Rectangular function

The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized boxcar function) is defined as:

:$$mathrm{rect}(t) = sqcap(t) = egin{cases}0 & mbox{if } |t| > frac{1}{2} \ [3pt] frac{1}{2} & mbox{if } |t| = frac{1}{2} \ [3pt] 1 & mbox{if } |t| < frac{1}{2}.end{cases}

Alternate definitions of the function define $$mathrm{rect}(pm egin{matrix} frac{1}{2} end{matrix}) to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, $$u(t):

:$$mathrm{rect}left(frac{t}{ au} ight) = u left( t + frac{ au}{2} ight) - u left( t - frac{ au}{2} ight),,

or, alternatively:

:$$mathrm{rect}(t) = u left( t + frac{1}{2} ight) cdot u left( frac{1}{2} - t ight).,

The unitary Fourier transforms of the rectangular function are:

:$$int_{-infty}^infty mathrm{rect}(t)cdot e^{-i 2pi f t} , dt=frac{sin(pi f)}{pi f} = mathrm{sinc}(f),,

and:

:$$frac{1}{sqrt{2piint_{-infty}^infty mathrm{rect}(t)cdot e^{-i omega t} , dt=frac{1}{sqrt{2picdot mathrm{sinc}left(frac{omega}{2pi} ight),,

where $$mathrm{sinc} is the normalized form.

We can define the triangular function as the convolution of two rectangular functions:

:$$mathrm{tri}(t) = mathrm{rect}(t) * mathrm{rect}(t).,

Viewing the rectangular function as a probability distribution function, its characteristic function is:

:$$varphi(k) = frac{sin(k/2)}{k/2},,

and its moment generating function is:

:$$M(k)=frac{mathrm{sinh}(k/2)}{k/2},,

where $$mathrm{sinh}(t) is the hyperbolic sine function.

ee also

*Fourier transform
*Square wave
*Triangular function