Triangular Pulse

The triangular pulse helps model signals that rise and fall linearly over a finite interval. Engineers apply it to design window functions for spectral analysis.

Core Concept

A triangular pulse is defined by its peak amplitude and its total duration. It remains exactly zero everywhere outside this active time interval. This shape appears often in linear interpolation.

Mathematically, you can generate a triangle wave by convolving two identical rectangular pulses. This relationship simplifies many complex signal operations. The sharp corners of the triangle contain high frequency components.

Because the signal has finite energy, it possesses a well defined Fourier transform. The transform results in the square of a sinc function. This means its frequency spectrum is entirely positive.

Key Formula

The standard triangular pulse function is written as tri(t/T) = 1 - |t|/T for |t| < T. It equals zero for all other values of time.

Given: x(t) = tri(t/2)\nFormula: X(w) = T * sinc^2(wT/(2*pi)) for a pulse of width 2T\nSteps:\n1. Identify T = 2 from the denominator.\n2. The base width is 4 units total.\n3. Substitute T = 2 into the standard transform.\nFinal Answer: X(w) = 2 * sinc^2(w/pi)

Exam Tip: The convolution of a rectangular pulse of width T with itself gives a triangular pulse of width 2T. Always double check your pulse bounds before applying the transform.

Properties Summary

• Even symmetry: tri(t) = tri(-t).
• Area: Integral from -T to T is T times amplitude.
• First derivative: Yields a positive and negative rectangular pulse.
• Fourier transform pair: tri(t/T) pairs with T * sinc^2(wT/(2*pi)).
• Energy: Total energy is (2/3) * T * A^2.

Quick Revision

• Models linear transitions.
• Bounded tightly by -T and T.
• Peak occurs exactly at the origin.
• Generated by rect convolution.
• Spectrum decays as 1/w^2.
• Exam trap: Forgetting that a triangle defined as tri(t/T) actually spans from -T to T, not -T/2 to T/2.