Discrete Time Fourier Transform (DTFT)

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

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

where $\tilde{\omega}\isdef \omega T\in[-\pi,\pi)$ denotes the continuous normalized radian frequency variable,^B.1 and

is the signal amplitude at sample number

The inverse DTFT is

$\displaystyle x(n) = \frac{1}{2\pi}\int_{-\pi}^\pi X(\tilde{\omega}) e^{j\tilde{\omega}n} d\tilde{\omega}$

which can be derived in a manner analogous to the derivation of the inverse DFT (see Chapter 6).

Instead of operating on sampled signals of length (like the DFT), the DTFT operates on sampled signals defined over all integers $n\in{\bf Z}$ . As a result, the DTFT frequencies form a continuum. That is, the DTFT is a function of continuous frequency $\tilde{\omega}\in[-\pi,\pi)$ , 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 points uniformly distributed along the unit circle in the complex plane (see Fig.6.1). Thus, as $N\to\infty$ , a continuous frequency axis must result in the limit along the unit circle in the plane. The axis is still finite in length, however, because the time domain remains sampled.