Rectangular Function
Main Concept
The rectangular function, also known as the gate function, unit pulse, or normalized boxcar function is defined as:
Recttτ = Πtτ ={0t > τ212t = τ21t<τ2
The rectangular function is a function that produces a rectangular-shaped pulse with a width of τ (where 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(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 sintt, and the rectangular function form a Fourier transform pair.
The Fourier transform of F(t) = Recttτ is:
Fω = ∫−∞∞Recttτ e−j ω t ⅆt = τ sincω τ2
Where:
ω = hertz
τ = a constant
j = imaginary number
Rect = rectangular function
sinc = sinc function sintt
The bandwidth or the range of frequency of the function is ≈ 2⋅πτ
Adjust the value of t to observe the change in the Fourier transform
τ =
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