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1854 lines (1762 loc) · 198 KB
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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Waveforms</title>
<script type="text/javascript" src="page.js"></script>
<style type="text/css">
div.linebreak {
width: 100%;
}
div.box {
overflow: hidden; /* prevents horizontal scrollbar in MSIE */
}
div.const_equation_box {
display: none;
width: 100%;
}
div.const_equation_box svg.equation {
margin-bottom: 1em;
}
div.hcontainer > div.slider_box {
-webkit-flex: 100 1 20em;
flex: 100 1 20em;
}
.harmonic_count {
margin: 0em;
}
button + button {
margin-left: 0em;
}
button {
margin-right: 1em;
}
button.active {
background: linear-gradient(#A4CC99, #E8FFDD);
color: #000000;
}
input[type="number"].error {
background-color: red;
}
.harmonic_slider {
display: block;
-webkit-appearance: none;
-moz-appearance: none;
appearance: none;
box-sizing: border-box;
width: 100%;
height: 2em;
padding: 0em;
margin: 0em;
border: 1px solid black;
border-radius: 0.5em;
background: linear-gradient(#A4CC99, #E8FFDD);
}
.harmonic_slider::-webkit-slider-thumb {
-webkit-appearance: none;
appearance: none;
width: 2em;
height: 2em;
border: 1px solid black;
border-radius: 0.5em;
cursor: pointer;
background: linear-gradient(#FFFAF4, #DDBB99);
}
.harmonic_slider::-moz-range-thumb {
-moz-appearance: none;
appearance: none;
width: 2em;
height: 2em;
border: 1px solid black;
border-radius: 0.5em;
cursor: pointer;
background: linear-gradient(#FFFAF4, #DDBB99);
}
div.graph_frame {
position: relative;
box-sizing: content-box;
width: 100%;
padding: 0% 0px 0px 0px; /* padding-top is set from code and is relative to parent width, so this assures fixed width/height ratio */
height: 0px;
}
svg.graph {
position: absolute;
left: 0px;
top: 0px;
box-sizing: border-box;
width: 100%;
height: 100%;
background-color: white;
}
.axis {
stroke: #000000;
stroke-width: 1px;
}
.gridline {
stroke: #E0E0E0;
stroke-width: 1px;
}
text {
font-weight: bold;
font-size: 20px;
font-family: sans-serif;
text-anchor: middle;
fill:#000000;
}
tspan.sub {
font-size: 15px;
}
text.legend {
text-anchor: start;
}
text.legend tspan.ideal {
fill: #80FF80;
}
text.legend tspan.harmonic_sum {
fill: #FF0000;
}
text.legend tspan.harmonic {
fill: #FF00FF;
}
.curve {
fill: none;
}
.curve.ideal {
stroke: #80FF80;
stroke-width: 3px;
}
.curve.harmonic_sum {
stroke: #FF0000;
stroke-width: 1px;
}
.curve.harmonic {
stroke: #FF00FF;
stroke-width: 1px;
}
</style>
<script type="text/javascript">
"use strict";
var waveform_select_group_elem;
var harmonic_count_elem;
var harmonic_slider_elem;
var graph1_frame_elem;
var graph2_frame_elem;
class Graph
{
frame_elem;
nbr_curves;
main_min_x;
main_max_x;
main_min_y;
main_max_y;
center_x;
center_y;
x_unit;
y_unit;
total_width;
total_height;
constructor(frame_elem, nbr_curves, legend_texts, curve_styles, default_style, main_x_range, main_y_range, main_x_unit, main_y_unit)
{
this.frame_elem = frame_elem;
this.nbr_curves = nbr_curves;
this.main_min_x = main_x_range[0];
this.main_max_x = main_x_range[1];
this.main_min_y = main_y_range[0];
this.main_max_y = main_y_range[1];
this.center_x = 0;
this.center_y = 0;
this.x_unit = main_x_unit;
this.y_unit = main_y_unit;
this.total_width = this.main_max_x - this.main_min_x + 20;
this.total_height = this.main_max_y - this.main_min_y + 20;
this.frame_elem.style.paddingTop = (100 * this.total_height / this.total_width).toFixed(4) + "%";
var s = "<svg class='graph' viewBox='" + (this.main_min_x - 1) + " " + (this.main_min_y - 19) + " "
+ this.total_width + " " + this.total_height + "'>"
+ this.svg_grid()
+ this.svg_legend(legend_texts, curve_styles, default_style);
// Prepare empty curves
for (var curve_idx = 0; curve_idx < this.nbr_curves; curve_idx++)
{
s += "<path id='" + this.frame_elem.id + "_curve" + curve_idx + "' class='curve "
+ ((curve_idx >= curve_styles.length) ? default_style : curve_styles[curve_idx])
+ "' d='' \/>";
}
s += "<\/svg>";
this.frame_elem.innerHTML = s;
}
arrow_head_d_str(length, half_width, angle)
{
const dl0 = 0;
const dw0 = -half_width;
const dl1 = length;
const dw1 = half_width;
const dl2 = -length;
const dw2 = half_width;
return 'm 0,0 ' + ((angle == 0) ?
