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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; bottom: 0px; top: 0px; left: 0px; right: 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; }
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#write.first-line-indent li { margin-left: 2em; }
.for-image #write { padding-left: 8px; padding-right: 8px; }
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}
#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; }
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*, ::after, ::before { box-sizing: border-box; }
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.footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; }
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.md-math-block:not(:empty)::after { display: none; }
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[contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; }
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.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; }
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.md-toc-h6 .md-toc-inner { margin-left: 10em; }
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.md-toc-h3 .md-toc-inner { margin-left: 3.5em; }
.md-toc-h4 .md-toc-inner { margin-left: 5em; }
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.md-toc-h6 .md-toc-inner { margin-left: 8em; }
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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; }
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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; }
.md-inline-math .MathJax_SVG .noError { display: none !important; }
.html-for-mac .inline-math-svg .MathJax_SVG { vertical-align: 0.2px; }
.md-fences-math .MathJax_SVG_Display, .md-math-block .MathJax_SVG_Display { text-align: center; margin: 0px; position: relative; text-indent: 0px; max-width: none; max-height: none; min-height: 0px; min-width: 100%; width: auto; overflow-y: visible; display: block !important; }
.MathJax_SVG_Display, .md-inline-math .MathJax_SVG_Display { width: auto; margin: inherit; display: inline-block !important; }
.MathJax_SVG .MJX-monospace { font-family: var(--monospace); }
.MathJax_SVG .MJX-sans-serif { font-family: sans-serif; }
.MathJax_SVG { display: inline; font-style: normal; font-weight: 400; line-height: normal; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; zoom: 90%; }
#math-inline-preview-content { zoom: 1.1; }
.MathJax_SVG * { transition: none 0s ease 0s; }
.MathJax_SVG_Display svg { vertical-align: middle !important; margin-bottom: 0px !important; margin-top: 0px !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; }
mark .md-meta { color: rgb(0, 0, 0); }
@media print {
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<div id='write' class=''><h1 id='time-and-frequency-analysis-methods-on-gw-signals'><span>Time and Frequency Analysis Methods on GW Signals</span></h1><h6 id='background-notes-by-x-li'><span>Background Notes by X. Li</span></h6><h3 id='part-i-basic-transform-methods'><code>Part I: Basic Transform Methods</code></h3><h4 id='11-revisit-continuous-fourier-transform-and-discrete-fourier-transform'><span>1.1 Revisit Continuous Fourier Transform and Discrete Fourier Transform</span></h4><h5 id='continuous-fourier-transform'><span>Continuous Fourier Transform:</span></h5><p><span> </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n7" cid="n7" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.546ex" height="5.844ex" viewBox="0 -1476.6 10998.8 2516.3" role="img" focusable="false" style="vertical-align: -2.415ex; max-width: 100%;"><defs><path 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transform="translate(0,1150)"><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-4E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMAIN-2212" x="888" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMAIN-31" x="1666" y="0"></use></g></g><use xlink:href="#E61-MJMATHI-78" x="5182" y="0"></use><use xlink:href="#E61-MJMAIN-28" x="5754" y="0"></use><use xlink:href="#E61-MJMATHI-6E" x="6143" y="0"></use><use xlink:href="#E61-MJMAIN-29" x="6743" y="0"></use><g transform="translate(7132,0)"><use xlink:href="#E61-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E61-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-6A" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMAIN-32" x="1189" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-3C0" x="1689" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-6B" x="2262" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-6E" x="2784" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMAIN-2F" x="3384" y="0"></use><use transform="scale(0.707)" xlink:href="#E61-MJMATHI-4E" x="3884" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-2">X(k)=\sum_{n=1}^{N-1} x(n)e^{-j2\pi kn/N}</script></div></div><p><span>where we let </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n11" cid="n11" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.273ex" height="5.21ex" viewBox="0 -1476.6 3131.6 2243.2" role="img" focusable="false" 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430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1008" height="60" x="0" y="220"></rect><use xlink:href="#E62-MJMATHI-6B" x="243" y="676"></use><use xlink:href="#E62-MJMATHI-4E" x="60" y="-686"></use></g><use xlink:href="#E62-MJMAIN-2261" x="1525" y="0"></use><use xlink:href="#E62-MJMATHI-66" x="2581" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3">\frac{k}{N}\equiv f</script></div></div><p><span>and</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n13" cid="n13" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.329ex" height="1.912ex" viewBox="0 -712 2294.6 823.2" role="img" focusable="false" style="vertical-align: -0.258ex; max-width: 100%;"><defs><path stroke-width="0" id="E63-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E63-MJMAIN-2261" d="M56 444Q56 457 70 464H707Q722 456 722 444Q722 430 706 424H72Q56 429 56 444ZM56 237T56 250T70 270H707Q722 262 722 250T707 230H70Q56 237 56 250ZM56 56Q56 71 72 76H706Q722 70 722 56Q722 44 707 36H70Q56 43 56 56Z"></path><path stroke-width="0" id="E63-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E63-MJMATHI-6E" x="0" y="0"></use><use xlink:href="#E63-MJMAIN-2261" x="877" y="0"></use><use xlink:href="#E63-MJMATHI-74" x="1933" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-4">n \equiv t</script></div></div><h4 id='12-chirp-z-transform'><span>1.2 Chirp Z Transform</span></h4><p><span>The </span><strong><span>chirp Z-transform</span></strong><span> (</span><strong><span>CZT</span></strong><span>) is a generalization of the </span><a href='https://en.wikipedia.org/wiki/Discrete_Fourier_transform'><span>discrete Fourier transform</span></a><span> (DFT). While the DFT samples the </span><a href='https://en.wikipedia.org/wiki/Z-transform'><span>Z plane</span></a><span> at uniformly-spaced points along the unit circle, the chirp Z-transform samples along spiral arcs in the Z-plane, corresponding to straight lines in the </span><a href='https://en.wikipedia.org/wiki/S_plane'><span>S plane</span></a><span>.