2b:null 2c:["$","title",null,{"children":"Frequency response | aallman | Observable"}] 31:T4fc,getButterworthCoefficients = function(poles, sampleFrequency, cutOffFrequency) { if (poles > 2) return null; // it is what it is const A0 = [0, 1, 1, 1, 1 , 1]; // I know a zero pole filter doesn't make sense const B0 = [0, 1, 1, 1, 1, 1]; const B1 = [0, 1, 1.4142136, 2, 2.6131259, 3.236068]; const B2 = [0, 0, 1, 2, 3.4142136, 5.236068]; const B3 = [0, 0, 0, 1, 2.6131259, 5.236068]; const B4 = [0, 0, 0, 0, 1, 3.236068]; const B5 = [0, 0, 0, 0, 0, 1,]; const A1 = [0, 1, 2, 3, 4, 5]; const A2 = [0, 0, 1, 3, 6, 10]; const A3 = [0, 0, 0, 1, 4, 10]; const A4 = [0, 0, 0, 0, 1, 5,]; const A5 = [0, 0, 0, 0, 0, 1,]; const C = 1/(Math.tan(Math.PI*cutOffFrequency/sampleFrequency)); let k, a0, a1, a2, a3, a4, a5, b0, b1, b2, b3, b4, b5; switch (poles) { case 1: k = B0[poles] + B1[poles]*C; a0 = A0[poles]/k; a1 = A1[poles]/k; b0 = (B0[poles]-B1[poles]*C)/k; return {a0:a0, a1:a1, b0:b0}; break; case 2: k = B0[poles] + B1[poles]*C + B2[poles]*C*C; a0 = A0[poles]/k; a1 = A1[poles]/k; a2 = A2[poles]/k; b0 = (2*B0[poles]-2*B2[poles]*C*C)/k; b1 = (B0[poles]-B1[poles]*C+B2[poles]*C*C)/k; return {a0:a0, a1:a1, a2:a2, b0:b0, b1:b1}; break; } } 32:[] 2f:{"id":"26b677194ae7f1cf","slug":"frequency-response","title":"Frequency response","thumbnail":"e8edfd561885a257717402a76675a62c84e6d6edb0755bcb6a7976c8b81e537e","access_level":"public","create_time":"$D2018-04-25T02:42:08.751Z","update_time":"$D2022-05-06T01:39:59.148Z","view_time":null,"first_public_time":"$D2018-04-27T03:57:31.835Z","fork":true,"stars":0,"starred":false,"owner":{"type":"team","id":"ed4d001e00049698","login":"aallman","name":"","avatar_url":"https://avatars-next.observableusercontent.com/3e12326494164eb7b07e4e2586bc8100a17b022344acc0e296c608cb34f8453b","tier":"starter_2024"},"version":746,"publish_version":746,"role":null,"authored":false,"authors":[{"type":"individual","id":"0baa8c389b124206","login":"aallman","name":"","avatar_url":"https://avatars-next.observableusercontent.com/3e12326494164eb7b07e4e2586bc8100a17b022344acc0e296c608cb34f8453b","tier":null}],"members":[{"type":"individual","id":"0baa8c389b124206","login":"aallman","name":"","avatar_url":"https://avatars-next.observableusercontent.com/3e12326494164eb7b07e4e2586bc8100a17b022344acc0e296c608cb34f8453b","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Frequency response","cells":[{"id":0,"value":"md`# Frequency response`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":699,"value":"md`Here are the original signals:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":85,"value":"vegalite({\n data: {values: sine1},\n mark: {type: \"line\", interpolate: \"step\"},\n width:700,\n height: 100,\n autosize: \"fit\",\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\",\n axis: {tickCount: 3, title: \"Sec\", format: \"%s.%L\"}\n },\n y: {\n field: \"value\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n }\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":470,"value":"vegalite({\n data: {values: square1},\n mark: {type: \"line\", interpolate: \"step\"},\n width:700,\n height: 100,\n autosize: \"fit\",\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\",\n axis: {tickCount: 3, title: \"Sec\", format: \"%s.%L\"}\n },\n y: {\n field: \"value\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n }\n }\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":236,"value":"md`### Low-pass digital filter results\nThis section displays the output of a few low-pass filters. You can select which source signal to use, automagically sweep through a range of signal frequencies, manually adjust the frequency, etc. \n\nThe current frequency is: ${signalFrequency} Hz.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":495,"value":" viewof signalSelection = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":239,"value":"viewof cutOffFrequencyHz = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":390,"value":" viewof signalFrequencyAdjust = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":357,"value":"viewof runSweep = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":269,"value":"vegalite({\n data: {values: filtered1TwoCycles},\n width:700,\n height: 100,\n layer:[\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\",\n axis: {tickCount: 3, title: \"Sec\", format: \"%s.