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Rectangular function

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

Alternate definitions of the function define to be 0, 1, or undefined.

Relation to the Heaviside step function

The rectangular function may be expressed in terms of the Heaviside step function as

For a duration , we have

Relation to the sign function

For a duration , we have

Relation to the boxcar function

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

where the function is centered at and has duration .

Fourier transform of the rectangular function

The unitary Fourier transforms of the rectangular function are

and

where is the unnormalized sinc function and is the normalized sinc function.

Note that as long as the definition of the pulse function is only motivated by the time-domain experience of it, 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, such as the infinite bandwidth requirement incurred by the indefinitely-sharp edges in the time-domain definition.

Relation to the triangular function

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

Use in probability

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

and its moment generating function is

where is the hyperbolic sine function.

Rational approximation

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

Demonstration of validity

First, we consider the case where . Notice that the term is always positive for integer . However, and hence approaches zero for large .

It follows that

Second, we consider the case where . Notice that the term is always positive for integer . However, and hence grows very large for large .

It follows that

Third, we consider the case where . We may simply substitute in our equation

We see that it satisfies the definition of the pulse function

See also