[</span><a href='https://en.wikipedia.org/wiki/Chirp_Z-transform#cite_note-Shilling-1'><span>1</span><span>]</span></a><span>[</span><a href='https://en.wikipedia.org/wiki/Chirp_Z-transform#cite_note-2'><span>2</span><span>]</span></a><span> The DFT, real DFT, and zoom DFT can be calculated as special cases of the CZT. ( - Wikipedia)</span></p><p><span>To obtain DCZT from DFT, we introduce</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n17" cid="n17" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.829ex" height="2.673ex" viewBox="0 -1039.7 7245.8 1150.9" role="img" focusable="false" style="vertical-align: -0.258ex; max-width: 100%;"><defs><path stroke-width="0" id="E64-MJMATHI-57" d="M436 683Q450 683 486 682T553 680Q604 680 638 681T677 682Q695 682 695 674Q695 670 692 659Q687 641 683 639T661 637Q636 636 621 632T600 624T597 615Q597 603 613 377T629 138L631 141Q633 144 637 151T649 170T666 200T690 241T720 295T759 362Q863 546 877 572T892 604Q892 619 873 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16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E64-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E64-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E64-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E64-MJMATHI-6A" 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637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E64-MJMATHI-57" x="0" y="0"></use><g transform="translate(1079,412)"><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-6E" x="521" y="0"></use></g><use xlink:href="#E64-MJMAIN-3D" x="2249" y="0"></use><g transform="translate(3305,0)"><use xlink:href="#E64-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-6A" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-32" x="1189" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-3C0" x="1689" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-6B" x="2262" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-6E" x="2784" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMAIN-2F" x="3384" y="0"></use><use transform="scale(0.707)" xlink:href="#E64-MJMATHI-4E" x="3884" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-5">W^{kn}=e^{-j2\pi kn/N}</script></div></div><p><span>where</span><span> </span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n19" cid="n19" mdtype="math_block"><div class="md-rawblock-container md-math-container" 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305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E65-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E65-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 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y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMATHI-6E" x="521" y="0"></use></g><use xlink:href="#E65-MJMAIN-3D" x="2249" y="0"></use><g transform="translate(3305,0)"><use xlink:href="#E65-MJMATHI-57" x="0" y="0"></use><g transform="translate(1079,412)"><use transform="scale(0.707)" xlink:href="#E65-MJMATHI-6B" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E65-MJMAIN-32" x="736" y="595"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-2F" x="974" y="0"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-32" x="1474" y="0"></use></g></g><use xlink:href="#E65-MJMAIN-2B" x="6103" y="0"></use><g transform="translate(7103,0)"><use xlink:href="#E65-MJMATHI-57" x="0" y="0"></use><g transform="translate(1079,412)"><use transform="scale(0.707)" xlink:href="#E65-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E65-MJMAIN-32" x="848" y="513"></use><use transform="scale(0.707)" xlink:href="#E65-MJMAIN-2F" x="1053" 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x="4408" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">W^{kn}=W^{k^{2}/2}+ W^{n^{2}/2} + W^{-(k-n)^{2}/2}</script></div></div><p><span>thus, the DFT becomes</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n21" cid="n21" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="39.847ex" height="7.113ex" viewBox="0 -1804.3 17156.4 3062.5" role="img" focusable="false" style="vertical-align: -2.922ex; max-width: 100%;"><defs><path stroke-width="0" id="E66-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 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y="0"></use><use transform="scale(0.5)" xlink:href="#E69-MJMAIN-32" x="550" y="674"></use></g><use transform="scale(0.707)" xlink:href="#E69-MJMAIN-2F" x="3908" y="0"></use><use transform="scale(0.707)" xlink:href="#E69-MJMAIN-32" x="4408" y="0"></use></g></g><use xlink:href="#E69-MJMAIN-29" x="23846" y="0"></use><use xlink:href="#E69-MJMAIN-5D" x="24235" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-10">X(k)=W^{k^{2}/2}IFFT[FFT(x(n)W^{n^{2}/2})FFT(W^{-(k-n)^{2}/2})]</script></div></div><p><span>Some notes:</span></p><p><span>When computing using FFT, the signal in the block is treated as periodic**(to be verified), however, the input signal itself does not need to be.</span></p><p><span>Fresnel integral approximation?</span></p><h4 id='13-gabor-transform'><span>1.3 Gabor Transform</span></h4><p><span>The </span><strong><span>Gabor transform</span></strong><span> is a special case of the </span><a href='https://en.wikipedia.org/wiki/Short-time_Fourier_transform'><span>short-time Fourier transform</span></a><span>. It is used to determine the </span><a href='https://en.wikipedia.org/wiki/Sine_wave'><span>sinusoidal</span></a><span> </span><a href='https://en.wikipedia.org/wiki/Frequency'><span>frequency</span></a><span> and </span><a href='https://en.wikipedia.org/wiki/Phase_(waves)'><span>phase</span></a><span> content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a </span><a href='https://en.wikipedia.org/wiki/Gaussian_function'><span>Gaussian function</span></a><span>, which can be regarded as a </span><a href='https://en.wikipedia.org/wiki/Window_function'><span>window function</span></a><span>, and the resulting function is then transformed with a Fourier transform to derive the </span><a href='https://en.wikipedia.org/wiki/Time-frequency_analysis'><span>time-frequency analysis</span></a><span>. 