Physically realizable, linear time-invariant systems can be described by a set of linear differential equations (LDEs):

Figure: Graphical description of a basic linear time-invariant system with an input, f (t) and an output, y (t).

Equivalently,

With an = 1.

y (t) = yi (t) + ys (t)

We refer to yi (t) as the zero-input response – the homogeneous solution due only to the initial conditions of the system. We refer to ys (t) as the zero-state response – the particular solution in response to the input f (t). We now discuss how to solve for each of these components of the system's response.

Finding the Zero-Input Response:

The zero-input response, yi (t), is the system response due to initial conditions only.

Close the switch in the circuit pictured in Figure at time t=0 and then leave everything else alone. The voltage response is shown in Figure

Finding the Zero-State Response:

Solving a linear differential equation

Given a specific input f (t) is a difficult task in general. More importantly, the method depends entirely on the nature of f (t); if we change the input signal, we must completely re-solve the system of equations to find the system response.

Convolution helps to bypass these difficulties. We explain how convolution helps to determine the system's output, given only the input f (t) and the system's impulse response h(t).

Before deriving the convolution procedure, we show that a system's impulse response is easily derived from its linear, differential equation (LDE). We will show the derivation for the LDE below, where m <>:

We can rewrite as

QD [y (t)] = PD [f (t)]

Where QD [.] is an operator that maps y (t) to the left hand side

And PD [.] maps f (t) to the right hand side. Impulse response of the system described by the equation is given by:

h (t) = bn δ(t) + PD [yn (t)] μ(t)

Where for m <>we have bn = 0. Also, yn equals the zero input response with initial conditions.

{y^{n-1} (0)=1,y^{n-2} (0)=1,…,y(0)=0}