Dirac comb

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Dirac comb (also known as sha function, impulse train or sampling function) is a <a href="/facts/Periodic_function/Sm8lJQsF">periodic</a> <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> with the formula

Ш
          
          
             
            T
          
        
        ⁡
        (
        t
        )
         
        :=
        
          ∑
          
            k
            =
            −
            ∞
          
          
            ∞
          
        
        δ
        (
        t
        −
        k
        T
        )
      
    
    {\displaystyle \operatorname {\text{Ш}} _{\ T}(t)\ :=\sum _{k=-\infty }^{\infty }\delta (t-kT)}

for some given period 
 
 
 
 T
 
 
 {\displaystyle T}
 
. Here t is a real variable and the sum extends over all <a href="/facts/Integer/PUwwrawx">integers</a> k. The <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
 and the Dirac comb are <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">tempered distributions</a>. The graph of the function resembles a <a href="/facts/Comb/lIutLoTP">comb</a> (with the 
 
 
 
 δ
 
 
 {\displaystyle \delta }
 
s as the comb's teeth), hence its name and the use of the comb-like <a href="/facts/Cyrillic_script/0btN3oWC">Cyrillic</a> letter <a href="/facts/Sha_(Cyrillic)/D6hcCyhN">sha</a> (Ш) to denote the function.
The symbol 
 
 
 
 
 Ш
 
 
 
 (
 t
 )
 
 
 {\displaystyle \operatorname {\text{Ш}} \,\,(t)}
 
, where the period is omitted, represents a Dirac comb of unit period. This implies

Ш
          
          
             
            T
          
        
        ⁡
        (
        t
        )
         
        =
        
          
            1
            T
          
        
        
          Ш
        
        ⁡
         
        
        
        
        
          (
          
            
              t
              T
            
          
          )
        
        .
      
    
    {\displaystyle \operatorname {\text{Ш}} _{\ T}(t)\ ={\frac {1}{T}}\operatorname {\text{Ш}} \ \!\!\!\left({\frac {t}{T}}\right).}

Because the Dirac comb function is periodic, it can be represented as a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> based on the <a href="/facts/Dirichlet_kernel/eKdT65eP">Dirichlet kernel</a>:

Ш
          
          
             
            T
          
        
        ⁡
        (
        t
        )
        =
        
          
            1
            T
          
        
        
          ∑
          
            n
            =
            −
            ∞
          
          
            ∞
          
        
        
          e
          
            i
            2
            π
            n
            
              
                t
                T
              
            
          
        
        .
      
    
    {\displaystyle \operatorname {\text{Ш}} _{\ T}(t)={\frac {1}{T}}\sum _{n=-\infty }^{\infty }e^{i2\pi n{\frac {t}{T}}}.}

The Dirac comb function allows one to represent both <a href="/facts/Continuous_function/sKbl02pB">continuous</a> and <a href="/facts/Discrete_mathematics/XdRUnzUL">discrete</a> phenomena, such as <a href="/facts/Sampling_(signal_processing)/xxck7aLN">sampling</a> and <a href="/facts/Aliasing/Pzg84HL0">aliasing</a>, in a single framework of <a href="/facts/Fourier_transform/XXhKJtc8">continuous Fourier analysis</a> on tempered distributions, without any reference to Fourier series. The <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of a Dirac comb is another Dirac comb. Owing to the <a href="/facts/Convolution_theorem/E7zhjIx2">Convolution Theorem</a> on tempered distributions which turns out to be the <a href="/facts/Poisson_summation_formula/CDM4uRPs">Poisson summation formula</a>, in <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, the Dirac comb allows modelling sampling by <a href="/facts/Multiplication/VtcUjpNv">multiplication</a> with it, but it also allows modelling periodization by <a href="/facts/Convolution/PVPrdz9J">convolution</a> with it.

Dirac comb open-in-new

Dirac comb