{"id":46919,"date":"2024-04-15T22:19:22","date_gmt":"2024-04-15T22:19:22","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=46919"},"modified":"2024-04-15T22:19:22","modified_gmt":"2024-04-15T22:19:22","slug":"the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/","title":{"rendered":"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 &#8211; {3 \\over 4}{z^{ &#8211; 1}}} \\over {1 &#8211; {3 \\over 4}{z^{ &#8211; 1}} + {1 \\over 8}{z^{ &#8211; 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| > {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| < {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} < \\left| z \\right| < {1 \\over 2}$$ Which one of the following statements is valid?"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a974beadf324\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    Both S1 and S2 are true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    Both S2 and S3 true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    Both S1 and S3 are true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    S1, S2 and S3 are all true                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is: C. Both S1 and S3 are true.<\/p>\n<p>The system is stable if all poles of the transfer function lie inside the unit circle. The poles of the transfer function are at $z=-1$ and $z=2$. Since both poles lie inside the unit circle, the system is stable.<\/p>\n<p>The system is causal if the impulse response of the system is zero for all time $t&lt;0$. The impulse response of the system is given by<\/p>\n<p>$$h(t) = {2 \\over 4} \\delta(t) &#8211; {3 \\over 8} \\delta(t-1) + {1 \\over 8} \\delta(t-2)$$<\/p>\n<p>Since the impulse response <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> of the system is not zero for all time $t&lt;0$, the system is not causal.<\/p>\n<p>The region of convergence (ROC) of a transfer function is the set of all values of $z$ for which the transfer function is defined and converges. The ROC <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> of the transfer function is given by<\/p>\n<p>$$\\left| z \\right| &gt; {1 \\over 2}$$<\/p>\n<p>Since the ROC of the transfer function includes the unit circle, the system is stable. However, since the ROC of the transfer function does not include the origin, the system is not causal.<\/p>\n<p>Therefore, both S1 and S3 are true.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[959],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 - {3 \\over 4}{z^{ - 1}}} \\over {1 - {3 \\over 4}{z^{ - 1}} + {1 \\over 8}{z^{ - 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| &gt; {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| &lt; {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} &lt; \\left| z \\right| &lt; {1 \\over 2}$$ Which one of the following statements is valid?<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 - {3 \\over 4}{z^{ - 1}}} \\over {1 - {3 \\over 4}{z^{ - 1}} + {1 \\over 8}{z^{ - 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| &gt; {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| &lt; {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} &lt; \\left| z \\right| &lt; {1 \\over 2}$$ Which one of the following statements is valid?\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T22:19:22+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 - {3 \\over 4}{z^{ - 1}}} \\over {1 - {3 \\over 4}{z^{ - 1}} + {1 \\over 8}{z^{ - 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| > {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| < {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} < \\left| z \\right| < {1 \\over 2}$$ Which one of the following statements is valid?","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/","og_locale":"en_US","og_type":"article","og_title":"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 - {3 \\over 4}{z^{ - 1}}} \\over {1 - {3 \\over 4}{z^{ - 1}} + {1 \\over 8}{z^{ - 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| > {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| < {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} < \\left| z \\right| < {1 \\over 2}$$ Which one of the following statements is valid?","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T22:19:22+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/","name":"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 - {3 \\over 4}{z^{ - 1}}} \\over {1 - {3 \\over 4}{z^{ - 1}} + {1 \\over 8}{z^{ - 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| > {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| < {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} < \\left| z \\right| < {1 \\over 2}$$ Which one of the following statements is valid?","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T22:19:22+00:00","dateModified":"2024-04-15T22:19:22+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-transfer-function-of-a-discrete-time-lti-system-is-given-by-hleft-z-right-2-3-over-4z-1-over-1-3-over-4z-1-1-over-8z-2-consider-the-follo\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"Signal processing","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/signal-processing\/"},{"@type":"ListItem","position":3,"name":"The transfer function of a discrete time LTI system is given by $$H\\left( z \\right) = {{2 &#8211; {3 \\over 4}{z^{ &#8211; 1}}} \\over {1 &#8211; {3 \\over 4}{z^{ &#8211; 1}} + {1 \\over 8}{z^{ &#8211; 2}}}}$$ Consider the following statements: S1 : The system is stable and causal for $$ROC:\\left| z \\right| > {1 \\over 2}$$ S2 : The system is stable but not causal for $$ROC:\\left| z \\right| < {1 \\over 4}$$ S3 : The system is neither stable nor causal for $$ROC:{1 \\over 4} < \\left| z \\right| < {1 \\over 2}$$ Which one of the following statements is valid?"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d97f17072bfa490596c8f78363955d55?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/46919"}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=46919"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/46919\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=46919"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=46919"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=46919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}