The DFT equation is

$X(K) = \sum_{n=0}^{N-1} x(n) W_N^{nK} \hspace{1cm} K=0,1,2,…..N-1$

In decimation-in-time technique ,we split the input x(n) into odd and even parts.

$X(K) = \sum_{n→even} x(n) W_N^{nK} + \sum_{n→odd} x(n) W_N^{nK} \\ X(K) = \sum_{n=0}^{N/2-1} x(2n) W_N^{2nK} + \sum_{n=0}^{N/2-1} x(2n) W_N^{2n+1}K \\ X(K) = \sum_{n=0}^{N/2-1} x(2n) W_N^{2nK} + …