eg 8 4 2 1 with 3 stages With the input sequence in natural order computations

Eg 8 4 2 1 with 3 stages with the input sequence in

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arrive at 1-point transforms that correspond to the actual DFT. (e.g. 8 4 2 1, with 3 stages) With the input sequence in natural order, computations can be done, but the DFT result is in bit-reversed order and must be reordered. (001 100) For a 4-point input, binary indices :{00, 01, 10, 11} bit order :{ x [0], x [1], x [2], x [3]} 8-point input sequence binary indices :{000, 001, 010, 011, 100, 101, 110, 111} bit order :{ x [0], x [1], x [2], x [3], x [4], x [5], x [6], x [7]}
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12 Separating even and odd indices, and letting x [ n ] = x a and x [ n + N /2] = x b 1 , 2, 1, 0, = k , 2 2 2 1 0 2 N nk N n b a DFT W x x k X N 1 , 2, 1, 0, = k , 1 2 2 2 1 0 2 N nk N n N n b a DFT W W x x k X N
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13 The computations result in a butterfly structure with: Inputs : x [ n ] and x [ n + ] Outputs : X DFT [2 k ] = { x [ n ] + x [ n + ]} X DFT [2 k + 1] = { x [ n ] - x [ n + ]} 2 N 2 N 2 N n N W
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14 A typical butterfly for the decimation-in- frequency FFT algorithm
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15 The factors W t is called twiddle factors , appear only in the lower corners of the butterfly wings at each stage. Their exponents t have a definite order, described as follows for an N = 2 m -point FFT algorithm with m stages: Number P of distinct twiddle factors W t at i th stage P = 2 m - i . Values of t in the twiddle factors W t : t = 2 i - 1 Q with Q = 0, 1, 2, ..., P - 1.
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16 The first stage of decimation in frequency (DIF) FFT algorithm for N = 8 P = 2 m - i = 2 3 - 1 = 4. Thus Q = 0, 1, 2, 3. Stage 1 : i = 1, t = 2 i - 1 Q = 2 0 (0) = 0 t = 2 0 (1) = 1 t = 2 0 (2) = 2 t = 2 0 (3) = 3
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17 The second stage of decimation in frequency (DIF) FFT algorithm for N = 8. P = 2 m - i = 2 3-2 = 2. Thus Q = 0, 1. Stage 2 : i = 2, t = 2 i - 1 Q = 2 1 (0) = 0 t = 2 2 - 1 Q = 2 1 (1) = 2
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18 The third stage of decimation in frequency (DIF) FFT algorithm for N = 8 P = 2 m - i = 2 3 - 3 = 1. Thus Q = 0. Stage 3 : i = 3, t = 2 3 - 1 Q = 2 2 (0) = 0
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19 Example 4.3 Determine DFT for the signal, using 2-stages Fast Fourier Transform (FFT). } 1 , 0 , 2 , 4 { ] [ n y
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20 The Decimation in Time FFT Algorithm In the decimation in time (DIT) FFT algorithm, the process starts with N 1-point transforms, combine adjacent pairs at each successive stage into 2-point transforms, then 4-point transforms, and so on until we get a single N -point DFT result.
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  • Fall '16
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  • DFT, Fast Fourier transform, N samples of xp

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