// angle == 0 (pointing right)
dl0 + ',' + dw0 + ' ' + dl1 + ',' + dw1 + ' ' + dl2 + ',' + dw2 :
// angle == 90 (pointing up)
dw0 + ',' + -dl0 + ' ' + dw1 + ',' + -dl1 + ' ' + dw2 + ',' + -dl2) + ' z';
}
svg_grid()
{
const extra_x = 2;
const extra_y = 2;
const grid_left_x = this.main_min_x - extra_x;
const grid_right_x = this.main_max_x + extra_x;
const grid_top_y = this.main_min_y - extra_y;
const grid_bottom_y = this.main_max_y + extra_y;
const grid_delta_x = this.x_unit / 4;
const grid_delta_y = this.y_unit;
const axis_arrow_half_width = 6;
const axis_arrow_length = 12;
const axis_label_parallel_offset = 6;
const axis_label_orth_offset = 11;
const tick_label_parallel_offset = 6;
const tick_label_orth_offset = 10;
var s = '';
for (var x = Math.ceil(this.main_min_x / grid_delta_x) * grid_delta_x; x <= this.main_max_x; x += grid_delta_x)
{
s += "<path class='gridline' d='M " + x + "," + grid_bottom_y + " " + x + "," + grid_top_y + "' \/>";
}
for (var y = Math.ceil(this.main_min_y / grid_delta_y) * grid_delta_y; y <= this.main_max_y; y += grid_delta_y)
{
s += "<path class='gridline' d='M " + grid_left_x + "," + y + " " + grid_right_x + "," + y + "' \/>";
}
s +=
"<path id='" + this.frame_elem.id + "_x_axis' class='axis' d='M "
+ grid_left_x + "," + this.center_y + " " + grid_right_x + "," + this.center_y + " "
+ this.arrow_head_d_str(axis_arrow_length, axis_arrow_half_width, 0) + "' \/>" +
"<path id='" + this.frame_elem.id + "_y_axis' class='axis' d='M "
+ this.center_x + "," + grid_bottom_y + " " + this.center_x + "," + grid_top_y + " "
+ this.arrow_head_d_str(axis_arrow_length, axis_arrow_half_width, 90) + "' \/>" +
"<text id='" + this.frame_elem.id + "_x_label' x='" + (grid_right_x + axis_label_parallel_offset)
+ "' y='" + (this.center_y - axis_label_orth_offset) + "'>x<\/text>" +
"<text id='" + this.frame_elem.id + "_y_label' x='" + (this.center_x + axis_label_orth_offset)
+ "' y='" + (grid_top_y - axis_label_parallel_offset) + "'>y<\/text>" +
"<text id='" + this.frame_elem.id + "_x_tick_label' x='"
+ (this.center_x + this.x_unit - tick_label_parallel_offset)
+ "' y='" + (this.center_y - tick_label_orth_offset) + "'>1<\/text>" +
"<text id='" + this.frame_elem.id + "_y_tick_label' x='" + (this.center_x - tick_label_orth_offset)
+ "' y='" + (this.center_y - this.y_unit + tick_label_parallel_offset) + "'>1<\/text>";
return s;
}
svg_legend(legend_texts, curve_styles, default_style)
{
const legend_x = this.main_min_x;
const legend_y = this.main_min_y - 3;
var s = "<text id='" + this.frame_elem.id + "_legend' class='legend' x='" + legend_x + "' y='" + legend_y + "'>";
for (var legend_idx = 0; legend_idx < legend_texts.length; legend_idx++)
{
if (legend_idx > 0)
{
s += " ";
}
s += "<tspan class='" + ((legend_idx >= curve_styles.length) ? default_style : curve_styles[legend_idx])
+ "'>—<\/tspan> " + legend_texts[legend_idx];
}
s += "<\/text>";
return s;
}
update_curve(func, curve_idx)
{
