Discrete Fourier series
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:
(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,
References
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