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index.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
.mac-os-11 { --title-bar-height:28px; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
a.url { word-break: break-all; }
a:active, a:hover { outline: 0px; }
.in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); }
#write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; }
#write.first-line-indent p { text-indent: 2em; }
#write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; }
#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
body.typora-export { padding-left: 30px; padding-right: 30px; }
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.typora-export .task-list-item input { pointer-events: none; }
@media screen and (max-width: 500px) {
body.typora-export { padding-left: 0px; padding-right: 0px; }
#write { padding-left: 20px; padding-right: 20px; }
.CodeMirror-sizer { margin-left: 0px !important; }
.CodeMirror-gutters { display: none !important; }
}
#write li > figure:last-child { margin-bottom: 0.5rem; }
#write ol, #write ul { position: relative; }
img { max-width: 100%; vertical-align: middle; image-orientation: from-image; }
button, input, select, textarea { color: inherit; font: inherit; }
input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; }
*, ::after, ::before { box-sizing: border-box; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; }
#write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; }
p { line-height: inherit; }
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p { orphans: 4; }
h1 { font-size: 2rem; }
h2 { font-size: 1.8rem; }
h3 { font-size: 1.6rem; }
h4 { font-size: 1.4rem; }
h5 { font-size: 1.2rem; }
h6 { font-size: 1rem; }
.md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; }
.hidden { display: none; }
.md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; }
a { cursor: pointer; }
sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; }
sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; }
#write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; }
figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; }
figure > table { margin: 0px; }
tr { break-inside: avoid; break-after: auto; }
thead { display: table-header-group; }
table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; }
table.md-table td { min-width: 32px; }
.CodeMirror-gutters { border-right: 0px; background-color: inherit; }
.CodeMirror-linenumber { user-select: none; }
.CodeMirror { text-align: left; }
.CodeMirror-placeholder { opacity: 0.3; }
.CodeMirror pre { padding: 0px 4px; }
.CodeMirror-lines { padding: 0px; }
div.hr:focus { cursor: none; }
#write pre { white-space: pre-wrap; }
#write.fences-no-line-wrapping pre { white-space: pre; }
#write pre.ty-contain-cm { white-space: normal; }
.CodeMirror-gutters { margin-right: 4px; }
.md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; }
.md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; }
#write .md-fences.mock-cm { white-space: pre-wrap; }
.md-fences.md-fences-with-lineno { padding-left: 0px; }
#write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; }
.md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; }
.CodeMirror-line, twitterwidget { break-inside: avoid; }
.footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; }
.footnotes + .footnotes { margin-top: 0px; }
.md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; }
li div { padding-top: 0px; }
blockquote { margin: 1rem 0px; }
li .mathjax-block, li p { margin: 0.5rem 0px; }
li blockquote { margin: 1rem 0px; }
li { margin: 0px; position: relative; }
blockquote > :last-child { margin-bottom: 0px; }
blockquote > :first-child, li > :first-child { margin-top: 0px; }
.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
#write .footnote-line { white-space: pre-wrap; }
@media print {
body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; }
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.footnote-line { margin-top: 0.714em; font-size: 0.7em; }
a img, img a { cursor: pointer; }
pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; }
p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; }
#write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; }
p > .md-image:only-child { display: inline-block; width: 100%; }
#write .MathJax_Display { margin: 0.8em 0px 0px; }
.md-math-block { width: 100%; }
.md-math-block:not(:empty)::after { display: none; }
.MathJax_ref { fill: currentcolor; }
[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
.md-task-list-item { position: relative; list-style-type: none; }
.task-list-item.md-task-list-item { padding-left: 0px; }
.md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; }
.math { font-size: 1rem; }
.md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; }
.md-toc-content { position: relative; margin-left: 0px; }
.md-toc-content::after, .md-toc::after { display: none; }
.md-toc-item { display: block; color: rgb(65, 131, 196); }
.md-toc-item a { text-decoration: none; }
.md-toc-inner:hover { text-decoration: underline; }
.md-toc-inner { display: inline-block; cursor: pointer; }
.md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; }
.md-toc-h2 .md-toc-inner { margin-left: 2em; }
.md-toc-h3 .md-toc-inner { margin-left: 4em; }
.md-toc-h4 .md-toc-inner { margin-left: 6em; }
.md-toc-h5 .md-toc-inner { margin-left: 8em; }
.md-toc-h6 .md-toc-inner { margin-left: 10em; }
@media screen and (max-width: 48em) {
.md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
.md-toc-h4 .md-toc-inner { margin-left: 5em; }
.md-toc-h5 .md-toc-inner { margin-left: 6.5em; }
.md-toc-h6 .md-toc-inner { margin-left: 8em; }
}
a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; }
.footnote-line a:not(.reversefootnote) { color: inherit; }
