{"id":5132,"date":"2018-04-08T21:32:58","date_gmt":"2018-04-09T02:32:58","guid":{"rendered":"http:\/\/blog.espol.edu.ec\/telg1001\/?p=5132"},"modified":"2026-04-05T21:33:46","modified_gmt":"2026-04-06T02:33:46","slug":"s3eva2016ti_t4-rampa%cf%89-transformada-inversa-fourier","status":"publish","type":"post","link":"https:\/\/blog.espol.edu.ec\/algoritmos101\/ss-s3eva\/s3eva2016ti_t4-rampa%cf%89-transformada-inversa-fourier\/","title":{"rendered":"s3Eva2016TI_T4 rampa(\u03c9) - transformada inversa de Fourier"},"content":{"rendered":"\n

Ejercicio<\/strong>: 3Eva2016TI_T4 rampa(\u03c9) - transformada inversa de Fourier<\/a><\/p>\n\n\n\n

la funci\u00f3n de magnitud |X(\u03c9)|  es par en el eje vertical,<\/p>\n\n\n\n

\"3E2016TI<\/figure>\n\n\n\n

usando la derivada de |X(\u03c9)|,<\/p>\n\n\n\n

\"3E2016TI<\/figure>\n\n\n\n

la expresi\u00f3n de la gr\u00e1fica usando impulsos y rect\u00e1ngulos en dominio \u03c9 es,<\/p>\n\n\n \\frac{\\delta}{\\delta \\omega}X(\\omega) = \\delta( \\omega +\\omega_0) - \\frac{1}{\\omega_0} P_{\\omega_0\/2}\\Big( \\omega +\\frac{\\omega_0}{2}\\Big) <\/span>\n\n\n + \\frac{1}{\\omega_0} P_{\\omega_0\/2}\\Big( \\omega -\\frac{\\omega_0}{2}\\Big) - \\delta( \\omega - \\omega_0) <\/span>\n\n\n\n

aplicando la transformada inversa de Fourier<\/p>\n\n\n \\mathscr{F} ^{-1} \\Big[ \\frac{\\delta}{\\delta \\omega}X(\\omega) \\Big] = \\mathscr{F} ^{-1} \\Big[\\delta( \\omega +\\omega_0) - \\frac{1}{\\omega_0} P_{\\omega_0\/2}\\Big( \\omega +\\frac{\\omega_0}{2}\\Big) \\Big]<\/span>\n\n\n + \\mathscr{F} ^{-1} \\Big[\\frac{1}{\\omega_0} P_{\\omega_0\/2}\\Big( \\omega -\\frac{\\omega_0}{2}\\Big) - \\delta( \\omega - \\omega_0) \\Big] <\/span>\n\n\n\n

si se realiza por partes, y considerando solo la gr\u00e1fica de magnitud |X(\u03c9)|<\/p>\n\n\n\\mathscr{F} ^{-1} \\Big[ \\frac{\\delta}{\\delta \\omega}X(\\omega) \\Big] = -jtx_1(t)<\/span>\n\n\n \\mathscr{F} ^{-1} \\Big[\\delta( \\omega +\\omega_0)\\Big] = \\frac{1}{2\\pi} e^{-j\\omega_0 t}<\/span>\n\n\n \\mathscr{F} ^{-1} \\Big[\\frac{1}{\\omega_0} P_{\\omega_0\/2}\\Big( \\omega +\\frac{\\omega_0}{2}\\Big) \\Big] = \\frac{1}{\\omega_0} \\frac{1}{\\pi t} \\sin \\Big(\\frac{\\omega_0}{2} t \\Big) e^{-j\\frac{\\omega_0}{2}t}<\/span>\n\n\n\n

sustituyendo en la ecuaci\u00f3n principal,<\/p>\n\n\n -jtx_1(t) = \\frac{1}{2\\pi} e^{-j\\omega_0 t} - \\frac{1}{\\omega_0} \\frac{1}{\\pi t} \\sin \\Big(\\frac{\\omega_0}{2} t \\Big) e^{-j\\frac{\\omega_0}{2}t} <\/span>\n\n\n + \\frac{1}{\\omega_0} \\frac{1}{\\pi t} \\sin \\Big(\\frac{\\omega_0}{2} t \\Big) e^{j\\frac{\\omega_0}{2}t} - \\frac{1}{2\\pi} e^{j\\omega_0 t} <\/span>\n\n\n\n

