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  1. Copilot Answer
    Wikipedia
    https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem
    Nyquist–Shannon sampling theorem - Wikipedia
    Introduction

    Sampling is a process of converting a signal (for example, a function of continuous time or space) into a sequence of values (a function of discrete time or space). Shannon's version of the theorem states:
    A sufficient … See more

    Aliasing

    When is a function with a Fourier transform :
    Then the samples, of are sufficient to create a periodic summation of (see Discrete-time Fourier transform#Relation to Fourier Transform):
    which is a … See more

    Derivation as a special case of Poisson summation

    When there is no overlap of the copies (also known as "images") of , the term of Eq.1 can be recovered by the product:
    where:
    The sampling theorem is proved since uniquely de… See more

    Shannon's original proof

    Poisson shows that the Fourier series in Eq.1 produces the periodic summation of , regardless of and . Shannon, however, only derives the series coefficients for the case . Virtually quoting Shannon's original paper:
    Let be … See more

    Wikipedia text under CC-BY-SA license

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