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The Fast Fourier Transform (FFT)The sequences of (10.65) and (10.66) cannot be decomposed further. They justify the statementmade earlier, that each computation produces a vector where each component of this vector, forn = 1, 2, 3 , …,7 is obtained by one multiplication and one addition. This is often referred to as abutterfly operation.Substitution of (10.65) and (10.66) into (10.60), yields2mF even [ m] = F even1 [ m] + W N Feven2 [ m](10.67)Likewise, F odd [ m] can be decomposed into DFTs of length 2; then, Fm [ ] can be computed frommFm ( ) = F even ( m) + W N Fodd ( m) m = 012… , , , , 7(10.68)for N = 8 . Of course, this procedure can be extended for any N that is divisible by 2.Figure 10.5 shows the signal flow graph of a decimation in time, in−place FFT algorithm forN = 8 , where the input is shuffled in accordance with the above procedure. The subscript N inW has been omitted for clarity.Column 0 (x[n]) Column 1 (N /4 ) Column 2 (N / 2) Column 3 (N)X[m]Row 0Row 1Row 2Row 3Row 4Row 5Row 6Row 7x0 [ ]x4 [ ]x2 [ ]x6 [ ]x1 [ ]x5 [ ]x3 [ ]W 0 W 0W 0W 4 W 6 W 7W 4W 0W 4W 0W 4W 0W 2W 4W 6W 0W 2W 4W 1W 2W 3W 4W 5W 6x7 [ ] X7 [ ]Figure 10.5. Signal flow graph of a decimation in time, in-place FFT algorithm, forX0 [ ]X1 [ ]X2 [ ]X3 [ ]X4 [ ]X5 [ ]X6 [ ]N = 8Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications10−23