Dirac comb

It may be considered to be the discrete analog of the Dirac comb.

Using the Dirac comb distribution and its Fourier series:

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

The Dirac comb may also be expressed as a sum of exponentials, so we may write:

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform.

This series of sharp spectral lines is called a frequency comb or a frequency Dirac comb.

A fundamental wrapped distribution is the Dirac comb which is a wrapped delta function:

For Fourier transform purposes, sampling is modeled as a product between s(t) and a Dirac comb function.

Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with s(t).

When the time interval between adjacent samples is a constant (T), the sequence of delta functions is called a Dirac comb.

Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:

The scaling property of the Dirac comb follows from the properties of the Dirac delta function.

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

Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the "unwrapped" distribution and a Dirac comb:

The Fourier transform of a periodic function, s(t), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

Mathematically, that process is often modelled as the output of a lowpass filter whose input is a Dirac comb whose teeth have been weighted by the sample values.

Thus the DTFT of the s[n] sequence is also the Fourier transform of the modulated Dirac comb function.

For example, a sensitivity window comprising a Dirac comb combined with a rectangular pulse, is considered a multiple exposure, even though the sensitivity never goes to zero during the exposure.

Theoretically, the interpolation formula can be implemented as a low pass filter, whose impulse response is sinc(t/T) and whose input is which is a Dirac comb function modulated by the signal samples.

In directional statistics, the Dirac comb of period 2π is equivalent to a wrapped Dirac delta function, and is the analog of the Dirac delta function in linear statistics.

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.