• If we put an impulse into a system the output will be an impulse response.


Figure 18.2 Response of the system to a single pulse

• If we add all of the impulse responses together we will get a total system response. This operation is called convolution.


Figure 18.3 A set of pulses for a system gives summed responses to give the output

• Consider the unit impulse of a system with the given differential equation. Note: This method is only valid for trivial differential equations with only one homogeneous term. The preferred method is shown later.


• Note that the derivation of the unit impulse function assumed zero initial conditions, so the process of convolution must also assume systems start at rest and undeflected.