.md-attr { display: none; }
.md-fn-count::after { content: "."; }
code, pre, samp, tt { font-family: var(--monospace); }
kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; }
.md-comment { color: rgb(162, 127, 3); opacity: 0.8; font-family: var(--monospace); }
code { text-align: left; vertical-align: initial; }
a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; }
.os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; }
.md-diagram-panel > svg { max-width: 100%; }
[lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; }
[lang="mermaid"] .node text { font-size: 1rem; }
table tr th { border-bottom: 0px; }
video { max-width: 100%; display: block; margin: 0px auto; }
iframe { max-width: 100%; width: 100%; border: none; }
.highlight td, .highlight tr { border: 0px; }
mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); }
.md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; }
.md-expand mark .md-meta { opacity: 0.3 !important; }
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}
.md-diagram-panel .messageText { stroke: none !important; }
.md-diagram-panel .start-state { fill: var(--node-fill); }
.md-diagram-panel .edgeLabel rect { opacity: 1 !important; }
.md-fences.md-fences-math { font-size: 1em; }
.md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; }
.md-fences-advanced:not(.md-focus) { background: inherit; }
.typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; }
.typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; }
.typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; }
.typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; }
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.typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; }
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@media screen and (max-width: 800px) {
.typora-export-sidebar { display: none; }
}
.outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; }
.outline-content ul { margin-top: 0px; margin-bottom: 0px; }
.outline-content strong { font-weight: 400; }
.outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; }
.outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; }
.outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; }
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.CodeMirror-scroll { overflow: auto hidden; z-index: 3; }
.CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); }
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.cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); }
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</style><title>README</title>
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<body class='typora-export os-windows'><div class='typora-export-content'>
<div id='write' class=''><p><a href='https://badge.fury.io/py/TFchirp'><img src="https://badge.fury.io/py/TFchirp.svg" referrerpolicy="no-referrer" alt="PyPI version"></a></p><h2 id='time-frequency-transform-for-chirp-signals'><span>Time Frequency Transform for Chirp Signals</span></h2><p><span>Step 1: Quadratic chirp signal</span></p><p><span>Generate a quadratic chirp signal from 10 Hz to 120 Hz in 1 second with 10,000 sampling points.</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="Python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><span><span></span>x</span></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">scipy</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">matplotlib</span>.<span class="cm-property">pyplot</span> <span class="cm-keyword">as</span> <span class="cm-variable">plt</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Generate a quadratic chirp signal</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">dt</span> = <span class="cm-number">0.0001</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">rate</span> = <span class="cm-builtin">int</span>(<span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-variable">dt</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">ts</span> = <span class="cm-variable">np</span>.<span class="cm-property">linspace</span>(<span class="cm-number">0</span>, <span class="cm-number">1</span>, <span class="cm-builtin">int</span>(<span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-variable">dt</span>))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">data</span> = <span class="cm-variable">scipy</span>.<span class="cm-property">signal</span>.<span class="cm-property">chirp</span>(<span class="cm-variable">ts</span>, <span class="cm-number">10</span>, <span class="cm-number">1</span>, <span class="cm-number">120</span>, <span class="cm-variable">method</span>=<span class="cm-string">'quadratic'</span>)</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 206px;"></div><div class="CodeMirror-gutters" style="display: none; height: 206px;"></div></div></div></pre><p><span>Step 2: S Transform Spectrogram</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="Python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">TFchirp</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text="" cm-zwsp="">
