Rectangular Function

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

It is a simple step function.Alternate definitions of the function define <math>mathrm(pm tfrac)</math> to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, <math>u(t)</math>:

or, alternatively:

In more general form:

Where the function is centred at X and has duration Y.

The unitary Fourier transforms of the rectangular function are:

and:

where <math>mathrm</math> is the normalized form.

Note that as long as the definition of the pulse function is only motivated by the time-domain experience of it, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier...

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- <math>mathrm(t) = sqcap(t) = begin

It is a simple step function.Alternate definitions of the function define <math>mathrm(pm tfrac)</math> to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, <math>u(t)</math>:

- <math>mathrmleft(fracright) = u left( t + frac right) - u left( t - frac right),</math>

or, alternatively:

- <math>mathrm(t) = u left( t + frac right) - u left( t - frac right).</math>

In more general form:

- <math>operatornameleft(frac right) = u(t - X + Y/2) - u(t - X - Y/2)</math>

Where the function is centred at X and has duration Y.

The unitary Fourier transforms of the rectangular function are:

- <math>int_^infty mathrm(t)cdot e^ , dt

and:

- <math>fracint_^infty mathrm(t)cdot e^ , dt

where <math>mathrm</math> is the normalized form.

Note that as long as the definition of the pulse function is only motivated by the time-domain experience of it, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier...

Read More

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