{"id":44538,"date":"2024-04-15T21:45:10","date_gmt":"2024-04-15T21:45:10","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=44538"},"modified":"2024-04-15T21:45:10","modified_gmt":"2024-04-15T21:45:10","slug":"consider-a-six-point-decimation-in-time-fast-fourier-transform-fft-algorithm-for-which-the-signal-flow-graph-corresponding-to-xi-is-shown-in-the-figure-let-w_6-exp-left-fracj2","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/consider-a-six-point-decimation-in-time-fast-fourier-transform-fft-algorithm-for-which-the-signal-flow-graph-corresponding-to-xi-is-shown-in-the-figure-let-w_6-exp-left-fracj2\/","title":{"rendered":"Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \\exp \\left( { &#8211; \\frac{{j2\\pi }}{6}} \\right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?"},"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_6a96f2b7ce125\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    $${a_1} = 1,{a_2} = W_6^2,{a_3} = {W_6}$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $${a_1} = - 1,{a_2} = W_6^2,{a_3} = {W_6}$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    $${a_1} = - 1,{a_2} = {W_6},{a_3} = W_6^2$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    $${a_1} = 1,{a_2} = {W_6},{a_3} = W_6^2$$                <\/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 $\\boxed{{a_1} = 1,{a_2} = {W_6},{a_3} = W_6^2}$.<\/p>\n<p>The decimation-in-time FFT algorithm is a recursive algorithm that can be used to compute the discrete Fourier transform (DFT) of a sequence. The algorithm works by dividing the sequence into smaller subsequences, which are then transformed using the DFT. The results of the DFT of the subsequences are then combined to compute the DFT of the original sequence.<\/p>\n<p>The signal-flow graph shown in the figure is for a six-point decimation-in-time FFT algorithm. The coefficients $a_1$, $a_2$, and $a_3$ are the twiddle factors that are used to multiply the input sequence before it is transformed. The twiddle factors are chosen so that the DFT of the input sequence is computed correctly.<\/p>\n<p>The twiddle factors for a six-point decimation-in-time FFT algorithm are given by<\/p>\n<p>$$W_6 = \\exp \\left( { &#8211; \\frac{{j2\\pi }}{6}} \\right)$$<\/p>\n<p>$$W_6^2 = \\exp \\left( { &#8211; \\frac{{j4\\pi }}{6}} \\right) = <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n              <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>   <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> &#8211; W_6$$<\/p>\n<p>$$W_6^3 = \\exp \\left( { &#8211; \\frac{{j6\\pi }}{6}} \\right) = &#8211; W_6^2 = W_6$$<\/p>\n<p>Therefore, the values of the coefficients $a_1$, $a_2$, and $a_3$ in terms of $W_6$ are<\/p>\n<p>$$a_1 = 1$$<\/p>\n<p>$$a_2 = W_6$$<\/p>\n<p>$$a_3 = W_6^2$$<\/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>Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \\exp \\left( { - \\frac{{j2\\pi }}{6}} \\right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?<\/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\/consider-a-six-point-decimation-in-time-fast-fourier-transform-fft-algorithm-for-which-the-signal-flow-graph-corresponding-to-xi-is-shown-in-the-figure-let-w_6-exp-left-fracj2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \\exp \\left( { - \\frac{{j2\\pi }}{6}} \\right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/consider-a-six-point-decimation-in-time-fast-fourier-transform-fft-algorithm-for-which-the-signal-flow-graph-corresponding-to-xi-is-shown-in-the-figure-let-w_6-exp-left-fracj2\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T21:45:10+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":"Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \\exp \\left( { - \\frac{{j2\\pi }}{6}} \\right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?","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\/consider-a-six-point-decimation-in-time-fast-fourier-transform-fft-algorithm-for-which-the-signal-flow-graph-corresponding-to-xi-is-shown-in-the-figure-let-w_6-exp-left-fracj2\/","og_locale":"en_US","og_type":"article","og_title":"Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. 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