Discrete Time Fourier Transform (DTFT)

The Discrete Time Fourier Transform (DTFT) can be viewed as the limiting form of the DFT when its length $N$ is allowed to approach infinity:

$$X(\omega) \isdefs \sum_{n=-\infty}^\infty x(n) e^{-j\omega n},$$ (3.2)

where $\omega\in[-\pi,\pi)$ denotes the continuous radian frequency variable,^3.3 and $x(n)$ is the signal amplitude at sample number $n$ .

The inverse DTFT is

$$x(n) \eqsp \frac{1}{2\pi}\int_{-\pi}^\pi X(\omega) e^{j\omega n} d\omega,$$ (3.3)

which can be derived in a manner analogous to the derivation of the inverse DFT [264].

Instead of operating on sampled signals of length $N$ (like the DFT), the DTFT operates on sampled signals $x(n)$ defined over all integers $n\in\mathbb{Z}$ .

Unlike the DFT, the DTFT frequencies form a continuum. That is, the DTFT is a function of continuous frequency $\omega\in[-\pi,\pi)$ , while the DFT is a function of discrete frequency $\omega_k$ , $k\in[0,N-1]$ . The DFT frequencies $\omega_k = 2\pi k/N$ , $k=0,1,2,\ldots,N-1$ , are given by the angles of $N$ points uniformly distributed along the unit circle in the complex plane. Thus, as $N\to\infty$ , a continuous frequency axis must result in the limit along the unit circle. The axis is still finite in length, however, because the time domain remains sampled.