{"id":37745,"date":"2025-06-26T05:18:33","date_gmt":"2025-06-26T05:18:33","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=37745"},"modified":"2025-06-26T05:18:34","modified_gmt":"2025-06-26T05:18:34","slug":"for-time-invariance-sysxt-tau-yt-tau","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/for-time-invariance-sysxt-tau-yt-tau\/","title":{"rendered":"For time-invariance, Sys{x(t &#8211; \\tau)} = y(t &#8211; \\tau)"},"content":{"rendered":"\n<p> For time-invariance, Sys{x(t &#8211; \\tau)} = y(t &#8211; \\tau). Determine if the system is time- invariant or not: 1. y(t) = 5x(t) 2. y(t) = 2 cos (x(t)) 3. y(t) = 3 cos(t)x(t<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"838\" height=\"261\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-347.png\" alt=\"\" class=\"wp-image-37746\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-347.png 838w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-347-300x93.png 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-347-768x239.png 768w\" sizes=\"auto, (max-width: 838px) 100vw, 838px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To determine whether each system is time-invariant, we check if applying a time shift to the input results in an equivalent time shift in the output. Mathematically, a system is time-invariant if <strong>Sys{x(t\u202f\u2212\u202f\u03c4)} = y(t\u202f\u2212\u202f\u03c4)<\/strong> for any shift \u03c4.<\/p>\n\n\n\n<p>Let\u2019s evaluate each system one by one.<\/p>\n\n\n\n<p><strong>1. y(t) = 5x(t)<\/strong> Let us apply a time shift \u03c4 to the input: Input: x(t\u202f\u2212\u202f\u03c4) Output becomes: y\u2081(t) = 5x(t\u202f\u2212\u202f\u03c4) Compare it to y(t\u202f\u2212\u202f\u03c4): y(t\u202f\u2212\u202f\u03c4) = 5x(t\u202f\u2212\u202f\u03c4) Since they match, this system is <strong>time-invariant<\/strong>.<\/p>\n\n\n\n<p><strong>2. y(t) = 2 cos(x(t))<\/strong> Apply the time shift: Input: x(t\u202f\u2212\u202f\u03c4) Output becomes: y\u2082(t) = 2 cos(x(t\u202f\u2212\u202f\u03c4)) Compare to y(t\u202f\u2212\u202f\u03c4): y(t\u202f\u2212\u202f\u03c4) = 2 cos(x(t\u202f\u2212\u202f\u03c4)) Since they are identical, this system is <strong>time-invariant<\/strong>.<\/p>\n\n\n\n<p><strong>3. y(t) = 3 cos(t)\u00b7x(t)<\/strong> Apply the time shift: Input: x(t\u202f\u2212\u202f\u03c4) Output becomes: y\u2083(t) = 3 cos(t)\u00b7x(t\u202f\u2212\u202f\u03c4) Compare to y(t\u202f\u2212\u202f\u03c4): y(t\u202f\u2212\u202f\u03c4) = 3 cos(t\u202f\u2212\u202f\u03c4)\u00b7x(t\u202f\u2212\u202f\u03c4) These expressions differ because cos(t) is not equal to cos(t\u202f\u2212\u202f\u03c4), which means the system relies explicitly on the time variable t. Therefore, this system is <strong>not time-invariant<\/strong>.<\/p>\n\n\n\n<p><strong>Conclusion:<\/strong> Systems 1 and 2 are time-invariant because their outputs shift consistently with the input. However, system 3 is time-variant, as it contains an explicit dependence on time via the cos(t) term, which changes independently of the input shift. Time-invariance fails whenever system behavior varies with time in a fixed way not tied solely to the input signal.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"852\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-964.jpeg\" alt=\"\" class=\"wp-image-37747\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-964.jpeg 852w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-964-250x300.jpeg 250w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-964-768x923.jpeg 768w\" sizes=\"auto, (max-width: 852px) 100vw, 852px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>For time-invariance, Sys{x(t &#8211; \\tau)} = y(t &#8211; \\tau). Determine if the system is time- invariant or not: 1. y(t) = 5x(t) 2. y(t) = 2 cos (x(t)) 3. y(t) = 3 cos(t)x(t The Correct Answer and Explanation is: To determine whether each system is time-invariant, we check if applying a time shift to the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-37745","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37745","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=37745"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37745\/revisions"}],"predecessor-version":[{"id":37748,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/37745\/revisions\/37748"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=37745"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=37745"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=37745"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}