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Unformatted text preview: Ve451 Lecture Notes Dianguang Ma Summer 2010 Chapter 8 Efficient Computation of the DFT: Fast Fourier Transform Algorithms Introduction The DFT plays an important role in many applications of digital signal processing. A major reason for this importance is the existence of efficient algorithms for computing the DFT. Direct Computation of the DFT Basically, the computational problem for the DFT is to compute the sequence ( ) of complex valued numbers given another sequence of data ( ) of length , according to the formular ( ) ( ) kn N n X k N x n N X k x n W = = 1 , 0,1, , 1 In general, the data sequence ( ) is also assumed to be complex valued. N k N x n = K 1 Similarly, the IDFT becomes 1 ( ) ( ) , 0,1, , 1 Since the DFT and IDFT involve basically the same type of computations, our discussion of efficient computational algorithms for the DFT N kn N k x n X k W n N N = = = K applies as well to the efficient computation of the IDFT. 2 We observe that for each value of , direct computation of ( ) involves complex multiplications and 1 complex additions. Consequently, to compute all values of the DFT requires complex multi k X k N N N N plications and ( 1) complex additions. N N Direct computation of the DFT is basically inefficient, primarily because it does not exploit the symmetry and periodicity properties of the phase factor . In particular, these properties are Symmet N W /2 ry property: Periodicity property: k N k N N k N k N N W W W W + + =  = DivideandConquer Approach to Computation of the DFT The divideandconquer approach is based on the decomposition of an point DFT into successively smaller DFTs. This basic approach leads to a family of computationally efficient algorithms known coll N ectively as (FFT) algorithms. fast Fourier transform Let us consider the computation of an point DFT, where Now the sequence ( ) can be stored either in a onedimensional array indexed by , where 1, or as a twodimensional array indexed b N N LM x n n n N =  y and , where 0 1 and 1. l m l L m M   The sequence ( ) can be stored in a varity of ways, each of which depends on the mapping of index to the indices ( , ). For example, the mapping leads to a rowwise arrangement. On the other x n n l m n Ml m = + hand, the mapping leads to a columnwise arrangement. n l mL = + A similar arrangement can be used to store the computed DFT values. In particular, the mapping is from the index to a pair of indices ( , ), where 0 1 and 0 1. If we select the mapping k p q p L q M k Mp   = + the DFT is stored on a rowwise basis. On the other hand, the mapping results in a columnwise storage of ( ). q k qL p X k = + Now suppose that ( ) is mapped into the rectangular array ( , ) and ( ) is mapped into a corresponding rectangular array ( , )....
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 Summer '10
 DianguangMa
 Digital Signal Processing, Signal Processing

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