Properties of the Derivative Operator system:

1. Cascade of systems Suppose we give the output of the derivative operator system as input to another LSI system 'A'. Let y(t) be the output of the combined system for some given input x(t). Now suppose we give x(t) as input to the system 'A' first and then pass its output to the derivative operator system. Let the final output now be z(t). Then from the property that cascading of LSI systems is independent of the order of cascading, we get y(t) = z(t).

2. Memory of the System The system obviously possesses memory as the derivative operator requires a certain interval length to be defined in.

3. Causality of the System To answer this we must consider the left, right and center derivates separately. Clearly the left derivative is causal while the center and right derivatives may or may not be so. However for a differentiable function, all the three derivatives being equal, the system is indeed causal.

4. Stability of the System Consider the input signal shown below. Clearly we see that a bounded input does not lead to a bounded output which becomes obvious at points where the derivative of the input signals tends to infinity. Thus the system is not stable.

x(t)

Exercise: Give an example of a bounded input signal such that its derivative is not bounded as time tends to infinity?

5. Invertibility of the System Is the derivative operator invertible? No, because when we consider the class of constants as input then the output is always zero. Thus the derivative operator is not one is to one. However the system is invertible upto an additive constant.