29:null 2a:["$","title",null,{"children":"04. Discrete representations of generalized time domain functions | Andrzej Kubaszek | Observable"}] 2f:T80b,outPNT0xy = { let x_b_xy = []; let T0 = h1h2T0[T0ind]; if (T0 !== h1h2T0_hist[T0ind].par) //T0 must be checked first - it has influace for all graphs h1h2T0_hist[T0ind].changed = true; for (let k = T0ind; k--; ) { //h₁ or h₂ index is below T0ind if (h1h2T0_hist[T0ind].changed || h1h2T0[k] !== h1h2T0_hist[k].par) { h1h2T0_hist[k].changed = true; } } for (let k = h1h2T0.length; k--; ) { if (!h1h2T0_hist[k].changed && h1h2T0_hist[k].lastXY ) { x_b_xy[k] = h1h2T0_hist[k].lastXY } else { let h; if (k === T0ind) { //i.e. o.par === T0 h = (tMinMax[1]-T0) / (MAX_N-1); // takes whole t axis } else { h = h1h2T0[k]; // h₁ or h₂ } let p_ = new PN("c_P_TRAP", 1/h); let p_to2 = new PN("c_P_TRAP_PWR_2", 1/h); let argExp = p_to2.add(1).pow(0.5).mul(-T0); if (k === T0ind) // in graph #2, the pure delay is neutralized by the operator exp(-(-T0)p) //and then shown as an offset on the x-axis. In this way, a much better accuracy is achieved. argExp = argExp.sub(p_.mul(-T0)); let out = ( new PN(1/h) .div(p_to2.add(p_).add(4)) ) .mul( (T0 === 0) ? new PN(1) : argExp.exp() ); //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! x_b_xy[k] = //data for d3chartPN(); {yh:out.asList(), h:h, t_offs:(k===T0ind)? T0 : 0 ,nmHtml:`${(k!==T0ind)?` ‒○‒ x_b(t)|_{h_${k+1},T₀}` :` ‒‒‒ x_bT₀(t-T₀)|_T₀`}` ,color:colors_[k] ,pathAttrs:{"stroke-width": (k===T0ind)? 3.5 : 1.5} ,symbAttrs:(k===T0ind)?{}:{symb: "circle", r:3, "stroke-width": 1}} h1h2T0_hist[k].lastXY = x_b_xy[k]; h1h2T0_hist[k].par = h1h2T0[k]; h1h2T0_hist[k].changed = false; } } return (x_b_xy.reverse()); }30:[] 2d:{"id":"889dce8edb64fe79","slug":"discrete-representations-of-generalized-time-domain-func","title":"04. Discrete representations of generalized time domain functions","thumbnail":"b700983b937ca5982f775d72ad96643984d693b90089c9110aec2fefbb1cd853","access_level":"public","create_time":"$D2019-02-26T11:41:01.486Z","update_time":"$D2019-03-15T13:28:44.140Z","view_time":null,"first_public_time":"$D2019-02-26T12:49:36.126Z","has_unpublished_edits":null,"fork":true,"stars":0,"starred":false,"owner":{"type":"team","id":"12ed84ab2a5fb1d1","login":"andrzejq","name":"Andrzej Kubaszek","avatar_url":"https://avatars-next.observableusercontent.com/d5544e4f3e218de6ac8d753ce4dc79d57d9e2452dd35a57847ea297f0ad39698","tier":"starter_2024"},"version":1847,"publish_version":1847,"role":null,"authored":false,"authors":[{"type":"individual","id":"a26fa75e78b968fe","login":"andrzejq","name":"Andrzej Kubaszek","avatar_url":"https://avatars-next.observableusercontent.com/d5544e4f3e218de6ac8d753ce4dc79d57d9e2452dd35a57847ea297f0ad39698","tier":null}],"members":[{"type":"individual","id":"a26fa75e78b968fe","login":"andrzejq","name":"Andrzej Kubaszek","avatar_url":"https://avatars-next.observableusercontent.com/d5544e4f3e218de6ac8d753ce4dc79d57d9e2452dd35a57847ea297f0ad39698","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":{"id":"40fb5921db7af181","slug":"polynomial-numbers-pn","title":"Polynomial Numbers (PN)","owner":{"id":"12ed84ab2a5fb1d1","login":"andrzejq","name":"Andrzej Kubaszek"}},"type":"notebook","body":{"title":"04. Discrete representations of generalized time domain functions","cells":[{"id":861,"value":"md`\n# 04. Discrete representations of generalized time domain functions\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":0,"value":"md`\n\nStarting from [**expression for approximate sequence of samples**](https://observablehq.com/@andrzejq/02-laplace-transform-and-z-transform) of function ${tex`\\{ x(t) \\}`}:\n\n\n ${tex.block`\n\n\\underline{x} = {\\color{magenta}X (\\underline{p})} {\\color{brown} \\ \\cdot \\ \\underline{p} \\cdot (^\\sim0.5^\\sim, 1^\\sim1^\\sim1^\\sim \\dots ^\\sim) }\n `}\n\n\nwe get discrete samples of abstract functions, like derivative of non-smooth function, like \n ${tex`\n\\{x_a(t)\\} = \\{ \\cfrac{\\mathrm{d}}{\\mathrm{d}t} \\ \\delta(t) \\} = { \\color{brown} \\ p \\cdot \\{1\\} \\ \\cdot \\ } { \\color{magenta} p} \n`} , \nwhere Laplace transform ${tex` {\\color{magenta} X(p) = p} \\ `} \n\nor function containing negative delay, like \n ${tex`\n\\{x_b(t)\\} = { \\color{brown} p \\cdot \\{1\\} \\ \\cdot \\ } { \\color{magenta} \\cfrac{1}{p^2 + p + 4}\\ \\exp(-(-0.1)\\sqrt{p^2 +1}) } \n`} \n\n(see\n[Mikusinski's remarks about negative delay operator](http://www.pei.prz.edu.pl/~kubaszek/smacd06/JM_OperCalc.html)).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1083,"value":"html`

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