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]

Rectangular function

Alternative definitions of the function define to be 0,[3] 1,[4][5] or undefined.

Its periodic version is called a rectangular wave.

HistoryEdit

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.

Relation to the boxcar functionEdit

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

 

where   is the Heaviside function; the function is centered at   and has duration  , from   to  

Fourier transform of the rectangular functionEdit

 
Plot of normalised   function (i.e.  ) with its spectral frequency components.

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

 
using ordinary frequency f, where   is the normalized form of the sinc function and
 
using angular frequency  , where   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.)

Relation to the triangular functionEdit

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

 

Use in probabilityEdit

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 approximationEdit

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

 

Demonstration of validityEdit

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 alsoEdit

ReferencesEdit

  1. ^ Wolfram Research (2008). "HeavisidePi, Wolfram Language function". Retrieved October 11, 2022.
  2. ^ a b Weisstein, Eric W. "Rectangle Function". MathWorld.
  3. ^ Wang, Ruye (2012). Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis. Cambridge University Press. pp. 135–136. ISBN 9780521516884.
  4. ^ Tang, K. T. (2007). Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models. Springer. p. 85. ISBN 9783540446958.
  5. ^ Kumar, A. Anand (2011). Signals and Systems. PHI Learning Pvt. Ltd. pp. 258–260. ISBN 9788120343108.
  6. ^ Klauder, John R (1960). "The Theory and Design of Chirp Radars". Bell System Technical Journal. 39 (4): 745–808. doi:10.1002/j.1538-7305.1960.tb03942.x.
  7. ^ Woodward, Philipp M (1953). Probability and Information Theory, with Applications to Radar. Pergamon Press. p. 29.
  8. ^ Higgins, John Rowland (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press Inc. p. 4. ISBN 0198596995.
  9. ^ Zayed, Ahmed I (1996). Handbook of Function and Generalized Function Transformations. CRC Press. p. 507. ISBN 9780849380761.