{"id":4581,"date":"2025-03-28T11:23:36","date_gmt":"2025-03-28T05:53:36","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=4581"},"modified":"2025-07-29T17:28:07","modified_gmt":"2025-07-29T11:58:07","slug":"lti-system-properties","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties","title":{"rendered":"LTI SYSTEM PROPERTIES"},"content":{"rendered":"<h2 style=\"text-align: justify;\">LTI SYSTEM PROPERTIES AND THE IMPULSE RESPONSE<\/h2>\n<p style=\"text-align: justify;\">The impulse response completely characterizes the LTI system. Here, we will discuss system properties such as memory, causality, stability and invertibility related to impulse response of LTI system.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties\/#LTI-systems-with-and-without-memory\" >LTI systems with and without memory<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties\/#Causality-for-LTI-system\" >Causality for LTI system<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties\/#-Stability-of-LTI-system\" >\u00a0Stability of LTI system<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties\/#Invertibility-of-LTI-system\" >Invertibility of LTI system<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/lti-systems\/lti-system-properties\/#STEP-RESPONSE-OF-AN-LTI-SYSTEM\" >STEP RESPONSE OF AN LTI SYSTEM<\/a><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"LTI-systems-with-and-without-memory\"><\/span>LTI systems with and without memory<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">A system is memoryless if its output at any time depends only on the value of the input at the same time. The output of a discrete-time system<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4582 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/03\/discrete.jpg\" alt=\"Discrete\" width=\"662\" height=\"270\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/discrete.jpg 662w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/discrete-300x122.jpg 300w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Causality-for-LTI-system\"><\/span>Causality for LTI system<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">The output of a causal system depends only on the present and past values of the input.<\/p>\n<p style=\"text-align: justify;\">The convolution sum<\/p>\n<h3 style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4584 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/03\/convolution-2.jpg\" alt=\"convolution\" width=\"667\" height=\"387\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/convolution-2.jpg 667w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/convolution-2-300x174.jpg 300w\" sizes=\"auto, (max-width: 667px) 100vw, 667px\" \/><\/h3>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"-Stability-of-LTI-system\"><\/span>\u00a0Stability of LTI system<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">A system is stable if every bounded input results in bounded output.<br \/>\nLet x[n] be bounded in magnitude<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4585 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/03\/stability.jpg\" alt=\"Stability\" width=\"662\" height=\"431\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/stability.jpg 662w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/stability-300x195.jpg 300w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Invertibility-of-LTI-system\"><\/span>Invertibility of LTI system<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Consider a continuous-time LTI system with impulse response h(t). This system is invertible only if an inverse system exists that, when connected in series with the original system, produces an output equal to the input to the first system<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4586 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/03\/Invertibility.jpg\" alt=\"Invertibility\" width=\"656\" height=\"374\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/Invertibility.jpg 656w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/Invertibility-300x171.jpg 300w\" sizes=\"auto, (max-width: 656px) 100vw, 656px\" \/><\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"STEP-RESPONSE-OF-AN-LTI-SYSTEM\"><\/span>STEP RESPONSE OF AN LTI SYSTEM<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">The unit step response s(t) or s[n] is response corresponding to unit step that is x(t)= u(t) or x[n] = u[n]<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4589 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/03\/unit-step.jpg\" alt=\"Unit Step\" width=\"626\" height=\"265\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/unit-step.jpg 626w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/03\/unit-step-300x127.jpg 300w\" sizes=\"auto, (max-width: 626px) 100vw, 626px\" \/><\/p>\n<p style=\"text-align: justify;\">The unit step response can also be used to characterize an LTI system, since we can calculate the impulse response from it.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>LTI SYSTEM PROPERTIES AND THE IMPULSE RESPONSE The impulse response completely characterizes the LTI system. Here, we will discuss system<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1351,5],"tags":[1358,1356,1357],"class_list":["post-4581","post","type-post","status-publish","format-standard","hentry","category-lti-systems","category-ee","tag-causality-for-lti-system","tag-invertibility-of-lti-system","tag-stability-of-lti-system"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/4581","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=4581"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/4581\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=4581"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=4581"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=4581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}