{"id":52406,"date":"2024-04-15T23:38:54","date_gmt":"2024-04-15T23:38:54","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=52406"},"modified":"2024-04-15T23:38:54","modified_gmt":"2024-04-15T23:38:54","slug":"the-fourier-series-representation-of-an-impulse-train-denoted-by-sleft-t-right-sumlimits_n-infty-infty-delta-left-t-nt_0-right-rmisrmgiven","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-fourier-series-representation-of-an-impulse-train-denoted-by-sleft-t-right-sumlimits_n-infty-infty-delta-left-t-nt_0-right-rmisrmgiven\/","title":{"rendered":"The Fourier series representation of an impulse train denoted by $$s\\left( t \\right) = \\sum\\limits_{n = &#8211; \\infty }^\\infty {\\delta \\left( {t &#8211; n{T_0}} \\right)} \\,{\\rm{is}}\\,{\\rm{given}}\\,{\\rm{by}}$$"},"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_6a994789acc11\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    $${1 over {{T_0}}}sumlimits_{n = - infty }^infty {exp left( { - {{j2pi nt} over {{T_0}}}} \right)} $$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $${1 over {{T_0}}}sumlimits_{n = - infty }^infty {exp } left( { - {{jpi nt} over {{T_0}}}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    $${1 over {{T_0}}}sumlimits_{n = - infty }^infty {exp } left( {{{jpi nt} over {{T_0}}}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    $${1 over {{T_0}}}sumlimits_{n = - infty }^infty {exp } left( {{{j2pi nt} over {{T_0}}}} \right)$$                <\/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 $\\boxed{{1 \\over {{T_0}}}\\sum\\limits_{n = &#8211; \\infty }^\\infty {\\exp } \\left( {{{j\\pi nt} \\over {{T_0}}}} \\right)}$.<\/p>\n<p>The Fourier series of an impulse train is a sum of sine waves with frequencies that are integer multiples of the fundamental frequency. The fundamental frequency is equal to the reciprocal of the period of the impulse train. In this case, the period of the impulse train is $T_0$, so the fundamental frequency is $1\/T_0$.<\/p>\n<p>The Fourier series coefficients of an impulse train are given by $a_n = 1\/T_0$ for all $n$. This means that the Fourier series of an impulse train is given by<\/p>\n<p>$$s(t) = \\frac{1}{T_0} \\sum_{n=-\\infty}^{\\infty} <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> <\/a>\r\n    <\/div> \\exp \\left( j\\frac{\\pi n t}{T_0} \\right)$$<\/p>\n<p>This is the answer choice $\\boxed{{1 \\over {{T_0}}}\\sum\\limits_{n = &#8211; \\infty }^\\infty {\\exp } \\left( {{{j\\pi nt} \\over {{T_0}}}} \\right)}$.<\/p>\n<p>The other answer choices are incorrect because they do not include the factor of $1\/T_0$ in the Fourier series coefficients. This factor is necessary to ensure that the Fourier series converges to the impulse train.<\/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 Fourier series representation of an impulse train denoted by $$s\\left( t \\right) = \\sum\\limits_{n = - \\infty }^\\infty {\\delta \\left( {t - n{T_0}} \\right)} \\,{\\rm{is}}\\,{\\rm{given}}\\,{\\rm{by}}$$<\/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-fourier-series-representation-of-an-impulse-train-denoted-by-sleft-t-right-sumlimits_n-infty-infty-delta-left-t-nt_0-right-rmisrmgiven\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Fourier series representation of an impulse train denoted by $$s\\left( t \\right) = \\sum\\limits_{n = - 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