What is the Time Reversal Operation on Signals?

What is Time Reversal of a Signal?

The time reversal of a signal is folding of the signal about the time origin (or t = 0). The time reversal or folding of a signal is also called as the reflection of the signal about the time origin (or t = 0). Time reversal of a signal is a useful operation on signals in convolution.

Time Reversal of a Continuous-Time Signal

The time reversal of a continuous time signal x(t) is the rotation of the signal by 180° about the vertical axis. Mathematically, for the continuous time signal x(t), the time reversal is given as,

$$\mathrm{y(t) \:=\: x(−t)}$$

An arbitrary continuous-time signal x(t) and its time reversal x(-t) are shown in Figure-1.

Time Reversal of a Discrete-Time Sequence

For a discrete time sequence x(n), the time reversal is given by,

$$\mathrm{y(n) \:=\: x(−n)}$$

An arbitrary discrete-time signal x(n) and its time reversal x(-n) are shown in Figure-2.

Numerical Example

Sketch the following signals −

$$\mathrm{x(t) \:=\: 3u(−t)}$$

$$\mathrm{x(t) \:=\: 2r(−t)}$$

Solution

Given signal is,

$$\mathrm{x(t) \:=\: 3u(−t)}$$

The given signal [3u(−t)] can be obtained by first drawing the step signal 3u(t) and then time reversing the signal 3u(t) about the time origin (i.e., t = 0) to obtain the signal 3u(−t) as shown in Figure-3.

Given signal is,

$$\mathrm{x(t) \:=\: 2r(−t)}$$

The given signal [2r(−t)] can be obtained by first drawing the ramp signal 2r(t) as shown in Figure-4. Then time reversing or folding the signal 2r(t) about the time origin (i.e., t = 0) to obtain [2r(−t)] as shown in Figure-4.