If we consider the mapping n=Ml+m, then it leads to an arrangement in which the first row consists of the first M elements of x(n), the second row consists of the next M elements of x(n), and so on.

The decimation of the data sequence should be repeated again and again until the resulting sequences are reduced to one point sequences. For N=2v, this decimation can be performed v=log2N times. Thus the total number of complex additions is reduced to Nlog2N.

The expression for N point DFT is given as

The above given diagram is the basic butterfly computation in the decimation-in-frequency FFT algorithm.

We observe that the direct computation of F1(k) requires (N/2)2 complex multiplications. The same applies to the computation of F2(k). Furthermore, there are N/2 additional complex multiplications required to compute WNk. Hence it requires N(N+1)/2 complex multiplications to compute X(k).

We know that, X(k) = F1(k)+WNk F2(k)

For a complex valued sequence x(n) of N points, the DFT may be expressed as

In decimation-in-frequency FFT algorithm, the input is taken in order and the output is obtained in the bit reversal order.

The formula for calculating N point DFT is given as

The above given diagram is the basic butterfly computation in the decimation-in-time FFT algorithm.