const curve_elem = document.getElementById(this.frame_elem.id + '_curve' + curve_idx);
if (func === undefined)
{
curve_elem.style.display = 'none';
}
else
{
var s = '';
var new_start = true;
for (var x = this.main_min_x; x <= this.main_max_x; x++)
{
var f_y = func((x - this.center_x) / this.x_unit);
if (isFinite(f_y))
{
if (new_start) {
s += "M ";
new_start = false;
}
s += x + "," + (this.center_y - this.y_unit * f_y).toFixed(4) + " ";
}
else
{
new_start = true;
}
}
curve_elem.setAttribute('d', s);
curve_elem.style.display = 'block';
}
}
update_curves(func_list)
{
for (var curve_idx = 0; curve_idx < func_list.length; curve_idx++)
{
this.update_curve(func_list[curve_idx], curve_idx);
}
}
}
const TWO_PI = 2 * Math.PI;
function mod(x, y)
{
return x - Math.floor(x / y) * y;
}
function X(x)
{
return mod(x + 1/2, 1) - 1/2;
}
// bessel_j1_of_n_pi[n] = J1(n * pi)
const bessel_j1_of_n_pi = [
0.0,
0.28461534, -0.21238253, 0.1767252 , -0.15453082, 0.1390251 ,
-0.12740889, 0.1182885 , -0.11088192, 0.10471207, -0.09946917,
0.09494223, -0.09098187, 0.08747899, -0.08435181, 0.08153765,
-0.07898755, 0.07666264, -0.07453162, 0.07256896, -0.07075359,
0.06906798, -0.06749737, 0.06602925, -0.06465293, 0.06335924,
-0.06214022, 0.06098895, -0.05989939, 0.05886621, -0.05788471,
0.05695072, -0.05606053, 0.05521082, -0.05439861, 0.05362122,
-0.05287623, 0.05216146, -0.05147491, 0.05081477, -0.0501794 ,
0.04956728, -0.04897702, 0.04840737, -0.04785713, 0.04732525,
-0.04681071, 0.0463126 , -0.04583006, 0.04536229, -0.04490856];
const square_waveform = {
name: 'Square',
ideal_func: function (x)
{
return Math.PI * (Math.floor(X(x)) + 1/2) / 2;
},
harmonic_func: function (n, x)
{
// n'th harmonic
return (n & 1) * Math.sin(TWO_PI * n * x) / n;
},
equation_id: 'square_consts',
};
const triangle_waveform = {
name: 'Triangle',
ideal_func: function (x)
{
return Math.PI * Math.PI * (Math.abs(X(x + 1/4)) - 1/4) / 2;
},
harmonic_func: function (n, x)
{
// n'th harmonic
return (n & 1) * Math.sin(TWO_PI * (n * x + (n - 1) / 4)) / (n * n);
},
equation_id: 'triangle_consts',
};
const sawtooth1_waveform = {
name: 'Sawtooth \\',
ideal_func: function (x)
{
return -Math.PI * X(x - 1/2);
},
harmonic_func: function (n, x)
{
// n'th harmonic
return Math.sin(TWO_PI * n * x) / n;
},
equation_id: 'sawtooth1_consts',
};
const sawtooth2_waveform = {
name: 'Sawtooth /',
ideal_func: function (x)
{
return Math.PI * X(x);
},
harmonic_func: function (n, x)
{
// n'th harmonic
return Math.sin(TWO_PI * (n * x + (n - 1) / 2)) / n;
},
equation_id: 'sawtooth2_consts',
};
const semisine1_waveform = {
name: 'Semi-sine \u2229',
ideal_func: function (x)
{
return 3 * Math.PI * Math.abs(Math.sin(Math.PI * (x + 1/4))) / 4 - 3/2;
},
harmonic_func: function (n, x)
{
// n'th harmonic
return 3 * Math.sin(TWO_PI * (n * x + (n - 1) / 4)) / (4 * n * n - 1);
},
equation_id: 'semisine1_consts',
};
const semisine2_waveform = {
name: 'Semi-sine \u222A',
ideal_func: function (x)
