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**Anna university - ACEW**- Electronics and Communication Engineering
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Fatima Michael College of Engineering & Technology UNIT - I DISCRETE FOURIER TRANSFORMS (DFT) 1. DIRECT COMPUTATION 2. RADIX-2 FFT 3. DECIMATION-IN-TIME FFT 4. FLOWGRAPHS 5. BIT REVERSAL PERMUTATION 6. COMPLEXITY 7. DECIMATION-IN-FREQUENCY FFT Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology THE FFT A fast Fourier transform (FFT) is any fast algorithm for computing the DFT. The development of FFT algorithms had a tremendous impact on computational aspects of signal processing and applied science. The DFT of an N -point signal {x[n], 0 n N 1} is defined as X[k] = N X1 x[n] WN kn , 0kN n=0 where ✓ 2⇡ WN = e = cos N is the principal N -th root of unity. j 2⇡ N ◆ ✓ 2⇡ + j sin N 1 ◆ DIRECT DFT COMPUTATION Direct computation of X[k] for 0 k N (N 1 requires 1)2 complex multiplications N (N 1) complex additions Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology RADIX-2 FFT The radix-2 FFT algorithms are used for data vectors of lengths N = 2K . They proceed by dividing the DFT into two DFTs of length N/2 each, and iterating. There are several types of radix2 FFT algorithms, the most common being the decimation-in-time (DIT) and the decimation-in-frequency (DIF). This terminology will become clear in the next sections. Preliminaries The development of the FFT will call on two properties of WN . The first property is: WN2 = WN/2 which is derived as WN2 = e =e j 2⇡ N ·2 2⇡ j N/2 = WN/2 . More generally, we have nk WN2nk = WN/2 . The second property is: k+ N2 WN = WNk Fatima Michael College of Engineering & Technology

Fatima Michael College of Engineering & Technology which is derived as k+ N2 WN 2⇡ N = ej N (k+ 2 ) 2⇡ 2⇡ N = e j N k · ej N ( 2 ) 2⇡ = ej N k · ej⇡ 2⇡ = ej N k = WNk Fatima Michael College of Engineering & Technology

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