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Ejemplos de DTFT

Module by: Don Johnson. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: DTFT Examples by Don Johnson

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Summary: Como calcular las transformadas de Fourier discreta en el tiempo para secuencias que disminuyen.

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Ejemplo 1

Calculemos la transformada de Fourier en tiempo discreto para la secuencia del exponencial decadente sn=anun s n a n u n , donde un u n la secuencia del Escalón unitario. Al remplazar la expresión de la señal la formula de la transformada de Fourier.

Formula de la Transformada de Fourier

Sⅇⅈ2πf=∑n=-∞∞anunⅇ-ⅈ2πfn=∑n=0∞aⅇ-ⅈ2πfn S 2 f n a n u n 2 f n n 0 a 2 f n (1)

La suma es un caso especial de series geométricas.

Series Geométricas

∀α,|α|<1:∑n=0∞αn=11−α α α 1 n 0 α n 1 1 α (2)
Así, por mientras que |a|<1 a 1 , tenemos nuestra transformada de Fourier.
Sⅇⅈ2πf=11−aⅇ-ⅈ2πf S 2 f 1 1 a 2 f (3)

Usando la relación de Euler, podemos expresar la magnitud y el ángulo de este espectro.

|Sⅇⅈ2πf|=11−acos2πf2+a2sin22πf S 2 f 1 1 a 2 f 2 a 2 2 f 2 (4)
∠Sⅇⅈ2πf=-arctanasin2πf1−acos2πf S 2 f a 2 f 1 a 2 f (5)

No importa que valor de aa escojamos, las formulas anteriores demuestran claramente la naturaleza periódica del espectro de señales discretas en el tiempo. figura 1 muestra como el espectro es un función periódica. Tan solo tenemos que considerar el espectro entre -12 1 2 y 12 1 2 para definirla unambiguosamente. Cuando a>0 a 0 , tenemos un espectro de pasa bajas – el espectro desaparece cuando la frecuencia incrementa de 0 0 a 12 1 2 — con una a a incrementa nos lleva a un contenido mayor de frecuencias bajas; para a<0 a 0 ,tenemos un espectro de pasa altas. (figura 2).

Figura 1: El espectro de la señal exponencial ( a=0.5 a 0.5 ) es mostrado sobre el rango de frecuencias -22 -2 2 , claramente demostrando la periodicidad de toda la espectra discreta en el tiempo. EL ángulo tiene las unidades en grados.
Figura 1 (spectrum10.png)
Figura 2: El espectro de varias señales exponenciales es mostrado aquí . ¿Cual es la relación aparente entre el espectro de a=0.5 a 0.5 y a=-0.5 a -0.5 ?
Figura 2 (spectrum11.png)

Ejemplo 2

Análogo a una señal de pulso análogo encontramos el espectro de la secuencia de pulso de tamaño- N N pulse sequence.

sn=1if0≤n≤N−10otherwise s n 1 0 n N 1 0 (6)

La transformada de Fourier de esta secuencia tiene la forma de una serie geométrica truncada.

Sⅇⅈ2πf=∑n=0N−1ⅇ-ⅈ2πfn S 2 f n 0 N 1 2 f n (7)

Para las llamadas series geométricas finitas, sabemos que

Series Geométricas Finitas

∑n= n 0 N+ n 0 −1αn=α n 0 1−αN1−α n n 0 N n 0 1 α n α n 0 1 α N 1 α (8)
para todos los valores de α α .

Exercise 1

Derive esta formula para la formula de series geométrica finitas. El “truco” es el considerar la diferencia entre la suma de las series y la suma de las series multiplicada por α α .

Solution

α∑n= n 0 N+ n 0 −1αn−∑n= n 0 N+ n 0 −1αn=αN+ n 0 −α n 0 α n n 0 N n 0 1 α n n n 0 N n 0 1 α n α N n 0 α n 0 (9)
la cual, después de algunas manipulaciones, da la formula de la suma geométrica.

Aplicando este resultado da (figura 3.)

Sⅇⅈ2πf=1−ⅇ-ⅈ2πfN1−ⅇ-ⅈ2πf=ⅇ-ⅈπfN−1sinπfNsinπf S 2 f 1 2 f N 1 2 f f N 1 f N f (10)

El radio de las funciones de seno tiene la forma genérica de sinNxsinx N x x , que es mejor conocida como la función de sinc discreta , dsincx dsinc x . Por lo tanto, nuestra transformada puede ser expresada como Sⅇⅈ2πf=ⅇ-ⅈπfN−1dsincπf S 2 f f N 1 dsinc f . El espectro del pulso discreto contiene muchas ondulaciones, las cuales el numero incrementa con N N , la duración del pulso.

Figura 3: El espectro para un pulse de tamaño-diez es mostrado aquí. ¿Puedé usted explicar la apariencia complicada que el ángulo tiene?
Figura 3 (spectrum12.png)

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