Fourier Theorems for the DTFT

This section states and proves selected Fourier theorems for the DTFT. A more complete list of Fourier theorems for the Discrete Fourier Transform case (DFT) may be found in [232].^3.2

As introduced in §2.1 above, the Discrete-Time Fourier Transform (DTFT) is defined as

$$X(\omega) \isdef \sum_{n=-\infty}^\infty x(n) e^{-j\omega n},$$

where the symbol $\isdef$ means ``is defined as'' or ``equal by definition.'' We say that $X$ is the spectrum of $x$ .