1312 lines
32 KiB
HTML
Executable File
1312 lines
32 KiB
HTML
Executable File
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<!doctype html>
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<html lang="en" class="no-js">
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<head>
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width,initial-scale=1">
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<link rel="prev" href="../space_weather/imagers/">
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<link rel="next" href="../faq/">
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<link rel="icon" href="../assets/artemis_icon.ico">
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<meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.5.42">
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<title>Autocorrelation (ACF) - ARTEMIS</title>
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<link rel="stylesheet" href="../assets/stylesheets/main.0253249f.min.css">
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<link rel="stylesheet" href="../assets/stylesheets/palette.06af60db.min.css">
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<link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
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<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
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<style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
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<link rel="stylesheet" href="https://unpkg.com/katex@0/dist/katex.min.css">
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<script>__md_scope=new URL("..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
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</head>
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<body dir="ltr" data-md-color-scheme="default" data-md-color-primary="black" data-md-color-accent="indigo">
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<input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
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<input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
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<label class="md-overlay" for="__drawer"></label>
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<div data-md-component="skip">
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<a href="#autocorrelation-function-acf" class="md-skip">
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Skip to content
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</a>
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</div>
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<div data-md-component="announce">
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</div>
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<header class="md-header md-header--shadow" data-md-component="header">
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<nav class="md-header__inner md-grid" aria-label="Header">
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<a href=".." title="ARTEMIS" class="md-header__button md-logo" aria-label="ARTEMIS" data-md-component="logo">
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<img src="../assets/logo_small_white.svg" alt="logo">
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</a>
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<label class="md-header__button md-icon" for="__drawer">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
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</label>
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<div class="md-header__title" data-md-component="header-title">
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<div class="md-header__ellipsis">
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<div class="md-header__topic">
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<span class="md-ellipsis">
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ARTEMIS
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</span>
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</div>
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<div class="md-header__topic" data-md-component="header-topic">
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<span class="md-ellipsis">
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Autocorrelation (ACF)
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</span>
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</div>
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</div>
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</div>
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<form class="md-header__option" data-md-component="palette">
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<input class="md-option" data-md-color-media="" data-md-color-scheme="default" data-md-color-primary="black" data-md-color-accent="indigo" aria-label="Switch to dark mode" type="radio" name="__palette" id="__palette_0">
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<label class="md-header__button md-icon" title="Switch to dark mode" for="__palette_1" hidden>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2a7 7 0 0 0-7 7c0 2.38 1.19 4.47 3 5.74V17a1 1 0 0 0 1 1h6a1 1 0 0 0 1-1v-2.26c1.81-1.27 3-3.36 3-5.74a7 7 0 0 0-7-7M9 21a1 1 0 0 0 1 1h4a1 1 0 0 0 1-1v-1H9z"/></svg>
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</label>
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<input class="md-option" data-md-color-media="" data-md-color-scheme="slate" data-md-color-primary="black" data-md-color-accent="indigo" aria-label="Switch to light mode" type="radio" name="__palette" id="__palette_1">
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<label class="md-header__button md-icon" title="Switch to light mode" for="__palette_0" hidden>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2a7 7 0 0 1 7 7c0 2.38-1.19 4.47-3 5.74V17a1 1 0 0 1-1 1H9a1 1 0 0 1-1-1v-2.26C6.19 13.47 5 11.38 5 9a7 7 0 0 1 7-7M9 21v-1h6v1a1 1 0 0 1-1 1h-4a1 1 0 0 1-1-1m3-17a5 5 0 0 0-5 5c0 2.05 1.23 3.81 3 4.58V16h4v-2.42c1.77-.77 3-2.53 3-4.58a5 5 0 0 0-5-5"/></svg>
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</label>
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</form>
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<script>var palette=__md_get("__palette");if(palette&&palette.color){if("(prefers-color-scheme)"===palette.color.media){var media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']");palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent")}for(var[key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
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<label class="md-header__button md-icon" for="__search">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
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</label>
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<div class="md-search" data-md-component="search" role="dialog">
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<label class="md-search__overlay" for="__search"></label>
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<div class="md-search__inner" role="search">
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<form class="md-search__form" name="search">
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<input type="text" class="md-search__input" name="query" aria-label="Search" placeholder="Search" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
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<label class="md-search__icon md-icon" for="__search">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
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</label>
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<nav class="md-search__options" aria-label="Search">
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<button type="reset" class="md-search__icon md-icon" title="Clear" aria-label="Clear" tabindex="-1">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12z"/></svg>
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</button>
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</nav>
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</form>
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<div class="md-search__output">
