{"id":3687,"date":"2021-07-09T11:58:00","date_gmt":"2021-07-09T06:28:00","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=3687"},"modified":"2021-10-30T07:40:35","modified_gmt":"2021-10-30T07:40:35","slug":"what-is-fourier-transform","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-fourier-transform\/","title":{"rendered":"What is Fourier Transform?"},"content":{"rendered":"\n<p>The <strong>Fourier transform<\/strong> simply states that any non-periodic signal which has finite area under the curve can be represented into integrals of the sines and cosines after being multiplied by a certain weight. The Fourier transform is given by<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"342\" height=\"86\" src=\"https:\/\/lh3.googleusercontent.com\/oNvgNrKi97zmmVxMY4iHOMjNnsol0t0cPXTG8OsY8a0jcTziyeW3TlpNY9ijdqiulnxMbkjWv7tGeAAknIpq8LAUEZjlpCcDhSjxPZ7RJYfHu75lrcWkhYCOqeN9rlKM8EKDFMgu\"><\/p>\n\n\n\n<p>The inverse is given as<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"261\" height=\"83\" src=\"https:\/\/lh5.googleusercontent.com\/CeFjs0WTBcZddinQLcuxp5TX7WXwmYH8jfMT42K5Wo4tVLe61ozInrXvKDPat1b1AAtePHbJfs8xMx0H0OuY27KDeN2chkQJKdXr2oPfYChCPT872E6HGJsfeYwxwjDiXso_xJt8\"><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Conditions for Fourier Transform<\/strong><\/h2>\n\n\n\n<p>We can find Fourier transform of a function if it satisfies Dirichlet\u2019s condition. For the existence of its Fourier Transform. The conditions are<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>The function must have finite maxima and minima.<\/li><li>In a given interval of time the function should have finite discontinuities.<\/li><li>The function must be integrable at a given interval of time.<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"But what is the Fourier Transform?  A visual introduction.\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/spUNpyF58BY?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Discrete Time Fourier Transform<\/strong><\/h2>\n\n\n\n<p>The sequence in discrete time can be represented by x[n] and the sequence terms are of complex exponential terms as shown below.<\/p>\n\n\n\n<p>The DTFT sequence is as shown<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"271\" height=\"64\" src=\"https:\/\/lh3.googleusercontent.com\/m91VqYv9g9kWMwNss37y85NPmhBt6c2xS4j6e_iOS2tjHUNKtsZ6MP2UXNVIrxBgkxOvKBDy4dhuY1okGKWNgjAPfnMqV7vlPSVRsgd09B3997gAKgHNzCVDtpyMRSALd0WP76Xl\"><\/p>\n\n\n\n<p>The Inverse DTFT is as shown <\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"306\" height=\"86\" src=\"https:\/\/lh5.googleusercontent.com\/A9b3_AgHoGrV8QgUcIMEUM081SOtTXJv1CDNrjFolaMOGEq9ZN6B8YFDVDBdfpiwkuz54ocASpvLuBelV0WMrU4BiasfpYNUDA9gQkCTw7_6domdX1l7VlhJCjNrcCRRWwx7wnK7\"><\/p>\n\n\n\n<p><strong>Condition for Convergence<\/strong><\/p>\n\n\n\n<p>The term x[n] in the below series is always summable irrespective of the fact whether the series converges or not.<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"201\" height=\"40\" src=\"https:\/\/lh3.googleusercontent.com\/m91VqYv9g9kWMwNss37y85NPmhBt6c2xS4j6e_iOS2tjHUNKtsZ6MP2UXNVIrxBgkxOvKBDy4dhuY1okGKWNgjAPfnMqV7vlPSVRsgd09B3997gAKgHNzCVDtpyMRSALd0WP76Xl\"><\/p>\n\n\n\n<p>Then the sequence will have finite energy if it is absolutely summable and most importantly, finite energy sequence is not always absolutely summable.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Properties<\/strong><\/h2>\n\n\n\n<p>If x(t)\u2194 X(\u2375) and the output y(t)\u2194Y(\u2375). Then the properties of Fourier Transform are <\/p>\n\n\n\n<p><strong>Linearity property<\/strong><\/p>\n\n\n\n<p>Then this property follows the superposition theorem. Moreover the linearity property shows that<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"157\" height=\"73\" src=\"https:\/\/lh6.googleusercontent.com\/3ruKt9Yf4zwNSpIzJ0Sbdt7rtXf1a0H664ffRnKqFlq5XJYPioE_JOJaTj7YKxSBc4h8__Q749LMgYZxiF8hPbER6U2j61i96wfrRAEuaIeUjc8OY8Qm87rUHhUgstj1UziG-Nt-\"><\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"159\" height=\"52\" src=\"https:\/\/lh6.googleusercontent.com\/0ceKfAq8Cn5V2wxma1hoYA6CQU7csis22LQJMlnJ91gLP-rYEtTy6vvxUUjqs1qOtr2reL19aAslTpvRWVylkcp73Yui4HMzI5sOHp17Douzhm16c0EDqu0em_w289i1IQF4dG0e\"><\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"356\" height=\"49\" src=\"https:\/\/lh3.googleusercontent.com\/Yy5B98E72OpDOJJ7_N4Qfxvx8xoEpYNKIqdiJ08R7DqfoBsudWVkkITfS0rhHRRY2sapxAAM6bsX_DvT_jAHLjxSZbVkSgEHt7jiv6TFXcJeTWO4YJs_xWxEiN2sF96NySumwtwd\"><\/p>\n\n\n\n<p><strong>Time Shifting Property<\/strong><\/p>\n\n\n\n<p>The time-shifting property means that a shift in time corresponds to a phase rotation in the frequency domain.