{"id":92,"date":"2017-02-07T09:25:23","date_gmt":"2017-02-07T14:25:23","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/fiec05058\/?p=92"},"modified":"2026-01-30T06:42:20","modified_gmt":"2026-01-30T11:42:20","slug":"senales-escalon-e-impulso","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/ss-u01\/senales-escalon-e-impulso\/","title":{"rendered":"1.6 Se\u00f1ales Escal\u00f3n unitario \u03bc(t) e Impulso unitario \u03b4(t)"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<div class=\"wp-block-group has-medium-font-size is-layout-flex wp-block-group-is-layout-flex\">\n<p><a href=\"#escalonunitario\">escal\u00f3n<\/a> unitario<\/p>\n\n\n\n<p><a href=\"#impulsounitario\">impulso<\/a> unitario<\/p>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<p><strong><em>Referencia<\/em><\/strong>: Lathi 1.4.1 p83, Hsu 1.3 p6, Oppenheim 1.4 p30<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"escalonunitario\">1. Escal\u00f3n unitario \u03bc(t)<\/h2>\n\n\n\n<p>Muchos de los temas de curso usan sistemas causales que inician en t=0. La funci\u00f3n que describe \u00e9ste comportamiento es escal\u00f3n unitario \u03bc(t) descrita como:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\mu(t)= \\begin{cases} 1, &amp; t\\geq 0 \\\\ 0, &amp; t &lt;0\\end{cases} <\/span>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"553\" height=\"415\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/02\/senalmodelo02.png\" alt=\"se\u00f1al escal\u00f3n gr\u00e1fica\" class=\"wp-image-18218\" \/><\/figure>\n\n\n\n<p>Algunas se\u00f1ales pueden convenientemente ser descritas usando t\u00e9rminos de una funci\u00f3n \u03bc(t) como se muestra en los siguientes ejemplos.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Se\u00f1ales Escal\u00f3n e Impulso unitario, gr\u00e1ficas con Python\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Fec39sVGEwo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Algoritmo en Python<\/h3>\n\n\n\n<p>La funci\u00f3n \u03bc(t) se puede describir con la <code>np.piecewise()<\/code>o <code>np.heaviside(t,1)<\/code><\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# Se\u00f1ales modelo varias\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# INGRESO\n#u = lambda t: np.piecewise(t,t&gt;=0,&#x5B;1,0])\nu = lambda t: np.heaviside(t,1)\n\na = -10\nb = 10\ndt = 0.1\n\n# PROCEDIMIENTO\nti  = np.arange(a, b, dt)\nu0_i = u(ti)\n\n# SALIDA - GRAFICA\nprint('t:',ti)\nprint('t:',u0_i)\n\n# grafica\nplt.figure(1)\nplt.plot(ti,u0_i)\n\nplt.xlabel('t')\nplt.ylabel('escalon u(t)')\nplt.margins(dt)\nplt.grid()\nplt.show()\n<\/pre><\/div>\n\n\n<h2 class=\"wp-block-heading\">1.1 Se\u00f1al Causal con Escal\u00f3n unitario<\/h2>\n\n\n\n<p>Para convertir una se\u00f1al cualquiera a causal, que inicie en t=0, se multiplica la se\u00f1al por \u03bc(t).<\/p>\n\n\n\n<p>Por ejemplo, x(t)=e<sup>-\u03b1t<\/sup> se puede convertir a una se\u00f1al causal si se escribe como:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> x(t) = e^{-\\alpha t} \\mu(t) <\/span>\n\n\n\n<p>Para mostrar lo indicado y considerando \u03b1=1 tenemos que:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"554\" height=\"416\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/02\/senalmodelo03.png\" alt=\"se\u00f1al modelo causal gr\u00e1fica \" class=\"wp-image-18219\" \/><\/figure>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# se\u00f1al a causal\nx = lambda t: np.exp(-t)*u(t)\nxi = x(ti)\n\n# SALIDA - GRAFICA\nplt.figure(2)\nplt.plot(ti,xi)\n\nplt.xlabel('t')\nplt.ylabel('x(t)u(t)')\nplt.margins(dt)\nplt.grid()\nplt.show()\n<\/pre><\/div>\n\n\n<p>La funci\u00f3n escal\u00f3n unitario tambi\u00e9n permite realizar descripciones matem\u00e1ticas sobre diferentes segmentos del tiempo.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">1.2 Se\u00f1al Rectangular<\/h2>\n\n\n\n<p>Por ejemplo, una se\u00f1al rectangular se puede representar como la suma de dos se\u00f1ales \u03bc(t) desplazadas:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> x(t) = \\mu (t-2) - \\mu (t-4) <\/span>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"553\" height=\"416\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/02\/senalmodelo04.png\" alt=\"Se\u00f1al rectangular gr\u00e1fica\" class=\"wp-image-18221\" \/><\/figure>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# Rectangular como u(t-2)-u(t-4)\n# u2 = lambda t: np.piecewise(t,t&gt;=2,&#x5B;1,0])\n# u4 = lambda t: np.piecewise(t,t&gt;=4,&#x5B;1,0])\n\nu2 = lambda t: u(t-2)\nu4 = lambda t: u(t-4)\nrectangular = lambda t:u2(t) - u4(t)\nrect_i = rectangular(ti)\n\n# SALIDA - GRAFICA\nplt.figure(3)\nplt.plot(ti,rect_i)\n\nplt.xlabel('t')\nplt.ylabel('rectangular(t)')\nplt.margins(dt)\nplt.grid()\nplt.show()\n<\/pre><\/div>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<div class=\"wp-block-group has-medium-font-size is-layout-flex wp-block-group-is-layout-flex\">\n<p><a href=\"#escalonunitario\">escal\u00f3n<\/a> unitario<\/p>\n\n\n\n<p><a href=\"#impulsounitario\">impulso<\/a> unitario<\/p>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"impulsounitario\">2. Impulso unitario \u03b4(t)<\/h2>\n\n\n\n<p>El impulso unitario \u03b4(t) definida primero por Paul Dirac de la forma:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\delta(t)=0,t\\neq 0 <\/span>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\int_{-\\infty}^{\\infty} \\delta(t) dt = 1 <\/span>\n\n\n\n<p>Se puede ver al impulso como un pulso rectangular muy peque\u00f1o de \u00e1rea unitaria. El ancho del pulso rectangular es muy peque\u00f1o.