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Transform</span></h4><p><span>The </span><em><span>S</span></em><span> transform is a generalization of the </span><a href='https://en.wikipedia.org/wiki/Short-time_Fourier_transform'><span>short-time Fourier transform</span></a><span> (STFT), extending the </span><a href='https://en.wikipedia.org/wiki/Continuous_wavelet_transform'><span>continuous wavelet transform</span></a><span> and overcoming some of its disadvantages. Modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in </span><em><span>S</span></em><span> transform. The </span><em><span>S</span></em><span> transform doesn't have a cross-term problem and yields a better signal clarity than </span><a href='https://en.wikipedia.org/wiki/Gabor_transform'><span>Gabor transform</span></a><span>. 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transform="translate(11007,0)"><use xlink:href="#E79-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E79-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3C0" x="778" y="0"></use><g transform="translate(955,0)"><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3B1" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E79-MJMAIN-32" x="905" y="513"></use></g><use transform="scale(0.707)" xlink:href="#E79-MJMAIN-2F" x="2444" y="0"></use><g transform="translate(2082,0)"><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3C9" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E79-MJMAIN-32" x="879" y="513"></use></g></g></g><g transform="translate(14415,0)"><use xlink:href="#E79-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E79-MJMAIN-32" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3C0" x="911" y="0"></use><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3B1" x="1484" y="0"></use><use transform="scale(0.707)" xlink:href="#E79-MJMATHI-3C4" x="2125" y="0"></use></g></g><use xlink:href="#E79-MJMATHI-64" x="16849" y="0"></use><use xlink:href="#E79-MJMATHI-3B1" x="17372" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-20">S_x(\tau, \omega)=\int_{-\infty}^{\infty}X(\omega+\alpha)e^{-\pi \alpha^{2}/\omega^{2}}e^{j2\pi \alpha \tau}d\alpha</script></div></div><p><span>where X(t) represents the Fourier Transform. (Convolution is used to get this spectral form)</span></p><p><span>Let</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n62" cid="n62" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="29.653ex" height="2.546ex" viewBox="0 -821.2 12767.2 1096.3" role="img" focusable="false" style="vertical-align: -0.639ex; max-width: 100%;"><defs><path stroke-width="0" id="E80-MJMATHI-3C4" d="M39 284Q18 284 18 294Q18 301 45 338T99 398Q134 425 164 429Q170 431 332 431Q492 431 497 429Q517 424 517 402Q517 388 508 376T485 360Q479 358 389 358T299 356Q298 355 283 274T251 109T233 20Q228 5 215 -4T186 -13Q153 -13 153 20V30L203 192Q214 228 227 272T248 336L254 357Q254 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-192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"></path><path stroke-width="0" id="E80-MJMATHI-46" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></path><path stroke-width="0" id="E80-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E80-MJMATHI-3C4" x="0" y="0"></use><use xlink:href="#E80-MJMAIN-3D" x="794" y="0"></use><use xlink:href="#E80-MJMATHI-6E" 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y="-213"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-21">\tau = n\triangle_T, \omega = p\triangle_F,\alpha=p\triangle_T</script></div></div><p><span>We obtain the </span></p><h5 id='discrete-s-transform'><span>Discrete S Transform:</span></h5><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n65" cid="n65" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="53.083ex" height="7.493ex" viewBox="0 -1804.3 22855.3 3226.3" role="img" focusable="false" style="vertical-align: -3.303ex; max-width: 100%;"><defs><path stroke-width="0" id="E81-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 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xlink:href="#E81-MJMATHI-6D" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E81-MJMAIN-32" x="1241" y="513"></use></g></g></g><g transform="translate(19281,0)"><use xlink:href="#E81-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMAIN-32" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-3C0" x="911" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-70" x="1484" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-6D" x="1987" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMAIN-2F" x="2866" y="0"></use><use transform="scale(0.707)" xlink:href="#E81-MJMATHI-4E" x="3365" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-22">S_x(n\triangle_T, p\triangle_F)=\sum_{p=0}^{N-1}X[(p+m)\triangle_F]e^{-\pi p^{2}/m^{2}}e^{j2\pi pm/N}</script></div></div><h5 id='implementation'><span>Implementation:</span></h5><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="pseudocode" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="pseudocode"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 8.15973px;"><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-variable">Step1</span>.<span class="cm-variable">Compute</span> </span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">X</span><span class="cm-bracket">[</span><span class="cm-variable">p</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{</span><span class="cm-variable">F</span><span class="cm-bracket">}]</span>\,<span class="cm-bracket">}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">X</span><span class="cm-bracket">[</span><span class="cm-variable">p</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{{</span><span class="cm-variable">F</span><span class="cm-bracket">}}]</span>\, </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">loop</span><span class="cm-bracket">{</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Step2</span>.<span class="cm-variable">Compute</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">e</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-operator">-</span>\<span class="cm-variable">pi</span> <span class="cm-bracket">{</span>\<span class="cm-variable">frac</span> <span class="cm-bracket">{</span><span class="cm-variable">p</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}{</span><span class="cm-variable">m</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}}}}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">e</span><span class="cm-operator">^</span><span class="cm-bracket">{{</span><span class="cm-operator">-</span>\<span class="cm-variable">pi</span> <span class="cm-bracket">{</span>\<span class="cm-variable">frac</span> <span class="cm-bracket">{</span><span class="cm-variable">p</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}{</span><span class="cm-variable">m</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}}}}</span><span class="cm-keyword">for</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">f</span><span class="cm-operator">=</span><span class="cm-variable">m</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{</span><span class="cm-variable">F</span><span class="cm-bracket">}}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">f</span><span class="cm-operator">=</span><span class="cm-variable">m</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{{</span><span class="cm-variable">F</span><span class="cm-bracket">}}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Step3</span>.