Editing Nyquist–Shannon sampling theorem (section)

==Derivation as a special case of Poisson summation==
When there is no overlap of the copies (also known as "images") of <math>X(f)</math>, the <math>k=0</math> term of {{EquationNote|Eq.1}} can be recovered by the product:

<math display="block">X(f) = H(f) \cdot X_{1/T}(f),</math>

where:

<math display="block">H(f)\ \triangleq\ \begin{cases}1 & |f| < B \\ 0 & |f| > f_s - B. \end{cases}</math>

The sampling theorem is proved since <math>X(f)</math> uniquely determines <math>x(t)</math>.

All that remains is to derive the formula for reconstruction. <math>H(f)</math> need not be precisely defined in the region <math>[B,\ f_s-B]</math> because <math>X_{1/T}(f)</math> is zero in that region. However, the worst case is when <math>B=f_s/2,</math> the Nyquist frequency. A function that is sufficient for that and all less severe cases is''':'''

<math display="block">H(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) = \begin{cases}1 & |f| < \frac{f_s}{2} \\ 0 & |f| > \frac{f_s}{2}, \end{cases}</math>

where <math>\mathrm{rect}</math> is the [[rectangular function]]. Therefore:

:<math>X(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) \cdot X_{1/T}(f)</math>
:::<math> = \mathrm{rect}(Tf)\cdot \sum_{n=-\infty}^{\infty} T\cdot x(nT)\ e^{-i 2\pi n T f}</math>&nbsp; &nbsp; &nbsp; (from &nbsp;{{EquationNote|Eq.1}}, above).
:::<math> = \sum_{n=-\infty}^{\infty} x(nT)\cdot \underbrace{T\cdot \mathrm{rect} (Tf) \cdot e^{-i 2\pi n T f}}_{
\mathcal{F}\left \{
 \mathrm{sinc} \left( \frac{t - nT}{T} \right)
\right \}}.</math>&nbsp; &nbsp; &nbsp;{{efn-ua|group=bottom|The sinc function follows from rows 202 and 102 of the [[Table of Fourier transforms|transform tables]]}}

The inverse transform of both sides produces the [[Whittaker–Shannon interpolation formula]]:

:<math>x(t) = \sum_{n=-\infty}^{\infty} x(nT)\cdot \mathrm{sinc} \left( \frac{t - nT}{T}\right),</math>

which shows how the samples, <math>x(nT)</math>, can be combined to reconstruct <math>x(t)</math>.

* Larger-than-necessary values of <math>f_s</math> (smaller values of <math>T</math>), called ''oversampling'', have no effect on the outcome of the reconstruction and have the benefit of leaving room for a ''transition band'' in which <math>H(f)</math> is free to take intermediate values. [[Undersampling]], which causes aliasing, is not in general a reversible operation.
* Theoretically, the interpolation formula can be implemented as a [[low-pass filter]], whose impulse response is <math>\mathrm{sinc} (t/T)</math> and whose input is <math>\textstyle\sum_{n=-\infty}^{\infty} x(nT)\cdot \delta(t - nT),</math> which is a [[Dirac comb]] function modulated by the signal samples. Practical [[digital-to-analog converter]]s (DAC) implement an approximation like the [[zero-order hold]]. In that case, oversampling can reduce the approximation error.