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"552\" height=\"412\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/02\/senalmodelo05.png\" alt=\"se\u00f1al modelo05 impulso unitario\" class=\"wp-image-18223\" \/><\/figure>\n\n\n\n<p>Siguiendo el concepto, se puede representar como:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Algoritmo en Python<\/h3>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; title: ; notranslate\" title=\"\">\n# Se\u00f1ales impulso unitario\nimport numpy as np\nimport matplotlib.pyplot as plt\n\n# INGRESO\n# u = lambda t: np.piecewise(t,t&gt;=0,&#x5B;1,0])\nu = lambda t: np.heaviside(t,1)\n\ndt = 0.1  # tama\u00f1o de paso\nt0 = 0    # punto de impulso\n\nimpulso = lambda t,dt: u(t)-u(t-dt)\n\n# intervalo de observaci\u00f3n &#x5B;a,b]\na = -10\nb = 10\n\n# PROCEDIMIENTO\nti  = np.arange(a, b, dt)\nimpulso_i = impulso(ti,dt)\n\n# SALIDA - GRAFICA\n# plt.plot(ti,impulso_i)\nplt.stem(ti,impulso_i)\nplt.xlabel('t')\nplt.ylabel('impulso()')\nplt.grid()\nplt.show()\n<\/pre><\/div>\n\n\n<p>Otra forma de representar el impulso unitario es:<\/p>\n\n\n<span class=\"wp-katex-eq katex-display\" data-display=\"true\"> \\delta(t)= \\begin{cases} 1, &amp; t = 0 \\\\ 0, &amp; t \\neq 0\\end{cases} <\/span>\n\n\n\n<p>que como instrucci\u00f3n es,<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>t0 = 0  # punto de impulso\ntolera = 1e-10 # casicero\nimpulso = lambda t,t0,tolera: 1*(np.abs(t-t0)&lt;=tolera)<\/code><\/pre>\n\n\n\n<p>Para la gr\u00e1fica tambi\u00e9n se usa <code>stem(x,y)<\/code>:<\/p>\n\n\n<div class=\"wp-block-syntaxhighlighter-code \"><pre class=\"brush: python; gutter: false; title: ; notranslate\" title=\"\">\n# SALIDA - GRAFICA\nplt.figure(5)\nplt.stem(ti,impulso_i)\n\nplt.margins(dt)\nplt.xlabel('t')\nplt.ylabel('impulso(t)')\nplt.grid()\nplt.show()\n<\/pre><\/div>\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"554\" height=\"415\" src=\"http:\/\/blog.espol.edu.ec\/algoritmos101\/files\/2017\/02\/senalmodelo06.png\" alt=\"se\u00f1al impulso unitario en cero, discreto\" class=\"wp-image-18224\" \/><\/figure>\n\n\n\n<p>Tambi\u00e9n se puede usar la librer\u00eda Scipy para tener un impulso m\u00e1s general:<\/p>\n\n\n\n<p><a href=\"https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.signal.unit_impulse.html\">https:\/\/docs.scipy.org\/doc\/scipy\/reference\/generated\/scipy.signal.unit_impulse.html<\/a><\/p>\n\n\n\n<p>hay que importar la librer\u00eda Scipy para ese caso.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n\n\n\n<div class=\"wp-block-group has-medium-font-size is-layout-flex wp-block-group-is-layout-flex\">\n<p><a href=\"#escalonunitario\">escal\u00f3n<\/a> unitario<\/p>\n\n\n\n<p><a href=\"#impulsonunitario\">impulso<\/a> unitario<\/p>\n<\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\n","protected":false},"excerpt":{"rendered":"<p>escal\u00f3n unitario impulso unitario Referencia: Lathi 1.4.1 p83, Hsu 1.3 p6, Oppenheim 1.4 p30 1. Escal\u00f3n unitario \u03bc(t) Muchos de los temas de curso usan sistemas causales que inician en t=0. La funci\u00f3n que describe \u00e9ste comportamiento es escal\u00f3n unitario \u03bc(t) descrita como: Algunas se\u00f1ales pueden convenientemente ser descritas usando t\u00e9rminos de una funci\u00f3n \u03bc(t) [&hellip;]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-ss-unidades","format":"standard","meta":{"footnotes":""},"categories":[165],"tags":[],"class_list":["post-92","post","type-post","status-publish","format-standard","hentry","category-ss-u01"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/92","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/users\/8043"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/comments?post=92"}],"version-history":[{"count":9,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/92\/revisions"}],"predecessor-version":[{"id":21237,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/92\/revisions\/21237"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=92"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=92"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=92"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}