Total Response of Linear System

The system output for $t\ge 0$ is a result of two independent causes:

1. System state at time $t=0$ (initial condition).
2. The output $x(t)$ for $t\ge 0$.

Thus, we have:

$$\text{Total response}=\text{Zero-input response}+\text{Zero-state response}$$

where:

• ZIR is response due to the initial conditions at $t=0$ with zero input
• ZSR is response due to the input $x(t)$ for $t\ge 0$ when initial condition is zero