In mathematical terms, the desired frequency response is the rectangular function:

This function has the appearance of a smoothed top-hat or rectangular function.

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

The spike events are modelled in a discrete fashion with the wave form conventionally represented as a rectangular function.

We can define the triangular function as the convolution of two rectangular functions:

If is the rectangular function, then is the number of pixels in the mask which are within of the nucleus.

The unitary Fourier transforms of the rectangular function are:

These can be found in pulse waves, square waves, boxcar functions, and rectangular functions.

The pulse wave is also known as the rectangular wave, the periodic version of the rectangular function.

The normalized sinc function is the Fourier transform of the rectangular function with no scaling.

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with .

The filter's impulse response is a sinc function in the time domain, and its frequency response is a rectangular function.

It can also be defined with respect to the Heaviside step function u(t) or the rectangular function (t):

When the w(t) is a rectangular function, the transform is called Rec-STFT.

The rectangular function, the normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.

The simplest signal a pulse radar can transmit is a sinusoidal pulse of amplitude, and carrier frequency, , truncated by a rectangular function of width, .

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:

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged; its frequency response is a rectangular function and is a brick-wall filter.

In signal processing terms, this is spectral leakage in the spatial domain and is caused by application of a rectangular function as a window function on what would otherwise be an infinite array of speakers.

An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain.

He does not derive or prove the properties of the sinc function, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between rect (the rectangular function) and sinc was well known.