{"id":208628,"date":"2025-04-27T20:25:57","date_gmt":"2025-04-27T20:25:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=208628"},"modified":"2025-04-27T20:26:00","modified_gmt":"2025-04-27T20:26:00","slug":"a-continuous-time-linear-systems-with-input-xt-and-output-yt-yields-the-following-input-output-pairs","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/27\/a-continuous-time-linear-systems-with-input-xt-and-output-yt-yields-the-following-input-output-pairs\/","title":{"rendered":"A continuous-time linear systemS with input x(t) and output y(t) yields the following input-output pairs"},"content":{"rendered":"\n

A continuous-time linear systemS with input x(t) and output y(t) yields the following input-output pairs:<\/p>\n\n\n\n

x(t) = ej21 ~ y(t) = ej31,<\/p>\n\n\n\n

x(t) = e-j 21 ~ y(t) = e- j 31.<\/p>\n\n\n\n

(a) If x1 (t) = cos(2t), determine the corresponding output y1 (t) for systemS.<\/p>\n\n\n\n

(b) If x2(t) = cos(2(t \u2013 1\/2)), determine the corresponding output y2(t) for system<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Problem Overview<\/h3>\n\n\n\n

In this problem, we are dealing with a continuous-time linear system ( S ) that has two given input-output pairs and we need to determine the corresponding outputs for two different inputs: ( x_1(t) = \\cos(2t) ) and ( x_2(t) = \\cos(2(t – 1\/2)) ).<\/p>\n\n\n\n

The input-output relationships are given by:<\/p>\n\n\n\n

    \n
  1. ( x(t) = e^{j2t} ) yields ( y(t) = e^{j3t} ),<\/li>\n\n\n\n
  2. ( x(t) = e^{-j2t} ) yields ( y(t) = e^{-j3t} ).<\/li>\n<\/ol>\n\n\n\n

    System Behavior<\/h3>\n\n\n\n

    The system ( S ) appears to act on complex exponentials of the form ( e^{j\\omega t} ). Based on the given pairs:<\/p>\n\n\n\n

      \n
    1. For ( x(t) = e^{j2t} ), the output is ( y(t) = e^{j3t} )<\/strong>,<\/li>\n\n\n\n
    2. For ( x(t) = e^{-j2t} ), the output is ( y(t) = e^{-j3t} )<\/strong>.<\/li>\n<\/ol>\n\n\n\n

      This suggests that the system has a frequency shift property. Specifically, the system seems to map an input frequency ( \\omega ) to an output frequency ( \\omega + 1 ). This implies that the system acts as a frequency shifter<\/strong> that adds 1 to the frequency of the input signal.<\/p>\n\n\n\n

      Part (a): ( x_1(t) = \\cos(2t) )<\/h3>\n\n\n\n

      The input ( x_1(t) = \\cos(2t) ) can be expressed in terms of complex exponentials using Euler\u2019s formula:<\/p>\n\n\n\n

      [
      \\cos(2t) = \\frac{1}{2} \\left( e^{j2t} + e^{-j2t} \\right)
      ]<\/p>\n\n\n\n

      From the system’s behavior:<\/p>\n\n\n\n

        \n
      • For ( x(t) = e^{j2t} ), the output is ( y(t) = e^{j3t} ),<\/li>\n\n\n\n
      • For ( x(t) = e^{-j2t} ), the output is ( y(t) = e^{-j3t} ).<\/li>\n<\/ul>\n\n\n\n

        Therefore, applying the system\u2019s frequency shift to both parts of the cosine function:
        [
        y_1(t) = \\frac{1}{2} \\left( e^{j3t} + e^{-j3t} \\right) = \\cos(3t)
        ]<\/p>\n\n\n\n

        Thus, the corresponding output for ( x_1(t) = \\cos(2t) ) is:<\/p>\n\n\n\n

        [
        y_1(t) = \\cos(3t)
        ]<\/p>\n\n\n\n

        Part (b): ( x_2(t) = \\cos(2(t – 1\/2)) )<\/h3>\n\n\n\n

        Now, for ( x_2(t) = \\cos(2(t – 1\/2)) ), we first simplify the expression:<\/p>\n\n\n\n

        [
        x_2(t) = \\cos(2t – 1) = \\frac{1}{2} \\left( e^{j(2t – 1)} + e^{-j(2t – 1)} \\right)<\/p>\n","protected":false},"excerpt":{"rendered":"

        A continuous-time linear systemS with input x(t) and output y(t) yields the following input-output pairs: x(t) = ej21 ~ y(t) = ej31, x(t) = e-j 21 ~ y(t) = e- j 31. (a) If x1 (t) = cos(2t), determine the corresponding output y1 (t) for systemS. (b) If x2(t) = cos(2(t \u2013 1\/2)), determine the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-208628","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208628","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=208628"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/208628\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=208628"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=208628"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=208628"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}