What is an LTI system ? Why LTI systems are so important ?

Linear time-invariant (LTI) systems are systems that are both linear and time-invariant.

Linearity states that when a linear combination of the two inputs is fed to the system, the output of the system is a linear combination of the respective outputs.

Let x₁(t) and x₂(t) be any two signals. Suppose that the output of a system to x₁(t) is y₁(t) and the output of the system to x₂(t) is y₂(t). If this always implies that the output of the system to α₁ x₁(t)+α₂ x₂(t) is α₁ y₁(t) + α₂ y₂(t), then the system is linear and the superposition principle is said to hold.

A system is said to be time invariant if when y(t) is the output that corresponds to x(t), then for any τ, y (t − τ) is the output that corresponds to x (t − τ).

LTI systems are so important –

1.     Because many systems encountered in nature can be successfully modeled as LTI systems.

2.      For linear time invariant system, we only need to know the impulse response h(t) of the system (or equivalently frequency response H(ω)) in order to predict the output of the system in response  to any input. This is done by convoluting the input with the impulse response. So a linear time  invariant system is a lot easier to analyze. 

      So LTI systems can be analyzed in considerable detail, providing insight into their properties

      This is not true for nonlinear or time variant system.