Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function,[1] gate function, unit pulse, or the normalized boxcar function) is defined as[2]

Alternative definitions of the function define

Its periodic version is called a rectangular wave.

The rect function has been introduced by Woodward[6] in [7] as an ideal cutout operator, together with the sinc function[8][9] as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively.

The rectangular function is a special case of the more general boxcar function:

is the Heaviside step function; the function is centered at

The unitary Fourier transforms of the rectangular function are[2]

{\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} _{\pi }(f),}

is the normalized form[10] of the sinc function and

is the unnormalized form of the sinc function.

Note that as long as the definition of the pulse function is only motivated by its behavior in the time-domain experience, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans.

However, some aspects of the theoretical result may be understood intuitively, as finiteness in time domain corresponds to an infinite frequency response.

(Vice versa, a finite Fourier transform will correspond to infinite time domain response.)

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

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with

The characteristic function is

is the hyperbolic sine function.

The pulse function may also be expressed as a limit of a rational function:

Notice that the term

We may simply substitute in our equation:

We see that it satisfies the definition of the pulse function.

The rectangle function can be used to represent the Dirac delta function

around 0 in the function domain is calculated as,

and this can be written in terms of the Dirac delta function as,

The Fourier transform of the Dirac delta function

{\displaystyle \delta (f)=\int _{-\infty }^{\infty }\delta (t)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}{\frac {1}{a}}\int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}\operatorname {sinc} {(af)}.}

where the sinc function here is the normalized sinc function.

goes to infinity, the Fourier transform of

means that the frequency spectrum of the Dirac delta function is infinitely broad.

As a pulse is shorten in time, it is larger in spectrum.