Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as:

Alternative definitions of the function define ${\displaystyle \mathrm {rect} (\pm {\tfrac {1}{2}})}$ to be 0, 1, or undefined.

The rectangular function is a special case of the more general boxcar function:

Where u is the Heaviside function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2.

Another example is this: rect((t - (T/2)) / T ) goes from 0 to T, so in terms of Heaviside function u(t) - u((t-T) / T )

The unitary Fourier transforms of the rectangular function are

using ordinary frequency f, and

using angular frequency ω, where ${\displaystyle \mathrm {sinc} }$ is the unnormalized form of the sinc function.

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