{
return -3 * Math.PI * Math.abs(Math.sin(Math.PI * (x - 1/4))) / 4 + 3/2;
},
harmonic_func: function (n, x)
{
// n'th harmonic
return 3 * Math.sin(TWO_PI * (n * x - (n - 1) / 4)) / (4 * n * n - 1);
},
equation_id: 'semisine2_consts',
};
const parabola1_waveform = {
name: 'Parabola \u2229',
ideal_func: function (x)
{
return -Math.PI * Math.PI * (Math.pow(X(x - 1/4), 2) - 1/12);
},
harmonic_func: function (n, x)
{
// n'th harmonic
return Math.sin(TWO_PI * (n * x + (n - 1) / 4)) / (n * n);
},
equation_id: 'parabola1_consts',
};
const parabola2_waveform = {
name: 'Parabola \u222A',
ideal_func: function (x)
{
return Math.PI * Math.PI * (Math.pow(X(x + 1/4), 2) - 1/12);
},
harmonic_func: function (n, x)
{
// n'th harmonic
return Math.sin(TWO_PI * (n * x - (n - 1) / 4)) / (n * n);
},
equation_id: 'parabola2_consts',
};
const semiellipse1_waveform = {
name: 'Semi-ellipse \u2229',
ideal_func: function (x)
{
return (Math.sqrt(1 - 4 * Math.pow(X(x - 1/4), 2)) - Math.PI / 4) / bessel_j1_of_n_pi[1];
},
harmonic_func: function (n, x)
{
// n'th harmonic
return bessel_j1_of_n_pi[n] * Math.sin(TWO_PI * (n * x - (n - 1) / 4)) / (n * bessel_j1_of_n_pi[1]);
},
equation_id: 'semiellipse1_consts',
};
const semiellipse2_waveform = {
name: 'Semi-ellipse \u222A',
ideal_func: function (x)
{
return -(Math.sqrt(1 - 4 * Math.pow(X(x + 1/4), 2)) - Math.PI / 4) / bessel_j1_of_n_pi[1];
},
harmonic_func: function (n, x)
{
// n'th harmonic
return bessel_j1_of_n_pi[n] * Math.sin(TWO_PI * (n * x + (n - 1) / 4)) / (n * bessel_j1_of_n_pi[1]);
},
equation_id: 'semiellipse2_consts',
};
const waveforms = [
square_waveform, triangle_waveform, sawtooth1_waveform, sawtooth2_waveform,
semisine1_waveform, semisine2_waveform, parabola1_waveform, parabola2_waveform,
semiellipse1_waveform, semiellipse2_waveform
];
const MIN_N_HARMONICS = 1;
const MAX_N_HARMONICS = 50;
const DEFAULT_N_HARMONICS = 5;
var waveform_button_elems;
var graphs = [];
var active_waveform;
function sum(f, n, x)
{
var sm = 0;
for (var i = 1; i <= n; i++)
{
sm += f(i, x);
}
return sm;
}
function string_to_n_harmonics(s)
{
var v = (s.trim() == '') ? NaN : Math.round(s);
v = (v >= MIN_N_HARMONICS && v <= MAX_N_HARMONICS) ? v : NaN;
return v;
}
function update_graphs(ev)
{
const source_elem = (ev.target == harmonic_count_elem) ? harmonic_count_elem : harmonic_slider_elem;
const target_elem = (ev.target == harmonic_count_elem) ? harmonic_slider_elem : harmonic_count_elem;
const n_harmonics = string_to_n_harmonics(source_elem.value);
if (isNaN(n_harmonics))
{
harmonic_count_elem.classList.add('error');
}
else
{
harmonic_count_elem.classList.remove('error');
target_elem.value = n_harmonics;
const waveform_harmonic_sum_func = function(x) {return sum(active_waveform.harmonic_func, n_harmonics, x)};
graphs[0].update_curves([active_waveform.ideal_func, waveform_harmonic_sum_func]);
for (var i = 1; i <= graphs[1].nbr_curves; i++)
{
function waveform_harmonic_func(x)
{
