Twiddle Factor Notation

In FFT terminology, $W_N^k$ denotes the $k$ th ``twiddle factor,'' where $W_N$ is a primitive $N$ th root of unity:

$$W_N \mathrel{\stackrel{\mathrm{\Delta}}{=}}e^{-j2\pi/N}.$$

The aliasing expression can therefore be written as

$$\begin{eqnarray*} Y(z) &=& \frac{1}{N} \sum_{m=0}^{N-1} X\left(z^\frac{1}{N}e^{-jm\frac{2\pi}{N}} \right), \; z\in\mathbb{C}\\ [5pt] &=& \frac{1}{N} \sum_{m=0}^{N-1} X(W_N^m z^{1/N}). \end{eqnarray*}$$