Rectangular Function - Maple Programming Help

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Rectangular Function

Main Concept

The rectangular function, also known as the gate function, unit pulse, or normalized boxcar function is defined as:

Rectt&tau; &equals; Πt&tau; &equals;&lcub;0t &gt; &tau;212t  &equals; &tau;21t<&tau;2

The rectangular function is a function that produces a rectangular-shaped pulse with a width of &tau; (where &tau;&equals;1 in the unit function) centered at t = 0. The rectangular function pulse also has a height of 1.

 

Fourier transform

The Fourier transform usually transforms a mathematical function of time, f(t), into a new function usually denoted by F(&omega;) whose arguments is frequency with units of cycles/sec (hertz) or radians per second. This new function is known as the Fourier transform. The Fourier transform is a mathematical transformation used within many applications in physics and engineering. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces. 

 

The rectangular function can often be seen in signal processing as a representation of different signals. The sinc function, defined as sintt, and the rectangular function form a Fourier transform pair.

 

The Fourier transform of F(t) = Recttτ is:  

 

 

F&omega; &equals; Rectt&tau; ej &omega; t &DifferentialD;t &equals; &tau; sinc&omega; &tau;2

 

 

Where:

&omega; =  hertz

&tau;  = a constant

j = imaginary number

Rect = rectangular function

sinc = sinc function sintt

 

 The bandwidth or the range of frequency of the function is ≈  2&pi;&tau;

 

Adjust the value of t to observe the change in the Fourier transform

&tau; &equals;

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