{"id":64464,"date":"2025-05-12T09:40:45","date_gmt":"2025-05-12T06:40:45","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64464"},"modified":"2025-05-12T09:40:45","modified_gmt":"2025-05-12T06:40:45","slug":"inverse-discrete-fourier-transform","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/engineering\/inverse-discrete-fourier-transform\/","title":{"rendered":"Inverse Discrete Fourier Transform Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 class=\"text-lg font-display font-bold\">IDFT Time-Domain Signal Reconstructor<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"64464\" data-favorite-title=\"Inverse Discrete Fourier Transform\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/engineering\/inverse-discrete-fourier-transform\/\" data-favorite-excerpt=\"This calculator performs the Inverse Discrete Fourier Transform (IDFT) to convert a set of frequency domain components back into a discrete time-domain signal. It\u2019s useful in signal processing, digital communication, and audio engineering applications.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-electrical\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/engineering\/inverse-discrete-fourier-transform\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64464\" data-post=\"64464\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single has-term\" id=\"input-1\"><label for=\"field-1\">Frequency domain values<\/label><div class=\"term\">X[k]<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"frequency_domain_values\" id=\"field-1\" step=\"any\" value=\"1, 0, 0, 0\"\/><small>Comma-separated real or complex values (e.g. 1, 0, 0, 0 or 1+2j, 0, -1, 0)<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64464\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Time domain sequence x[n]<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"64464\" aria-live=\"polite\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">2<\/span><span class=\"reaction text-3xl\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<p><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>IDFT Mathematical Definition<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n$$x[n] = \\frac{1}{N} \\sum_{k=0}^{N-1} X[k] \\cdot e^{j 2\\pi kn \/ N}$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$x[n]$$ = output time-domain sample  <\/li>\n<li>$$X[k]$$ = frequency-domain component  <\/li>\n<li>$$N$$ = total number of samples  <\/li>\n<li>$$j$$ = imaginary unit  <\/li>\n<li>$$n$$ = sample index (0 to N-1)  <\/li>\n<li>$$k$$ = frequency index (0 to N-1)<\/li>\n<\/ul>\n<p>This is the standard definition of the IDFT, reconstructing the time-domain signal from its spectral representation.<br \/>\r\n<\/section><br \/>\n<section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Number of Turns \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<p>\n<strong>Given:<\/strong><\/p>\n<ul>\n<li>X[0] = 10, X[1] = -2 + 2j, X[2] = 0, X[3] = -2 &#8211; 2j  <\/li>\n<li>N = 4<\/li>\n<\/ul>\n<p><strong>Calculation:<\/strong><\/p>\n<ol>\n<li>$$x[0] = \\frac{1}{4} \\cdot \\left( X[0] + X[1] + X[2] + X[3] \\right)$$<\/li>\n<li>$$= \\frac{1}{4} \\cdot \\left( 10 + (-2 + 2j) + 0 + (-2 &#8211; 2j) \\right)$$<\/li>\n<li>$$= \\frac{1}{4} \\cdot 6 = 1.5$$<\/li>\n<\/ol>\n<p>\n\r\n<\/section><br \/>\nThe Inverse Discrete Fourier Transform (IDFT) is the fundamental tool in digital signal processing used to transform signals from the frequency domain back into the time domain. It\u2019s especially valuable when analyzing or synthesizing signals in communication systems, filtering, and audio processing. This calculator takes in a list of complex-valued frequency components and computes each corresponding sample in the time domain, supporting educational use and quick verifications.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator performs the Inverse Discrete Fourier Transform (IDFT) to convert a set of frequency domain components back into a discrete time-domain signal. It\u2019s useful in signal processing, digital communication, and audio engineering applications.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[4],"tags":[21],"class_list":["post-64464","post","type-post","status-publish","format-standard","hentry","category-engineering","tag-electrical"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64464","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=64464"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64464\/revisions"}],"predecessor-version":[{"id":64466,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64464\/revisions\/64466"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}