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Ecuación de Matriz para la DTFS

Module by: Roy Ha. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Matrix Equation for the DTFS by Roy Ha

Summary: Este modulo ve como se escribe la ecuación de matriz para la DTFS para hacer los calculos y mostrar las bases mas facilmente.

La DTFS es nada mas un cambio de bases en CN N . Para comenzar, tenemos f⁢n f n en términos de la base estándar.

f⁢n=f⁢0⁢ e 0 +f⁢1⁢ e 1 +…+f⁢N−1⁢ e N - 1 =∑k=0n−1f⁢k⁢δ⁢k−n f n f 0 e 0 f 1 e 1 … f N 1 e N - 1 k 0 n 1 f k δ k n
(1)
f⁢0f⁢1f⁢2⋮f⁢N−1=f⁢000⋮0+0f⁢10⋮0+00f⁢2⋮0+…+000⋮f⁢N−1 f 0 f 1 f 2 ⋮ f N 1 f 0 0 0 ⋮ 0 0 f 1 0 ⋮ 0 0 0 f 2 ⋮ 0 … 0 0 0 ⋮ f N 1
(2)
Tomando la DTFS, podemos escribir f⁢n f n en términos de la base de Fourier senosoidal
f⁢n=∑k=0N−1 c k ⁢ei⁢2⁢πN⁢k⁢n f n k 0 N 1 c k 2 N k n
(3)
f⁢0f⁢1f⁢2⋮f⁢N−1= c 0 ⁢111⋮1+ c 1 ⁢1ei⁢2⁢πNei⁢4⁢πN⋮ei⁢2⁢πN⁢(N−1)+ c 2 ⁢1ei⁢4⁢πNei⁢8⁢πN⋮ei⁢4⁢πN⁢(N−1)+… f 0 f 1 f 2 ⋮ f N 1 c 0 1 1 1 ⋮ 1 c 1 1 2 N 4 N ⋮ 2 N N 1 c 2 1 4 N 8 N ⋮ 4 N N 1 …
(4)
Podemos formar la matriz base (llamaremos esto WW envés deBB) al acomodar los vectores bases en las columnas obtenemos
W=( b 0 ⁢n b 1 ⁢n… b N - 1 ⁢n )=( 111…1 1ei⁢2⁢πNei⁢4⁢πN…ei⁢2⁢πN⁢(N−1) 1ei⁢4⁢πNei⁢8⁢πN…ei⁢2⁢πN⁢2⁢(N−1) ⋮⋮⋮⋮⋮ 1ei⁢2⁢πN⁢(N−1)ei⁢2⁢πN⁢2⁢(N−1)…ei⁢2⁢πN⁢(N−1)⁢(N−1) ) W b 0 n b 1 n … b N - 1 n 1 1 1 … 1 1 2 N 4 N … 2 N N 1 1 4 N 8 N … 2 N 2 N 1 ⋮ ⋮ ⋮ ⋮ ⋮ 1 2 N N 1 2 N 2 N 1 … 2 N N 1 N 1
(5)
con b k ⁢n=ei⁢2⁢πN⁢k⁢n b k n 2 N k n

note:

la entrada k-th fila y n-th columna es W j , k =ei⁢2⁢πN⁢k⁢n= W n , k W j , k 2 N k n W n , k
Así, aquí tenemos una simetría adicional (W=WT)⇒(WT¯=W¯=1N⁢W-1) W W W W 1 N W (ya que b k ⁢n b k n son ortonormales)

Ahora podemos rescribir la ecuación DTFS en forma de matriz, donde tenemos:

  • f f = señal (vector en CN N )
  • c c = coeficientes DTFS (vector en CN N )

Tabla 1
"synthesis" f=W⁢c f W c f⁢n=〈c, b n ¯〉 f n c b n
"analysis" c=WT¯⁢f=W¯⁢f c W f W f c⁢k=〈f, b k 〉 c k f b k

Encontrar (e invertir) la DFTS es nada mas una multiplicación de matrices.

Todo lo que se encuentra en CN N esta limpio: no se utilizan límites, no se usan preguntas de convergencia, nada mas se utilice aritmética de matrices.

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