agrupando,<\/p>\n\n\n -jtx_1(t) = \\frac{1}{2\\pi}\\Big( e^{-j\\omega_0 t} -e^{j\\omega_0 t}\\Big) <\/span>\n\n\n + \\frac{1}{\\omega_0 \\pi t} \\sin \\Big(\\frac{\\omega_0}{2} t \\Big)\\Big(- e^{-j\\frac{\\omega_0}{2} t} +e^{j\\frac{\\omega_0}{2} t}\\Big) <\/span>\n\n\n\n

para simplificar se divide ambos lados de la ecuaci\u00f3n para -jt<\/p>\n\n\n x_1(t) = \\frac{1}{\\pi t}\\Big( \\frac{e^{j\\omega_0 t}-e^{-j\\omega_0 t} }{2j}\\Big) <\/span>\n\n\n - \\frac{2}{\\omega_0 \\pi t^2 } \\sin \\Big(\\frac{\\omega_0}{2} t \\Big)\\Big(\\frac{e^{j\\frac{\\omega_0}{2} t}- e^{-j\\frac{\\omega_0}{2} t}}{2j}\\Big) <\/span>\n\n\n\n

que es la forma exponencial del seno y coseno<\/p>\n\n\n x_1(t) = \\frac{1}{\\pi t} \\sin (\\omega_0 t) - \\frac{2}{\\omega_0 \\pi t^2 } \\sin \\Big(\\frac{\\omega_0}{2} t \\Big) \\sin \\Big( \\frac{\\omega_0}{2} t \\Big) <\/span>\n\n\n x_1(t) = \\frac{1}{\\pi t}\\sin (\\omega_0 t) - \\frac{2}{\\omega_0 \\pi t^2 }\\sin^2 \\Big(\\frac{\\omega_0}{2} t \\Big) <\/span>\n\n\n x_1(t) = \\frac{\\omega_0}{\\pi}\\frac{\\sin (\\omega_0 t)}{\\omega_0 t} - \\frac{\\omega_0}{2\\pi}\\Bigg[\\frac{\\sin \\Big(\\frac{\\omega_0}{2} t \\Big)}{\\frac{\\omega_0}{2} t} \\Bigg]^2 <\/span>\n\n\n\n

\"3E2016TI<\/figure>\n\n\n\n

ahora, considerando la gr\u00e1fica de fase:
<\/p>\n\n\n\n

se tiene que,<\/p>\n\n\n x(t) = x_1(t-3) <\/span>\n\n\n x(t) = \\frac{\\omega_0}{\\pi}\\frac{\\sin (\\omega_0 (t-3))}{\\omega_0 (t-3)} - \\frac{\\omega_0}{2\\pi}\\Bigg[\\frac{\\sin \\Big(\\frac{\\omega_0}{2} (t-3) \\Big)}{\\frac{\\omega_0}{2} (t-3)}\\Bigg]^2 <\/span>\n","protected":false},"excerpt":{"rendered":"

Ejercicio: 3Eva2016TI_T4 rampa(\u03c9) - transformada inversa de Fourier la funci\u00f3n de magnitud |X(\u03c9)|  es par en el eje vertical, usando la derivada de |X(\u03c9)|, la expresi\u00f3n de la gr\u00e1fica usando impulsos y rect\u00e1ngulos en dominio \u03c9 es, aplicando la transformada inversa de Fourier si se realiza por partes, y considerando solo la gr\u00e1fica de magnitud […]<\/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":[189],"tags":[199],"class_list":["post-5132","post","type-post","status-publish","format-standard","hentry","category-ss-s3eva","tag-senalessistemas"],"_links":{"self":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/5132","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=5132"}],"version-history":[{"count":6,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/5132\/revisions"}],"predecessor-version":[{"id":23974,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/posts\/5132\/revisions\/23974"}],"wp:attachment":[{"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/media?parent=5132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/categories?post=5132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.espol.edu.ec\/algoritmos101\/wp-json\/wp\/v2\/tags?post=5132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}