{"id":2763,"date":"2017-12-02T08:05:06","date_gmt":"2017-12-02T13:05:06","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/telg1001\/?p=2763"},"modified":"2026-04-05T21:32:07","modified_gmt":"2026-04-06T02:32:07","slug":"s2eva2009tii_t2-lti-dt-hnyn-determine-xn","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/ss-s2eva\/s2eva2009tii_t2-lti-dt-hnyn-determine-xn\/","title":{"rendered":"s2Eva2009TII_T2 LTI DT Dado h[n], y[n] determine x[n]"},"content":{"rendered":"\n

Ejercicio<\/strong>: 2Eva2009TII_T2 LTI DT Dado h[n], y[n] determine X[n]<\/a><\/p>\n\n\n\n

literal a. Expresi\u00f3n de x[n]<\/h2>\n\n\n\n

Con los datos de h[n] y y[n], se obtienen las transformadas H[z] y Y[z],<\/p>\n\n\n h[n] = 2 \\Big( \\frac{1}{3} \\Big)^n \\mu [n-1] = 2 \\Big( \\frac{1}{3} \\Big)^1\\Big( \\frac{1}{3} \\Big)^{-1}\\Big( \\frac{1}{3} \\Big)^n \\mu [n-1] <\/span>\n\n\n h[n] = \\frac{2}{3} \\Big( \\frac{1}{3} \\Big)^{n-1} \\mu [n-1] <\/span>\n\n\n\n

la transformada z<\/a> es:<\/p>\n\n\n H[z] = \\frac{2}{3} \\frac{1}{z-\\frac{1}{3}} <\/span>\n\n\n\n

Para la se\u00f1al de salida Yc[z] conocida<\/strong>:<\/p>\n\n\n y_c[n] = (-2)^{n} \\mu [n-1] = (-2)^1(-2)^{n-1} \\mu [n-1] <\/span>\n\n\n y_c[n] = -2 (-2)^{n-1} \\mu [n-1] <\/span>\n\n\n\n

la transformada z<\/a> es:<\/p>\n\n\n Y_c[z] = -2\\frac{1}{z+2}<\/span>\n\n\n\n

La se\u00f1al de entrada x[n]<\/p>\n\n\n x[n] = a\\delta [n] + b (c)^{n-1} \\mu [n-1] <\/span>\n\n\n X[z] = a + b \\frac{1}{z-c} <\/span>\n\n\n\n

La se\u00f1al de salida  Y[z] esperada<\/strong> en dominio z se obtiene como Y[z] = H[z]X[z], la expresi\u00f3n se escribe como:<\/p>\n\n\n y_e [z] = \\Bigg[\\frac{2}{3} \\frac{1}{z-\\frac{1}{3}}\\Bigg] \\Bigg[a + b \\frac{1}{z-c}\\Bigg]<\/span>\n\n\n = \\frac{2}{3} \\Bigg[\\frac{a}{z-\\frac{1}{3}} + \\frac{b}{(z-c)(z-\\frac{1}{3})}\\Bigg]<\/span>\n\n\n = \\frac{2}{3} \\Bigg[\\frac{a(z-c) + b}{(z-c)(z-\\frac{1}{3})}\\Bigg] <\/span>\n\n\n y_e [z] = \\frac{2}{3} \\Bigg[\\frac{a(z-c) + b}{(z-c)(z-\\frac{1}{3})}\\Bigg] = \\frac{2}{3} \\Bigg[\\frac{az - ac + b}{(z-c)(z-\\frac{1}{3})}\\Bigg]<\/span>\n\n\n\n

igualando las expresiones con Y[z] conocida<\/strong> con Y[z] esperada<\/strong>:<\/p>\n\n\n Y_c[z] = Y_e[z] <\/span>\n\n\n -2\\frac{1}{z+2} = \\frac{2}{3} \\Bigg[ \\frac{az - ac + b}{(z-c)(z-\\frac{1}{3})} \\Bigg] <\/span>\n\n\n -3\\frac{1}{z+2} = \\frac{az - ac + b}{(z-c)(z-\\frac{1}{3})} <\/span>\n\n\n\n

