Discrete Fourier series
Appearance
In digital signal processing, a Discrete Fourier series (DFS) a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).
Introduction
Relation to Fourier series
The exponential form of Fourier series is given by:
which is periodic with an arbitrary period denoted by When continuous time is replaced by discrete time for integer values of and time interval the series becomes:
With constrained to integer values, we normally constrain the ratio to an integer value, resulting in an -periodic function:
Discrete Fourier series
(Eq.1) |
which are harmonics of a fundamental digital frequency
Due to the -periodicity of the kernel, the summation can be "folded" as follows:
- which is proportional to the inverse DFT of one cycle of the function we've denoted by the periodic summation,
Further reading
- Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). "4.2, 8.4". Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0-13-754920-2.
samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n]. ... The Fourier series coefficients can be interpreted as a sequence of finite length for k=0,...,(N-1), and zero otherwise, or as a periodic sequence defined for all k.
- Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (PDF) (1 ed.). Boca Raton,FL: CRC Press. pp. 72, 76. ISBN 978-1-4200-7046-0. Retrieved 4 October 2020.
the DFS coefficients for the periodized signal are a discrete set of values for its DTFT
- Nuttall, Albert H. (Feb 1981). "Some Windows with Very Good Sidelobe Behavior". IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91. doi:10.1109/TASSP.1981.1163506.