# The impulse response of a causal linear time-invariant system is given as h(t). Now consider the following two statements and answer which of the following is true?

The impulse response of a causal linear time-invariant system is given as h(t). Now consider the following two statements and answer which of the following is true? P: The system satisfies the superposition principle Q: h(t) = 0 for t < 0

### Right Answer is: P is true and Q is true

#### SOLUTION

A system is said to be a linear time-invariant (LTI) system if it follows the superposition principle.

Superposition Principle: If y1 and y2 are the responses to the input sequences x1 and x2, respectively, then the input ax1 + bx2 produces the response ay1 + by2.

The superposition principle allows us to study the behavior of a linear system starting from test signals such as impulses or sinusoids and obtaining the responses to complicated signals by weighted sums of the basic responses. A linear system is said to be linear time-invariant (LTI), if a time shift in the input results in the same time shift in the output or, in other words if it does not change its behavior in time.

Casual System

A system is said to be causal if the output of the system depends only on the input at the present time and/or in the past, but not the future value of the input. Thus, a causal system is non-anticipative, i.e. output cannot come before the input.

To have the given system to be causal, its impulse response must be zero for t < 0. The impulse occurs at t = 0 and there should not be output without input. Thus if h(t) is the system response to an impulse input then for causal system h(t) = 0 for t < 0. This is the required condition for the causal system.

h(t) = 0 for t < 0

Thus both the above condition is satisfied i.e

P: The system satisfies the superposition principle

Q: h(t) = 0 for t < 0

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