{"id":3768,"date":"2021-07-09T12:11:06","date_gmt":"2021-07-09T06:41:06","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=3768"},"modified":"2026-04-11T05:13:40","modified_gmt":"2026-04-10T23:43:40","slug":"what-is-fft","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-fft\/","title":{"rendered":"What is FFT?"},"content":{"rendered":"\n<p>Certain algorithms permit implementations of Discrete Fourier transform with considerable savings in computation time. These algorithms are known as Fast Fourier Transform. Also known as FFT.<\/p>\n\n\n\n<p>FFT algorithms are based on the fundamental principle of decomposing the computation of DFT of sequence length into successively smaller discrete Fourier transforms.<\/p>\n\n\n\n<p>They are basically two classes of FFT algorithms<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Decimation in time<\/li>\n\n\n\n<li>Decimation in frequency<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"The Fast Fourier Transform (FFT): Most Ingenious Algorithm Ever?\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/h7apO7q16V0?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>Decimation-in-time algorithm<\/strong> FFT<\/h2>\n\n\n\n<p>This algorithm is known as Radix-2 DIT-FFT algorithm which means the number of output points N can be expressed as a power of 2 that is N = 2<sup>M<\/sup>&nbsp; where M is an integer.<\/p>\n\n\n\n<p>Let x(n) be a sequence where N is assumed to be a power of 2. Decimate or break this sequence into two sequences of length N\/2 where one sequence consists of even-indexed values of x(n) and the other of odd-indexed values of x(n).<\/p>\n\n\n\n<p>xe(n)&nbsp; = x(2n) &nbsp; n= 0,1,\u2026\u2026\u2026.. N\/2-1<\/p>\n\n\n\n<p>xo(n) = x(2n+1)&nbsp; n= 0,1,\u2026\u2026\u2026.. N\/2 -1<\/p>\n\n\n\n<p>The N-point DFT of x(n) can be written as&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/07\/image-5-1.png?w=1170&#038;ssl=1\" alt=\"\" class=\"wp-image-3770\"\/><\/figure>\n\n\n\n<p>Separating x(n) into even and odd indexed values of x(n) we obtain&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.goseeko.com\/blog\/wp-content\/uploads\/2021\/07\/image-4-1.png?w=1170&#038;ssl=1\" alt=\"\" class=\"wp-image-3769\"\/><\/figure>\n\n\n\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" width=\"319\" height=\"347\" src=\"https:\/\/lh4.googleusercontent.com\/db0BcdsosJW8LQOwlLFn4X9E4RvudzPmZcxyVP62eAabGc9fwYZJeK9B-VisNHcedXfRJ0Ea6XtiNiijmh-BjmePLNNAoxw_7HURXxaVY58VCqv5FJ6ENtJTlVMVONwiJHOebj0q\"><\/p>\n\n\n\n<p>W<sub>8<\/sub><sup>4<\/sup> = -1<\/p>\n\n\n\n<p>and W<sub>8<\/sub> <sup>5<\/sup> = &#8211; W <sub>8<\/sub><sup>1<\/sup><\/p>\n\n\n\n<p>also W<sub>8<\/sub><sup>6<\/sup> = &#8211; W<sub>8<\/sub><sup>2<\/sup><\/p>\n\n\n\n<p>W<sub>8<\/sub><sup>7<\/sup> = &#8211; W<sub>8<\/sub><sup>3<\/sup><\/p>\n\n\n\n<p>G(0) \u2013 H(0) = x(4)<\/p>\n\n\n\n<p>G(1)&nbsp; &#8211; W <sub>8<\/sub><sup>1 <\/sup>H(1) = x(5)<\/p>\n\n\n\n<p>G(2) &#8211; W<sub>8<\/sub><sup>2 <\/sup>H(2) = x(6)<\/p>\n\n\n\n<p>G(3) &#8211; W<sub>8<\/sub><sup>3 <\/sup>&nbsp;H(3) = x(7)<\/p>\n\n\n\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" width=\"397\" height=\"253\" src=\"https:\/\/lh4.googleusercontent.com\/x61MnlkGyqNGb7qM04uGa7ykuiNw782FDlX1vtQKFs3x9vBu5op-IGpzPPEETvxJRfljOXHmYJPMMl3tWEb7pmjlBw5DPja9mtcB-qidD4r1E1jfo1MNWQEN8j1bZPnjxPc8_WfU\"><\/p>\n\n\n\n<p>Let&nbsp; us consider one example now, to understand the concept.