Dirac comb

The Dirac comb ( also Dirac impulse sequence or Shah function) describes a periodic sequence of Dirac shocks. Clearly he has the shape of a comb and is symbolized because of this similarity, often with the Cyrillic letter Ш ( Shah ).

Application is the Dirac comb in mathematics and signal processing by Fourier analysis.

Definition

The Dirac comb is a periodic Schwartz tempered distribution, which makes use of the Dirac delta distributions.

Clearly, for a period T. of the Dirac comb is composed therefore of infinitely many Dirac shocks that are spaced apart T.

So apply for the implementation of the Dirac comb on a test function

Fourier transform of the Dirac comb

The Poisson summation formula states that the Dirac comb ( of period 1 ) is a fixed point of the Fourier transform. More generally,

Wherein for the continuous Fourier transform of the standard in the literature to signal processing Convention

Is used.

Sampling and aliasing

With the help of the Dirac comb allows the sampling of a function described mathematically by multiplying the sampled function:

The multiplication of a smooth, fast -falling continuous signal with a Dirac comb is a model of an ideal sampler (English: sampler ) at the sampling rate T.

In the theory of signal processing, the Dirac comb is an elegant tool to prove and understand disturbing aliasing, the Nyquist -Shannon sampling theorem.