{ "metadata": { "name": "Decibelios" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Decibelios" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Definici\u00f3n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "El decibelio (dB) es una *unidad relativa*, es decir, no absoluta o de comparaci\u00f3n que expresa la relaci\u00f3n entre *dos magnitudes*:\n", "\n", "+ la de estudio y\n", "+ la de comparaci\u00f3n\n", "\n", "El decibelio es una unidad:\n", "\n", "+ logar\u00edtmica, \n", "+ adimensional y \n", "+ escalar\n", "\n", "Un belio equivale a 10 decibelios.\n", "\n", "Como es una unidad logar\u00edtmica en base 10, si 0 belios es la magnitud de referencia, entonces 10 decibelios expresa un aumento de *10* veces la potencia y 20 decibelios expresa un aumento de __100__ veces la potencia. Esto es, una diferencia de 40dB es una diferencia absoluta de 10000 unidades de medida ($10^4$).\n", "\n", "Una tabla de valores ser\u00eda:\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
$10 log X$$X$
4010000
301000
20100
1010
01
-100.1
-200.01
-300.001
-400.0001
\n", "\n" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Telecomunicaciones" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "En telecomunicaciones no hay un nivel fijo o de referencia. Usamos el decibelio para *comparar* dos cantidades ,posiblemente muy diferentes, y por ello usamos la escala logar\u00edtmica.\n", "\n", "Por ejemplo la ganancia, en decibelios, de un dispositivo es:\n", "\n", "$dB = 10 log_{10} \\frac{P_S}{P_E}$\n", "\n", "donde $P_E$ es la potencia de entrada y $P_S$ es la de salida. Si $dB>0$ el sistema es amplificador y si $dB<0$ es atenuador." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Suma de decibelios" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Como hemos dicho, al ser el decibelio una unidad relativa, no podemos sumar sin m\u00e1s dos magnitudes en estas unidades ya que podr\u00edan referirse a or\u00edgenes relativos distintos. Por ejemplo, si tenemos dos medidas de sonido de 21dB el total no son 42db si no 24dB. La forma de sumar magnitudes en decibelios es transformarlas a las unidades originales, sumar y luego volver a pasar a decibelios o bien aplicar:\n", "\n", "$dBTotales=10log_{10}(10^{\\frac{X_1}{10}}+10^{\\frac{X_2}{10}}+...)$\n", "\n", "Donde los $X_n$ son los diferentes valores a sumar en dB." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Algunos ejemplos" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "10*math.log10(10./5.)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 9, "text": [ "3.010299956639812" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Vemos que una diferencia en magnitud del doble son *3dB*" ] }, { "cell_type": "code", "collapsed": false, "input": [ "10*math.log10(5./10.)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 10, "text": [ "-3.010299956639812" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Del mismo modo, una diferencia en magnitud de la mitad son *-3dB*" ] }, { "cell_type": "code", "collapsed": false, "input": [ "10*math.log10(30./10.)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 12, "text": [ "4.771212547196624" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "... pero una diferencia del triple son **4.7dB**, no *6db*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Una antena que recibe 1W y produce una se\u00f1al de 100W tiene una ganancia, en dB, de " ] }, { "cell_type": "code", "collapsed": false, "input": [ "10*math.log10(100./1.)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 22, "text": [ "20.0" ] } ], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Y ahora, a ver si se dibujar (primera prueba) la curva $log_{10}$\n", "Notar que, al igual que matlab, lo que hacemos es samplear la curva y pasarle los puntos a la funci\u00f3n plot.\n", "El ejemplo lo he sacado de [aqui](https://github.com/calebmadrigal/FourierTalkOSCON)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "import scipy, math\n", "import matplotlib.pyplot as plt\n", "\n", "# Graphing helper function\n", "def setup_graph(title='', x_label='', y_label='', fig_size=None):\n", " fig = plt.figure()\n", " if fig_size != None:\n", " fig.set_size_inches(fig_size[0], fig_size[1])\n", " ax = fig.add_subplot(111)\n", " ax.set_title(title)\n", " ax.set_xlabel(x_label)\n", " ax.set_ylabel(y_label)\n", "\n", "freq = 1 #hz - cycles per second\n", "amplitude = 10\n", "time_to_plot = 2 # second\n", "sample_rate = 100 # samples per second\n", "num_samples = sample_rate * time_to_plot\n", "\n", "#afeixit per Pere per evitar representar log(0)\n", "time_origin=0.01\n", "\n", "t = scipy.linspace(time_origin, time_to_plot, num_samples)\n", "#signal = [amplitude * math.sin(freq * i * 2*math.pi) for i in t] # Explain the 2*pi\n", "\"\"\"\n", "ver que signal es una lista de\n", "num_samples\n", "con las muestras de la funcion a samplear \n", "\"\"\"\n", "signal = [amplitude * math.log10(i) for i in t]\n", "\n", "setup_graph(x_label='', y_label='', title='log10')\n", "plot(t, signal)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 19, "text": [ "[]" ] }, { "output_type": "display_data", "png": 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} ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Fuente: Wikipedia y otros\n", "Autor: Pere Vil\u00e1s 2013" ] } ], "metadata": {} } ] }