<span class="cm-variable">Move</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">X</span><span class="cm-bracket">[</span><span class="cm-variable">p</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{</span><span class="cm-variable">F</span><span class="cm-bracket">}]}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">X</span><span class="cm-bracket">[</span><span class="cm-variable">p</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{{</span><span class="cm-variable">F</span><span class="cm-bracket">}}]</span><span class="cm-variable">to</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">X</span><span class="cm-bracket">[(</span><span class="cm-variable">p</span><span class="cm-operator">+</span><span class="cm-variable">m</span><span class="cm-bracket">)</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{</span><span class="cm-variable">F</span><span class="cm-bracket">}]}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">X</span><span class="cm-bracket">[(</span><span class="cm-variable">p</span><span class="cm-operator">+</span><span class="cm-variable">m</span><span class="cm-bracket">)</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{{</span><span class="cm-variable">F</span><span class="cm-bracket">}}]</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Step4</span>.<span class="cm-variable">Multiply</span> <span class="cm-variable">Step2</span> <span class="cm-keyword">and</span> <span class="cm-variable">Step3</span> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">B</span><span class="cm-bracket">[</span><span class="cm-variable">m</span>,<span class="cm-variable">p</span><span class="cm-bracket">]</span><span class="cm-operator">=</span><span class="cm-variable">X</span><span class="cm-bracket">[(</span><span class="cm-variable">p</span><span class="cm-operator">+</span><span class="cm-variable">m</span><span class="cm-bracket">)</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{</span><span class="cm-variable">F</span><span class="cm-bracket">}]</span>\<span class="cm-variable">cdot</span> <span class="cm-variable">e</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-operator">-</span>\<span class="cm-variable">pi</span> <span class="cm-bracket">{</span>\<span class="cm-variable">frac</span> <span class="cm-bracket">{</span><span class="cm-variable">p</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}{</span><span class="cm-variable">m</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}}}}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span><span class="cm-bracket">[</span><span class="cm-variable">m</span>,<span class="cm-variable">p</span><span class="cm-bracket">]</span><span class="cm-operator">=</span><span class="cm-variable">X</span><span class="cm-bracket">[(</span><span class="cm-variable">p</span><span class="cm-operator">+</span><span class="cm-variable">m</span><span class="cm-bracket">)</span>\<span class="cm-variable">Delta</span> <span class="cm-variable">_</span><span class="cm-bracket">{{</span><span class="cm-variable">F</span><span class="cm-bracket">}}]</span>\<span class="cm-variable">cdot</span> <span class="cm-variable">e</span><span class="cm-operator">^</span><span class="cm-bracket">{{</span><span class="cm-operator">-</span>\<span class="cm-variable">pi</span> <span class="cm-bracket">{</span>\<span class="cm-variable">frac</span> <span class="cm-bracket">{</span><span class="cm-variable">p</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}{</span><span class="cm-variable">m</span><span class="cm-operator">^</span><span class="cm-bracket">{</span><span class="cm-number">2</span><span class="cm-bracket">}}}}}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Step5</span>.<span class="cm-variable">IDFT</span><span class="cm-bracket">(</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-bracket">{</span>\<span class="cm-variable">displaystyle</span> <span class="cm-variable">B</span><span class="cm-bracket">[</span><span class="cm-variable">m</span>,<span class="cm-variable">p</span><span class="cm-bracket">]}</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> </span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">B</span><span class="cm-bracket">[</span><span class="cm-variable">m</span>,<span class="cm-variable">p</span><span class="cm-bracket">])</span>.</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Repeat</span>.<span class="cm-bracket">}</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1054px;"></div><div class="CodeMirror-gutters" style="display: none; height: 1054px;"></div></div></div></pre><h4 id='15-constant-q-transform'><span>1.5 Constant Q Transform</span></h4><p><strong><span>CQT</span></strong><span> transforms a data series to the frequency domain. It is related to the </span><a href='https://en.wikipedia.org/wiki/Fourier_transform'><span>Fourier transform</span></a><span>[</span><a href='https://en.wikipedia.org/wiki/Constant-Q_transform#cite_note-b91-1'><span>1</span><span>]</span></a><span> and very closely related to the complex </span><a href='https://en.wikipedia.org/wiki/Morlet_wavelet'><span>Morlet wavelet</span></a><span> transform.[</span><a href='https://en.wikipedia.org/wiki/Constant-Q_transform#cite_note-2'><span>2</span><span>]</span></a></p><p><span>The transform can be thought of as a series of filters </span><em><span>f**k</span></em><span>, logarithmically spaced in frequency, with the </span><em><span>k</span></em><span>-th filter having a </span><a href='https://en.wikipedia.org/wiki/Spectral_width'><span>spectral width</span></a><span> </span><em><span>δf**k</span></em><span> equal to a multiple of the previous filter's width:</span></p><p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4620127b4c5a85de8a9849fe3a8e23274db0d9" referrerpolicy="no-referrer" alt="{\displaystyle \delta f_{k}=2^{1/n}\cdot \delta f_{k-1}=\left(2^{1/n}\right)^{k}\cdot \delta f_{\text{min}},}"></p><p><span>where </span><em><span>δf**k</span></em><span> is the bandwidth of the </span><em><span>k</span></em><span>-th filter, </span><em><span>f</span></em><span>min is the central frequency of the lowest filter, and </span><em><span>n</span></em><span> is the number of filters per </span><a