Dirac comb

The Dirac crest (also Dirac thrust sequence or Shah function ) describes a periodic sequence of Dirac thrusts . It clearly has the shape of a comb and is often symbolized with the Cyrillic letter Ш (Shah) because of this similarity .

The Dirac comb is used in mathematics and signal processing using Fourier analysis .

definition

The Dirac ridge represents a periodic temperature-controlled distribution that makes use of the Dirac delta distribution .

{\ displaystyle \ Delta _ {T} (t) = \ sum _ {n \ in \ mathbb {Z}} \ delta (t-nT)}

for a period . The Dirac comb is clearly composed of an infinite number of Dirac joints that are at a distance from one another. ${\ displaystyle T> 0}$ ${\ displaystyle T}$

For the application of the Dirac comb on a test function applies ${\ displaystyle \ phi \ in C_ {c} ^ {\ infty} (\ mathbb {R}) = {\ mathcal {D}} (\ mathbb {R})}$

{\ displaystyle \ Delta _ {T} \ phi: = \ sum _ {n \ in \ mathbb {Z}} \ phi (nT)}

.

Fourier transform of the Dirac comb

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

{\ displaystyle {\ mathcal {F}} \ {\ Delta _ {T} \} = {\ frac {1} {T}} \, \ Delta _ {\ frac {1} {T}},}

where for the continuous Fourier transformation the usual convention in the literature for signal processing

{\ displaystyle {\ mathcal {F}} \ {f (t) \} = \ int _ {- \ infty} ^ {\ infty} f (x) \, e ^ {- 2 \ pi \ mathrm {i} tx} \, \ mathrm {d} x}

is used.

Sampling and aliasing effects

With the help of the Dirac comb, the sampling of a function can be described mathematically by multiplication with the function to be sampled:

Sampling by multiplication with a Dirac comb

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

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

literature

Hans Dieter Lüke : Signal transmission . 11th edition. Springer, 2010, ISBN 978-3-642-10199-1 .