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 B.1 summarizes all four Fourier-transform cases corresponding to discrete or continuous time and/or frequency.

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