Notes on Digital Signal Processing: Practical Recipes for Design, Analysis and Implementation

Note 12. Discrete-Time Fourier Transform

The discrete-time Fourier transform (DTFT) is the appropriate Fourier technique to use in order to obtain a continuous-frequency spectrum for a signal that is a function of discrete time. The continuous-frequency spectrum obtained from the DTFT is periodic, with a period equal to T^–1, where T is the discrete-time sampling interval. The DTFT finds widespread use within DSP, primarily because a digital filter’s unit sample response and frequency response comprise a DTFT pair.

The DTFT is defined by

12.1

and the corresponding inverse is given by

12.2

where ω is the continuous radian frequency and T is ...