{"id":1772,"date":"2016-10-21T11:43:52","date_gmt":"2016-10-21T16:43:52","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/estg1003\/?p=1772"},"modified":"2026-04-05T16:43:25","modified_gmt":"2026-04-05T21:43:25","slug":"s3eva2017tii_t3-autocorrelacion-ampm","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/stp-ejemplos\/s3eva2017tii_t3-autocorrelacion-ampm\/","title":{"rendered":"s3Eva2017TII_T3 Autocorrelacion AM\/PM"},"content":{"rendered":"\n

Ejercicio<\/strong>:3Eva2017TII_T3 Autocorrelacion AM\/PM<\/a><\/p>\n\n\n\n X(t) = A_1 \\cos (\\omega _0 t + \\theta _1) + A_2 \\cos (\\sqrt{2}\\omega_0 t + \\theta _2) <\/span>\n\n\n\n

Las variables aleatorias \u0398 son uniformes en [0,2\u03c0], por lo que sus funciones de densidad de probabilidad son iguales a:<\/p>\n\n\n\n f_{\\theta _1} (\\theta _1) = f_{\\theta _2}(\\theta _2) = \\frac{1}{2\\pi} <\/span>\n\n\n\n

a) valor esperado:<\/p>\n\n\n E[X(t)] = E[A_1 \\cos (\\omega _0 t + \\theta _1) + A_2 \\cos (\\sqrt{2}\\omega_0 t + \\theta _2) ] <\/span>\n\n\n =E[A_1 \\cos (\\omega _0 t + \\theta _1)] + E[A_2 \\cos (\\sqrt{2}\\omega_0 t + \\theta _2) ] <\/span>\n\n\n\n

las variables A1<\/sub> y \u03981<\/sub> son independientes, de la misma forma A2<\/sub> y \u03982<\/sub>, lo que permite separar: E[XY] = E[X][Y]<\/p>\n\n\n =E[A_1]E[\\cos (\\omega _0 t + \\theta _1)] +E[A_2]E[\\cos (\\sqrt{2}\\omega_0 t + \\theta _2)] <\/span>\n\n\n\n

y evaluar:<\/p>\n\n\n E[\\cos (\\omega _0 t + \\theta _1)] = \\int_{0}^{2\\pi} \\cos (\\omega _0 t + \\theta _1) \\frac{1}{2\\pi} d\\theta _1 <\/span>\n\n\n E[g(x)] = \\int_{0}^{2\\pi} g(x) f(x) dx <\/span>\n\n\n = \\frac{1}{2\\pi}\\int_{0}^{2\\pi} \\cos (\\omega _0 t + \\theta _1) d\\theta _1 <\/span>\n\n\n\n

que aplicando:<\/p>\n\n\n cos(x \\pm y)= cos(x)cos(y) \\mp sen(x)sen(y) <\/span>\n\n\n = \\frac{1}{2\\pi}\\int_{0}^{2\\pi} \\Big[\\cos (\\omega _0 t) \\cos(\\theta _1) - \\sin(\\omega _0 t)\\sin(\\theta _1)\\Big] d\\theta _1 <\/span>\n\n\n = \\frac{1}{2\\pi}\\cos (\\omega _0 t) \\int_{0}^{2\\pi} \\cos(\\theta _1) d\\theta _1 - \\frac{1}{2\\pi}\\sin(\\omega _0 t) \\int_{0}^{2\\pi} \\sin(\\theta _1) d\\theta _1 <\/span>\n\n\n = \\frac{1}{2\\pi}\\cos (\\omega _0 t) \\sin(\\theta _1)\\Big|_{0}^{2\\pi} - \\frac{1}{2\\pi}\\sin(\\omega _0 t) [-\\cos(\\theta _1)] \\Big|_{0}^{2\\pi} <\/span>\n\n\n = \\frac{1}{2\\pi}\\cos (\\omega _0 t) [\\sin(2\\pi)-sin(0)] + \\frac{1}{2\\pi}\\sin(\\omega _0 t) [\\cos(2\\pi) - \\cos(0)] <\/span>\n\n\n E[\\cos (\\omega _0 t + \\theta _1)] = 0 <\/span>\n\n\n\n

por lo que, aplicando lo mismo para el siguiente t\u00e9rmino del integral del valor esperado, se obtiene que:<\/p>\n\n\n E[X(t)] = 0 <\/span>\n\n\n\n

