{"id":1014,"date":"2021-12-19T19:48:12","date_gmt":"2021-12-19T11:48:12","guid":{"rendered":"https:\/\/blog.cauchyschwarz.com\/?p=1014"},"modified":"2021-12-19T19:48:15","modified_gmt":"2021-12-19T11:48:15","slug":"dft","status":"publish","type":"post","link":"https:\/\/blog.cauchyschwarz.com\/?p=1014","title":{"rendered":"DFT"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-right counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69e0c3b7e2b42\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69e0c3b7e2b42\" checked aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#Basic_Mathematics\" >Basic Mathematics<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#%E6%AD%A3%E5%BC%A6%E6%B3%A2Sinewave\" >\u6b63\u5f26\u6ce2(Sinewave)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#%E7%82%B9%E7%A7%AFScalar_Product_Dot_Product\" >\u70b9\u79ef(Scalar Product, Dot Product)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#%E5%8D%B7%E7%A7%AFConvolution\" >\u5377\u79ef(Convolution)<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E5%85%AC%E5%BC%8F\" >\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u7684\u516c\u5f0f<\/a><ul class='ez-toc-list-level-2' ><li class='ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#%E5%85%B3%E4%BA%8E%E7%9B%B8%E4%BD%8D\" >\u5173\u4e8e\u76f8\u4f4d<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-1'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/blog.cauchyschwarz.com\/?p=1014\/#Inverse_DFT\" >Inverse DFT<\/a><\/li><\/ul><\/nav><\/div>\n<h1 class=\"wp-block-heading\" id=\"basic-mathematics\"><span class=\"ez-toc-section\" id=\"Basic_Mathematics\"><\/span>Basic Mathematics<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"%E6%AD%A3%E5%BC%A6%E6%B3%A2sinewave\"><span class=\"ez-toc-section\" id=\"%E6%AD%A3%E5%BC%A6%E6%B3%A2Sinewave\"><\/span>\u6b63\u5f26\u6ce2(Sinewave)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$ x[n] = A\\cos {(\\omega nT+\\phi)} = A\\cos{(2\\pi fnT+\\phi)} $<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\\( A \\) : amplitude<\/li><li>\\( \\omega \\) : angular frequency in radians\/seconds<\/li><li>\\( f=\\omega \/ 2\\pi \\) : frequency in Hertz(cycles\/seconds)<\/li><li>\\( \\phi \\) : initial phase in radians<\/li><li>\\( n \\) : time index<\/li><li>\\( T=1\/f_s \\) : sampling period in seconds\\( t=nT=n\/f_s \\)<\/li><\/ul>\n\n\n\n<p>\u8fd9\u91cc\u7684\\(T\\)\u4e0d\u662f\u6307\u7684\u6b64\u4e09\u89d2\u51fd\u6570\u7684\u5468\u671f\uff0c\u800c\u662f\u6307\u7684\u91c7\u6837\u70b9\u4e4b\u95f4\u95f4\u9694\u7684\u65f6\u95f4\u3002\u6240\u4ee5\u8fd9\u91cc\u7684\\(f\\)\u548c\\(f_s\\)\u4e5f\u662f\u4e0d\u540c\u7684\u6982\u5ff5\uff0c\\(f\\)\u548c\\(\\omega\\)\u6307\u7684\u662f\u6b64\u4e09\u89d2\u51fd\u6570\u81ea\u8eab\u7684\u9891\u7387\uff0c\u800c\\(f_s\\)\u6307\u7684\u662f\u91c7\u6837\u7684\u9891\u7387\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u8fd9\u4e2a\u516c\u5f0f\u7684\u542b\u4e49\u5c31\u662f\u5bf9\u4e8e\u4e00\u4e2a\u632f\u5e45\u4e3a\\(A\\)\uff0c\u521d\u59cb\u76f8\u4f4d\u4e3a\\(\\phi\\)\uff0c\u89d2\u901f\u5ea6\u4e3a\\(\\omega\\)\u6216\u8005\u8bf4\u9891\u7387\u4e3a\\(f\\)\u7684\\(cos\\)\u8fd9\u79cd\u5f62\u5