Signal Notation

• $x \in \mathbb{C}^N$ means $x$ is a length $N$ complex sequence
• $x = x(\cdot)$
• $X = \hbox{\sc DFT}(x) \in \mathbb{C}^N$
• $X(k) = \hbox{\sc DFT}_k(x)$
• $X(k) = \hbox{\sc DFT}_{N,k}(x)$ denotes the $k$ th bin of a length $N$ DFT
• $X(k) \in \mathbb{C}$
• $x(n) = \hbox{\sc IDFT}_n(X)$
• $x(n)\in\mathbb{R}$ or $\mathbb{C}$
• $n,k \in \mathbb{Z}$ or $n,k \in \mathbb{Z}_N$ (integers modulo $N$ )
• For $x\in \mathbb{C}^\infty$ , $X = \hbox{\sc DTFT}(x) \in \mathbb{C}_{2\pi}^{\infty}$
• $\overline{x} =$ conjugate of $x$
• $\angle x =$ phase of $x$