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Par de la Transformada de Fourier Discreta en el Tiempo

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

Based on: Discrete-Time Fourier Transform Pair by Don Johnson

Summary: Calculando frecuencias discretas en el tiempo utilizando las transformadas de Fourier

Cuando obtenemos una señal discreta a muestrear una señal análoga, la frecuencia Nyquist corresponde a la frecuencia discreta 12 1 2 . Para demostrar esto, note que un senosoidal en la frecuencia Nyquist 12⁢ T s 1 2 T s tiene una forma muestreada que iguala

Senosoidal en la Frecuencia Nyquist de 1/2T

cos2⁢π×12⁢ T s ⁢n⁢ T s =cosπ⁢n=−1n 2 1 2 T s n T s n 1 n
(1)

El exponencial en la DTFT en la frecuencia 12 1 2 igual e−(j⁢2⁢π⁢n)2=e−(j⁢π⁢n)=−1n 2 n 2 n 1 n , lo que significa que la correspondencia entre una frecuencia análoga y una frecuencia discreta es establecida:

Relación para la Frecuencia Discreta en el Tiempo Análogo

f D = f A ⁢ T s f D f A T s
(2)

onde f D f D y f A f A representan las variables de frecuencia análoga y frecuencia discreta, respectivamente. La figura de aliasing provee otra manera para derivar este resultado. Conforme la duración de cada punto en la señal de muestreo periódica p T s ⁢t p T s t se hace mas pequeña, las amplitudes de las repeticiones del espectro de la señal, que son gobernadas por los coeficientes de la series de Fourier de p T s ⁢t p T s t , se vuelve iguales. 1 Así, el espectro muestrario de la señal se convierte periódico con periodo 1 T s 1 T s . Así la frecuencia Nyquist 12⁢ T s 1 2 T s corresponde a la frecuencia 12 1 2 .

La transformada inversa de Fourier discreta en el tiempo se deriva fácilmente en la siguiente relación:

∫−1212e−(j⁢2⁢π⁢f⁢m)⁢ej⁢π⁢f⁢ndf={1  if  m=n0  if  m≠n 1 2 1 2 f 2 f m f n 1 m n 0 m n
(3)

Así como encontramos que

∫−1212S⁢ej⁢2⁢π⁢f⁢ej⁢2⁢π⁢f⁢ndf=∫−1212∑mms⁢m⁢e−(j⁢2⁢π⁢f⁢m)⁢ej⁢2⁢π⁢f⁢ndf=∑mms⁢m⁢∫−1212e(−(j⁢2⁢π⁢f))⁢(m−n)df=s⁢n f 1 2 1 2 S 2 f 2 f n f 1 2 1 2 m m s m 2 f m 2 f n m m s m f 1 2 1 2 2 f m n s n
(4)

Los pares para la transformada de Fourier discretos en el tiempo son

Pares de la Transformada de Fourier en Tiempo Discreto

S⁢ej⁢2⁢π⁢f=∑nns⁢n⁢e−(j⁢2⁢π⁢f⁢n) S 2 f n n s n 2 f n
(5)

Pares de la Transformada de Fourier en Tiempo Discreto

s⁢n=∫−1212S⁢ej⁢2⁢π⁢f⁢ej⁢2⁢π⁢f⁢ndf s n f 1 2 1 2 S 2 f 2 f n
(6)

Footnotes

  1. Examinar la señal de pulso periodica revela que cuando Δ Δ se hace pequeña, el valor de c 0 c 0 , el coeficiente mas grande de Fourier, decae al valor cero: | c 0 |=A⁢ΔT c 0 A Δ T . Así, para mantener un teorema de muestreo viable en términos matemáticos, la amplitud A A debe incrementar a 1Δ 1 Δ , convirtiéndose infinitamente grande conformela duración del pulso disminuye. Sistemas prácticos usan un valor pequeño de Δ Δ , digamos 0.1⁢ T s 0.1 T s y usan amplificadores para reescalar la señal.

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