Derive the sampling theorem

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The sampling theorem states that in order to accurately reconstruct a continuous signal from its samples, the sampling frequency must be at least twice the highest frequency component of the signal.

The sampling theorem, also known as Nyquist-Shannon theorem, was developed by Claude Shannon and Harry Nyquist in the 1940s.
It states that in order to accurately reconstruct a continuous signal from its samples, the sampling frequency must be at least twice the highest frequency component of the signal.
If the sampling frequency is lower than the Nyquist rate, aliasing occurs, leading to distortion in the reconstructed signal.
For example, in audio applications, the human hearing range is typically up to 20 kHz, so a sampling frequency of at least 40 kHz is required to accurately capture and reproduce the audio signal.

Derive the sampling theorem

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