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Una introducción a las propiedades generales de las series de Fourier.

Empezaremos por refrescar su memoria sobre las ecuaciones básicas de las series de Fourier :

f t n c n ω 0 n t
c n 1 T t 0 T f t ω 0 n t
Deje · describen la transformación de f t a los coeficientes de Fourier f t n n c n · grafica funciones con valores complejos a secuencias de números complejos .


· es una transformación linear .

Si f t c n y g t d n . Entonces α α α f t α c n y f t g t c n d n

Muy fácil, nada mas es la linealidad del integral.

f t g t n n t 0 T f t g t ω 0 n t n n 1 T t 0 T f t ω 0 n t 1 T t 0 T g t ω 0 n t n n c n d n c n d n


Desplazamiento en el tiempo es igual a un desplazamiento angular de los coeficientes de Fourier

f t t 0 ω 0 n t 0 c n si c n c n c n , entonces ω 0 n t 0 c n ω 0 n t 0 c n c n ω 0 t 0 n c n ω 0 t 0 n

f t t 0 n n 1 T t 0 T f t t 0 ω 0 n t n n 1 T t t 0 T t 0 f t t 0 ω 0 n t t 0 ω 0 n t 0 n n 1 T t t 0 T t 0 f t ~ ω 0 n t ~ ω 0 n t 0 n n ω 0 n t ~ c n

La relaciÓN de parseval

t 0 T f t 2 T n c n 2
La relación de Parseval nos permite calcular la engría de la señal de sus series de Fourier.
Parseval nos dice que las series de Fourier grafican maps L 0 T 2 a l 2 .

¿Pará f t poder tener“energía finita,”que es lo que c n hace cuando n ?

c n 2 para f t tener energía finita.

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¿Sí n n 0 c n 1 n , es f L 0 T 2 ?

Si, por que c n 2 1 n 2 , la cual se puede sumar.

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Ahora ,¿sí n n 0 c n 1 n , es f L 0 T 2 ?

No, por que c n 2 1 n , la cual no se puede sumar.

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El radio de descomposición de una serie de fourier determina si f t tiene energía finita .

DiferenciaciÓN en el dominio de fourier

f t c n t f t n ω 0 c n

Ya que

f t n c n ω 0 n t
t f t n c n t ω 0 n t n c n ω 0 n ω 0 n t
Un diferenciador atenúa las frecuencias bajas f t y acentúa las frecuencias altas. Remueve rasgos generales y acentúaáreas con variaciones básicas.
Una manera común para medir matemáticamente que la suavidad de la función f t es el ver cuantas derivadas tienen energía finita.
Esto se hace al observar los coeficientes de fourier de una señal, específicamente el que tan rápido se descomponen cuando n .Si f t c n y c n tiene la forma 1 n k , entonces t m f t n ω 0 m c n tiene la forma n m n k .Entonces para que la m th derivada tenga energía finita, necesitamos n n m n k 2 por lo tanto n m n k se descompone mas rápido que 1 n lo cual implica que 2 k 2 m 1 o k 2 m 1 2 El radio de descomposición de las series de fourier determina la suavidad.

IntegraciÓN en el dominio de fourier


f t c n
τ t f τ 1 ω 0 n c n
Si c 0 0 , esta expresión no tiene ningún sentido.

Integración acentúa frecuencias bajas y atenúa frecuencias altas. Integradores muestran las cosas generales de las señales y suprimen variaciones de corto plazo (lo cual es ruido en muchos casos). Integradores son mejores que diferenciadores.

MultiplicaciÓN de seÑAles

Dado a una señal f t con coeficientes de Fourier c n y una señal g t con coeficientes d n , podemos definir una nueva señal como, y t , donde y t f t g t . Descubrimos que la representación de series de Fourier de y t , e n , es tal que e n k c k d n - k . Esto es para decir que la multiplicación de señales en el dominio del tiempo es equivalente a la convolución discreta en el dominio de la frecuencia. La prueba es la siguiente

e n 1 T t 0 T f t g t ω 0 n t 1 T t 0 T k c k ω 0 k t g t ω 0 n t k c k 1 T t 0 T g t ω 0 n k t k c k d n - k

Questions & Answers

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Señales y sistemas. OpenStax CNX. Sep 28, 2006 Download for free at http://cnx.org/content/col10373/1.2
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