Delay-Doppler Communications

Appendix B: Some useful matrix properties

B.1 The DFT matrix

The N-point discrete Fourier transform (DFT) matrix is defined as

${\mathbf{F}}_N=\frac{1}{\sqrt{N}}{\left\{{\omega }_N^{kl}\right\}}_{k,l=0}^{N-1}=\frac{1}{\sqrt{N}}\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill {\omega }_N^{1\cdot 1}\hfill & \hfill {\omega }_N^{1\cdot 2}\hfill & \hfill \cdots \hfill & \hfill {\omega }_N^{(N-1)\cdot 1}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 1\hfill & \hfill {\omega }_N^{(N-1)\cdot 1}\hfill & \hfill {\omega }_N^{(N-1)\cdot 2}\hfill & \hfill \cdots \hfill & \hfill {\omega }_N^{(N-1)(N-1)}\hfill \end{array}\right),$

where ${\omega }_N={\mathrm{e}}^{-j2\pi /N}$. Given a vector a of length N we denote its DFT as $\tilde{\mathbf{a}}={\mathbf{F}}_N\mathbf{a}$. The IDFT matrix is given by ${\mathbf{F}}_N^{-1}={\mathbf{F}}_N^{†}$ and $\mathbf{a}={\mathbf{F}}_N^{†}\tilde{\mathbf{a}}$.

The columns of ${\mathbf{F}}_{}$