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<div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
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<div class="md-search-result" data-md-component="search-result">
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<div class="md-search-result__meta">
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Initializing search
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</div>
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<ol class="md-search-result__list" role="presentation"></ol>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div class="md-header__source">
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<a href="https://github.com/AresValley/Artemis" title="Go to repository" class="md-source" data-md-component="source">
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<div class="md-source__icon md-icon">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.6.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M439.55 236.05 244 40.45a28.87 28.87 0 0 0-40.81 0l-40.66 40.63 51.52 51.52c27.06-9.14 52.68 16.77 43.39 43.68l49.66 49.66c34.23-11.8 61.18 31 35.47 56.69-26.49 26.49-70.21-2.87-56-37.34L240.22 199v121.85c25.3 12.54 22.26 41.85 9.08 55a34.34 34.34 0 0 1-48.55 0c-17.57-17.6-11.07-46.91 11.25-56v-123c-20.8-8.51-24.6-30.74-18.64-45L142.57 101 8.45 235.14a28.86 28.86 0 0 0 0 40.81l195.61 195.6a28.86 28.86 0 0 0 40.8 0l194.69-194.69a28.86 28.86 0 0 0 0-40.81"/></svg>
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</div>
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<div class="md-source__repository">
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GitHub
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</div>
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</a>
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</div>
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</nav>
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</header>
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<div class="md-container" data-md-component="container">
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<main class="md-main" data-md-component="main">
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<div class="md-main__inner md-grid">
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<div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
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<div class="md-sidebar__scrollwrap">
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<div class="md-sidebar__inner">
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<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
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<label class="md-nav__title" for="__drawer">
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<a href=".." title="ARTEMIS" class="md-nav__button md-logo" aria-label="ARTEMIS" data-md-component="logo">
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<img src="../assets/logo_small_white.svg" alt="logo">
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</a>
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ARTEMIS
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</label>
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<div class="md-nav__source">
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<a href="https://github.com/AresValley/Artemis" title="Go to repository" class="md-source" data-md-component="source">
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<div class="md-source__icon md-icon">
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.6.0 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M439.55 236.05 244 40.45a28.87 28.87 0 0 0-40.81 0l-40.66 40.63 51.52 51.52c27.06-9.14 52.68 16.77 43.39 43.68l49.66 49.66c34.23-11.8 61.18 31 35.47 56.69-26.49 26.49-70.21-2.87-56-37.34L240.22 199v121.85c25.3 12.54 22.26 41.85 9.08 55a34.34 34.34 0 0 1-48.55 0c-17.57-17.6-11.07-46.91 11.25-56v-123c-20.8-8.51-24.6-30.74-18.64-45L142.57 101 8.45 235.14a28.86 28.86 0 0 0 0 40.81l195.61 195.6a28.86 28.86 0 0 0 40.8 0l194.69-194.69a28.86 28.86 0 0 0 0-40.81"/></svg>
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</div>
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<div class="md-source__repository">
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GitHub
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</div>
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</a>
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</div>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href=".." class="md-nav__link">
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<span class="md-ellipsis">
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Welcome
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
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<label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
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<span class="md-ellipsis">
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Getting Started
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_2">
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<span class="md-nav__icon md-icon"></span>
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Getting Started
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../installation/" class="md-nav__link">
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<span class="md-ellipsis">
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Installation
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../run_from_source/" class="md-nav__link">
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<span class="md-ellipsis">
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Run from source
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../build_package/" class="md-nav__link">
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<span class="md-ellipsis">
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Build Package
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</span>
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item">
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<a href="../basic_operations/" class="md-nav__link">
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<span class="md-ellipsis">
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Basic Operations
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
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<label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
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<span class="md-ellipsis">
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Database
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4">
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<span class="md-nav__icon md-icon"></span>
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Database
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../database/db_overview/" class="md-nav__link">
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<span class="md-ellipsis">
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Overview
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</span>
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</a>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4_2" >
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<label class="md-nav__link" for="__nav_4_2" id="__nav_4_2_label" tabindex="0">
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<span class="md-ellipsis">
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Structure
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</span>
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<span class="md-nav__icon md-icon"></span>
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</label>
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<nav class="md-nav" data-md-level="2" aria-labelledby="__nav_4_2_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_4_2">
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<span class="md-nav__icon md-icon"></span>
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Structure
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../database/db_info/" class="md-nav__link">
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<span class="md-ellipsis">
