Signals and Systems: Causal, Non-Causal, and Anti-Causal Signals

Causal Signal

A continuous time signal 𝑥(𝑡) is called causal signal if the signal 𝑥(𝑡) = 0 for 𝑡 < 0. Therefore, a causal signal does not exist for negative time. The unit step signal u(t) is an example of causal signal as shown in Figure-1.

Similarly, a discrete time sequence x(n) is called the causal sequence if the sequence x(n) = 0 for n < 0.

Anti-Causal Signal

A continuous-time signal x(t) is called the anti-causal signal if x(t) = 0 for t > 0. Hence, an anti-causal signal does not exist for positive time. The time reversed unit step signal u(-t) is an example of anti-causal signal (see Figure-2).

Similarly, a discrete time sequence x(n) is said to be anti-causal sequence if the sequence x(n) = 0 for 𝑡 > 0.

Non-Causal Signal

A signal which is not causal is called the non-causal signal. Hence, by the definition, a signal that exists for positive as well as negative time is neither causal nor anti-causal, it is non-causal signal. The sine and cosine signals are examples of non-causal signal (see Figure-3).

Important – All the anti-causal signals are non-causal signals but the converse is not true.

Numerical Example

Find which of the following signals are causal or anti-causal or non-causal −

• $\mathrm{x(t)=e^{3t}u(t-2)}$

• $\mathrm{x(t)=\sin 5t\: u(t)}$

• $\mathrm{x(t)=4u(-t)}$

• $\mathrm{x(n)=u(-n)}$

• $\mathrm{x(t)=\cos 3t}$

• $\mathrm{x(n)=u(n+3)-u(n-3)}$

Solution

• The given signal is,

$$\mathrm{x(t)=e^{3t}u(t-2)}$$

Here, 𝑢(𝑡 − 2) = 0 for t < 0, thus, the signal x(t) = 0 for t < 0. Therefore, the given signal x(t) is a causal signal.

• Given,

$$\mathrm{x(t)=\sin 5t\: u(t)}$$

As the unit step signal u(t) does not exist for negative time. Thus, the signal x(t) is a causal signal because x(t) = 0 for t < 0.

• Given signal is,

$$\mathrm{x(t)=4u(-t)}$$

The given signal x(t) exists only for negative time, i.e., t < 0. Therefore, it is an anti-causal signal. Also, it can be called non-causal signal.

• Given

$$\mathrm{x(n)=u(-n)}$$

The given signal x(n) exists only for negative time, i.e., n < 0. Hence, it is anticausal. It can also be called non-causal signal.

• The given signal is

$$\mathrm{x(t)=\cos 3t}$$

The given signal x(t) is a cosine signal which exists from (−∞ to ∞). Thus, it is a non-causal signal.

• Given

$$\mathrm{x(n)=u(n+3)-u(n-3)}$$

The given signal x(n) exists from 𝑛 = −3 to 𝑛 = 3, i.e., the signal exists for both positive and negative time. Hence, it is a non-causal signal.