f0f\u7684\u4e09\u89d2\u51fd\u6570\uff0c\u6211\u4eec\u6309\u7167\u91c7\u6837\u95f4\u9694\u4e3a\\(T\\)\u6216\u8005\u8bf4\u91c7\u6837\u9891\u7387\u4e3a\\(f_s\\)\u7684\u65b9\u5f0f\u53bb\u91c7\u6837\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4e0b\u6807\u4e3a\\(n\\)\u7684\u91c7\u6837\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u8fd9\u4e2a\u516c\u5f0f\u53bb\u8ba1\u7b97\u5176\u91c7\u6837\u503c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"%E7%82%B9%E7%A7%AFscalar-product-dot-product\"><span class=\"ez-toc-section\" id=\"%E7%82%B9%E7%A7%AFScalar_Product_Dot_Product\"><\/span>\u70b9\u79ef(Scalar Product, Dot Product)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$ &lt;x,y&gt; = \\sum\\limits_{n=0}^{N-1}{x[n]\\times y^{*}[n]} $&nbsp;<strong>\u8fd9\u91cc\u7684\u661f\u53f7\u8868\u793a\u53d6\u5171\u8f6d\uff0c\u5373\u4e24\u4e2a\u5e8f\u5217\u7684\u70b9\u79ef\u7b49\u4e8e\u524d\u4e00\u4e2a\u5e8f\u5217\u548c\u540e\u4e00\u4e2a\u5e8f\u5217\u7684\u5171\u8f6d\u5e8f\u5217\u7684\u5185\u79ef\u3002<\/strong>&nbsp;\u5982\u679c\u70b9\u79ef\u4e3a\\(0\\)\uff0c\u90a3\u4e48\u8868\u793a\u8fd9\u4e24\u4e2a\u5e8f\u5217\u662f\u6b63\u4ea4\u7684\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"%E5%8D%B7%E7%A7%AFconvolution\"><span class=\"ez-toc-section\" id=\"%E5%8D%B7%E7%A7%AFConvolution\"><\/span>\u5377\u79ef(Convolution)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$ y[n]=(x_1[n]*x_2[n])<em>n = \\sum\\limits<\/em>{m=0}^{N-1}x_1[m]x_2[n-m] $<\/p>\n\n\n\n<h1 class=\"wp-block-heading\" id=\"%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E5%85%AC%E5%BC%8F\"><span class=\"ez-toc-section\" id=\"%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E5%85%AC%E5%BC%8F\"><\/span>\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u7684\u516c\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>$ X[k] = \\sum\\limits_{n=0}^{N-1}x[n]e^{-j2\\pi kn\/N} k=0,&#8230;,N-1 $<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\\(n\\): discrete time index (normalized time, \\(T=1\\))<\/li><li>\\(k\\): discrete frequency index<\/li><li>\\(\\omega_k=2\\pi k\/N\\): frequency in radians<\/li><li>\\(f_k=f_sk\/N\\): frequency in Hz(\\(fs\\):sampling rate)<\/li><\/ul>\n\n\n\n<p>\u53ef\u4ee5\u8fd9\u6837\u7406\u89e3\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u6211\u4eec\u6709\u4e00\u7cfb\u5217\u7684\u57fa\u672c\u6b63\u5f26\u6ce2\uff1a\\(s[n] = e^{j2\\pi kn\/N} = e^{j2\\pi k\/N \\times n}\\)\uff0c\u8fd9\u4e9b\u6b63\u5f26\u6ce2\u7684\u9891\u7387\u4ece\\(0\\)\u5f00\u59cb\uff0c\u6309\u7167\\(f_s\/N\\)\u7684\u95f4\u9694\u9012\u589e\uff0c\u4e00\u76f4\u5230\\(f_s(N-1)\/N\\)\u3002\u9891\u7387\u4e3a\\(0\\)\u53ef\u4ee5\u7406\u89e3\u4e3a\u5e38\u6570\uff0c\u6216\u8005\u8bf4\u8be5\u6b63\u5f26\u6ce2\u7684\u9891\u7387\u7b49\u4e8e\u91c7\u6837\u9891\u7387\uff0c\u8fd9\u6837\u6bcf\u6b21\u91c7\u6837\u51fa\u6765\u7684\u503c\u90fd\u662f\u76f8\u7b49\u7684\u3002\u56e0\u6b64\u8fd9\u7ec4\u6b63\u5f26\u6ce2\u7684\u9891\u7387\u5230\\(f_s(N-1)\/N\\)\u5c31\u622a\u6b62\u4e86\u3002\u81f3\u4e8e\u9891\u7387\u5927\u4e8e\\(f_s\\)\u7684\u6b63\u5f26\u6ce2\uff0c\u7531\u4e8e\u5176\u9891\u7387\u5927\u4e8e\u4e86\u91c7\u6837\u9891\u7387\uff0c\u56e0\u6b