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{"name": "Prove that the Fourier transform of a rectangular pulse is the sinc function.", "extensions": [], "custom_part_types": [], "resources": [["question-resources/pulse_train.png", "/srv/numbas/media/question-resources/pulse_train.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Prove that the Fourier transform of a rectangular pulse is the sinc function.", "tags": ["eg-247", "Fourier transform"], "metadata": {"description": "A simple test of knowledge of the EFS of rectangular pulse train with a given duty cycle which leads to the Fourier transform as the period T tends to infinity.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the periodic signal $\\tilde{x}(t)$ with amplitude $A=1$, period $T =$ {T} ms and duty-cycle $\\eta$ = {eta} ({eta*100}%) illustrated below.</p>\n<p></p>\n<p><img src=\"resources/question-resources/pulse_train.png\"/></p>\n<p>Given that the pulse width is $w = 2t_0$ s, the duty cycle is $\\eta = 2t_0/T = w/T$ ($\\eta\\% = w/T\\times 100$), and the fundametal frequency of&nbsp;the waveform is \\[\\Omega_0=\\frac{2\\pi}{T}\\,\\mathrm{rad/s}.\\]</p>\n<p>Compute the following:</p>", "advice": "<ul>\n<li>The pulse width $w =&nbsp;\\eta T =$ {eta*T} ms.</li>\n<li>$w = 2t_0$ so $t_0$ = {eta*T/2} ms.</li>\n<li>$\\Omega_0 = \\frac{2\\pi}{T} =$ {(2000/T)}$\\pi$ rad/s.</li>\n</ul>\n<p>\\[C_k = -\\frac{1}{j2\\pi k}\\left(\\exp\\left(-jk \\pi \\frac{2t_0}{T}\\right)-\\exp\\left(jk\\pi \\frac{2t_0}{T}\\right)\\right)\\]</p>\n<p>so&nbsp;</p>\n<p>\\[C_k = -\\frac{1}{\\pi k}\\left(\\frac{\\exp\\left(-jk \\pi \\frac{2t_0}{T}\\right)-\\exp\\left(jk\\pi \\frac{2t_0}{T}\\right)}{j2}\\right)\\]</p>\n<p>\\[C_k = \\frac{1}{\\pi k}\\left(\\frac{\\exp\\left(jk \\pi \\frac{2t_0}{T}\\right)-\\exp\\left(-jk\\pi \\frac{2t_0}{T}\\right)}{j2}\\right)\\]</p>\n<p>\\[C_k = \\frac{1}{\\pi k}\\sin\\left(k \\pi \\frac{2t_0}{T}\\right)\\]</p>\n<p>Converting this to the sinc function:&nbsp;</p>\n<p>\\[C_k =\\frac{T}{2t_0}\\mathrm{sinc}\\left(k \\frac{2t_0}{T}\\right)\\] or</p>\n<p>\\[C_k =\\frac{1}{\\eta}\\mathrm{sinc}\\left(k \\eta\\right)\\]</p>\n<p>When $k = $ {8}, $k2t_0/T = k\\eta$ = {k*eta} and</p>\n<p>$C_k =$ {(1/eta)*sin(pi*eta)/(pi*eta)}</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"T": {"name": "T", "group": "Ungrouped variables", "definition": "random(1000,500,200,100,50,20,10,5,2,1)", "description": "<p>The period of a periodic pulse chain in milliseconds.</p>", "templateType": "anything"}, "eta": {"name": "eta", "group": "Ungrouped variables", "definition": "random(0.05..0.45#0.05)", "description": "<p>width factor - a number less than 100.</p>", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(-15..15#2)", "description": "", "templateType": "anything"}, "t0": {"name": "t0", "group": "Ungrouped variables", "definition": "T*eta/2", "description": "<p>t0 in ms.</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["T", "eta", "k", "t0"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>The pulse-width $w$ in ms.</p>", "minValue": "T*eta", "maxValue": "T*eta", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>the limit $t_0$ in the exponential Fourier series coefficient</p>\n<p>\\[C_k = \\frac{1}{T}\\int_{-t_0}^{t_0} e^{-jk\\Omega_0 t}\\,dt\\]</p>\n<p>in ms.</p>", "minValue": "t0", "maxValue": "t0", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Determine the periodic frequency $\\Omega_0$ rad/s.</p>", "minValue": "2000*pi/T", "maxValue": "2000*pi/T", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>The result of the integration is</p>\n<p>\\[C_k = -\\frac{1}{j2\\pi k}\\left(\\exp\\left(-jk \\pi \\frac{2t_0}{T}\\right)-\\exp\\left(jk\\pi \\frac{2t_0}{T}\\right)\\right)\\]</p>\n<p></p>\n<p>Which&nbsp;simpler function is this?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["\\[-\\frac{1}{k\\pi}\\sin\\left(k\\pi \\frac{2t_0}{T}\\right)\\]", "\\[\\frac{1}{k\\pi}\\sin\\left(k\\pi \\frac{2t_0}{T}\\right)\\]", "\\[\\frac{j}{k\\pi}\\sin\\left(k\\pi \\frac{2t_0}{T}\\right)\\]", "\\[-\\frac{j}{jk\\pi}\\sin\\left(k\\pi \\frac{2t_0}{T}\\right)\\]"], "matrix": ["0", "1", 0, 0], "distractors": ["", "", "", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>The sinc function is defined as \\[\\mathrm{sinc}(x) = \\frac{\\sin(\\pi x)}{\\pi x}.\\]&nbsp;</p>\n<p>From the previous result, which expression which uses the sinc function to define the coefficients $C_k$ is correct?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["\\[\\frac{T}{k2t_0}\\mathrm{sinc}\\left(k\\frac{2t_0}{T}\\right)\\]", "\\[\\frac{2t_0}{kT}\\mathrm{sinc}\\left(k\\frac{2t_0}{T}\\right)\\]", "\\[\\frac{2t_0}{T}\\mathrm{sinc}\\left(k\\frac{2t_0}{T}\\right)\\]", "\\[\\frac{T}{2t_0}\\mathrm{sinc}\\left(k\\frac{2t_0}{T}\\right)\\]"], "matrix": ["0", 0, "1", 0], "distractors": ["", "", "", ""]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>What is the value of $k(2t_0/T)$ when $k = $ {k}?</p>", "minValue": "k*2*t0/T", "maxValue": "k*2*t0/T", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Determine the value of $C_k$ for&nbsp;$k =$ {k}.</p>", "minValue": "(2*t0/T)*sin(pi*k*2*t0/T)/(pi*k*2*t0/T)", "maxValue": "(2*t0/T)*sin(pi*k*2*t0/T)/(pi*k*2*t0/T)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}]}]}], "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}]}