href='https://en.wikipedia.org/wiki/Octave_(electronics)'><span>octave</span></a><span>. ( - Wikipedia)</span></p><p><strong><span>Shifted (m) STFT:</span></strong></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n74" cid="n74" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.421ex" height="7.493ex" viewBox="0 -1804.3 16112 3226.3" role="img" focusable="false" style="vertical-align: -3.303ex; max-width: 100%;"><defs><path stroke-width="0" id="E82-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 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xlink:href="#E82-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMAIN-32" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-3C0" x="911" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-70" x="1484" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-6D" x="1987" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMAIN-2F" x="2866" y="0"></use><use transform="scale(0.707)" xlink:href="#E82-MJMATHI-4E" x="3365" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-23">X(k,m)=\sum_{p=0}^{N-1}W[n-m]x[n]e^{j2\pi pm/N}</script></div></div><p><span>given data with sampling frequency fs = 1/T, we have filter width </span><em><span>δf_k</span></em><span> and quality factor Q such that</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n76" cid="n76" 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x="0" y="0"></use><use xlink:href="#E85-MJMAIN-5B" x="888" y="0"></use><use xlink:href="#E85-MJMATHI-6B" x="1166" y="0"></use><use xlink:href="#E85-MJMAIN-5D" x="1687" y="0"></use><use xlink:href="#E85-MJMAIN-2212" x="2187" y="0"></use><use xlink:href="#E85-MJMAIN-31" x="3187" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-26">W[k,n]=\alpha-(1-\alpha)cos\frac{2\pi n}{N[k]-1}</script></div></div><p><span>with</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n83" cid="n83" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="28.737ex" height="2.673ex" viewBox="0 -821.2 12372.8 1150.9" 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x="3054" y="0"></use></g><g transform="translate(482,-361)"><use transform="scale(0.5)" xlink:href="#E87-MJMATHI-4E" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E87-MJMAIN-5B" x="888" y="0"></use><use transform="scale(0.5)" xlink:href="#E87-MJMATHI-6B" x="1166" y="0"></use><use transform="scale(0.5)" xlink:href="#E87-MJMAIN-5D" x="1687" y="0"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-28">X[k]=\frac{1}{N[k]}\sum_{n=0}^{N[k]-1}W[k,n]x[n]e^{\frac{-j2\pi Qn}{N[k]}}</script></div></div><p><span>Notes:</span></p><p><span>CQT is well suited to music data as the out put of the transform is effectively amplitude/phase against log frequency which is useful when the frequency spans several octaves. However, in the case of gravitational waves</span></p><h4 id='16-implementation-on-chirp-signal-windowed-fft---tukey-window'><span>1.6 Implementation on Chirp Signal (Windowed FFT - </span><em><span>Tukey window</span></em><span>)</span></h4><p><span>In [1]:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 8.15973px;"><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"><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">matplotlib</span>.<span class="cm-property">pyplot</span> <span class="cm-keyword">as</span> <span class="cm-variable">plt</span></span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 46px;"></div><div class="CodeMirror-gutters" style="display: none; height: 46px;"></div></div></div></pre><p><span>In [2]:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 8.15973px;"><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-variable">dt</span> = <span class="cm-number">0.001</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">t</span> = <span class="cm-variable">np</span>.<span class="cm-property">arange</span>(<span class="cm-number">0</span>,<span class="cm-number">3</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">f0</span> = <span class="cm-number">50</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">f1</span> = <span class="cm-number">250</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">t1</span> = <span class="cm-number">2</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">x</span> = <span class="cm-variable">np</span>.<span class="cm-property">cos</span>(<span class="cm-number">2</span><span class="cm-operator">*</span><span class="cm-variable">np</span>.<span class="cm-property">pi</span><span class="cm-operator">*</span><span class="cm-variable">t</span><span class="cm-operator">*</span>(<span class="cm-variable">f0</span> <span class="cm-operator">+</span> (<span class="cm-variable">f1</span> <span class="cm-operator">-</span> <span class="cm-variable">f0</span>)<span class="cm-operator">*</span><span class="cm-variable">np</span>.<span class="cm-property">power</span>(<span class="cm-variable">t</span>, <span class="cm-number">2</span>)<span class="cm-operator">/</span>(<span class="cm-number">3</span><span class="cm-operator">*</span><span class="cm-variable">t1</span><span class="cm-operator">**</span><span class="cm-number">2</span>)))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">fs</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 cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#Chirp signal</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">t</span>, <span class="cm-variable">x</span>)</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 252px;"></div><div class="CodeMirror-gutters" style="display: none; height: 252px;"></div></div></div></pre><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/chirp_signal.png" referrerpolicy="no-referrer"></p><p><span>In [3]:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 8.15973px;"><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"><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">from</span> <span class="cm-variable">scipy</span> <span class="cm-keyword">import</span> <span class="cm-variable">signal</span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 23px;"></div><div class="CodeMirror-gutters" style="display: none; height: 23px;"></div></div></div></pre><p><span>In [4]:</span></p><pre class="md-fences md-end-block ty-contain-cm modeLoaded" spellcheck="false" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 9.45833px; left: 8.15973px;"><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-variable">freqs</span>, <span class="cm-variable">times</span>, <span class="cm-variable">spectrogram</span> = <span class="cm-variable">signal</span>.<span class="cm-property">spectrogram</span>(<span class="cm-variable">x</span>)</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#Scipy PSD Power Spectral Density</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">freqs</span>, <span class="cm-variable">psd</span> = <span class="cm-variable">signal</span>.