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==Aliasing==
{{Main|Aliasing}}
[[File:CPT-sound-nyquist-thereom-1.5percycle.svg|thumb|250px|The samples of two sine waves can be identical when at least one of them is at a frequency above half the sample rate.]]
When <math>x(t)</math> is a function with a [[Fourier transform]] <math>X(f)</math>''':'''
:<math display="block">X(f)\ \triangleq\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ {\rm d}t,</math>
the [[Poisson summation formula]] indicates that the samples, <math>x(nT)</math>, of <math>x(t)</math> are sufficient to create a [[periodic summation]] of <math>X(f)</math>. The result is:
:<math display="block">X_s(f)\ \triangleq \sum_{k=-\infty}^{\infty} X\left(f - k f_s\right) = \sum_{n=-\infty}^{\infty} T\cdot x(nT)\ e^{-i 2\pi n T f}</math> ({{EquationRef|Eq.1}})
[[File:AliasedSpectrum.png|thumb|400pxupright=1.8|right|''<math>X''(''f'')</math> (top blue) and ''X''<sub>A</submath>X_A(''f'')</math> (bottom blue) are continuous Fourier transforms of two {{em|different}} functions, <math>x(t)</math> and <math>x_A(t)</math> (not shown). When the functions are sampled at rate <math>f_s</math>, the images (green) are added to the original transforms (blue) when one examines the discrete-time Fourier transforms (DTFT) of the sequences. In this hypothetical example, the DTFTs are identical, which means {{em|the sampled sequences are identical}}, even though the original continuous pre-sampled functions are not. If these were audio signals, <math>x(t)</math> and <math>x_A(t)</math> might not sound the same. But their samples (taken at rate ''f''<submath>sf_s</submath>) are identical and would lead to identical reproduced sounds; thus ''x''<sub>A</submath>x_A(''t'')</math> is an alias of ''<math>x''(''t'')</math> at this sample rate.]]
which is a periodic function and its equivalent representation as a [[Fourier series]], whose coefficients are <math>T\cdot x(nT).</math>. This function is also known as the [[discrete-time Fourier transform]] (DTFT) of the sample sequence.
As depicted, copies of <math>X(f)</math> are shifted by multiples of the sampling rate <math>f_s</math> and combined by addition. For a band-limited function <math>(X(f) = 0, \text{ for all } |f| \ge B)</math> and sufficiently large <math>f_s,</math> it is possible for the copies to remain distinct from each other. But if the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous <math>X(f).</math> Any frequency component above <math>f_s/2</math> is indistinguishable from a lower-frequency component, called an ''alias'', associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), <math>x(t)</math> is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a [[lowpass filter]], and in this application it is called an [[anti-aliasing filter]].
[[File:ReconstructFilter.png|thumb|right|400pxupright=1.8|Spectrum, ''X<submath>sX_s(f)</submath>''(''f''), of a properly sampled bandlimited signal (blue) and the adjacent DTFT images (green) that do not overlap. A ''brick-wall'' low-pass filter, ''<math>H''(''f'')</math>, removes the images, leaves the original spectrum, ''<math>X''(''f'')</math>, and recovers the original signal from its samples.]]
[[File:Nyquist sampling.gif|500pxupright=1.8|thumb|right|The figure on the left shows a function (in gray/black) being sampled and reconstructed (in gold) at steadily increasing sample-densities, while the figure on the right shows the frequency spectrum of the gray/black function, which does not change. The highest frequency in the spectrum is ½half the width of the entire spectrum. The width of the steadily-increasing pink shading is equal to the sample-rate. When it encompasses the entire frequency spectrum it is twice as large as the highest frequency, and that is when the reconstructed waveform matches the sampled one.]]
==Derivation as a special case of Poisson summation==
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