Fourier Transforms for Continuous/Discrete Time/Frequency

The Fourier transform can be defined for signals which are

• discrete or continuous in time, and
• finite or infinite in duration.

This results in four cases. Quite naturally, the frequency domain has the same four cases,

• discrete or continuous in frequency, and
• finite or infinite in bandwidth.

When time is discrete, the frequency axis is finite, and vice versa.

This book has been concerned almost exclusively with the discrete-time, discrete-frequency case (the DFT), and in that case, both the time and frequency axes are finite in length. In the following sections, we briefly summarize the other three cases. Table E summarizes all four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

Table E.1: Four cases of sampled/continuous time and frequency.