Signals and Systems – Causal and Non-Causal System

Causal System

A system whose output or response at any time instant (t) depends only on the present and past values of the input but not on the future values of the input is called the causal system. For a causal system, the output or response does not begin before the input signal is applied. This is why, a causal system is also called a non-anticipated system.

The causal systems are real time systems and they can be physically realised. For a causal system, the impulse response of the system is zero for negative time (i.e., t < 0) because the impulse signal δ(t) or δ(n) exists only at t = 0 or n = 0, i.e.,

ℎ(𝑡) = 0; 𝑡 < 0 … for continuous time system

ℎ(𝑛) = 0; 𝑛 < 0 … for discrete time system

A resistor is an example of a continuous time causal system. Some more examples of causal systems are given below −

𝑦(𝑡) = 𝑥(𝑡 − 3) + 3𝑥(𝑡)

𝑦(𝑡) = 𝑡𝑥(𝑡)

𝑦(𝑛) = 𝑥(𝑛 − 1) + 𝑥(𝑛)

𝑦(𝑛) = 𝑛𝑥(𝑛)

Where, y(t) or y(n) and x(t) or x(n) are the output and input of the system respectively.

Non-Causal System

A system whose output or response at any time instant (t) depends upon future values of the input is called the non-causal system. The non-causal systems can generate an output before the input is given, hence they are also known as anticipative systems.

The non-causal systems do not exist in real time. Also, these systems are not physically realisable. The image processing systems are the examples of noncausal systems.

Some examples of the non-causal systems are given below −

𝑦(𝑡) = 𝑥(𝑡 + 3) + 2𝑥(𝑡)

𝑦(𝑡) = 𝑥²(𝑡) + 𝑡𝑥(𝑡 + 3)

𝑦(𝑛) = 𝑥(𝑛) + 3𝑥(3𝑛)

𝑦(𝑛) = 𝑥²(𝑛) + 2𝑥(𝑛 + 1)

Solved Example

Find whether the following systems are causal or non-causal system −

• 𝑦(𝑡) = 𝑥²(𝑡) + 𝑥(𝑡 − 3)
• 𝑦(𝑡) = 𝑥(3 − 𝑡) + 𝑥(𝑡 − 2)
• 𝑦(𝑛) = 𝑥(2𝑛)
• 𝑦(𝑛) = sin[𝑥(𝑛)]
• $\mathrm{y(t)=\int_{-\alpha }^{2t}\, x(u)du}$

Solution

• Given system is,

  𝑦(𝑡) = 𝑥²(𝑡) + 𝑥(𝑡 − 3)

  Causality of the given system can be determined by considering different values of t as follows −

  𝑡 = 0 → 𝑦(0) = 𝑥²(0) + 𝑥(−3)

  𝑡 = (−2) → 𝑦(−2) = 𝑥²(−2) + 𝑥(−5)

  𝑡 = 2 → 𝑦(2) = 𝑥²(2) + 𝑥(−1)

  Hence, for all the values of t, the output depends only on the present and past values of the input. Thus, the given system is a causal system.

• Given System is,

  𝑦(𝑡) = 𝑥(3 − 𝑡) + 𝑥(𝑡 − 2)

  Causality of the given system can be determined by considering different values of t as follows −

  𝑡 = 0 → 𝑦(0) = 𝑥(3) + 𝑥(−2)

  𝑡 = (−1) → 𝑦(−1) = 𝑥(4) + 𝑥(−3)

  𝑡 = 1 → 𝑦(1) = 𝑥(2) + 𝑥(−1)

  It is clear that for some values of t, the output of the system depends on the future values of the input. Hence, the given system is a non-causal system.

• Given,

  𝑦(𝑛) = 𝑥(3𝑛)

  The output of the system at different time instants is

  𝑛 = 0 → 𝑦(0) = 𝑥(0)

  𝑛 = −1 → 𝑦(−1) = 𝑥(−3)

  𝑛 = 1 → 𝑦(1) = 𝑥(3)

  𝑛 = 2 → 𝑦(2) = 𝑥(6)

  Hence, for positive values of n, the output of the system depends upon the future values of the input. Therefore, the given system is a non-causal system.

• Given system is,

  𝑦(𝑛) = sin[𝑥(𝑛)]

  The output of the system at different values of n is

  𝑛 = 0 → 𝑦(0) = sin[𝑥(0)]

  𝑛 = −3 → 𝑦(−3) = sin[𝑥(−3)]

  𝑛 = 3 → 𝑦(3) = sin[𝑥(3)]

  Hence, for all values of n, the output of the system depends only on the present input values. Therefore, the system is a causal system.

• The given system is,

  $$\mathrm{y(t)=\int_{-\alpha }^{2t}\, x(u)du}$$

  For different values of t, the output the system is,

  $$\mathrm{t=0\rightarrow y(0)=\int_{-\alpha }^{0}x(u)du=[x(0)-x(-\alpha )]}$$ $$\mathrm{t=-1\rightarrow y(-1)=\int_{-\alpha }^{-2}x(u)du=[x(-2)-x(-\alpha )]}$$ $$\mathrm{t=1\rightarrow y(1)=\int_{-\alpha }^{2}x(u)du=[x(2)-x(-\alpha )]}$$ $$\mathrm{t=2\rightarrow y(2)=\int_{-\alpha }^{4}x(u)du=[x(4)-x(-\alpha )]}$$

Hence, for the positive values of t, the output of the system depends upon the future values of the input. Therefore, the given system is a non-causal system.

Manish Kumar Saini

Updated on: 10-Nov-2021

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