<span class="cm-property">welch</span>(<span class="cm-variable">x</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></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">figure</span>(<span class="cm-variable">figsize</span>=(<span class="cm-number">5</span>, <span class="cm-number">4</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">semilogx</span>(<span class="cm-variable">freqs</span>, <span class="cm-variable">psd</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">'PSD: power spectral density'</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">xlabel</span>(<span class="cm-string">'Frequency'</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">ylabel</span>(<span class="cm-string">'Power'</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">tight_layout</span>()</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#Scipy Spectrogram</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">freqs</span>, <span class="cm-variable">times</span>, <span class="cm-variable">spectrogram</span> = <span class="cm-variable">signal</span>.<span class="cm-property">spectrogram</span>(<span class="cm-variable">x</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></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">figure</span>(<span class="cm-variable">figsize</span>=(<span class="cm-number">5</span>, <span class="cm-number">4</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-variable">spectrogram</span>, <span class="cm-variable">aspect</span>=<span class="cm-string">'auto'</span>, <span class="cm-variable">cmap</span>=<span class="cm-string">'hot_r'</span>, <span class="cm-variable">origin</span>=<span class="cm-string">'lower'</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">'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">ylabel</span>(<span class="cm-string">'Frequency band'</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">xlabel</span>(<span class="cm-string">'Time window'</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">tight_layout</span>()</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 481px;"></div><div class="CodeMirror-gutters" style="display: none; height: 481px;"></div></div></div></pre><p> </p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/chirp_psd.png" referrerpolicy="no-referrer" alt="img"></p><p> </p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/chirp_spectrogram_scipy.png" referrerpolicy="no-referrer" alt="img"></p><p> </p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/chirp_spectrogram_matplot.png" referrerpolicy="no-referrer" alt="chirp_spectrogram_matplot"></p><h4 id='17-on-the-implementation-of-time-frequency-analysis-for-cnn'><span>1.7 On the implementation of Time-Frequency Analysis for CNN</span></h4><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/compare_Gabor_CQT.png" referrerpolicy="no-referrer"></p><p><span>Constant Q Transform's high resolution at low frequencies is desirable in many cases for accurate analysis of the signal in both the time and frequency domain. However, when applying CQT for the purpose of identifying specific features on a spectrogram, high resolution in certain directions also means less obvious feature representation in certain cases (</span><strong><span>this assumption is highly dependent on the type of data you have</span></strong><span>). The Q range also affects the result significantly.</span></p><p><span>Some examples on GW chirp signal:</span></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/GW_CQT_verylargeQ.png" referrerpolicy="no-referrer" alt="GW_CQT_verylargeQ"></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/GW_CQT_largeQ.png" referrerpolicy="no-referrer" alt="GW_CQT_largeQ"></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/GW_CQT_smallQ.png" referrerpolicy="no-referrer" alt="GW_CQT_smallQ"></p><p><span>Thus, the time and frequency sampling width/length need to be adjusted accordingly for performance and resolution.</span></p><p><span>A </span><strong><span>spectrogram</span></strong><span> can also be </span><strong><span>inverse transformed</span></strong><span> to time-domain. Taking the generated chirp signal as an example:</span></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/original_chirp.png" referrerpolicy="no-referrer" alt="original_chirp"></p><p><span>is the original chirp signal.</span></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/chirp_spectrogram_cqnsgt.png" referrerpolicy="no-referrer" alt="chirp_spectrogram_cqnsgt"></p><p><span>is the spectrogram using invertible Constant Q - Nonstationary Gabor Transform (Matlab CQT).</span></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/inverse_transform_chirp.png" referrerpolicy="no-referrer" alt="inverse_transform_chirp"></p><p><span>is the inverse transformed chirp signal.</span></p><p><img src="https://raw.githubusercontent.com/xli2522/GW-SignalGen/main/img/original_inverse_difference.png" referrerpolicy="no-referrer" alt="original_inverse_difference"></p><p><span>is the magnitude difference between the original chirp signal and the inverse transformed chirp signal.</span></p><h4 id='18-resources-on-cqt'><span>1.8 Resources on CQT</span></h4><p><span>Invertible CQT: </span><a href='https://www.univie.ac.at/nonstatgab/cqt/index.php' target='_blank' class='url'>https://www.univie.ac.at/nonstatgab/cqt/index.php</a><span> (Useful for the Spectrogram Denoising and Time-series parameter estimation)</span></p><p><span>Gabor Wavelets (bob package): </span><a href='https://pythonhosted.org/bob.ip.gabor/guide.html#gabor-wavelets' target='_blank' class='url'>https://pythonhosted.org/bob.ip.gabor/guide.html#gabor-wavelets</a></p><p><span>Window Functions (Wikipedia): </span><a href='https://en.wikipedia.org/wiki/Window_function' target='_blank' class='url'>https://en.wikipedia.org/wiki/Window_function</a></p><p><span>On the </span><strong><span>discrete Gabor transform</span></strong><span> and the </span><strong><span>discrete Zak transform</span></strong><span>: </span><a href='http://www.martinbastiaans.org/pdfs/sigpro.pdf' target='_blank' class='url'>http://www.martinbastiaans.org/pdfs/sigpro.pdf</a></p><h3 id='part-ii-transform-methods-targeting-chirp-signals'><code>Part II: Transform Methods Targeting Chirp Signals</code></h3><h4 id='21-fractional-fourier-transform-frft'><span>2.1 Fractional Fourier Transform (FrFT)</span></h4><p><span>The Fractional Fourier Transform is a family of linear transforms generalizing the Fourier Transform. It can be thought of as the Fourier Transform to the n-th power, where n need not be an integer. Thus it can transform a function to any intermediate domain between time and frequency. Its applications range from </span><em><span>filter design</span></em><span> and signal analysis to phase retrieval and pattern recognition.