return active_waveform.harmonic_func(i, x);
}
graphs[1].update_curve((i <= n_harmonics) ? waveform_harmonic_func : undefined, i - 1);
}
}
}
function waveform_button_clicked(ev)
{
active_waveform = ev.target.associated_obj;
for (var i = 0; i < waveform_button_elems.length; i++)
{
const button_elem = waveform_button_elems[i];
const waveform = button_elem.associated_obj;
const equation_elem = waveform.equation_elem;
if (button_elem == ev.target)
{
button_elem.classList.add('active');
equation_elem.style.display = 'block';
}
else
{
button_elem.classList.remove('active');
equation_elem.style.display = 'none';
}
}
update_graphs(ev);
}
function init()
{
waveform_select_group_elem = document.getElementById('waveform_select_group');
harmonic_count_elem = document.getElementById('harmonic_count');
harmonic_slider_elem = document.getElementById('harmonic_slider');
graph1_frame_elem = document.getElementById('graph1_frame');
graph2_frame_elem = document.getElementById('graph2_frame');
waveform_button_elems = [];
for (var i = 0; i < waveforms.length; i++)
{
const waveform = waveforms[i];
const button_elem = document.createElement('button');
button_elem.type = 'button';
button_elem.associated_obj = waveform;
button_elem.appendChild(document.createTextNode(waveform.name));
button_elem.addEventListener('click', waveform_button_clicked, false);
waveform_button_elems.push(button_elem);
waveform_select_group_elem.appendChild(button_elem);
waveform.equation_elem = document.getElementById(waveform.equation_id);
}
harmonic_count_elem.min = MIN_N_HARMONICS;
harmonic_count_elem.max = MAX_N_HARMONICS;
harmonic_slider_elem.min = MIN_N_HARMONICS;
harmonic_slider_elem.max = MAX_N_HARMONICS;
harmonic_slider_elem.value = DEFAULT_N_HARMONICS;
harmonic_count_elem.addEventListener('input', update_graphs, false);
harmonic_slider_elem.addEventListener('input', update_graphs, false);
harmonic_slider_elem.addEventListener('change', update_graphs, false); // needed for MSIE
graphs.push(new Graph(graph1_frame_elem, 2,
["w<tspan class='sub' dy='4'>∞<\/tspan><tspan dy='-4'>(x)<\/tspan>", "w<tspan class='sub' dy='4'>N<\/tspan><tspan dy='-4'>(x)<\/tspan>"],
['ideal', 'harmonic_sum'], '', [-200, 600], [-300, 300], 400, 100));
graphs.push(new Graph(graph2_frame_elem, MAX_N_HARMONICS,
["h<tspan class='sub' dy='4'>1<\/tspan><tspan dy='-4'>(x)<\/tspan>, ..., h<tspan class='sub' dy='4'>N<\/tspan><tspan dy='-4'>(x)<\/tspan>"],
[], 'harmonic', [-200, 600], [-100, 100], 400, 100));
waveform_button_elems[0].click();
}
window.addEventListener('load', init, false);
</script>
</head>
<body>
<div class="hcontainer vcenter">
<h1>Waveforms</h1>
<p class="page_subtitle">(Fourier Series)</p>
<div class="box">
<h2>Introduction</h2>
<p>If we have a fundamental sine wave (1<sup>st</sup> harmonic) of the following form
(for time-domain waves, x may be replaced by ft, where f is the fundamental frequency and t is the time point):</p>
