The , also known as '''Whittaker–Shannon-Kotelnikov sampling theorem''', is a fundamental result in the field of Information Theory , in particular Telecommunication s. In addition to E. T. Whittaker (statistical theorem published 1915), Claude Shannon and Harry Nyquist , it is also attributed to V. A. Kotelnikov , and sometimes referred to as, simply, the ''sampling theorem''.
Sampling is the process of converting a signal (e.g., a function of continuous time or space) into a numeric sequence (a function of discrete time or space). The theorem states conditions under which the samples represent no loss of information and can therefore be used to reconstruct the original signal with arbitrarily good fidelity. It states that the signal must be Bandlimited and that the Sampling Frequency must be at least twice the signal bandwidth.
A signal that is bandlimited is constrained in terms of how fast it can change and therefore how much detail it can convey in between discrete moments of time. The sampling theorem means that the discrete samples are a complete representation of the signal if the bandwidth is less than half the sampling rate, which is referred to as the Nyquist Frequency . Frequency components that are above the Nyquist frequency are subject to a phenomenon called Aliasing , which is undesirable in most applications. The severity of the problem depends on the relative strength of the aliased components.
To formalize these concepts, let represent a Real-valued Continuous-time signal and let represent its unitary Fourier Transform (to the domain of ordinary frequency, Hz ). I.e.
:
The figure depicts a bandlimited whose highest frequency is . I.e.
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0 \mbox{ for } f \ge f_\mathrm{H} \ </math>,
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"0" align="right"
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0 for all ''f'' > ''f<sub>H</sub>'') '''and''' consider what condition would allow no overlap of the tails of adjacent images ''X''(''f''):
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T \mathrm{rect} \left( rac{f}{f_s}
ight) = \begin{cases}T & f < rac{f_s}{2} \ 0 & f \ge rac{f_s}{2} \end{cases}</math>
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The bandlimited property also reduces the inverse transform to this form, where again we rely on
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Therefore, by comparison we can observe that
, which shows that
can be fully represented by just the discrete time samples of
. Of course, that means
can also be represented by its samples. (Q.E.D.)
And of course that representation is again the
Nyquist–Shannon Interpolation Formula . To derive it, we substitute
into the Fourier series and substitute the series for
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\sum_{n=-\infty}^\infty x(nT) \int_{-\infty}^{\infty} T\cdot \operatorname{rect}\left[ rac{f}{1/T}
ight] \cdot e^{j (2\pi f) (t-nT)}\df</math>
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\sum_{n=-\infty}^\infty x(nT)\cdot \mathcal{F}^{-1} \{ T\cdot \operatorname{rect}(T f) \}(t-nT)\,</math>
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\sum_{n=-\infty}^\infty x(nT)\cdot \operatorname{sinc} \left (t-nT)/T
ight \,</math>
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\mathcal{F} \{ x \}(f) = 0</math> for <math>f \,</math> outside the interval <math> {Link without Title} \,</math>,
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