
$${\hat{z}}_{k}={a}_{k}+i{b}_{k}=\frac{1}{\sqrt{n}}\sum _{j=0}^{n1}{z}_{j}\times \mathrm{exp}\left(i\frac{2\pi jk}{n}\right)\text{, \hspace{1em}}k=0,1,\dots ,n1\text{.}$$ 
$${\hat{w}}_{k}=\frac{1}{\sqrt{n}}\sum _{j=0}^{n1}{z}_{j}\times \mathrm{exp}\left(+i\frac{2\pi jk}{n}\right)\text{,}$$ 
On entry,  ${\mathbf{N}}\le 1$. 