// Numbas version: exam_results_page_options {"name": "Week 6: Fourier Transforms", "metadata": {"description": "
Retrieval practivce after pre-class activities
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "The Definition of the Fourier Transform", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"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}, "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}], "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": "Consider the periodic signal $\\tilde{x}(t)$ with amplitude $A=1$, period $T =$ {T} ms and duty-cycle $\\eta$ = {eta} ({eta*100}%) illustrated below.
\n\n\nGiven 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 the waveform is \\[\\Omega_0=\\frac{2\\pi}{T}\\,\\mathrm{rad/s}.\\]
\nCompute the following:
", "advice": "\\[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)\\]
\nso
\n\\[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)\\]
\n\\[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)\\]
\n\\[C_k = \\frac{1}{\\pi k}\\sin\\left(k \\pi \\frac{2t_0}{T}\\right)\\]
\nConverting this to the sinc function:
\n\\[C_k =\\frac{T}{2t_0}\\mathrm{sinc}\\left(k \\frac{2t_0}{T}\\right)\\] or
\n\\[C_k =\\frac{1}{\\eta}\\mathrm{sinc}\\left(k \\eta\\right)\\]
\nWhen $k = $ {8}, $k2t_0/T = k\\eta$ = {k*eta} and
\n$C_k =$ {(1/eta)*sin(pi*eta)/(pi*eta)}
\n", "rulesets": {}, "variables": {"T": {"name": "T", "group": "Ungrouped variables", "definition": "random(1000,500,200,100,50,20,10,5,2,1)", "description": "The period of a periodic pulse chain in milliseconds.
", "templateType": "anything"}, "eta": {"name": "eta", "group": "Ungrouped variables", "definition": "random(0.05..0.45#0.05)", "description": "width factor - a number less than 100.
", "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": "t0 in ms.
", "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": "The pulse-width $w$ in ms.
", "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": "the limit $t_0$ in the exponential Fourier series coefficient
\n\\[C_k = \\frac{1}{T}\\int_{-t_0}^{t_0} e^{-jk\\Omega_0 t}\\,dt\\]
\nin ms.
", "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": "Determine the periodic frequency $\\Omega_0$ rad/s.
", "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": "The result of the integration is
\n\\[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)\\]
\n\nWhich simpler function is this?
", "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": "The sinc function is defined as \\[\\mathrm{sinc}(x) = \\frac{\\sin(\\pi x)}{\\pi x}.\\]
\nFrom the previous result, which expression which uses the sinc function to define the coefficients $C_k$ is correct?
", "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": "What is the value of $k(2t_0/T)$ when $k = $ {k}?
", "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": "Determine the value of $C_k$ for $k =$ {k}.
", "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"}]}]}, {"name": "Properties of the Fourier Transform", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Fourier transform properties - identifying", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Jobling", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4013/"}], "tags": ["eg-247", "Fourier transforms", "Properties"], "metadata": {"description": "Retrieval practice test", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "These questions test your knowledge of Fourier transform properties and Fourier transform tables.
", "advice": "Refer to theTable of Fourier transform properties give in the notes.
", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Identify the following named Fourier transform properties
", "minMarks": 0, "maxMarks": "11", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "choices": ["Linearity", "Symmetry", "Time scaling", "Time shifting\nThese questions test your knowledge of Fourier transform properties and Fourier transform tables.
", "advice": "Refer to theTable of Fourier transform properties give in the notes.
", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Name the following Fourier transform properties
", "minMarks": 0, "maxMarks": "11", "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "choices": ["\\[a_1f_1(t)+a_2f_2(t) + \\cdots a_nf_n(t)\\]", "\\[F(t)\\]\nThese questions test your knowledge of Fourier transform properties and Fourier transform tables.
", "advice": "Refer to the Table of Fourier transform properties give in the notes.
", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Map the following Fourier transform properties from the time-domain $f(t)$ on the left to the frequency domain $F(\\omega)$ on the right.
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