The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized
boxcar function
In mathematics, a boxcar function is any function which is zero over the entire
real line except for a single interval where it is equal to a constant, ''A''. The boxcar function can be expressed in terms of the uniform distribution as
\operatorn ...
) is defined as
Alternative definitions of the function define
to be 0, 1, or undefined.
History
The ''rect'' function has been introduced by
Woodward in as an ideal
cutout operator, together with the
''sinc'' function as an ideal
interpolation operator, and their counter operations which are
sampling (
''comb'' operator) and
replicating (
''rep'' operator), respectively.
Relation to the boxcar function
The rectangular function is a special case of the more general
boxcar function
In mathematics, a boxcar function is any function which is zero over the entire
real line except for a single interval where it is equal to a constant, ''A''. The boxcar function can be expressed in terms of the uniform distribution as
\operatorn ...
:
where
is the
Heaviside function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
; the function is centered at
and has duration
, from
to
Fourier transform of the rectangular function
The
unitary Fourier transforms of the rectangular function are
using ordinary frequency , where
is the normalized form of the
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
and
using angular frequency
, where
is the unnormalized form of the
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
.
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.)
Relation to the triangular function
We can define the
triangular function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle. Often this is an isosceles triangle of height 1 and base 2 in which case it is referred to as ''th ...
as the
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of two rectangular functions:
Use in probability
Viewing the rectangular function as a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
, it is a special case of the
continuous uniform distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies betw ...
with
The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
is
and its
moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is
where
is the
hyperbolic sine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
function.
Rational approximation
The pulse function may also be expressed as a limit of a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
:
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. Therefore,
See also
*
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
*
Square wave
A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions b ...
*
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only ...
*
Top-hat filter
The name Top-hat filter refers to several real-space or Fourier space filtering techniques (not to be confused with the top-hat transform). The name top-hat originates from the shape of the filter, which is a rectangle function, when viewed in ...
References
{{DEFAULTSORT:Rectangular Function
Special functions