In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic generalized function for some given period . Here is a real variable and the sum extends over all integers . The Dirac delta function and the Dirac comb are tempered distributions. The graph of the function resembles a combs as the comb's 'teeth'), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function.

The symbol , where the period This implies

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

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. The Fourier transform of a Dirac comb is another Dirac comb. 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.

Dirac-comb identity

The Dirac comb can be constructed in two ways, either by using the comb operator (performing sampling) applied to the constant function , or, alternatively, by using the rep operator (performing periodization) applied to the Dirac delta . Formally, this yields the following:

In signal processing, this property on one hand allows sampling by multiplication with , and on the other hand it also allows the periodization by convolution with . The Dirac comb identity is a particular case of the Convolution Theorem for tempered distributions.

Scaling

The scaling property of the Dirac comb follows from the properties of the Dirac delta function for positive real numbers instead of negative ones is not a restriction because the negative sign would only reverse the order of the summation within , which does not affect the result.

Fourier series

is periodic with period for all where (using distribution theory

All Fourier coefficients are

This is a divergent series, when understood as a series of ordinary complex numbers, but becomes convergent in the sense of distributions.

A "square root" of the Dirac comb is employed in some applications to physics, specifically: However this is not a distribution in the ordinary sense.

Fourier transform

The Fourier transform expressed in frequency domain transforms into a rescaled Dirac comb of period , i.e. for

is thus an eigenfunction to the eigenvalue .

This result can be established defined by

is a convergent series of Gaussian functions, and Gaussians transform into Gaussians also results in a series of Gaussians, and explicit calculation establishes that

each Gaussian spike becomes an infinitely sharp Dirac impulse and eventually become indistinguishable from 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:

1. and

Since , we obtain in this limit the result to be demonstrated:

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 its Fourier transform

1. obeys:

and nearly everywhere with the only possible exceptions lying at and for each . This can be summarised by interpreting the Dirac comb as a limit of the Dirichlet kernel such that, at the positions point into the same direction and add constructively. In other words, the continuous Fourier transform of periodic functions leads to

1. with ,

and

The Fourier series when , i.e.

in angular frequency domain (radian/s).

As mentioned, the specific rule depends on the convention for the used Fourier transform. Indeed, when using the scaling property

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

Sampling and aliasing

Due to the self-transforming property of the Dirac comb and the convolution theorem

is equivalent to shifting the function by , convolution with the Dirac comb corresponds to replication or periodic summation:

This leads to a natural formulation of the Nyquist–Shannon sampling theorem contains no frequencies higher than B (i.e., its spectrum is nonzero only in the interval 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.

In time domain, this "multiplication with the rect function" is equivalent to "convolution with the sinc function". 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 , Theorem 22 for details.

Use in directional statistics

In directional statistics is equivalent to a wrapped Dirac delta function and is the analog of the Dirac delta function in linear statistics.

over the unit circle yields the value of that function at zero.

See also

• Comb filter

• Frequency comb

• Poisson summation formula

• Theta function

Notes

References

Further reading