{"id":209808,"date":"2025-04-29T12:29:28","date_gmt":"2025-04-29T12:29:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=209808"},"modified":"2025-04-29T12:29:29","modified_gmt":"2025-04-29T12:29:29","slug":"express-a-periodic-triangular-waveform-as-an-infinite-sum-of-shifted-versions-of-trit","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/04\/29\/express-a-periodic-triangular-waveform-as-an-infinite-sum-of-shifted-versions-of-trit\/","title":{"rendered":"Express a periodic triangular waveform as an infinite sum of shifted versions of tri(t)"},"content":{"rendered":"\n<p>Express a periodic triangular waveform as an infinite sum of shifted versions of tri(t), with the first component starting at t = 0. The waveform should have a maximum height of 2, a period of To = 1 s, and the component triangle functions should be adjacent to each other.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To express a <strong>periodic triangular waveform<\/strong> as an <strong>infinite sum of shifted versions<\/strong> of the standard triangular pulse function <code>tri(t)<\/code>, you can use the following expression:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer:<\/strong><\/h3>\n\n\n\n<p>[<br>x(t) = \\sum_{n=-\\infty}^{\\infty} 2 \\cdot \\text{tri}(t &#8211; n)<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (\u2248300 words):<\/strong><\/h3>\n\n\n\n<p>The goal is to construct a <strong>periodic triangular waveform<\/strong> using shifted and scaled copies of the <strong>triangular function<\/strong>, denoted <code>tri(t)<\/code>. The standard <code>tri(t)<\/code> function is typically defined as:<\/p>\n\n\n\n<p>[<br>\\text{tri}(t) =<br>\\begin{cases}<br>1 &#8211; |t|, &amp; \\text{if } |t| \\leq 1 \\<br>0, &amp; \\text{otherwise}<br>\\end{cases}<br>]<\/p>\n\n\n\n<p>This triangular function is centered at ( t = 0 ), has a peak value of 1 at ( t = 0 ), and a base width of 2 (from ( t = -1 ) to ( t = 1 )).<\/p>\n\n\n\n<p>However, in this problem:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The desired <strong>maximum height<\/strong> is <strong>2<\/strong>, so we scale the amplitude by 2.<\/li>\n\n\n\n<li>The desired <strong>period<\/strong> is ( T_0 = 1 ) s, so each triangle must begin right after the previous ends, i.e., with no gaps or overlaps.<\/li>\n\n\n\n<li>The triangle pulses must be <strong>adjacent<\/strong>, meaning the base width must equal the period.<\/li>\n<\/ul>\n\n\n\n<p>So, instead of the standard <code>tri(t)<\/code> centered at 0, we use <code>tri(t - n)<\/code> to <strong>shift<\/strong> the triangle to start at ( t = n ), making each copy of the triangle start at successive integer values of ( t ): 0, 1, 2, etc. To make it <strong>evenly cover all time<\/strong>, we sum over all integers ( n \\in \\mathbb{Z} ) (i.e., from (-\\infty) to (+\\infty)).<\/p>\n\n\n\n<p>Each triangle has a base from ( t = n &#8211; 1 ) to ( t = n + 1 ), meaning two adjacent triangles meet exactly at their bases, ensuring continuity.<\/p>\n\n\n\n<p>The amplitude factor of 2 ensures that each triangle reaches a <strong>peak height of 2<\/strong> instead of the default 1 of <code>tri(t)<\/code>.<\/p>\n\n\n\n<p>This representation is <strong>compact<\/strong>, mathematically elegant, and takes advantage of the inherent <strong>symmetry and periodicity<\/strong> of the triangle waveform.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Express a periodic triangular waveform as an infinite sum of shifted versions of tri(t), with the first component starting at t = 0. The waveform should have a maximum height of 2, a period of To = 1 s, and the component triangle functions should be adjacent to each other. The correct answer and explanation [&hellip;]<\/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-209808","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209808","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=209808"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/209808\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=209808"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=209808"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=209808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}