</span></p><p><span>A completely different meaning for fractional Fourier Transform was introduced by Bailey and Swartztrauber as essentially another name for a </span><em><span>z transform</span></em><span>, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points. ( - Wikipedia)</span></p><p><span>The FrFT provides a continuous representation of a signal from the time to the frequency domain at intermediate domains by means of the fractional order of the transform that changes from -pi/2 to pi/2. 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x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E94-MJMAIN-31" x="778" y="0"></use></g></g><use xlink:href="#E94-MJMAIN-28" x="5245" y="0"></use><g transform="translate(5634,0)"><g transform="translate(120,0)"><rect stroke="none" width="1163" height="60" x="0" y="220"></rect><use xlink:href="#E94-MJMAIN-31" x="331" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E94-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E94-MJMATHI-3B3" x="500" y="0"></use></g></g></g><use xlink:href="#E94-MJMAIN-29" x="7037" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-35">\alpha = - tan ^{-1}(\frac{1}{2\gamma})</script></div></div><p><span> </span><span>More info see The DLCT and its Applications 2013</span></p><h4 id='22-discrete-chirp-fourier-transform'><span>2.2 Discrete Chirp-Fourier Transform </span></h4><p><strong><span>Discrete Chirp-Fourier Transform</span></strong><span> </span></p><div contenteditable="false" 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Therefore, when using the DCFT to detect a chirp signal, the discrete chirp rate r0 of the signal should be an integer to guarantee that the parameter can be matched and that the peak will not be lost. This restriction affects the practical applications of the DCFT. (page 5)</span></p><p><span> </span><span>More info see The DCFT and its Applications 2013</span></p><h4 id='23-linear-chirp-transform'><strong><span>2.3 Linear Chirp Transform</span></strong></h4><p><span>The DLCT uses discrete complex linear chirp bases. It is not a time-frequency but rather a </span><em><span>frequency chirp-rate</span></em><span> transformation, implementable using Fast Fourier Transform. The Discrete Fourier Transform is a special case of the DLCT which has the properties of modulation and duality in time and frequency. 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(page 15)</span></p><p> </p><p><strong><span>Properties of the Discrete Linear Chirp Transform (page 17):</span></strong></p><p><span>Properties of the DLCT are similar to those of the DFT. </span></p><ul><li><span>Modulation property...</span></li><li><span>Duality property...</span></li></ul><p><strong><em><span>Implementation with the FFT</span></em></strong><span> (page 19)</span></p><p><span>The </span><em><span>DLCT</span></em><span> can be implemented using the FFT. 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xlink:href="#E112-MJMAIN-28" x="8994" y="0"></use><use xlink:href="#E112-MJMATHI-6B" x="9383" y="0"></use><use xlink:href="#E112-MJMAIN-2C" x="9904" y="0"></use><use xlink:href="#E112-MJMATHI-6D" x="10349" y="0"></use><use xlink:href="#E112-MJMAIN-29" x="11227" y="0"></use><use xlink:href="#E112-MJMAIN-7D" x="11616" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-53">=\Re e\{ exp(-j\frac{\pi k}{2N}) H(k,m)\}</script></div></div><p><span>where</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n211" cid="n211" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-54-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="26.296ex" height="2.673ex" viewBox="0 -821.2 11321.9 1150.9" role="img" focusable="false" style="vertical-align: -0.766ex; max-width: 100%;"><defs><path stroke-width="0" id="E113-MJMATHI-48" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 637T455 637Q416 636 405 634T387 623Q384 619 355 500Q348 474 340 442T328 395L324 380Q324 378 469 378H614L615 381Q615 384 646 504Q674 619 674 627T617 637Q594 637 587 639T580 648Q580 650 582 660Q586 677 588 679T604 682Q609 682 646 681T740 680Q802 680 835 681T871 682Q888 682 888 672Q888 645 876 638H874Q872 638 869 638T862 638T853 637T844 637Q805 636 794 634T776 623Q773 618 704 340T634 58Q634 51 638 51Q646 48 692 46H723Q729 38 729 37T726 19Q722 6 716 0H701Q664 2 567 2Q533 2 504 2T458 2T437 1Q420 1 420 10Q420 15 423 24Q428 43 433 45Q437 46 448 46H454Q481 46 514 49Q520 50 522 50T528 55T534 64T540 82T547 110T558 153Q565 181 569 198Q602 330 602 331T457 332H312L279 197Q245 63 245 58Q245 51 253 49T303 46H334Q340 38 340 37T337 19Q333 6 327 0H312Q275 2 178 2Q144 2 115 2T69 2T48 1Q31 1 31 10Q31 12 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z"></path><path stroke-width="0" id="E113-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E113-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E113-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E113-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E113-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E113-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 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661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E113-MJMAIN-7B" d="M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z"></path><path stroke-width="0" id="E113-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E113-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E113-MJMAIN-7D" d="M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z"></path></defs><g 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111 337Q111 355 69 410T27 496ZM562 628Q504 628 443 507L435 491L436 479Q437 471 437 446Q437 396 432 351L529 389L602 426Q673 462 673 463H672Q644 470 637 483T622 553Q608 628 562 628Z"></path><path stroke-width="0" id="E114-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E114-MJMAIN-7B" d="M434 -231Q434 -244 428 -250H410Q281 -250 230 -184Q225 -177 222 -172T217 -161T213 -148T211 -133T210 -111T209 -84T209 -47T209 0Q209 21 209 53Q208 142 204 153Q203 154 203 155Q189 191 153 211T82 231Q71 231 68 234T65 250T68 266T82 269Q116 269 152 289T203 345Q208 356 208 377T209 529V579Q209 634 215 656T244 698Q270 724 324 740Q361 748 377 749Q379 749 390 749T408 750H428Q434 744 434 732Q434 719 431 716Q429 713 415 713Q362 710 332 689T296 647Q291 634 291 499V417Q291 370 288 353T271 314Q240 271 184 255L170 250L184 245Q202 239 220 230T262 196T290 137Q291 131 291 1Q291 -134 296 -147Q306 -174 339 -192T415 -213Q429 -213 431 -216Q434 -219 434 -231Z"></path><path stroke-width="0" id="E114-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E114-MJMATHI-4C" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 