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Info
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</span>
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</a>
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</li>
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<li class="md-nav__item">
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<h1 id="autocorrelation-function-acf">Autocorrelation Function (ACF)</h1>
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<h2 id="theoretical-introduction">Theoretical Introduction</h2>
|
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<p>Correlation functions are valuable mathematical tools utilized across various scientific disciplines, including engineering, physics, and chemistry. The cross-correlation function, commonly defined as a sliding inner (dot) product, quantifies the similarity between two signals as a function of a temporal shift applied to one of them:</p>
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<div class="arithmatex">\[
|
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\left[ k*l \right]\equiv \int_{-\infty}^{+\infty}k^{*}(t)l(t+\tau)dt
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\]</div>
|
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<p>where <span class="arithmatex">\(k\)</span> and <span class="arithmatex">\(l\)</span> are two general non-discrete functions without discontinuity, <span class="arithmatex">\(k^{*}\)</span> is the complex conjugate of <span class="arithmatex">\(k\)</span> and <span class="arithmatex">\(\tau\)</span> is the lag (time delay for signal analysis). This representation is closely related to the convolution theorem by the following relationship:</p>
|
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<div class="arithmatex">\[
|
|
k(t)*l(t)\equiv k^{*}(-t)*l(t)
|
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\]</div>
|
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<p>In our case, the signal must be compared with itself to identify periodic patterns or detect anomalous high degrees of similarity within a given time interval. To achieve this objective, we can employ the Autocorrelation Function (ACF), which is essentially the cross-correlation of a signal with itself, where <span class="arithmatex">\(k(t)=l(t)\)</span>. The following equation defines the ACF, derived from a straightforward modification of the initial equation:</p>
|
|
<div class="arithmatex">\[
|
|
\left[ k*k \right](\tau)\equiv \int_{-\infty}^{+\infty}k^{*}(t)k(t+\tau)dt
|
|
\]</div>
|
|
<p>Within a discrete data set composed by <span class="arithmatex">\(Y(t)\)</span> records we can easily calculate autocorrelation from:</p>
|
|
<div class="arithmatex">\[
|
|
\rho_{\tau}=\frac{\sum_{i=1}^{N-\tau}(Y_i-\bar{Y})(Y_{i+\tau}-\bar{Y})}{\sum_{i=1}^{N}(Y_i-\bar{Y})^2}
|
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\]</div>
|
|
<p>where <span class="arithmatex">\(Y_{i+τ}\)</span> is a lagged data by <span class="arithmatex">\(τ\)</span> of <span class="arithmatex">\(Y_i\)</span> and <span class="arithmatex">\(\bar{Y}\)</span> is the average value of the original data set. Entire denominator is used to keep the normalization condition, indeed ρτ can assume a value between -1 (exact anticorrelation) and +1 (exact correlation). A complete correlation can be only achieved with a perfect overlap of the analyzed function with it self; the simple way to verify that is when <span class="arithmatex">\(τ=0\)</span></p>
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<p><img alt="Screenshot" src="../assets/acf_1.png" /></p>
|
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<h2 id="example-acf-analysis">Example: ACF Analysis</h2>
|
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<h3 id="stanag-4285">STANAG 4285</h3>
|
|
<p>The first example will be a NATO standard known as STANAG 4285. You can download a .wav sample from <a href="../assets/stanag_4285.wav"><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M5 20h14v-2H5m14-9h-4V3H9v6H5l7 7z"/></svg></span> HERE</a>. </p>
|
|
<div align="center">
|
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<audio controls>
|
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<source src="https://raw.githubusercontent.com/AresValley/Artemis/master/docs/assets/stanag_4285.wav" type="audio/wav">
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Your browser does not support the audio player.
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</audio>
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</div>
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|
<p>We start to collect some information about the structure of this signal: a good starting point is the official declassified NATO document dated 1989. The main frame structure can be found at <strong>Annex A-3/5</strong> with a graphical view at <strong>Annex A-7</strong>, also reported below:</p>
|
|
<ul>
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<li><a href="../assets/acf_2.png">STANAG 4285 (i)</a></li>
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<li><a href="../assets/acf_3.png">STANAG 4285 A-3</a></li>
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<li><a href="../assets/acf_4.png">STANAG 4285 A-4</a></li>
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<li><a href="../assets/acf_5.png">STANAG 4285 A-5</a></li>
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<li><a href="../assets/acf_6.png">STANAG 4285 A-7</a></li>
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</ul>
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<div class="admonition note">
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<p class="admonition-title">Summary</p>
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<ul>
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<li>1 x 80 bits of synchronization pattern</li>
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<li>4 x 32 bits of data</li>
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<li>3 x 16 bits of reference</li>
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</ul>
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</div>
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<p>Therefore, a full sequence will be composed as the sum of all symbols reported above: 256 symbols recursively transmitted at the ratio of 2400 bauds means a redundancy of 106.66 ms.
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To analyze an unknown signal I strongly recommend <strong>Signal Analyzer (SA)</strong>. SA is actually one of the most valuable and priceless software available by Sergey Makarov (Макаров Сергей Михайлович, sadly passed away on 29 May 2012) and it used to perform several tests on data sample coded as .wav or .mp3 file. The signal waterfall is reported below:</p>
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<p><img alt="Screenshot" src="../assets/acf_7.png" /></p>
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<p>When an Autocorrelation function is applied on selected data frame (see above), the result shows a clear recursive pattern with a gap of <strong>106.244 ms</strong> between every peak as expected. An ACF refinement can be done into <strong>WaveForm</strong> windows where we can also notice an identical structure as reported by the specifications</p>
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<p><img alt="Screenshot" src="../assets/acf_8.png" />
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<img alt="Screenshot" src="../assets/acf_9.png" /></p>
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<div class="admonition success">
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<p class="admonition-title">Summary</p>
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<ol>
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<li>Synchronization pattern is composed by <span class="arithmatex">\(2400*33.013E-3=79.2\simeq 80\)</span> symbols</li>
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<li>Data group contain <span class="arithmatex">\(2400*13.348E-3=32.0=32\)</span> symbols</li>
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<li>Reference sequence is formed by <span class="arithmatex">\(2400*6.674E-3=16.02\simeq 16\)</span> symbols</li>
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</ol>
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