Dirac comb

[1] Here t is a real variable and the sum extends over all integers k. The Dirac delta function

s as the comb's teeth), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function.

Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel:[1]

The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on tempered distributions, without any reference to Fourier series.

Owing to the Convolution Theorem on tempered distributions which turns out to be the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling by multiplication with it, but it also allows modelling periodization by convolution with it.

, or, alternatively, by using the rep operator (performing periodization) applied to the Dirac delta

In signal processing, this property on one hand allows sampling a function

[7] The Dirac comb identity is a particular case of the Convolution Theorem for tempered distributions.

A "square root" of the Dirac comb is employed in some applications to physics, specifically:[9]

expressed in frequency domain (Hz) the Dirac comb

also results in a series of Gaussians, and explicit calculation establishes that The functions

are thus each resembling a periodic function consisting of a series of equidistant Gaussian spikes

whose respective "heights" (pre-factors) are determined by slowly decreasing Gaussian envelope functions which drop to zero at infinity.

each Gaussian spike becomes an infinitely sharp Dirac impulse centered respectively at

converge to the same function and this limit function is a series of infinite equidistant Gaussian spikes, each spike being multiplied by the same pre-factor of one, i.e., the Dirac comb for unit period: Since

can be found by exploiting the scaling property of the Fourier transform, Another manner to establish that the Dirac comb transforms into another Dirac comb starts by examining continuous Fourier transforms of periodic functions in general, and then specialises to the case of the Dirac comb.

In order to also show that the specific rule depends on the convention for the Fourier transform, this will be shown using angular frequency with

For the special case of the Fourier transform of the Dirac comb, the Fourier series integral over a single period covers only the Dirac function at the origin and thus gives

This can be summarised by interpreting the Dirac comb as a limit of the Dirichlet kernel such that, at the positions

As mentioned, the specific rule depends on the convention for the used Fourier transform.

Indeed, when using the scaling property of the Dirac delta function, the above may be re-expressed in ordinary frequency domain (Hz) and one obtains again:

Finally, the Dirac comb is also an eigenfunction of the unitary continuous Fourier transform in angular frequency space to the eigenvalue 1 when

Due to the self-transforming property of the Dirac comb and the convolution theorem, this corresponds to convolution with the Dirac comb in the frequency domain.

, convolution with the Dirac comb corresponds to replication or periodic summation: This leads to a natural formulation of the Nyquist–Shannon sampling theorem.

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.

This is due to undetermined outcomes of the multiplication product at the interval boundaries.

It is smooth at the interval boundaries, hence it yields determined multiplication products everywhere, see Lighthill 1958, p. 62, Theorem 22 for details.

is usually distributed over the real-number line, or some subset thereof, and the probability density of

is a function whose domain is the set of real numbers, and whose integral from

is a function whose domain is some interval of the real numbers of length