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FFT ALGORITHMS FOR COMPOSITE N

FFT ALGORITHMS FOR COMPOSITE N. efficient computation of the DFT is still possible if the sequence length may be written a power of 2. However, the sequence length may be written as a product of factors. Suppose that N may be factored as follows: N = N . N 2.

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FFT ALGORITHMS FOR COMPOSITE N

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  1. FFT ALGORITHMS FOR COMPOSITE N efficient computation of the DFT is still possible if the sequence length may be written a power of 2. However, the sequence length may be written as a product of factors . Suppose that N may be factored as follows: N = N . N 2 We then decompose x(n) into N2 sequences of length N 1 and arrange these sequences in an array as follows

  2. By defining the index map for both n,k as The sequence x(n)is mapped into the rectangular array x(n1,n2),consequently X(k) is mapped into a corresponding rectangular array X(k1,k2)

  3. The N-point DFT is expressed as

  4. Step1: compute the N1- point DFT for each of the row n2=0, 1,….,N2 -1 (i.e. for N2 sequences) Step 2: compute new rectangular array Where Step 3: compute the N2-point DFT for each column n1=0, 1…... N1-1 of the array

  5. Computational speed of index mapping FFT algorithm Complex multiplication = Complex addition =N

  6. Example : derive the necessary 15-point DFT by using mixed radix , decimation in time FFT algorithm. Solution :For a sequence of length N = 15. with N1 = 3 and N2 = 5, the sequence x(n) may be decimated into five sequences of length three, and these sequences may then be arranged in a two-dimensional array as follows:

  7. Alternatively, if we let N1= 5 and N2 = 3, x(n) may be decimated into three sequences of length five and arranged in a two-dimensional array of three rows and five columns, Now. we compute the three-point DFTs for each of the five rows. This leads to the following 5 x 3 array:

  8. The next step is to multiply each of the terms F(n2,k1) by the phase factors . This computation results in the 5 x 3 array: The final step is to compute the five-point DFTs for each of the three columns. This computation yields the desired values of the DFT in the form

  9. When the input sequence x(n) and the output DFT X (k) in the two-dimensional arrays are read across from row 1 through row 5, we obtain the following sequences: We observe that the input data sequence is shuffled from the normal order in the computation of the DFT. On the other hand, the output sequence occurs in normal order. In this case the rearrangement of the input data array is due to segmentation of the one-dimensional array into a rectangular array and the order in which the DFTs are computed. This shuffling of either the input data sequence or the output Data sequence is a characteristic of most algorithms.

  10. the algorithm that we have introduced involves the following computations: Algorithm 1 1. Store the signal column-wise. 2. Compute the N1-point DFT of each row. 3. Multiply the resulting array by the phase factors wN A Compute the N2-point DFT of each column 5. Read the resulting array row-wise.

  11. Computation of N = 15-point DFT by means of 3-point and 5-point DFTs.

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