64\u5728\u8fd9\u91cc\u8003\u8651\u8fd9\u4e9b\u6b63\u5f26\u6ce2\u662f\u6ca1\u6709\u610f\u4e49\u7684\u3002<\/li><li>\u6211\u4eec\u6709\u4e86\u8fd9\u7ec4\u57fa\u672c\u7684\u6b63\u5f26\u6ce2\u4e4b\u540e\uff0c\u7136\u540e\u7531\u4e8e&#8221;\u70b9\u79ef\u6b63\u597d\u53ef\u4ee5\u8ba1\u7b97\u51fa\u539f\u59cb\u4fe1\u53f7\u4e2d\u6bcf\u4e2a\u6b63\u5f26\u6ce2\u6240\u5360\u7684\u6bd4\u4f8b&#8221;\uff0c\u4e8e\u662f\u5c31\u6709\u4e86\u8fd9\u4e2a\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u7684\u516c\u5f0f\u3002<\/li><\/ol>\n\n\n\n<p>\u5bf9\u4e8e\u4e0a\u9762\u8fd9\u4e2a\u516c\u5f0f\uff0c\u7531\u4e8e\u5df2\u7ecf\u8bf4\u4e86\u4f7f\u7528<em>normalized time, T=1<\/em>\uff0c\u56e0\u6b64\u91c7\u6837\u9891\u7387\\(f_s\\)\u4e5f\u4e3a1\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"%E5%85%B3%E4%BA%8E%E7%9B%B8%E4%BD%8D\"><span class=\"ez-toc-section\" id=\"%E5%85%B3%E4%BA%8E%E7%9B%B8%E4%BD%8D\"><\/span>\u5173\u4e8e\u76f8\u4f4d<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u5bf9\u4e8e\u7b97\u51fa\u6765\u7684\\(X[k]\\)\uff0c\u5982\u679c\u6211\u4eec\u53d6\u6a21(\\(X[k]\\)\u662f\u4e00\u4e2a\u590d\u6570\uff0c\u90a3\u4e48\u7b97\u51fa\u6765\u7684\u5c31\u662f\u8be5\u6b63\u5f26\u6ce2\u6240\u5360\u7684\u80fd\u91cf\/Amplitude\uff0c\u5982\u679c\u7b97\u5e45\u89d2\uff0c\u90a3\u4e48\u7b97\u51fa\u6765\u7684\u5c31\u662f\u8be5\u6b63\u5f26\u6ce2\u5728\u8fd9\u4e2a\u4fe1\u53f7\u4e2d\u7684\u4f4d\u7f6e\/Phase\/\u521d\u59cb\u76f8\u4f4d\u3002<\/p>\n\n\n\n<h1 class=\"wp-block-heading\" id=\"inverse-dft\"><span class=\"ez-toc-section\" id=\"Inverse_DFT\"><\/span>Inverse DFT<span class=\"ez-toc-section-end\"><\/span><\/h1>\n\n\n\n<p>$ x[n] = \\frac{1}{N} \\sum\\limits_{k=0}^{N-1}{X[k]s_k[n]} = \\frac{1}{N}\\sum\\limits_{k=0}^{N-1}{X[k]e^{j2\\pi kn\/N}} n=0,1,&#8230;,N-1 $ \u9006\u5411\u79bb\u6563\u5085\u91cc\u53f6\u548c\u6b63\u5411\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362\u7684\u533a\u522b\u6709\u4e24\u70b9\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u4f7f\u7528\u7684\u662f\\(e^{j2\\pi kn\/N}\\)<\/li><li>\u6c42\u548c\u7ed3\u679c\u9700\u8981\u9664\u4ee5\\(N\\)<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Basic Mathematics \u6b63\u5f26\u6ce2(Sinewave) $ x[n] = A\\cos {(\\omega nT+\\phi)} = A\\cos{(2\\pi fnT+\\phi)} $ \\( A \\)&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[115,114],"class_list":["post-1014","post","type-post","status-publish","format-standard","hentry","category-10","tag-audio-signal-processing","tag-coursera"],"_links":{"self":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/1014","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1014"}],"version-history":[{"count":1,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/1014\/revisions"}],"predecessor-version":[{"id":1015,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=\/wp\/v2\/posts\/1014\/revisions\/1015"}],"wp:attachment":[{"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1014"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1014"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.cauchyschwarz.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1014"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}