Nyquist–Shannon sampling theorem: Difference between revisions

Revision as of 10:55, 14 October 2023

Revision as of 11:08, 14 October 2023

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==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_s(f),</math>&nbsp; &nbsp; &nbsp; where''':'''

where:

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

:<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>

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_s(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''':'''

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_s(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>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>

:<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 rect(•) is the [[rectangular function]].&nbsp; Therefore''':'''

\rightwhere <math>\mathrm{rect}}.</math>&nbsp; &nbsp;is the [[rectangular function]]. &nbsp;Therefore:{{efn-ua|group=bottom|The sinc function follows from rows 202 and 102 of the [[Table of Fourier transforms|transform tables]]}}

:<math>X(f) = \mathrm{rect} \left(\frac{f}{f_s} \right) \cdot X_s(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 display="block">\begin{align}

:::<math> = \sum_{n=-\infty}^{\infty} x(nT)\cdot \underbrace{T\cdot \mathrm{rect} (Tf) \cdot e^{-i 2\pi n T f}}_{

:<math>X(f) &= \mathrm{rect} \left(\frac{f}{f_s} \right) \cdot X_s(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;\text{ (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 \}}.

\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]]}}

\end{align}</math>

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

:<math display="block">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 ''f<submath>sf_s</submath>'' (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.

==Shannon's original proof==