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However, in case of discrete Fourier series, the summation goes from n=0 to N-1, where N is the fundamental period of the signal. This is because a discrete signal has a finite number of samples, while a continuous signal has an infinite number of values.

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The summation goes from n=0 to infinity in case of Fourier series in terms of sin(nx) and cos(nx). In terms of eAhmad Kishki said:

A discrete Fourier series is a mathematical tool used to represent a periodic signal as a sum of sinusoidal functions. It is a way to break down a complex signal into simpler components, making it easier to analyze and manipulate.

A discrete Fourier series is used to represent signals that are sampled at discrete time intervals, whereas a continuous Fourier series is used for signals that are continuous in time. This means that a discrete Fourier series can only represent signals that are periodic with a finite number of samples.

The Nyquist frequency is the maximum frequency that can be accurately represented in a discrete Fourier series. It is important because if a signal is sampled at a rate less than twice the Nyquist frequency, aliasing will occur and the signal cannot be accurately reconstructed.

A discrete Fourier series is used to analyze and manipulate signals in signal processing. It is commonly used in applications such as filtering, noise reduction, and compression. It is also used in digital signal processing to convert analog signals into digital signals.

A discrete Fourier series has a wide range of applications in various fields such as physics, engineering, mathematics, and computer science. It is used for image and sound processing, data compression, pattern recognition, and solving differential equations, to name a few.

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