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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,
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,
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 −
- 𝑥(𝑡) = 3𝑢(−𝑡)
- 𝑥(𝑡) = 2𝑟(−𝑡)
Solution
Given signal is,
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,
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.