\n\n\n\n

b) Para el caso de autocorrelaci\u00f3n se tiene que:<\/p>\n\n\n\n E[X(t_1)X(t_2)] = <\/span>\n\n\n\n = E\\Big[ [A_1 \\cos (\\omega _0 t_1 + \\theta _1) + A_2 \\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2)] . <\/span>\n\n\n . [A_1 \\cos (\\omega _0 t_2 + \\theta _1) + A_2 \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] \\Big] <\/span>\n\n\n\n

multiplicando t\u00e9rmino a t\u00e9rmino<\/p>\n\n\n = E[A_1^2 \\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\omega _0 t_2 + \\theta _1)] + <\/span>\n\n\n E[A_1 A_2 \\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] + <\/span>\n\n\n E[A_2 A_1 \\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2) \\cos (\\omega _0 t_2 + \\theta _1)] + <\/span>\n\n\n E[A_2^2 \\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] = <\/span>\n\n\n\n

recordando que, las variables A1<\/sub> y \u03981<\/sub> son independientes, de la misma forma A2<\/sub> y \u03982<\/sub>, lo que permite separar: E[XY] = E[X][Y]<\/p>\n\n\n = E[A_1^2] E[\\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\omega _0 t_2 + \\theta _1)] + <\/span>\n\n\n E[A_1 A_2] E[\\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] + <\/span>\n\n\n E[A_2 A_1] E[\\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2) \\cos (\\omega _0 t_2 + \\theta _1)] + <\/span>\n\n\n E[A_2^2] E[\\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] = <\/span>\n\n\n\n

Se desarrolla para una parte del segundo t\u00e9rmino de la suma:
E[\\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] = <\/span><\/p>\n\n\n\n

que siendo independiente \u03981<\/sub> y \u03982<\/sub> E[XY] = E[X][Y]<\/p>\n\n\n = E[\\cos (\\omega _0 t_1 + \\theta _1)] E[\\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] = 0 <\/span>\n\n\n\n

cuyo primer t\u00e9rmino es cero, usando el resultado del literal a, y extendido el resultado para el segundo t\u00e9rmino, lo que reduce la expresi\u00f3n de la autocorrelaci\u00f3n a:<\/p>\n\n\n E[X(t_1)X(t_2)] = <\/span>\n\n\n = E[A_1^2] E[\\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\omega _0 t_2 + \\theta _1)] + <\/span>\n\n\n + E[A_2^2] E[\\cos (\\sqrt{2}\\omega_0 t_1 + \\theta _2) \\cos (\\sqrt{2}\\omega_0 t_2 + \\theta _2)] <\/span>\n\n\n\n

para el primer t\u00e9rmino:<\/p>\n\n\n E[\\cos (\\omega _0 t_1 + \\theta _1) \\cos (\\omega _0 t_2 + \\theta _1)] = <\/span>\n\n\n =\\frac{1}{2}E\\Big[\\cos [(\\omega _0 t_1 + \\theta _1) - (\\omega _0 t_2 + \\theta _1)] + \\cos [(\\omega _0 t_1 + \\theta _1) +(\\omega _0 t_2 + \\theta _1)\\Big] <\/span>\n\n\n =\\frac{1}{2}E\\Big[\\cos (\\omega _0 (t_1 - t_2)) + \\cos (\\omega _0 (t_1 + t_2) + 2 \\theta _1)\\Big] <\/span>\n\n\n =\\frac{1}{2}E\\Big[\\cos (\\omega _0 (t_1 - t_2)) \\Big] + E\\Big[\\cos (\\omega _0 (t_1 + t_2) + 2 \\theta _1)\\Big] <\/span>\n\n\n\n

del literal a, el segundo t\u00e9rmino se vuelve cero:<\/p>\n\n\n =\\frac{1}{2}E[\\cos (\\omega _0 (t_1 - t_2)) ] <\/span>\n\n\n\n

extendiendo el resultado para el segundo t\u00e9rmino de la autocorrelaci\u00f3n:<\/p>\n\n\n E[X(t_1)X(t_2)] = \\frac{1}{2} E\\Big[A_1^2\\Big] E\\Big[\\cos (\\omega _0 (t_1 - t_2)) \\Big] + <\/span>\n\n\n + \\frac{1}{2} E\\Big[A_2^2\\Big] E\\Big[\\cos (\\sqrt{2}\\omega _0 (t_1 - t_2)) \\Big] <\/span>\n\n\n\n

que se resume que la media es cero, constante, y que la autocorrelaci\u00f3n depende solo de la diferencias de tiempo |t1<\/sub>-t2<\/sub>| = \u03c4, por lo que el proceso se clasifica como estacionario en el sentido amplio WSS.<\/p>\n\n\n\n

Recuerde que A1<\/sub> y A2<\/sub> son tipo Gaussianas, por lo que E[A1<\/sub>2<\/sup>] es una constante.<\/p>\n","protected":false},"excerpt":{"rendered":"

Ejercicio:3Eva2017TII_T3 Autocorrelacion AM\/PM Las variables aleatorias \u0398 son uniformes en [0,2\u03c0], por lo que sus funciones de densidad de probabilidad son iguales a: a) valor esperado: las variables A1 y \u03981 son independientes, de la misma forma A2 y \u03982, lo que permite separar: E[XY] = E[X][Y] y evaluar: que aplicando: por lo que, aplicando […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-stp-ejercicios","format":"standard","meta":{"footnotes":""},"categories":[203],"tags":[58,237],"class_list":["post-1772","post","type-post","status-publish","format-standard","hentry","category-stp-ejemplos","tag-ejemplos-python","tag-pestocasticos"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1772","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=1772"}],"version-history":[{"count":4,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1772\/revisions"}],"predecessor-version":[{"id":23547,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/1772\/revisions\/23547"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=1772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=1772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=1772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}