<\/p>\n\n\n\n<p>Consider the sequence x[n]={ 2,1,-1,-3,0,1,2,1}. Calculate the FFT.<\/p>\n\n\n\n<p>Arrange the sequence as x(0) x(4) x(2) x(6) x(1) x(5) x(3) x(7)<\/p>\n\n\n\n<p>Since N=8 find the values W<sub>8<\/sub> <sup>0 <\/sup>to W <sub>8<\/sub><sup>7<\/sup><\/p>\n\n\n\n<p>Apply the butterfly diagram to obtain the&nbsp; values.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" width=\"409\" height=\"293\" src=\"https:\/\/lh5.googleusercontent.com\/yjIoIOsujsH_FMfK2eE_wi8eS0lrKvAR5vYwk_oiO-QIXExt-lKa7zpvtCEjO3BhTgn8zuig9YQaqP5V3CPBJdoUBC4wUfloXxU-Fr0XOFbai5iCgUEm_eNsj4H0PbM0ephHnvwg\"><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Decimation in Frequency<\/strong><\/h2>\n\n\n\n<p>Apart from time sequence, we can represent an N-point sequence in frequency. Let us take a four-point sequence to understand it better.<\/p>\n\n\n\n<p>Let the sequence be x[0],x[1],x[2],x[3]\u2026\u2026\u2026..x[7]<\/p>\n\n\n\n<p>Mathematically, this sequence will be as follows <\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"485\" height=\"28\" src=\"https:\/\/lh3.googleusercontent.com\/qTq5WZ3_llSMwwntSKyY_PG0eD945XgO_BjMxi0DS2UJBFDRIQknvBrPS3R-q4jssYcwmsYhzB3WxW0bv5DUaWuFpv6t1fB6ze29nnNk2ZW9mvPkEfE0YaHAZcviVwc1AHock41g\"><\/p>\n\n\n\n<p>Now let us make one group of sequence numbers 0 to 3 and another group of sequence 4 to 7. Now, mathematically this will be as follows <\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"412\" height=\"49\" src=\"https:\/\/lh5.googleusercontent.com\/X1n2C4zuCu7qddy2nZsrwFam6ybuVWk6_ABh2c3g-LgKvDm4xUkEWV0VBHWpatycg2mjH2oRHYUZdz7MCRECImj4DSPspymXliQtWFuSN1ud9xwRE7zL0SAQseppaZHhuB9R5Bi7\"><\/p>\n\n\n\n<p>Let us replace n by r, where r = 0, 1 , 2\u2026.N\/2\u22121.&nbsp;<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"212\" height=\"41\" src=\"https:\/\/lh3.googleusercontent.com\/wpoDcwj9kL3gKCSUXxQRhqHIi5XAYSsa37VT0yLXm4HzXnximaBSzA8CkySImiI6fl7yNrFSfsFft5i58njwohL62pRfKLcAR-84IQ6ZUSdi9Erf71Sw_Xkv_S11PfhkktyYbhTk\"><\/p>\n\n\n\n<p>We take the first four points&nbsp; x[0],x[1],x[2],x[3] initially, and try to represent them mathematically as follows \u2013<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"424\" height=\"196\" src=\"https:\/\/lh6.googleusercontent.com\/zqcK33NyK-bF5a9WxhVDEW1HNjiCMHM_W4uLf26iyU_RZtlZzvNHGKboeUWeObzFPgAER5QXMjJqNXXvSEFz5fEHpWHfGdZDIbPa_T0ZPFah9oI8_8EvFQZyxZ9H7-sSGz7H6c2p\"><\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" width=\"624\" height=\"35\" src=\"https:\/\/lh6.googleusercontent.com\/-1XVFcI29HvEcBQI-gfyae_2vTV1Vdg8s1utU81XzbbGRyZ2MJvXcY8K_xqHaA7ui3jpDKABjeVeTdWji9k43TK6ziRpKhSoD-dXkzZbi2DmXLsRACzQDTESXj7UqorpAfoQwjb5\"><\/p>\n\n\n\n<p><strong>Interested in learning about similar topics? 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