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{h_1(x)=\sin(2{\pi}x)}" width='94.230171pt' height='12.664796pt' viewBox='-.239051 -.241837 94.230171 12.664796'>
<defs>
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<use id='svg1_g6-40' xlink:href='#svg1_g4-40' transform='scale(1.5)'/>
<use id='svg1_g6-41' xlink:href='#svg1_g4-41' transform='scale(1.5)'/>
<use id='svg1_g6-50' xlink:href='#svg1_g4-50' transform='scale(1.5)'/>
<use id='svg1_g6-61' xlink:href='#svg1_g4-61' transform='scale(1.5)'/>
<use id='svg1_g6-105' xlink:href='#svg1_g4-105' transform='scale(1.5)'/>
<use id='svg1_g6-110' xlink:href='#svg1_g4-110' transform='scale(1.5)'/>
<use id='svg1_g6-115' xlink:href='#svg1_g4-115' transform='scale(1.5)'/>
<path id='svg1_g4-40' d='M1.873919-5.788505C1.079886-4.748321 .579645-3.30318 .579645-2.056547C.579645-.801974 1.079886 .643167 1.873919 1.683351H2.310638C1.611888 .547883 1.222812-.786093 1.222812-2.056547C1.222812-3.31906 1.611888-4.660977 2.310638-5.788505H1.873919Z'/>
<path id='svg1_g4-41' d='M.738451 1.683351C1.532485 .643167 2.032726-.801974 2.032726-2.048607C2.032726-3.30318 1.532485-4.748321 .738451-5.788505H.301733C1.000482-4.653037 1.389559-3.31906 1.389559-2.048607C1.389559-.786093 1.000482 .555824 .301733 1.683351H.738451Z'/>
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<path id='svg1_g4-50' d='M4.01781-.690809H1.056065C1.127528-1.151349 1.381618-1.445141 2.072428-1.850098L2.866461-2.278876C3.652554-2.707655 4.057512-3.287299 4.057512-3.978108C4.057512-4.446588 3.866944-4.883307 3.533449-5.185039S2.787058-5.629698 2.255055-5.629698C1.540425-5.629698 1.008423-5.375607 .69875-4.899187C.500241-4.597454 .412897-4.24808 .397017-3.676375H1.095766C1.119587-4.057512 1.167229-4.287781 1.262513-4.470409C1.445141-4.811844 1.810397-5.018292 2.231234-5.018292C2.866461-5.018292 3.342881-4.565693 3.342881-3.962228C3.342881-3.517569 3.08085-3.136433 2.580609-2.850581L1.850098-2.437683C.674929-1.770695 .333494-1.238692 .269971 0H4.01781V-.690809Z'/>
<path id='svg1_g4-61' d='M4.240139-2.802939H.397017V-2.247115H4.240139V-2.802939ZM4.240139-1.437201H.397017V-.881377H4.240139V-1.437201Z'/>
<path id='svg1_g4-105' d='M1.19105-4.160736H.532002V0H1.19105V-4.160736ZM1.19105-5.788505H.524062V-4.95477H1.19105V-5.788505Z'/>
<path id='svg1_g4-110' d='M.555824-4.160736V0H1.222812V-2.294757C1.222812-3.144373 1.667471-3.700196 2.350339-3.700196C2.874402-3.700196 3.207896-3.382583 3.207896-2.882342V0H3.866944V-3.144373C3.866944-3.835182 3.350822-4.279841 2.548848-4.279841C1.929502-4.279841 1.532485-4.041631 1.167229-3.461986V-4.160736H.555824Z'/>
