Editing Dirac comb (section)

==Sampling and aliasing==

Multiplying any function by a Dirac comb transforms it into a train of impulses with integrals equal to the value of the function at the nodes of the comb. This operation is frequently used to represent sampling.
<math display="block"> (\operatorname{\text{Ш}}_{\ T} x)(t) = \sum_{k=-\infty}^{\infty} \!\! x(t)\delta(t - kT) = \sum_{k=-\infty}^{\infty}\!\! x(kT)\delta(t - kT).</math>

Due to the [[Dirac comb#Fourier transform|self-transforming]] property of the Dirac comb and the [[convolution theorem]], this corresponds to convolution with the Dirac comb in the frequency domain.
<math display="block"> \operatorname{\text{Ш}}_{\ T} x \ \stackrel{\mathcal{F}}{\longleftrightarrow}\  \frac{1}{T}\operatorname{\text{Ш}}_\frac{1}{T} * X</math>

Since convolution with a delta function <math>\delta(t-kT)</math> is equivalent to shifting the function by <math>kT</math>, convolution with the Dirac comb corresponds to replication or [[periodic summation]]:
:<math> (\operatorname{\text{Ш}}_{\ \frac{1}{T}}\! * X)(f) =\! \sum_{k=-\infty}^{\infty} \!\!X\!\left(f - \frac{k}{T}\right) </math>

This leads to a natural formulation of the [[Nyquist–Shannon sampling theorem]]. If the spectrum of the function <math>x</math> contains no frequencies higher than B  (i.e., its spectrum is nonzero only in the interval <math>(-B, B)</math>) then samples of the original function at intervals <math>\tfrac{1}{2B}</math> are sufficient to reconstruct the original signal. It suffices to multiply the spectrum of the sampled function by a suitable [[rectangle function]], which is equivalent to applying a brick-wall [[lowpass filter]].
:<math> \operatorname{\text{Ш}}_{\ \!\frac{1}{2B}} x\ \ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \ 2B\, \operatorname{\text{Ш}}_{\ 2B} * X</math>

:<math> \frac{1}{2B}\Pi\left(\frac{f}{2B}\right) (2B \,\operatorname{\text{Ш}}_{\ 2B} * X) = X</math>

In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function."{{sfn|Woodward|1953|pp=33-34}} Hence, it restores the original function from its samples. This is known as the [[Whittaker–Shannon interpolation formula]].

'''Remark''': Most rigorously, multiplication of the rect function with a generalized function, such as the Dirac comb, fails. This is due to undetermined outcomes of the multiplication product at the interval boundaries. As a workaround, one uses a Lighthill unitary function instead of the rect function. It is smooth at the interval boundaries, hence it yields determined multiplication products everywhere, see {{harvnb|Lighthill|1958|p=62}}, Theorem 22 for details.