Rectangular function

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Rectangular function

The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized boxcar function
Boxcar function

In mathematics, a boxcar function is any function which is zero over the entirereal line except for a single interval where it is equal to a constant, A....
) is defined as:

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

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Heaviside step function

The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
, :

or, alternatively:

The unitary Fourier transforms

Continuous Fourier transform

In mathematics, the Fourier transform is an operation that Transform one complex number-valued function of a real variable into another. The new function, often called the frequency domain representation of the original function, describes which frequencies are present in the original function....
of the rectangular function are:

and:

where

Sinc function

In mathematics, the sinc function, denoted by and sometimes as , has two definitions. In digital signal processing and information theory, the normalized sinc function is commonly defined by...
is the normalized form.

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.

We can define the triangular function

Triangular function

The triangular function is defined either as:'or, equivalently, as the convolution of two identical unit rectangular functions:'...
as the convolution of two rectangular functions:

Viewing the rectangular function as a probability distribution

Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval ....
function, its characteristic function

Characteristic function (probability theory)

In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real number line it is given by the following formula, where X is any random variable with the distribution in question:...
is:

and its moment generating function is:

where is the hyperbolic sine function.

Rectangular function

See also