Differential and Difference Equations

We look at a special class of LSI systems that are frequently encountered in real life applications, namely those that are described by differential and difference equations. We first consider the derivative operator as a system:

The above system is an LSI system.

Proof:

The given system is obviously linear due to the linearity of the derivative operator. Also shift invariance can be easily shown as below:

Let h(t) = t-t₀ and g(t) = x(t-t₀) = x(h(t)). Let the output of the system to x(t) be y(t).    ( y(t) = d x(t) / dt   )

Now g'(t) = x'(h(t)).h'(t) = x'(t-t₀)

Thus input x(t-t₀) gives an output y(t-t₀).

Hence the system is LSI.

Note : Also it is now clearly seen that if the input to an LSI system is differentiated, then the output of that system is also differentiated. This property may be proved by taking the limit of the expression: {x(t+h)-x(t)}/h as 'h' tends to zero and using the linearity and shift-invariance of LSI systems.