<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"273\" height=\"60\" src=\"https:\/\/lh5.googleusercontent.com\/ebnI3bzMVeHA1X1Dtl3FRwmBs9QkeLYV8c2E3weQsuM3-yExOxNRCRozqvWD2wb2UAiizpZQebZcNvbQwm3yRsgmNUCJKfIy-8nHtgpDZU3vD3_mf5tOK9vd1QOSK6UodPIiXLI7\"><\/p>\n\n\n\n<p><strong>Frequency Shifting Property<\/strong><\/p>\n\n\n\n<p>Secondly, This property shows that multiplication in frequency domain results in frequency shifting<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"274\" height=\"70\" src=\"https:\/\/lh5.googleusercontent.com\/lTj0BDLpBDVhSMHHaydR8Z69Zhh72IeuPZS7tmvPUKYkle7qMGURRzgb5pZZOP59MKwVGk1lHMfz5R0PusqZp9cldXCAhc7Kq2dd_qQL4ygSNGwlz97jUdEY4vH-n5jSOtgyKCM2\"><\/p>\n\n\n\n<p><strong>Time Reversal Property<\/strong><\/p>\n\n\n\n<p>Lastly the time reversal property is as shown<\/p>\n\n\n\n<p><strong><img loading=\"lazy\" decoding=\"async\" width=\"202\" height=\"70\" src=\"https:\/\/lh3.googleusercontent.com\/zga6XfBR0ZgHspotBJ8U8b3S3GJrhvH2kGYY8jtRLxvV14erpkqq-Fh4RjwLGMZlpU6ehzfqqfNwWD-SynEMNDMG-3bkcLhCDkD70cHnsUjwPCGc2_WmZXGlBExMKDxxZPaeYs_2\"><\/strong><\/p>\n\n\n\n<p><strong>Time Scaling Property<\/strong><\/p>\n\n\n\n<p>This property shows that if a function is expanded in time by a quantity a , the F.T is compressed in frequency by the same amount. <\/p>\n\n\n\n<p>Hence, we can write the property is as shown in equation.<\/p>\n\n\n\n<p><strong><img loading=\"lazy\" decoding=\"async\" width=\"142\" height=\"68\" src=\"https:\/\/lh5.googleusercontent.com\/SY4A9vPQWdRUjsMZZvPDKQ4nJpKil93Y-TjOX0K_jBOluahhBK_FL2U62d6O-7SxRjz6JrMIJAuUdlDCat4T2NvhylcLhJO90ZDlLQ30El56ZMvZgryESx9GIkJ1aOgzwfZW-duo\"><\/strong><\/p>\n\n\n\n<p><strong>Differentiation Property<\/strong><\/p>\n\n\n\n<p><strong><img loading=\"lazy\" decoding=\"async\" width=\"238\" height=\"84\" src=\"https:\/\/lh3.googleusercontent.com\/RX6QV-yjF49yquzssE78lQjWTyeP-1NURXScHi5Lr4y5kejzJYHDK7v0odD5fsCSgbWyyDvEqiLoyc8rqcvWsNIJhtwBYvKgroYVeO4oc0o0hPL8iQEoPn9cVNB1L_F2sz7yoUQj\"><\/strong><\/p>\n\n\n\n<p><strong>Integration Property<\/strong><\/p>\n\n\n\n<p><strong><img loading=\"lazy\" decoding=\"async\" width=\"248\" height=\"58\" src=\"https:\/\/lh3.googleusercontent.com\/HXTI5Yg80XYUWeCvE38R0Oh5-J1fNLksnniXq3a4tWyd8CJicrNgfOZI6DL7Kh65pSWqS39Vsr-n5nT2IKwI0UHwD3CQ9fZlYbP-xHVUyTzRtGLxh27em9wwgoybJ7tDNoal9j_N\"><\/strong><\/p>\n\n\n\n<p><strong>Convolution Property<\/strong><\/p>\n\n\n\n<p><strong><img loading=\"lazy\" decoding=\"async\" width=\"320\" height=\"59\" src=\"https:\/\/lh6.googleusercontent.com\/xxN_eqTxRNR_U7HIuGHeQLjtV2Tu44_8UXkYkcIfZYtkZN8KqmNRTa4VETVBsIOTPX_ZqSVzekefF0jiyBDWUsqcVCqb_WpbqNAVMxphPnRqII864BwGeqg8_YqgPaUA7w_SoSIg\"><\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Application<\/strong><\/h2>\n\n\n\n<p>Their main application is in&nbsp; image compression, filtering and image analysis.<\/p>\n\n\n\n<p><strong>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-uninterrupted-power-supply-ups\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is UPS?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-are-flip-flops-in-sequential-circuits\/\" target=\"_blank\" rel=\"noreferrer noopener\">What are Flip-Flops?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-are-plcs\/\" target=\"_blank\" rel=\"noreferrer noopener\">Working of PLCs<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-5g\/\" target=\"_blank\" rel=\"noreferrer noopener\">What is 5G?<\/a><\/li><li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-a-mosfet\/\" target=\"_blank\" rel=\"noreferrer noopener\">Characteristics of MOSFET<\/a><\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>The Fourier transform states that non-periodic signal having finite area under the curve can be represented into integrals of sine and cosine<\/p>\n","protected":false},"author":3,"featured_media":3716,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[37],"tags":[],"class_list":["post-3687","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electronics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is Fourier Transform? 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