{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# TRANSFORMADA Z\n", "## SISTEMAS Y SEÑALES \n", "### Ingenieria de Telecomunicaciones \n", "### Universidad Pontificia Bolivariana \n", "### Por: Jose R. Zapata - [https://joserzapata.github.io/](https://joserzapata.github.io/) \n", "**joser.zapata@upb.edu.co**" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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Contenido

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" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Importar librerias basicas\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import sympy as sym\n", "\n", "%matplotlib inline\n", "plt.style.use('bmh') # estilo de las graficas\n", "from IPython.display import Latex # para visualizar ecuaciones en jupyter" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Definición\n", "la transformada $Z$ es una transformación que representa una señal discreta $x[k]$ en el dominio espectral. Se basa en la función exponencial compleja $z^{-k}$ con $z \\in \\mathbb{C}$ como señal base.\n", "\n", "## Transformada Bilateral Z\n", "\n", "$$\n", "\\large X(z) = \\sum_{k = -\\infty}^{\\infty} x[k] \\, z^{-k}\n", "$$\n", "\n", "## Transformada Unilateral Z\n", "Para señales causales tenemos:\n", "$$\n", "\\large \\boxed{X(z) = \\sum_{k = 0}^{\\infty} x[k] \\, z^{-k}}\n", "$$\n", "\n", "## Transformada Z inversa \n", "$$\n", "\\large \\boxed{x[k] = \\frac{1}{2 \\pi j} \\oint_{C} X(z) \\, z^{k - 1} \\; dz}\n", "$$\n", "\n", "\n", "# Región de Convergencia\n", "La transformada Z converge si es absolutamente sumable\n", "\n", "$$\n", " \\sum_{k = -\\infty}^{\\infty} | x[k] \\cdot z^{- k} | = \\sum_{k = -\\infty}^{\\infty} | x[k] | \\cdot | z |^{- k} < \\infty\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Propiedades\n", "\n", "|  | $x[k]$ | $X(z) = \\mathcal{Z} \\{ x[k] \\}$ | Region de convergencia (ROC) |\n", "|:---|:---:|:---:|:---|\n", "| Linealidad | $A \\, x_1[k] + B \\, x_2[k]$ | $A \\, X_1(z) + B \\, X_2(z)$ | $\\supseteq \\text{ROC}\\{x_1[k]\\} \\cap \\text{ROC}\\{x_2[k]\\}$ |\n", "| Conjugacion | $x^*[k]$ | $X^*(z^*)$ | $\\text{ROC}\\{ x[k] \\}$ |\n", "| Señales Reales | $x[k] = x^*[k]$ | $X(z) = X^*(z^*)$ | |\n", "| Convolucion Lineal | $x[k] * h[k]$ | $X(z) \\cdot H(z)$ | $\\supseteq \\text{ROC}\\{x[k]\\} \\cap \\text{ROC}\\{h[k]\\}$ |\n", "| Desplazamiento en el tiempo | $x[k - \\kappa]$ | $z^{- \\kappa} \\cdot X(z)$ | $\\supseteq \\text{ROC}\\{x[k]\\} \\setminus \\{0, \\infty \\}$ |\n", "| Modulacion | $z_0^k \\cdot x[k]$ | $X\\left( \\frac{z}{z_0} \\right)$ | $\\{z: \\frac{z}{z_0} \\in \\text{ROC} \\{ x[k] \\} \\}$ |\n", "| Inversion | $x[-k]$ | $X \\left( \\frac{1}{z} \\right)$ | $\\{z: \\frac{1}{z} \\in \\text{ROC} \\{ x[k] \\} \\}$ |\n", " \n", "Donde $A, B, z_0 \\in \\mathbb{C}$ y $\\kappa \\in \\mathbb{Z}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Tabla de Transformadas Básicas\n", "\n", "| $x[k]$ | $X(z) = \\mathcal{Z} \\{ x[k] \\}$ | ROC |\n", "|:---:|:---:|:---|\n", "| $\\delta[k]$ | $1$ | $\\mathbb{C}$ |\n", "| $\\epsilon[k]$ | $\\frac{z}{z-1}$ | $|z| > 1$ |\n", "| $k \\epsilon[k]$ | $\\frac{z}{(z-1)^2}$ | $|z| > 1$ |\n", "| $z_0^{k} \\epsilon[k]$ | $\\frac{z}{z - z_0}$ | $|z| > |z_0|$ |\n", "| $-z_0^{k} \\epsilon[-k-1]$ | $\\frac{z}{z - z_0}$ | $|z| < |z_0|$ |\n", "| $\\sin(\\Omega_0 k) \\epsilon[k]$ | $\\frac{z \\sin(\\Omega_0)}{z^2 - 2 z \\cos(\\Omega_0) + 1}$ | $|z| > 1$ |\n", "| $\\cos(\\Omega_0 k) \\epsilon[k]$ | $\\frac{z ( z - \\cos(\\Omega_0))}{z^2 - 2 z \\cos(\\Omega_0) + 1}$ | $|z| > 1$ |\n", "\n", "Donde $z_0 \\in \\mathbb{C}$, $\\Omega_0 \\in \\mathbb{R}$ y $n \\in \\mathbb{N}$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# REFERENCIAS\n", "- https://nbviewer.jupyter.org/github/spatialaudio/signals-and-systems-lecture/blob/master/z_transform/definition.ipynb\n", "- https://nbviewer.jupyter.org/github/spatialaudio/signals-and-systems-lecture/blob/master/z_transform/properties.ipynb\n", "- https://nbviewer.jupyter.org/github/spatialaudio/signals-and-systems-lecture/blob/master/z_transform/theorems.ipynb\n", "- https://nbviewer.jupyter.org/github/spatialaudio/signals-and-systems-lecture/blob/master/z_transform/table_theorems_transforms.ipynb" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Phd. Jose R. Zapata**\n", "- [https://joserzapata.github.io/](https://joserzapata.github.io/)\n", "- https://twitter.com/joserzapata\n", "" ] } ], "metadata": { "hide_input": false, "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Contenido", "title_sidebar": "Contenido", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": false }, "varInspector": { "cols": { "lenName": 16, "lenType": 16, "lenVar": 40 }, "kernels_config": { "python": { "delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())" }, "r": { "delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) " } }, "types_to_exclude": [ "module", "function", "builtin_function_or_method", "instance", "_Feature" ], "window_display": false } }, "nbformat": 4, "nbformat_minor": 2 }