</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Compute S Transform Spectrogram</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">spectrogram</span> = <span class="cm-variable">TFchirp</span>.<span class="cm-property">sTransform</span>(<span class="cm-variable">data</span>, <span class="cm-variable">sample_rate</span>=<span class="cm-variable">rate</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">imshow</span>(<span class="cm-builtin">abs</span>(<span class="cm-variable">spectrogram</span>), <span class="cm-variable">origin</span>=<span class="cm-string">'lower'</span>, <span class="cm-variable">aspect</span>=<span class="cm-string">'auto'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">'Original Spectrogram'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 160px;"></div><div class="CodeMirror-gutters" style="display: none; height: 160px;"></div></div></div></pre><p><img src="https://github.com/xli2522/S-Transform/blob/main/img/original_spectrogram.png?raw=true" referrerpolicy="no-referrer" alt="Original Spectrogram"></p><p><span>Step 3: Quick recovery of full ts from S transform * 0 frequency row*</span></p><p><span>(This recovered ts is computed based on the fact that the 0 frequency row always contain the full FFT result of the ts in this program by design.)</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="Python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Quick Recovery of ts from S Transform 0 frequency row</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">recovered_ts</span> = <span class="cm-variable">TFchirp</span>.<span class="cm-property">recoverS</span>(<span class="cm-variable">spectrogram</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">plot</span>(<span class="cm-variable">recovered_ts</span><span class="cm-operator">-</span><span class="cm-variable">data</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">'Time Series Reconstruction Error'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="display: none; height: 115px;"></div></div></div></pre><p><img src="https://github.com/xli2522/S-Transform/blob/main/img/reconstruction_error.png?raw=true" referrerpolicy="no-referrer" alt="Reconstruction Error"></p><p><span>Step 4: Recovered spectrogram:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="Python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Compute S Transform Spectrogram on the recovered time series</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">recoveredSpectrogram</span> = <span class="cm-variable">TFchirp</span>.<span class="cm-property">sTransform</span>(<span class="cm-variable">recovered_ts</span>, <span class="cm-variable">sample_rate</span>=<span class="cm-variable">rate</span>, <span class="cm-variable">frange</span>=[<span class="cm-number">0</span>,<span class="cm-number">500</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">imshow</span>(<span class="cm-builtin">abs</span>(<span class="cm-variable">recoveredSpectrogram</span>), <span class="cm-variable">origin</span>=<span class="cm-string">'lower'</span>, <span class="cm-variable">aspect</span>=<span class="cm-string">'auto'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">'Recovered Specctrogram'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="display: none; height: 115px;"></div></div></div></pre><p><img src="https://github.com/xli2522/S-Transform/blob/main/img/recovered_spectrogram.png?raw=true" referrerpolicy="no-referrer" alt="Recovered"></p><p><span>Step 5: The real inverse S transform</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Quick Inverse of ts from S Transform</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">inverse_ts</span>, <span class="cm-variable">inverse_tsFFT</span> = <span class="cm-variable">TFchirp</span>.<span class="cm-property">inverseS</span>(<span class="cm-variable">spectrogram</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">plot</span>(<span class="cm-variable">inverse_ts</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">plot</span>(<span class="cm-variable">inverse_ts</span><span class="cm-operator">-</span><span class="cm-variable">data</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">'Time Series Reconstruction Error'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">legend</span>([<span class="cm-string">'Recovered ts'</span>, <span class="cm-string">'Error'</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 160px;"></div><div class="CodeMirror-gutters" style="display: none; height: 160px;"></div></div></div></pre><p><img src="https://github.com/xli2522/S-Transform/blob/main/img/recovered_ts_error.png?raw=true" referrerpolicy="no-referrer" alt="Recovered ts and Error"></p><p><span>Step 6: Recovered spectrogram on the </span><em><span>real</span></em><span> inverse S transform ts</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python"><div class="CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 7.98611px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># Compute S Transform Spectrogram on the recovered time series</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">inverseSpectrogram</span> = <span class="cm-variable">TFchirp</span>.<span class="cm-property">sTransform</span>(<span class="cm-variable">inverse_ts</span>, <span class="cm-variable">sample_rate</span>=<span class="cm-variable">rate</span>, <span class="cm-variable">frange</span>=[<span class="cm-number">0</span>,<span class="cm-number">500</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">imshow</span>(<span class="cm-builtin">abs</span>(<span class="cm-variable">inverseSpectrogram</span>), <span class="cm-variable">origin</span>=<span class="cm-string">'lower'</span>, <span class="cm-variable">aspect</span>=<span class="cm-string">'auto'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">title</span>(<span class="cm-string">'Recovered Specctrogram'</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">plt</span>.<span class="cm-property">show</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 115px;"></div><div class="CodeMirror-gutters" style="display: none; height: 115px;"></div></div></div></pre><p><img src="https://github.com/xli2522/S-Transform/blob/main/img/real_recovered_spectrogram.png?raw=true" referrerpolicy="no-referrer" alt="Recovered Spectrogram"></p><p> </p></div></div>
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