Sigma approximation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, σ-approximation adjusts a <a href="/facts/Fourier_series/s3fsxPAT">Fourier summation</a> to greatly reduce the <a href="/facts/Gibbs_phenomenon/JCXxWvcD">Gibbs phenomenon</a>, which would otherwise occur at <a href="/facts/Discontinuity_(mathematics)/TGLMsQom">discontinuities</a>.
An m-1-term, σ-approximated summation for a series of period T can be written as follows:

s
        (
        θ
        )
        =
        
          
            1
            2
          
        
        
          a
          
            0
          
        
        +
        
          ∑
          
            k
            =
            1
          
          
            m
            −
            1
          
        
        
          
            (
            
              sinc
              ⁡
              
                
                  k
                  m
                
              
            
            )
          
          
            p
          
        
        ⋅
        
          [
          
            
              a
              
                k
              
            
            cos
            ⁡
            
              (
              
                
                  
                    
                      2
                      π
                      k
                    
                    T
                  
                
                θ
              
              )
            
            +
            
              b
              
                k
              
            
            sin
            ⁡
            
              (
              
                
                  
                    
                      2
                      π
                      k
                    
                    T
                  
                
                θ
              
              )
            
          
          ]
        
        ,
      
    
    {\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}\cdot \left[a_{k}\cos \left({\frac {2\pi k}{T}}\theta \right)+b_{k}\sin \left({\frac {2\pi k}{T}}\theta \right)\right],}

in terms of the normalized <a href="/facts/Sinc_function/lSvDtHy0">sinc function</a>:

sinc
        ⁡
        x
        =
        
          
            
              sin
              ⁡
              π
              x
            
            
              π
              x
            
          
        
        .
      
    
    {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.}

a
 
 k
 
 
 
 
 {\displaystyle a_{k}}
 
 and 
 
 
 
 
 b
 
 k
 
 
 
 
 {\displaystyle b_{k}}
 
 are the typical Fourier Series coefficients, and p, a non negative parameter, determines the amount of smoothening applied, where higher values of p further reduce the Gibbs phenomenon but can overly smoothen the representation of the function.
The term

(
            
              sinc
              ⁡
              
                
                  k
                  m
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \left(\operatorname {sinc} {\frac {k}{m}}\right)^{p}}

is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the 
 
 
 
 sinc
 
 
 {\displaystyle \operatorname {sinc} }
 
 function to rolloff the higher frequency Fourier Series coefficients.
As is known by the <a href="/facts/Uncertainty_principle/UUSoSp4T">Uncertainty principle</a>, having a sharp cutoff in the frequency domain (cutting off the Fourier Series abruptly without adjusting coefficients) causes a wide spread of information in the time domain (lots of ringing).
This can also be understood as applying a <a href="/facts/Window_function/5Xj7h2ah">Window function</a> to the Fourier series coefficients to balance maintaining a fast rise time (analogous to a narrow transition band) and small amounts of ringing (analogous to stopband attenuation).

Sigma approximation open-in-new

Sigma approximation