Unit doublet

In mathematics, the unit doublet is the derivative of the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a>. It can be used to differentiate <a href="/facts/Signal_(electronics)/41YqUJ8c">signals</a> in <a href="/facts/Electrical_engineering/YYXW1KLW">electrical engineering</a>: If u1 is the unit doublet, then

(
        x
        ∗
        
          u
          
            1
          
        
        )
        (
        t
        )
        =
        
          
            
              d
              x
              (
              t
              )
            
            
              d
              t
            
          
        
      
    
    {\displaystyle (x*u_{1})(t)={\frac {dx(t)}{dt}}}

where 
 
 
 
 ∗
 
 
 {\displaystyle *}
 
 is the <a href="/facts/Convolution/PVPrdz9J">convolution</a> operator.
The function is zero for all values except zero, where its behaviour is interesting. Its integral over any interval enclosing zero is zero. However, the integral of its absolute value over any region enclosing zero goes to infinity. The function can be thought of as the <a href="/facts/Limiting_case_(mathematics)/UEehGCom">limiting case</a> of two rectangles, one in the second <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">quadrant</a>, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero.

Unit doublet open-in-new

Unit doublet