AL_103.000_Transformando__Impulsos.ipynb

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    "# Transformando Impulsos \n",
    "\n",
    "## Transformaciones  $T:\\mathbb{R}^n \\rightarrow \\mathbb{R}^m$\n",
    "\n",
    "### Ejemplo:\n",
    "Sea \n",
    "$$T \\left( \\begin{bmatrix}u_0 \\\\ u_1 \\\\ u_2 \\end{bmatrix} \\right) =\\begin{bmatrix}u_0 + u_1\\\\ u_1 + u_2  \\end{bmatrix} $$\n",
    "\n",
    "El dominio de $T$ es $\\mathbb{R}^3$ y el codominio de $T$ es $\\mathbb{R}^2$.\n",
    "\n",
    "$$T \\left( \\begin{bmatrix}1 \\\\ 0 \\\\ 0 \\end{bmatrix} \\right) =\\begin{bmatrix}1 \\\\ 0  \\end{bmatrix},\n",
    "\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n",
    "T \\left( \\begin{bmatrix}0 \\\\ 1 \\\\ 0 \\end{bmatrix} \\right) =\\begin{bmatrix}1 \\\\ 1  \\end{bmatrix},\n",
    "\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n",
    "T \\left( \\begin{bmatrix}0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\right) =\\begin{bmatrix}0 \\\\ 1  \\end{bmatrix} $$\n",
    "\n",
    "Con los anteriores valores se puede volver a escribir la transformación.\n",
    "\n",
    "$\\begin{align}\n",
    "T\\left( \\begin{bmatrix}u_0 \\\\ u_1 \\\\ u_2 \\end{bmatrix} \\right) \n",
    "&=u_0 T \\left( \\begin{bmatrix}1 \\\\ 0 \\\\ 0 \\end{bmatrix} \\right)\n",
    "+ u_1 T \\left( \\begin{bmatrix}0 \\\\ 1 \\\\ 0 \\end{bmatrix} \\right)\n",
    "+ u_2 T \\left( \\begin{bmatrix}0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\right)\\\\\n",
    "&=u_0 \\begin{bmatrix}1 \\\\ 0  \\end{bmatrix}\n",
    "+ u_1 \\begin{bmatrix}1 \\\\ 1  \\end{bmatrix}\n",
    "+ u_2 \\begin{bmatrix}0 \\\\ 1  \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}u_0 \\\\ 0  \\end{bmatrix}\n",
    "+  \\begin{bmatrix}u_1 \\\\ u_1  \\end{bmatrix}\n",
    "+  \\begin{bmatrix}0 \\\\ u_2  \\end{bmatrix}\\\\\n",
    "&= \\begin{bmatrix}u_0 + u_1 \\\\ u_1 + u_2  \\end{bmatrix}\n",
    "\\end{align}$\n",
    "\n",
    "Las transformaciones de la forma  $T:\\mathbb{R}^n \\rightarrow \\mathbb{R}^m$ se pueden escribir como una multiplicación matricial. En donde las columnas de la identidad transformadas, forman la matriz de tamaño $m \\times n$. En el caso del ejemplo, la transformación se puede escribir de la siguiente forma.\n",
    "$$T(u)=Au=\\begin{bmatrix} T \\left( \\begin{bmatrix}1 \\\\ 0 \\\\ 0 \\end{bmatrix} \\right) \n",
    "& T \\left( \\begin{bmatrix}0 \\\\ 1 \\\\ 0 \\end{bmatrix} \\right) \n",
    "& T \\left( \\begin{bmatrix}0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\right)  \\end{bmatrix} \\begin{bmatrix}u_0 \\\\ u_1 \\\\ u_2 \\end{bmatrix}\n",
    "=\\begin{bmatrix} 1 &1 & 0\\\\0 & 1 & 1\\end{bmatrix} \\begin{bmatrix}u_0 \\\\ u_1 \\\\ u_2 \\end{bmatrix}  $$\n",
    "\n",
    "## Transformaciones $T:\\mathcal{l}^2 \\rightarrow \\mathcal{l}^2$\n",
    "\n",
    "Por cada $i \\in \\mathbb{Z}$ se define el vector __impulso unitario__ $\\delta_i[k]$ del espacio vectorial de $\\mathbb{R}^\\mathbb{Z}$  \n",
    "\n",
    "$$ \\delta_i[k] = \\left\\{ \\begin{aligned}\n",
    "1 \\text{ si } k = i\\\\\n",
    "0 \\text{ si } k \\neq i\\\\\n",
    "\\end{aligned} \\right. $$\n",
    "\n",
    "\n",
    "los impulsos unitarios en $\\mathbb{R}^\\mathbb{Z}$ hecen las veces de $\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}$ y $\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$ en  $\\mathbb{R}^2$. \n",
    "\n",