%L\"}\n },\n y: {\n field: \"value\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n }\n }\n },\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\"\n },\n y: {\n field: \"filteredValue\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n },\n color: {value:\"red\"}\n }\n }\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":413,"value":"vegalite({\n data: {values: filtered2TwoCycles},\n width:700,\n height: 100,\n layer:[\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\",\n axis: {tickCount: 3, title: \"Sec\", format: \"%s.%L\"}\n },\n y: {\n field: \"value\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n }\n }\n },\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\"\n },\n y: {\n field: \"filteredValue\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n },\n color: {value:\"red\"}\n }\n }\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":441,"value":"vegalite({\n data: {values: filtered2aTwoCycles},\n width:700,\n height: 100,\n layer:[\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\",\n axis: {tickCount: 3, title: \"Sec\", format: \"%s.%L\"}\n },\n y: {\n field: \"value\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n }\n }\n },\n {\n mark: {type: \"line\", interpolate: \"linear\"},\n encoding: {\n x: {\n field: \"time\",\n type: \"temporal\"\n },\n y: {\n field: \"filteredValue\",\n type: \"quantitative\",\n axis: {format: \"f\", title: \"Signal\"}\n },\n color: {value:\"red\"}\n }\n }\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":702,"value":"md`### Filter code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":263,"value":"firstOrderFilter = function(points,sampleFrequency, cutOffFrequency) {\n let coeff = getButterworthCoefficients(1, sampleFrequency, cutOffFrequency);\n let result = [];\n let z0 = 0;\n for (var i = 0; i < points.length; i++) {\n let point = points[i];\n let temp = point.value - coeff.b0*z0;\n let out = coeff.a0*temp + coeff.a1*z0;\n result.push({time:point.time, value: point.value, filteredValue: out});\n z0 = temp;\n }\n return result;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":403,"value":"secondOrderFilter = function(points, sampleFrequency, cutOffFrequency) {\n let coeff = getButterworthCoefficients(2, sampleFrequency, cutOffFrequency);\n let result = [];\n let z0 = 0;\n let z1 = 0;\n for (var i = 0; i < points.length; i++) {\n let point = points[i];\n let temp = point.value - coeff.b0*z0- coeff.b1*z1;\n let out = coeff.a0*temp + coeff.a1*z0 + coeff.a2*z1;\n result.push({time:point.time, value: point.value, filteredValue: out});\n z1 = z0;\n z0 = temp;\n }\n return result;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":734,"value":"md`\nThe next filter is supposed to implement this transfer function:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":730,"value":"tex`\nH(z) = k\\frac{z^2+b_1z+b_2}{z^2+a_1z+a_2}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":735,"value":"md`where:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":737,"value":"tex`\n\\alpha=\\tan(\\frac{\\omega_d}{2}),\\quad\nk=\\frac{\\alpha^2}{1+\\sqrt{2}\\alpha+\\alpha^2},\\quad\na_1 = \\frac{2(\\alpha^2-1)}{1+\\sqrt{2}\\alpha+\\alpha^2},\\quad\na_2=\\frac{1-\\sqrt{2}\\alpha+\\alpha^2}{1+\\sqrt{2}\\alpha+\\alpha^2},\\quad\nb_1=2,\\quad b_2=1\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":739,"value":"md`Giving this for the actual output value:`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":742,"value":"tex`y(n) = kx(n) + kb_1x(n-1) + kb_2x(n-2) - a_1y(n-1) - a_2y(n-2)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":426,"value":"yasof = function(points, sampleFrequency, cutOffFrequency) {\n // yet another second order filter\n \n // Calculate coefficients\n const a = Math.tan(2*Math.PI*cutOffFrequency/sampleFrequency/2);\n const sq2 = Math.sqrt(2);\n const k = (a*a)/(1 + sq2*a + a*a);\n const a1 = 2*(a*a - 1)/(1 + sq2*a + a*a);\n const a2 = (1 - sq2*a + a*a)/(1 + sq2*a + a*a);\n const b1 = 2;\n const b2 = 1;\n \n // Setup ongoing values\n let x1 = 0; // Previous input\n let x2 = 0; // the one before that\n let y1 = 0; // Previous output\n let y2 = 0; // ditto\n \n let result = [];\n for (var i = 0; i < points.length; i++) {\n let point = points[i];\n let x = point.value;\n let y = k*x + k*b1*x1 + k*b2*x2 - a1*y1 - a2*y2;\n x2 = x1;\n x1 = x;\n y2 = y1;\n y1 = y;\n result.push({time:point.time, value: point.value, filteredValue: y});\n }\n \n return result;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":17,"value":"firstOrderPhaseShiftMSec = getAmplitudeAndPhaseResponse(filtered1,sampleRateHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":321,"value":"signalFrequencySweeper = {\n let i=1;\n while (true) {\n if (runSweep) {\n if (i > 50) {\n i = 1;\n } else {\n i++;\n }\n }\n yield Promises.delay(200, i)\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":11,"value":"signalAmplitude = 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":34,"value":"sampleRateHz = 1000","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":73,"value":"sampleLengthSec = 2.0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":393,"value":"signalFrequency = signalFrequencySweeper + signalFrequencyAdjust","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":61,"value":"tMSec = _.range(sampleLengthSec*sampleRateHz).map(x => x*1000/sampleRateHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":53,"value":"sine1 = tMSec.map(t => ({time:t, value:signalAmplitude*Math.sin(2*Math.PI*signalFrequency*t/1000)}))\n\n \n \n ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":466,"value":"square1 = sine1.map(s => ({time:s.time, value:(s.value >= 0) ? 1 : 0}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":501,"value":"signal = (signalSelection==\"sine\") ? sine1 : square1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":265,"value":"filtered1 = firstOrderFilter(signal, sampleRateHz, cutOffFrequencyHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":345,"value":"filtered1TwoCycles = filtered1.slice(0,displayPoints)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":405,"value":"filtered2 = secondOrderFilter(signal, sampleRateHz, cutOffFrequencyHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":411,"value":"filtered2TwoCycles = filtered2.slice(0,displayPoints)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":437,"value":"filtered2a = yasof(signal, sampleRateHz, cutOffFrequencyHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":439,"value":"filtered2aTwoCycles = filtered2a.slice(0,displayPoints)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":364,"value":"getAmplitudeAndPhaseResponse = function(points, sampleRate) {\n let valueCrossingMSec = 0;\n let filterCrossingMSec = 0;\n let crossings = [];\n \n for (var i=1; i= 0) {\n //valueCrossingMSec = i*1000/sampleRate; // don't want to assume it's 1KHz\n valueCrossingMSec=i;\n crossings.push(i);\n }\n \n if (filterCrossingMSec == 0 && points[i].filteredValue < 0 && points[i-1].filteredValue >= 0) {\n //filterCrossingMSec = i*1000/sampleRate; // don't want to assume it's 1KHz\n filterCrossingMSec = i;\n crossings.push(i);\n }\n \n if (valueCrossingMSec > 0 && filterCrossingMSec > 0) {\n break;\n }\n \n }\n \n return (\n {\n crossingsIndexes:crossings,\n filterCrossingMSec:filterCrossingMSec,\n valueCrossingMSec:valueCrossingMSec,\n lag:filterCrossingMSec-valueCrossingMSec\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":326,"value":"displayPoints = sampleRateHz * sampleLengthSec / signalFrequency","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":254,"value":"getButterworthCoefficients(2, sampleRateHz, cutOffFrequencyHz)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":242,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"md`### Libraries used in this notebook (or planned to be used):`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":84,"value":"vegalite = require(\"@observablehq/vega-lite@0.0\")\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":196,"value":"_ = require(\"https://cdn.jsdelivr.net/npm/lodash@4.17.5/lodash.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"