Rectangular mask short-time Fourier transform

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Fourier_analysis/sUA8JP6q">Fourier analysis</a>, a rectangular mask short-time Fourier transform (rec-STFT) has the simple form of <a href="/facts/Short-time_Fourier_transform/XHo1t2m6">short-time Fourier transform</a>. Other types of the STFT may require more computation time than the rec-STFT.
The <a href="/facts/Rectangular_function/cVm8QG4F">rectangular mask function</a> can be defined for some bound (B) over time (t) as

w
        (
        t
        )
        =
        
          
            {
            
              
                
                   
                  1
                  ;
                
                
                  
                    |
                  
                  t
                  
                    |
                  
                  ≤
                  B
                
              
              
                
                   
                  0
                  ;
                
                
                  
                    |
                  
                  t
                  
                    |
                  
                  >
                  B
                
              
            
            
          
        
      
    
    {\displaystyle w(t)={\begin{cases}\ 1;&|t|\leq B\\\ 0;&|t|>B\end{cases}}}

We can change B for different <a href="/facts/Trade-off/xY3nXnJN">tradeoffs</a> between desired time resolution and frequency resolution.
Rec-STFT

X
        (
        t
        ,
        f
        )
        =
        
          ∫
          
            t
            −
            B
          
          
            t
            +
            B
          
        
        x
        (
        τ
        )
        
          e
          
            −
            j
            2
            π
            f
            τ
          
        
        
        d
        τ
      
    
    {\displaystyle X(t,f)=\int _{t-B}^{t+B}x(\tau )e^{-j2\pi f\tau }\,d\tau }

Inverse form

x
 (
 t
 )
 =
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 X
 (
 
 t
 
 1
 
 
 ,
 f
 )
 
 e
 
 j
 2
 π
 f
 t
 
 
 
 d
 f
 
  where 
 
 t
 −
 B
 <
 
 t
 
 1
 
 
 <
 t
 +
 B
 
 
 {\displaystyle x(t)=\int _{-\infty }^{\infty }X(t_{1},f)e^{j2\pi ft}\,df{\text{ where }}t-B<t_{1}<t+B}

Rectangular mask short-time Fourier transform open-in-new

Rectangular mask short-time Fourier transform