Rectangular and Triangular Pulse Function

The unit rectangular pulse function is mathematically represented by a rectangular function centered at the origin with a height of one unit. This function is defined by two parameters: T, which specifies the center location of the pulse along the time axis, and τ, which determines the pulse duration.

For example, consider a rectangular pulse with a 5V amplitude, a 3-second duration, and centered at t=2 seconds. This pulse can be expressed using the rectangular function, written as,

Synthesizing the rectangular pulse can be demonstrated graphically by adding two time-shifted step functions sequentially. In general terms, a unit rectangular function can always be expressed using the unit step function as follows:

The unit triangular function is mathematically expressed via the triangular function. It has unit height and is centered at the origin. For instance, consider a triangular pulse centered at t=3 seconds, with a magnitude of 2 and a width of 2 seconds. To express this triangular pulse, replace every t with t−3 and set the width equal to 2. The defined signal can be written as,

This triangular pulse function can be illustrated graphically, showing how its height reaches 2 at the center and tapers off to zero at the edges, spanning a total width of 2 seconds.

Both unit rectangular and triangular functions are fundamental in signal processing for representing various waveform shapes and are used in multiple applications for modeling and analyzing signals and systems. These functions are essential for understanding more complex signal behaviors and operations.