para que el denominador quede (z+2), se iguala el t\u00e9rmino con (z-c), con lo que c=-2<\/p>\n\n\n -3\\frac{1}{z+2} = \\frac{az-a(-2) + b}{(z+2)(z-\\frac{1}{3})}<\/span>\n\n\n -3 = \\frac{az+2a + b}{(z-\\frac{1}{3})}<\/span>\n\n\n -3 (z-\\frac{1}{3}) = az +2a + b<\/span>\n\n\n -3 z + 1 = az + 2a + b<\/span>\n\n\n\n

comparando el t\u00e9rmino z, se tiene que a=-3, quedando solo la parte constante para determinar el valor de b<\/p>\n\n\n 2a + b = 1<\/span>\n\n\n 2(-3) + b = 1<\/span>\n\n\n\n

se tiene que b= 7<\/p>\n\n\n\n

teniendo la expresi\u00f3n de la entrada como:<\/p>\n\n\n x[n] = -3\\delta [n] + 7 (-2)^{n-1} \\mu [n-1] <\/span>\n\n\n\n

literal b. Ecuaci\u00f3n de diferencias H[z]<\/h2>\n\n\n\n

Se inicia con los datos de H[z]<\/p>\n\n\n H[z] = \\frac{X[z]}{Y[z]} = \\frac{2}{3} \\frac{1}{z-\\frac{1}{3}} <\/span>\n\n\n \\Big(z-\\frac{1}{3} \\Big) Y[z] = \\frac{2}{3}X[z]<\/span>\n\n\n zY[z]-\\frac{1}{3} Y[z] = \\frac{2}{3}X[z]<\/span>\n\n\n y[n+1]-\\frac{1}{3} y[n] = \\frac{2}{3}x[n]<\/span>\n\n\n\n

literal c. Diagrama de bloques<\/h2>\n\n\n\n

Para el diagrama de bloques se desplaza para despejar y[n]<\/p>\n\n\n y[n]-\\frac{1}{3} y[n-1] = \\frac{2}{3}x[n-1]<\/span>\n\n\n y[n]= \\frac{1}{3} y[n-1] + \\frac{2}{3}x[n-1]<\/span>\n\n\n\n

\"2Eva2009TII_T2_CoficientesXn01\"<\/figure>\n\n\n H[z] = \\frac{X[z]}{Y[z]} = \\frac{2}{3} \\frac{1}{z-\\frac{1}{3}} <\/span>\n\n\n\n

Observaciones:<\/p>\n\n\n\n

Las ra\u00edces caracter\u00edsticas o frecuencias naturales del sistema H[z]  se encuentran dentro del c\u00edrculo de radio unitario. El sistema es asint\u00f3ticamente estable, que implica que es BIBO estable.<\/p>\n\n\n\n

 <\/p>\n\n\n\n

 <\/p>\n\n\n\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Ejercicio: 2Eva2009TII_T2 LTI DT Dado h[n], y[n] determine X[n] literal a. Expresi\u00f3n de x[n] Con los datos de h[n] y y[n], se obtienen las transformadas H[z] y Y[z], la transformada z es: Para la se\u00f1al de salida Yc[z] conocida: la transformada z es: La se\u00f1al de entrada x[n] La se\u00f1al de salida  Y[z] esperada en […]<\/p>\n","protected":false},"author":8043,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"wp-custom-template-entrada-ss-ejercicios","format":"standard","meta":{"footnotes":""},"categories":[187],"tags":[199],"class_list":["post-2763","post","type-post","status-publish","format-standard","hentry","category-ss-s2eva","tag-senalessistemas"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2763","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=2763"}],"version-history":[{"count":3,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2763\/revisions"}],"predecessor-version":[{"id":23972,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/2763\/revisions\/23972"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=2763"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=2763"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=2763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}