Fourier Theorems for the DTFT

This section states and proves selected Fourier theorems for the DTFT. A more complete list for the DFT case is given in [264].^3.4Since this material was originally part of an appendix, it is relatively dry reading. Feel free to skip to the next chapter and refer back as desired when a theorem is invoked.

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

$$X(\omega) \isdef \sum_{n=-\infty}^\infty x(n) e^{-j\omega n}.$$ (3.8)

We say that $X$ is the spectrum of $x$ .