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 .
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.
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
The unitary Fourier transforms of the rectangular function are [2]
using ordinary frequency f, where is the normalized form [10] of the sinc function and
using angular frequency , where is the unnormalized form of the sinc function.
For , its Fourier transform is
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.)
We can define the triangular function as the convolution of two rectangular functions:
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.
The pulse function may also be expressed as a limit of a rational function:
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,
The rectangle function can be used to represent the Dirac delta function . [11] Specifically,
For a function , its average over the width around 0 in the function domain is calculated as,
To obtain , the following limit is applied,
and this can be written in terms of the Dirac delta function as,
The Fourier transform of the Dirac delta function is
where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at and goes to infinity, the Fourier transform of is
means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.
In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, to model the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
A Fourier series is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Common forms of the Fourier series below.
In signal processing, a sinc filter can refer to either a sinc-in-time filter whose impulse response is a sinc function and whose frequency response is rectangular, or to a sinc-in-frequency filter whose impulse response is rectangular and whose frequency response is a sinc function. Calling them according to which domain the filter resembles a sinc avoids confusion. If the domain is unspecified, sinc-in-time is often assumed, or context hopefully can infer the correct domain.
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The th partial Fourier series of the function produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere except points of discontinuity.
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(x) and the related function Bi(x), are linearly independent solutions to the differential equation
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values.
In mathematics, a Dirac comb is a periodic function with the formula
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form is a cosine function, 'raised' up to sit above the (horizontal) axis.
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication.
In functional analysis, the Shannon wavelet is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.
In discrete calculus the indefinite sum operator, denoted by or , is the linear operator, inverse of the forward difference operator . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as
The Mehler kernel is a complex-valued function found to be the propagator of the quantum harmonic oscillator.
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.