So far, we have discussed the Fourier series and Fourier transform on the assumption that we know a continuous signal wave in the infinite time domain. However, in practical experiments, the data acquired, converted from the data measured by an analog-to-digital converter, are sequences of data 
( n = 0, 1, 2, …, n -1, n ) that are “discrete” and with “finite number”. To perform spectrum analysis using these finite numbers of discrete data, we must use the discrete Fourier transform (DFT). This DFT is defined as follows: Given N data sampled with the interval 
the DFT is defined as a series expansion on the assumption that the original signal is periodic function with the period
(although the original signal is not necessary periodic). However, various problems occur in the course of this processing. The first is the aliasing problem. When the signal is sampled with interval ,
information about the components with frequencies higher than
is lost. Therefore, we must pay attention to valid range of the spectra obtained. The second is the problem of the coincidence of periods. It is impossible to know the correct period of the original signal before the measurement. Therefore, the period of the original signal and the period of DFT do not coincide, and this difference produces the leakage error. We will discuss this leakage error and its countermeasure later. The third is the problem about the length of measurement. In the case of an isolated signal
x(t), we cannot have data in an infinite time range. However, since the Fourier coefficients 
and Fourier transform
by connecting the values of
smoothly. In the following, we explain how to compute DFT.