Decimation in Time

When $N$ is even, the DFT summation can be split into sums over the odd and even indexes of the input signal:

$$X(\omega_k) \isdef \hbox{\sc DFT}_{N,k}\{x\} \isdef \sum_{n=0}^{N-1} x(n) e^{-j\omega_k n T}$$

$$= \sum_{{\stackrel{n=0}{\vspace{2pt}\mbox{\tiny$n$\ even}}}}^{N-2} ... ...stackrel{n=0}{\vspace{2pt}\mbox{\tiny$n$\ odd}}}}^{N-1} x(n) e^{-j\omega_k n T}$$

$$= \sum_{n=0}^{\frac{N}{2}-1} x(2n) e^{-j2\pi {k\over N/2} nT} + e^{j2\pi\frac{k}{N}}\sum_{n=0}^{\frac{N}{2}-1} x(2n+1) e^{-j2\pi {k\over N/2} nT},$$

$$\isdef \hbox{\sc DFT}_{\frac{N}{2},k}\{\hbox{\sc Downsample}_2(x)\}$$

$$\mathop{\quad} +\;W_N^k\cdot\hbox{\sc DFT}_{\frac{N}{2},k}\{\hbox{\sc Downsample}_2[\hbox{\sc Shift}_1(x)]\} \protect$$ (H.1)

Thus, the length $N$ DFT is computable using two length $N/2$ DFTs. The complex factors $W_N^k=e^{j\omega_k}=\exp(j2\pi k/N)$ are called twiddle factors. The splitting into sums over even and odd time indexes is called decimation in time. (For decimation in frequency, the inverse DFT of the spectrum $X(\omega_k)$ is split into sums over even and odd bin numbers $k$.)