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) is defined as Alternative definitions of the function define to be 0, 1, or undefined. Its periodic version is called a rectangular wave. 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: where is the Heaviside function; 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 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 function We can define the triangular function as the convolution of two rectangular functions: Use in probability 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 The pulse function may also be expressed as a limit of a rational function: 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 Square wave Step