What is the Time Shifting Operation on Signals?

What is Time Shifting?

Time shifting or Shifting of a signal in time means that the signal may be either delayed in the time axis or advanced in the time axis.

Time Shifting of Continuous-Time Signal

The time shifting of a continuous time signal x(t) is represented as,

𝑦(𝑡) = 𝑥(𝑡 − 𝑡₀)

The time-shifting of a signal results in the time delay or time advancement. The above expression shows that the signal y(t) can be obtained by time shifting the signal x(t) by t₀ units. If t₀ is positive in the above expression, then the shift of the signal is to the right and hence the time shifting delays the signal. On the other hand, if t₀ is negative, then the shift of the signal is to the left and hence the shifting advances the signal in the time axis.

The time shifting of a continuous-time signal x(t) is shown in Figure-1.

Time Shifting of a Discrete-Time Signal

The time shifting operation of a discrete time signal x(n) is represented as

𝑦(𝑛) = 𝑥(𝑛 − 𝑛₀)

This equation shows that the signal y(n) can be obtained by time-shifting the signal x(n) by 𝑛₀ units. If the value of 𝑛₀ is positive, then the shift of the signal is to the right and the resulting signal is the delayed version. Whereas, if the value of 𝑛₀ is negative, then the shift of the signal is to the left and it is the time advanced version of the signal.

Figure-2 shows an arbitrary discrete-time signal x(n). Here, the signal x(n – 2) is shown which is obtained by shifting of the signal x(n) to the right by 2 units, i.e., the signal x(n) is delayed by 2 units. Also, the signal x(n + 2) is obtained by shifting x(n) to the left by 2 units, i.e., the signal x(n) is advanced by 2 units.

Numerical Example

Sketch the following signals −

• (𝑡) = 3𝑟(𝑡 − 3)
• (𝑡) = Π(𝑡 + 2)

Solution

• Given signal is,

  𝑥(𝑡) = 3𝑟(𝑡 − 3)

  The signal [3𝑟(𝑡 − 3)] can be obtained by first drawing the ramp signal [3r(t)] with slope of 3 as shown in Figure-3. Then, the shifting of the signal to the right by 3 units to obtain the given signal as shown in Figure-3.

  

• Given signal is,

  𝑥(𝑡) = Π(𝑡 + 2)

  The given signal [Π(𝑡 + 2)] can be obtained by first drawing Π(𝑡) as shown in Figure-4 and then shifting it to the left by 2 units to obtained the given signal as shown in Figure-4.