{"id":56949,"date":"2024-04-16T00:53:35","date_gmt":"2024-04-16T00:53:35","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=56949"},"modified":"2024-04-16T00:53:35","modified_gmt":"2024-04-16T00:53:35","slug":"it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/","title":{"rendered":"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { &#8211; \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]"},"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_6a975d3804aef\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    a = -1, b = 1                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    a = 0, b = 1                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    a = 1, b = 1                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    a = 0, b = -1                <\/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 $\\boxed{\\text{A}}$.<\/p>\n<p>The output of a linear time-invariant (LTI) system is given by the convolution of the input signal and the impulse response of the system. In this case, the input signal is $x[n] = {c_1}\\exp \\left( { &#8211; \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right)$ and the impulse response is <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> $h[n] = \\left{ {1,a,b} \\right}$. The output is therefore given by<\/p>\n<p>\\begin{align<em>}<br \/>\ny[n] &amp;= \\sum_{k=-\\infty}^{\\infty} x[k] h[n-k] \\\\ &amp;= {c_1}\\exp \\left( { &#8211; \\frac{{j\\pi n}}{2}} \\right) \\sum_{k=-\\infty}^{\\infty} h[n-k] + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right) \\sum_{k=-\\infty}^{\\infty} h[n-k] \\\\ &amp;= {c_1}\\exp \\left( { &#8211; \\frac{{j\\pi n}}{2}} \\right) \\left( h[0] + h[n-1] + h[n-2] \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right) \\left( h[0] + h[n-1] + h[n-2] \\right) \\\\ &amp;= \\left( {c_1} + {c_2} \\right) h[0] + \\left( {c_1} &#8211; {c_2} \\right) e^{-j\\pi n\/2} h[1] + \\left( {c_1} + {c_2} \\right) e^{j\\pi n\/2} h[2] \\\\ &amp;= \\left( {c_1} + {c_2} \\right) + \\left( {c_1} &#8211; {c_2} \\right) e^{-j\\pi n\/2} a + \\left( {c_1} <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> + {c_2} \\right) e^{j\\pi n\/2} b.<br \/>\n\\end{align<\/em>}<\/p>\n<p>We are given that the output is zero for all $n$, so we must have<\/p>\n<p>\\begin{align<em>}<br \/>\ny[0] &amp;= \\left( {c_1} + {c_2} \\right) + \\left( {c_1} &#8211; {c_2} \\right) a + \\left( {c_1} + {c_2} \\right) b = 0 \\\\<br \/>\ny[1] &amp;= \\left( {c_1} &#8211; {c_2} \\right) e^{-j\\pi} a + \\left( {c_1} + {c_2} \\right) e^{j\\pi} b = 0 \\\\<br \/>\ny[2] &amp;= \\left( {c_1} + {c_2} \\right) + \\left( {c_1} &#8211; {c_2} \\right) e^{-j2\\pi} a + \\left( {c_1} + {c_2} \\right) e^{j2\\pi} b = 0.<br \/>\n\\end{align<\/em>}<\/p>\n<p>Solving these equations, we find that $a = -1$ and $b = 1$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel 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>It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { - \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n &lt; 0 or n &gt; 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]<\/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\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { - \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n &lt; 0 or n &gt; 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]\" \/>\n<meta property=\"og:description\" content=\"Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-16T00:53:35+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=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { - \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]","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\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/","og_locale":"en_US","og_type":"article","og_title":"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { - \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]","og_description":"Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-16T00:53:35+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/","url":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/","name":"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { - \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-16T00:53:35+00:00","dateModified":"2024-04-16T00:53:35+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp\/#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":"It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \\[x\\left[ n \\right] = {c_1}\\exp \\left( { &#8211; \\frac{{j\\pi n}}{2}} \\right) + {c_2}\\exp \\left( {\\frac{{j\\pi n}}{2}} \\right),\\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \\[\\xrightarrow{{x\\left[ n \\right]}}\\boxed{\\begin{array}{*{20}{c}} {n = 0} \\\\ \\downarrow \\\\ {h\\left[ n \\right] = \\left\\{ {1,a,b} \\right\\}} \\end{array}}\\xrightarrow{{y\\left[ n \\right] = 0}}\\]"}]},{"@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\/56949"}],"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=56949"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/56949\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=56949"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=56949"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=56949"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}