# DFT, DFT and Fourier Series.

Cuthbert Nyack
The application of both the Fourier Series and the DFT result in discrete spectra. The Fourier Series uses a continuous periodic function in the time domain and gives a nonperiodic discrete spectrum. The discrete periodic function used by the DFT results in a discrete periodic spectrum. The difference between the 2 is in the periodicity of the spectrum. There is also a numerical difference because the Fourier series uses areas and an averaging factor, while the DFT uses samples and no averaging factor.

The applet below illustrates the relation between the 2. If the sampling frequency is high enough so there is no aliasing, then the DFT spectrum between 0 and ws/2 multiplied by the "averaging factor" 2/Ns (Ns is the number of samples) is equal to the Fourier Series. Increasing Nf2 to 100 show the usual periodicity of the DFT spectra, while the Fourier Series spectra remains nonperiodic.

The eg parameters (3.0, 4.0, 0.01, 11.39, 40, NA, 100, 150.0, 26.0, 4.0) show that aliasing has started to affect accuracy of the equivalence. Reducing the sampling frequency by reducing Ns introduces more aliasing. Leaving Ns at 40 and changing T2 to 12.26 removes the spectral leakage and restores the equivalence.

For the Fourier Series, the lines in the spectrum are given by wn = n2p/T rad/s where T is the period.
For the DFT the lines are given by wm = m2p/NT rad/s where T is the sampling time and N is the number of samples.