# 11.1 Transformación discreta de fourier

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Este modulo cubre los fundamentos de las Transformada Discreta de Fourier.

## N-puinto punto transformada discreta de fourier (dft)

$X(k)=\sum_{n=0}^{N-1} x(n)e^{-i\frac{2\pi }{n}kn}\forall k, k=\{0, \dots , N-1\}$
$x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi }{n}kn}\forall n, n=\{0, \dots , N-1\}$

Note que:

• $X(k)$ es la DTFT evaluado en $\omega =\frac{2\pi }{N}k\forall k, k=\{0, \dots , N-1\}$
• Completar con ceros $x(n)$ a $M$ muestras antes de sacar el DFT, da como resultado una versión muestreada de $M$ -puntos uniformes del DTFT :
$X(e^{i\frac{2\pi }{M}k})=\sum_{n=0}^{N-1} x(n)e^{-i\frac{2\pi }{M}k}$
$X(e^{i\frac{2\pi }{M}k})=\sum_{n=0}^{N-1} {x}_{\mathrm{zp}}(n)e^{-i\frac{2\pi }{M}k}$ $X(e^{i\frac{2\pi }{M}k})={X}_{\mathrm{zp}}(k)\forall k, k=\{0, \dots , M-1\}$
• La $N$ -pt DFT es suficiente para reconstruir toda la DTFT de una secuencia de $N$ -pt:
$X(e^{i\omega })=\sum_{n=0}^{N-1} x(n)e^{-i\omega n}$
$X(e^{i\omega })=\sum_{n=0}^{N-1} \frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi }{N}kn}e^{-i\omega n}$ $X(e^{i\omega })=\sum_{k=0}^{N-1} X(k)\frac{1}{N}\sum_{k=0}^{N-1} e^{-i(\omega -\frac{2\pi }{N}k)n}$ $X(e^{i\omega })=\sum_{k=0}^{N-1} X(k)\frac{1}{N}\frac{\sin \left(\frac{\omega N-2\pi k}{2}\right)}{\sin \left(\frac{\omega N-2\pi k}{2N}\right)}e^{-i(\omega -\frac{2\pi }{N}k)\frac{N-1}{2}}$

• DFT tiene una representación en forma de matriz muy conveniente. Definiendo ${W}_{N}=e^{-i\frac{2\pi }{N}}$ ,
$\begin{pmatrix}X(0)\\ X(1)\\ ⋮\\ X(N-1)\\ \end{pmatrix}=\begin{pmatrix}{W}_{N}^{0} & {W}_{N}^{0} & {W}_{N}^{0} & {W}_{N}^{0} & \dots \\ {W}_{N}^{0} & {W}_{N}^{1} & {W}_{N}^{2} & {W}_{N}^{3} & \dots \\ {W}_{N}^{0} & {W}_{N}^{2} & {W}_{N}^{4} & {W}_{N}^{6} & \dots \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮\\ \end{pmatrix}\begin{pmatrix}x(0)\\ x(1)\\ ⋮\\ x(N-1)\\ \end{pmatrix}$
donde $X=W(x)$ respectivamente. $W$ tiene las siguientes propiedades:
• $W$ es Vandermonde: La $n$ th columna de $W$ es un polinomio en ${W}_{N}^{n}$
• $W$ es simetrico: $W=W^T$
• $\frac{1}{\sqrt{N}}W$ es unitaria: $\frac{1}{\sqrt{N}}W(\frac{1}{\sqrt{N}}W)^{H}=(\frac{1}{\sqrt{N}}W)^{H}\frac{1}{\sqrt{N}}W=I$
• $\frac{1}{N}\overline{W}=W^{-1}$ , es la matriz y DFT.
• • Para $N$ un poder de 2, la FFT se puede usar para calcular la DFT usando $\frac{N}{2}\log_{2}N$ en vez de $N^{2}$ operaciones.

$N$ $\frac{N}{2}\log_{2}N$ $N^{2}$
16 32 256
64 192 4096
256 1024 65536
1024 5120 1048576

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