    "El especificar una trasnformacion $T(\\delta_i[k])$ para cada $i \\in \\mathbb{Z}$ se puede reducir a especificar solamente  $T(\\delta_0[k])$s si la transformación $T:\\mathcal{l}^2 \\rightarrow \\mathcal{l}^2$ es __invariante en el tiempo__, es decir\n",
    "$$\\text{si } y[k]=T(\\delta_0[k]) \\text{ entonces } T(\\delta_i[k])=y[k-i]$$\n",
    "\n",
    "Estas transformaciones usualmente se describen mediante la transformada Z.\n",
    "\n",
    "Ejemplo:\n",
    "Supongamos una transformación invariante en el tiempo, tal que $$T(\\delta_0[k])=\\mu_0[k]$$ Donde   \n",
    "\n",
    "$$ \\mu_0[k] \\left\\{ \\begin{aligned}\n",
    "1 \\text{ si } k \\geq 0\\\\\n",
    "0 \\text{ si } k < 0\\\\\n",
    "\\end{aligned} \\right. $$\n",
    "\n",
    "Esta transformación se representa mediante su transformada Z.\n",
    "$$\\mathcal{Z}\\left(\\mu_0[k]\\right)=\n",
    "\\frac{z}{z-1}$$\n",
    "\n",
    "Si pudieramos representar en una columna los valores $\\begin{bmatrix} \\vdots \\\\ u[0] \\\\ u[1] \\\\ u[2] \\\\ u[3] \\\\ \\vdots \\end{bmatrix}$ la transformación invariante en el tiempo haría los siguiente.\n",
    "\n",
    "$$\\begin{matrix}\n",
    "T\\left(\\begin{bmatrix}\\vdots \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix}\\right)=\\begin{bmatrix}\\vdots \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ \\vdots \\end{bmatrix} \n",
    "&T\\left(\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ \\vdots \\end{bmatrix}\\right)=\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ \\vdots \\end{bmatrix} \n",
    "&T\\left(\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ \\vdots \\end{bmatrix}\\right)=\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 1 \\\\ \\vdots \\end{bmatrix} \n",
    "&T\\left(\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ \\vdots \\end{bmatrix}\\right)=\\begin{bmatrix}\\vdots \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ \\vdots \\end{bmatrix} \n",
    "\\end{matrix}$$\n",
    "\n",
    "## Transformaciones $T:\\mathcal{L}^2 \\rightarrow \\mathcal{L}^2$\n",
    "\n",
    "\n",
    "La __'función' impulso__ o __delta de Kronequer__ $\\delta(t)$ que se define en el espacio conrinuuo $\\mathbb{R}^\\mathbb{R}$, cumple las siguientes propiedades:\n",
    "\\begin{align}\n",
    "& \\text{ Si } t \\neq 0 \\text{ entonces } \\delta(t)=0\\\\\n",
    "& \\int_{-\\infty}^{\\infty} \\delta(t) dt=1\n",
    "\\end{align}\n",
    "\n",
    "Una transfoemación $T:\\mathcal{L}^2 \\rightarrow \\mathcal{L}^2$ es __invariante en el tiempo__ si $$\\text{si } y(t)=T(\\delta(t)) \\text{ entonces } T(\\delta(t- \\tau))=y(t-\\tau)$$\n",
    "\n",
    "\n",
    "Para definir una transformación invariante en el tiempo es suficiemte definir la transformación para el delta de Kronequer.\n",
    "\n",
    "Estas transformaciones usualmente se describen mediante la transformada de Laplace.\n",
    "\n",
    "Ejemplo:\n",
    "Supongamos una transformación invariante en el tiempo, tal que $$T(\\delta(t))=\\mu(t)$$ Donde   \n",
    "\n",
    "$$ \\mu(t) \\left\\{ \\begin{aligned}\n",
    "1 \\text{ si } t \\geq 0\\\\\n",
    "0 \\text{ si } t < 0\\\\\n",
    "\\end{aligned} \\right. $$\n",
    "\n",
    "Esta transformación se representa mediante su transformada Laplace.\n",
    "$$\\mathcal{L}\\left(\\mu(t)\\right)=\n",
    "\\frac{1}{s}$$\n"
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