{"id":51675,"date":"2024-04-15T23:28:14","date_gmt":"2024-04-15T23:28:14","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=51675"},"modified":"2024-04-15T23:28:14","modified_gmt":"2024-04-15T23:28:14","slug":"an-is-a-real-valued-periodic-sequence-with-a-period-n-xn-and-xk-form-n-point-discrete-fourier-transform-dft-pairs-the-dft-yk-of-the-sequence-yleft-n-right-frac1nsumli","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/an-is-a-real-valued-periodic-sequence-with-a-period-n-xn-and-xk-form-n-point-discrete-fourier-transform-dft-pairs-the-dft-yk-of-the-sequence-yleft-n-right-frac1nsumli\/","title":{"rendered":"{a(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence $$y\\left( n \\right) = \\frac{1}{N}\\sum\\limits_{r = 0}^{N &#8211; 1} {x\\left( r \\right)} x\\left( {n + r} \\right)$$ is"},"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_6a9736ba89f20\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    $${left| {Xleft( k \right)} \right|^2}$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $$\frac{1}{2}sumlimits_{r = 0}^{N - 1} {Xleft( r \right)X&#039;left( {k + r} \right)} $$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    $$\frac{1}{2}sumlimits_{r = 0}^{N - 1} {Xleft( r \right)Xleft( {k + r} \right)} $$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"true\">\r\n                    0                <\/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 $\\boxed{\\text{C. }\\frac{1}{2}\\sum\\limits_{r = 0}^{N &#8211; 1} {X\\left( r \\right)X\\left( {k + r} \\right)}}$.<\/p>\n<p>The DFT of a sequence $x(n)$ is defined as<\/p>\n<p>$$X(k) = \\sum_{n=0}^{N-1} x(n) e^{-j2\\pi <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> nk\/N}$$<\/p>\n<p>where $N$ is the length of the sequence.<\/p>\n<p>The sequence $y(n)$ is given by<\/p>\n<p>$$y(n) = \\frac{1}{N}\\sum_{r=0}^{N-1} x(r) x(n+r)$$<\/p>\n<p>The DFT of $y(n)$ is given by<\/p>\n<p>$$Y(k) = \\frac{1}{N}\\sum_{n=0}^{N-1} y(n) e^{-j2\\pi nk\/N}$$<\/p>\n<p>Substituting the expression for $y(n)$ into the above equation, we get<\/p>\n<p>$$Y(k) = \\frac{1}{N}\\sum_{n=0}^{N-1} \\frac{1}{N}\\sum_{r=0}^{N-1} x(r) x(n+r) e^{-j2\\pi nk\/N}$$<\/p>\n<p>$$= \\frac{1}{N^2}\\sum_{r=0}^{N-1} <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> \\sum_{n=0}^{N-1} x(r) x(n+r) e^{-j2\\pi nk\/N}$$<\/p>\n<p>$$= \\frac{1}{N^2}\\sum_{r=0}^{N-1} x(r) \\left(\\sum_{n=0}^{N-1} x(n) e^{-j2\\pi rn\/N}\\right) e^{-j2\\pi nk\/N}$$<\/p>\n<p>$$= \\frac{1}{N^2}\\sum_{r=0}^{N-1} x(r) X(r) e^{-j2\\pi nk\/N}$$<\/p>\n<p>$$= \\frac{1}{N}\\sum_{r=0}^{N-1} X(r) e^{-j2\\pi nk\/N} \\left(\\frac{x(r)}{N}\\right)$$<\/p>\n<p>$$= \\frac{1}{N}\\sum_{r=0}^{N-1} X(r) e^{-j2\\pi nk\/N} \\left(\\frac{1}{N}\\sum_{n=0}^{N-1} x(n) e^{j2\\pi nr\/N}\\right)$$<\/p>\n<p>$$= \\frac{1}{N}\\sum_{r=0}^{N-1} X(r) e^{-j2\\pi nk\/N} \\left(X(r)\\right)^*$$<\/p>\n<p>$$= \\frac{1}{2}\\sum_{r=0}^{N-1} X(r) X(k+r)$$<\/p>\n<p>Therefore, the DFT of the sequence $y(n)$ is given by<\/p>\n<p>$$Y(k) = \\frac{1}{2}\\sum_{r=0}^{N-1} X(r) X(k+r)$$<\/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>{a(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point Discrete Fourier Transform (DFT) pairs. 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