Given a discrete signal $x_n\(, and its Discrete Fourier Transform \)X(kf_ s)=\mathbf{DFT}_N [x(nT_s)]\(, where \)n \in [0,N]$ , and \(N\) is a composite numer: $N=N_1 \cdot N_2\(, the following two equations hold \)\forall \space k \in [0,N]$
$$X_k = \mathbf{DFT}_ N[x_n] =\sum_ {i=0}^ {N_2-1} W_N^ {i\cdot k} \cdot \mathbf{DFT}[x_ {(i + j \cdot N_2)}] \qquad\quad j=0,1,\dots,N_1-1$$
$$X_k = \mathbf{DFT}_ N[x_n] =\sum_ {i=0}^ {N_2-1} W_N^ {k \cdot i \cdot N_1} \cdot \mathbf{DFT}[x_ {(i\cdot N_1 +j)}] \qquad\quad j=0,1,\dots,N_1-1$$
Where:
$W_N^k=e^{-j\cdot2\pi\frac{k}{N}}$ are called Twiddle Factors
The operation of dividing \(N\) points into $N_2$ subsets having $N_1$ points is called decimation of order $N_2$.
The decimation done in the first equation is called decimation in time (DIT).
The decimaton done in the second equation is called decimation in frequency (DIF).