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Note for Digital Signal Processing - DSP By ANNA SUPERKINGS

  • Digital Signal Processing - DSP
  • Note
  • Anna university - ACEW
  • Electronics and Communication Engineering
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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 , 0kN 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

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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

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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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Fatima Michael College of Engineering & Technology DECIMATION-IN-TIME FFT Consider an N -point signal x[n] of even length. The derivation of the DIT radix-2 FFT begins by splitting the sum into two parts — one part for the even-indexed values x[2n] and one part for the odd-indexed values x[2n + 1]. Define two N/2-point signals x1 [n] and x2 [n] as x0 [n] = x[2n] x1 [n] = x[2n + 1] for 0  n  N/2 1. The DFT of the N -point signal x[n] can be written as X[k] = N X1 x[n] WN nk N X1 + n=0 x[n] WN nk n=0 n even n odd which can be written as N/2 1 X[k] = X N/2 1 x[2n] WN 2nk + n=0 N/2 1 = X = n=0 N/2 1 = X n=0 x[2n + 1] WN (2n+1)k n=0 N/2 1 x0 [n] WN 2nk + n=0 N/2 1 X X X x1 [n] WN (2n+1)k n=0 x0 [n] WN 2nk k N/2 1 + WN · X x1 [n] WN 2nk n=0 N/2 1 nk x0 [n] WN/2 + WN k · X nk x1 [n] WN/2 n=0 where we used the first identity above. Recognizing that the N 2 -pont Fatima Michael College of Engineering & Technology

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