(...) sen(-2πnk/N) m[kT]*sen(...)
110
.0=0la señal sen(10t)
aportó 0 de amplituda la señal analizada
√−0.02002+ 0.00002≈0
1N∑k=0
N−1
m [kT ](cos(−2π nkN
)+ sen(−2π nkN
) j)
![Page 203: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/203.jpg)
Reuniendo todas las componentespodemos expresar analíticamente que...
![Page 204: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/204.jpg)
Señal (t )=2.sen(2t)+ 1.sen (6t)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-5.00
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
4.00
5.00
![Page 205: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/205.jpg)
Si representamos los resultadosen un Ecualizador Gráfico...
![Page 206: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/206.jpg)
![Page 207: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/207.jpg)
y = 2*sen(2t)
![Page 208: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/208.jpg)
y = 1*sen(6t)
![Page 209: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/209.jpg)
Pero, ¿cómo y por quéfunciona la Transformada de Fourier?
![Page 210: Cómo funciona la Transformada de Fourier](https://reader033.vdocuments.site/reader033/viewer/2022052904/55800abbd8b42a693c8b504f/html5/thumbnails/210.jpg)
Fourier (m,n ,N ,T )=1N ∑
k=0
N−1
m [kT ]e−2πn k
Nj