Rectangular and Triangular Pulse Function

A rectangular pulse 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. These parameters allow you to position and scale the pulse to match specific signal requirements in applications.

A unit rectangular pulse can be synthesized by sequentially adding two time-shifted step functions. This graphical method demonstrates that any rectangular pulse can be expressed using the unit step function, providing a fundamental relationship between these basic continuous time signals.

The unit triangular function has unit height and is centered at the origin. It is mathematically expressed via the triangular function, with its peak at the center and tapering linearly to zero at both edges, creating a symmetric triangular waveform.

To sketch a triangular pulse, replace every t with (t minus the center time) and set the width parameter accordingly. For example, a triangular pulse centered at t=3 seconds with magnitude 2 and width 2 seconds is expressed by replacing t with (t-3) and setting width equal to 2.

A rectangular pulse can have a 5V amplitude, a 3-second duration, and be centered at t=2 seconds. This pulse is expressed using the rectangular function with parameters T=2 seconds and τ=3 seconds, demonstrating how real-world signals are mathematically represented and applied in signal processing.

Both rectangular and triangular pulse functions are fundamental in signal processing for representing various waveform shapes and modeling signals. They serve as building blocks for understanding more complex signal behaviors and are essential for analyzing and designing systems that process real-world signals.

Time-shifting a triangular pulse moves its center location along the time axis. By replacing t with (t minus T), where T is the desired center time, you reposition the pulse while maintaining its shape, magnitude, and width, allowing flexible placement of the signal.