The input is an aperiodic sequence with length n points, and the output is also an aperiodic sequence with length n points, but the value of each point is different. It converts time domain signals into frequency domain for analysis and processing. The input is each value on the time axis, and the output is the value of each frequency component of the input sequence. It should be noted that the input is an aperiodic sequence of n points, but in the process of operation, we take n as the periodic extension and then perform the operation. The output should be periodic, but only one period is needed to represent the whole sequence, so we only take n points of one period. When doing other operations on FFT, we need to pay attention to its implied periodicity.

This is an epoch-making discovery in the field of digital signal processing, which reduces the calculation of discrete Fourier transform by several orders of magnitude and makes it possible for computers to realize real-time processing. Since the publication of Cooley and Tuji's paper on fast Fourier transform calculation method, digital signal processing has been independent from continuous signal processing, forming a complete system. It is the foundation of the rapid development of modern computer technology.

As for the complex sequence, you can put the complex number into the exponent of the natural constant e, which is the sampling of this function.