Rectangular function
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Relation to the boxcar function
The rectangular function is a special case of the more general boxcar function:
rect ( t − − X Y ) = H ( t − − ( X − − Y / 2 ) ) − − H ( t − − ( X + Y / 2 ) ) = H ( t − − X + Y / 2 ) − − H ( t − − X − − Y / 2 ) {\displaystyle \operatorname {rect} \left({\frac {t-X}{Y}}\right)=H(t-(X-Y/2))-H(t-(X+Y/2))=H(t-X+Y/2)-H(t-X-Y/2)}
where H ( x ) {\displaystyle H(x)} is the Heaviside step function; the function is centered at X {\displaystyle X} and has duration Y {\displaystyle Y} , from X − − Y / 2 {\displaystyle X-Y/2} to X + Y / 2. {\displaystyle X+Y/2.}
Fourier transform of the rectangular function