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 to be 0,[3] 1,[4][5] or undefined.
Its periodic version is called a rectangular wave.
History[edit]
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 function[edit]
The rectangular function is a special case of the more general boxcar function:
where is the Heaviside step function; the function is centered at and has duration , from to
Fourier transform of the rectangular function[edit]
The unitary Fourier transforms of the rectangular function are[2]
For , its Fourier transform is
Relation to the triangular function[edit]
We can define the triangular function as the convolution of two rectangular functions:
Use in probability[edit]
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[edit]
The pulse function may also be expressed as a limit of a rational function:
Demonstration of validity[edit]
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,
Dirac delta function[edit]
The rectangle function can be used to represent the Dirac delta function .[11] Specifically,
See also[edit]
References[edit]
- ^ Wolfram Research (2008). "HeavisidePi, Wolfram Language function". Retrieved October 11, 2022.
- ^ a b Weisstein, Eric W. "Rectangle Function". MathWorld.
- ^ Wang, Ruye (2012). Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis. Cambridge University Press. pp. 135–136. ISBN 9780521516884.
- ^ Tang, K. T. (2007). Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models. Springer. p. 85. ISBN 9783540446958.
- ^ Kumar, A. Anand (2011). Signals and Systems. PHI Learning Pvt. Ltd. pp. 258–260. ISBN 9788120343108.
- ^ 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.
- ^ Woodward, Philipp M (1953). Probability and Information Theory, with Applications to Radar. Pergamon Press. p. 29.
- ^ Higgins, John Rowland (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press Inc. p. 4. ISBN 0198596995.
- ^ Zayed, Ahmed I (1996). Handbook of Function and Generalized Function Transformations. CRC Press. p. 507. ISBN 9780849380761.
- ^ Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html
- ^ Khare, Kedar; Butola, Mansi; Rajora, Sunaina (2023). "Chapter 2.4 Sampling by Averaging, Distributions and Delta Function". Fourier Optics and Computational Imaging (2nd ed.). Springer. pp. 15–16. doi:10.1007/978-3-031-18353-9. ISBN 978-3-031-18353-9.