Time Reversal

Time reversal flips a signal perfectly across the vertical axis. Evaluating cross correlation functions requires this exact folding operation to match mathematical definitions.

Core Concept

Reversing time creates a perfect mirror image of the waveform. Whatever happened last now happens first. It physically represents playing an audio recording backward.

Mathematically, we replace the time variable with its negative counterpart. The vertical amplitude axis acts as the stationary hinge for this folding operation.

This operation is fundamental to computing the convolution integral. Before sliding one signal past another, one of the signals must be completely folded.

Key Formula

The time reversed signal is defined as y(t) = x(-t). Every point previously located at time t now resides at time -t.

Given: Signal x(t) is a triangle from t = 1 to t = 5, peaking at t = 3. Find y(t) = x(-t).\nFormula: Substitute t with -t to find new coordinate bounds.\nSteps:\n1. Left edge moves from 1 to -1.\n2. Right edge moves from 5 to -5.\n3. The new bounds are t = -5 to t = -1.\n4. The peak moves from 3 to -3.\nFinal Answer: y(t) is a triangle from -5 to -1, peaking at t = -3.

Exam Tip: When dealing with a combined operation like x(-t + 2), apply the shift to the left by 2 first, then fold. Folding first changes the direction of your shift.

Properties Summary

• Reflection: The signal reflects symmetrically across the amplitude axis.
• Even signals: For an even function, time reversal has absolutely zero visual effect.
• Odd signals: For an odd function, time reversal equals amplitude reversal.
• Frequency domain: Reversing time simply reverses the frequency axis in the transform.
• Power and Energy: The total signal power or energy remains completely unaltered.

Quick Revision

• Folds the graph horizontally.
• Flips around t equals zero.
• Crucial step for continuous convolution.
• Preserves exact shape and area.
• Does not change total energy.
• Exam trap: Forgetting to distribute the negative sign when dealing with expressions like x(-(t - 3)).