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like the FrFT can be used to convert non-sparse signals into sparse signals in time or frequency. (page 27)</span></p><p><strong><span>Sparsity</span></strong></p><p><span>Sparsity or compressibility reflects the fact that information carried by certain signal is much smaller than their bandwidth. Most signals are not sparse in the time domain, so linear transformation are used to make the sparse in either time or frequency using certain basis. Stationary signals, such as sinusoids or quasi-periodic speech segments, are well represented by the Discrete Cosine Transform. The DCT can be used to obtain a sparse representation in frequency for such signals. However, non-stationary signals, such as chirps may not be sparse in either time or frequency, but rather in an intermediate domain. (page 27)</span></p><p><strong><span>Resolution</span></strong></p><p><span>A critical point of the time-frequency analysis and signal separation is the resolution of the transform. The DLCT and the DFrFT have been used to separate linear chirps in the time-frequency plane by projecting them and then followed by a filtering or a windowing procedure. If the resolution of the transform is good, even very close harmonics can be separated easily and vice versa. (details see page 31)</span></p><p><strong><span>Peak Location</span></strong></p><p><span>All the algorithms that use the DLCT or the DFrFT for parametric characterization of chirps depend on searching for peaks for all possible chirp rates or fractional orders to obtain the optimal chirp rate or the optimal fractional order that maximizes the |DLCT{x(n)}| or equivalently |DFrFT{x(n)}|. Therefore, it is obvious that the peaks should occur at the corresponding chirp rates and frequencies. (page 32, </span><strong><span>+ example</span></strong><span>)</span></p><h4 id='32-estimation-of-linear-chirp-parameters'><strong><span>3.2 Estimation of Linear Chirp Parameters</span></strong></h4><p><span>Kalman Filtering is used to estimate parameters of chirps in -</span></p><p><span>[35] W. El Kaakour, M. Guglielmi, J. M. Piasco, and Le Carpentier, “Two identification methods of chirp parameters using state space models,” in Proc. of IEEE International Conference on Digital Signal Processing, Greece, vol. 2, Jul. 1997, pp. 903-906. </span></p><p><span>[36] M. Adjrad, A. Belouchrani, and A. Ouldali, “Estimation of chirp signal parameters using state space modelization by incorporating spatial information,” in Proc. of IEEE Seventh International Symposium on Signal Processing and Its Applications, France, vol. 2, Jul. 2003, pp. 531-534.</span></p><p><span>... (page 34)</span></p><p><strong><span>Chirplet Decomposition</span></strong></p><h4 id='33-sparsity-and-compression'><strong><span>3.3 Sparsity</span></strong><span> and Compression</span></h4><p><span>Sparsity in frequency is a key characteristic we are looking for. (High SNR, High CR, examples page 40)</span></p><h3 id='part-iv-types-of-chirp-signals-and-gw-signal-compression'><code>Part IV: Types of Chirp Signals and GW Signal Compression</code></h3><ul><li><p><span>Linear Chirp</span></p><ul><li><p><span>A linear chirp is a function whose frequency changes linearly with time. 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This is one of the reasons we need to use linear chirp bases instead of the classical Fourier bases because they are more suitable for representing the frequency changes of non-stationary signals. (page 10)</span></p><p> </p></li></ul></li><li><p><span>Quadratic Chirp</span></p></li><li><p><span>Logarithmic Chirp</span></p></li></ul><h3 id='part-v-decomposition-of-non-stationary-signals'><code>Part V: Decomposition of Non-stationary Signals</code></h3><h4 id='51-empirical-mode-decomposition'><strong><span>5.1 Empirical Mode Decomposition</span></strong></h4><p><span>The Hilbert-Huang transform method was developed to represent non-stationary signals in the time-frequency plane without assistance of window functions. It is a combined approach of Hilbert transformation and the empirical mode decomposition. The EMD is used to decompose the signal into a set of functions called Intrinsic Mode Functions (IMFs). It does not have priori defined basis function, unlike the Fourier and Wavelet transform, the whole decomposition is adaptive and depends on the local oscillation of the data.</span></p><p><span>The decomposition is based on the local characteristics time scale of the data and, therefore, it is applicable to </span><strong><span>nonlinear</span></strong><span> and </span><strong><span>non-stationary processes</span></strong><span>. </span></p><p><span>Hilbert Transform is applied to each intrinsic mode function for the purpose of providing the global time-frequency distribution of the underlying signal to estimate its instantaneous frequency. The application of the HHT method to audio and speech signals have already been done. (The HHT is used to estimate the IF of biomedical signals)</span></p><h4 id='52-instantaneous-frequency-estimation'><strong><span>5.2 Instantaneous Frequency Estimation</span></strong></h4><p><span>The Hilbert Transform is used to compute the IF of each of the signal components. (page 53)</span></p><p><span>Compare IF estimation results of DLCT and EMD. (page 54) </span></p><p><span>The performance of the EMD as an IF estimator is very much affected by the presence of the noise. Comparing the estimated IDs based on the DLCT and the EMD of both experiments with the actual IFs, we conclude the DLCT decomposition attains better results than EMD.</span></p><h4 id='53-time-frequency-analysis-using-dlct'><strong><span>5.3 Time-Frequency Analysis using DLCT</span></strong></h4><p><span>Time-frequency distributions (TFDs) are frequently used for IF estimation based on peak detection techniques. The most frequently TFD used for linear chirps is the Wigner-Ville distribution (WVD) due to its ideal representation for such signals. However, in the case of multi-component signals, Wigner-Ville distribution does not perform well because of the presence of extraneous cross-terms. (page 56)</span></p><p><em><span>An algorithm that combines the DLCT with the Wigner-Ville distribution to obtain a time-frequency representation with high resolution.</span></em></p><p><span>Locally the DLCT approximates the signal as a sum of linear chirps, for each of which the WVD provides optimal representations. Superposing these WVDs we obtain a time-frequency representation of the whole signal without interfering cross-terms. (page 57)</span></p><p> </p></div></div>
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