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In digital signal processing, a discrete Fourier series (DFS) is 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).
For example, several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform based on this reduced number ...
The Fourier series coefficients are: [] ... at intervals of / and performing an inverse discrete Fourier transform (DFT) on samples (see ...
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform. In the latter case, the exponential form of Fourier series synthesizes a discrete-time Fourier transform where variable represents frequency instead of time.
These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued.
In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit, [16] which has been described as the first formula for the DFT, [17] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string. [17]
13.1 Discrete Fourier transforms and fast Fourier ... the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.
In this case, if we make a very large matrix with complex exponentials in the rows (i.e., cosine real parts and sine imaginary parts), and increase the resolution without bound, we approach the kernel of the Fredholm integral equation of the 2nd kind, namely the Fourier operator that defines the continuous Fourier transform. A rectangular ...