<path id='svg1_g4-115' d='M3.477867-3.001447C3.469927-3.819302 2.929984-4.279841 1.969203-4.279841C1.000482-4.279841 .373196-3.7796 .373196-3.009387C.373196-2.35828 .70669-2.048607 1.691292-1.810397L2.310638-1.65953C2.771177-1.548365 2.953805-1.381618 2.953805-1.079886C2.953805-.690809 2.564728-.428778 1.985084-.428778C1.627769-.428778 1.326036-.532002 1.159289-.70669C1.056065-.825795 1.008423-.9449 .968721-1.238692H.269971C.301733-.277912 .841676 .182628 1.929502 .182628C2.977626 .182628 3.644614-.333494 3.644614-1.135468C3.644614-1.754814 3.295239-2.096249 2.469444-2.294757L1.834218-2.445623C1.294275-2.572669 1.064005-2.747356 1.064005-3.041149C1.064005-3.422285 1.405439-3.668435 1.945382-3.668435C2.477385-3.668435 2.763237-3.438165 2.779118-3.001447H3.477867Z'/>
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<path id='svg1_g1-120' d='M4.168676-3.180104L6.943824-6.241104H5.788505L3.799451-3.954287L2.751326-6.241104H1.64365L3.049089-3.180104L.202479 0H1.369708L3.442136-2.382101L4.514081 0H5.633668L4.168676-3.180104Z'/>
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<use x='56.413267' y='65.753425' xlink:href='#svg1_g1-104'/>
<use x='63.060269' y='67.546688' xlink:href='#svg1_g4-49'/>
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<use x='122.583141' y='65.753425' xlink:href='#svg1_g2-25'/>
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<use x='136.75371' y='65.753425' xlink:href='#svg1_g6-41'/>
</g>
</svg>
<p>and add n<sup>th</sup> harmonics of the form:</p>
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{h_n(x)=A_n\sin(2\pi(nx+B_n))}" width='156.490805pt' height='12.664796pt' viewBox='-.239051 -.241837 156.490805 12.664796'>
<defs>
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<p>(where A<sub>1</sub> = 1 and B<sub>1</sub> = 0 for the fundamental) then the waveform resulting from summing the
first N harmonics (including the fundamental) can be expressed as:</p>
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{w_N(x)=\sum_{n=1}^{N}h_n(x)}" width='100.892016pt' height='39.454343pt' viewBox='-.239051 -.232908 100.892016 39.454343'>
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<p>We define w<sub>∞</sub> to be the limiting case when an infinite number of harmonics are added:</p>
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{w_\infty(x)=\lim_{N\rightarrow\infty}w_N(x)}" width='114.368756pt' height='18.696071pt' viewBox='-.239051 -.241837 114.368756 18.696071'>
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</g>
</svg>
<h2>Examples</h2>
<div id="waveform_select_group"></div>
</div>
<div id="square_consts" class="box const_equation_box">
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{A_n=\frac{n\bmod2}{n}\text{~~,~~}B_n=0}" width='140.110254pt' height='28.217431pt' viewBox='-.239051 -.227778 140.110254 28.217431'>
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</svg>
<svg class='equation' data-svg-src="https://latex.codecogs.com/svg.latex?\fn_phv{w_\infty(x)=\frac{\pi}{2}\left(\bigl\lfloor{X(x)}\bigr\rfloor+\frac{1}{2}\right)}" width='146.866556pt' height='32.408907pt' viewBox='-.239051 -.22797 146.866556 32.408907'>
<defs>
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<div id="triangle_consts" class="box const_equation_box">
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