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:

 

The rectangular function is a function that produces a rectangular-shaped pulse with a width of $\mathrm{τ}$ (where $\mathrm{τ}=1$ 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($\mathrm{ω}$) 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 $\frac{\mathrm{sin}\left(t\right)}{t}$, and the rectangular function form a Fourier transform pair.

The Fourier transform of F(t) = $\mathrm{Rect}\left(\frac{t}{\mathrm{\tau }}\right)$ is:



 Where: $\mathrm{ω}$ =  hertz  = a constant j = imaginary number Rect = rectangular function sinc = sinc function $\left(\frac{\mathrm{sin}\left(t\right)}{t}\right)$

The bandwidth or the range of frequency of the function is ≈ 

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

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