Consider a discrete-time system.

We apply the unit step function to it. And we observe a response. And we conclude based on the response that the system is linear.

For example, we see that the response is s(n) = constant * u(n). So it is linear.

Based on this observation, can we conclude that the system is linear for all possible inputs? For example, if I apply the sine wave, would I obtain a response from which I can conclude too that the system is linear?

Put it in other words, are there systems that are linear for only particular inputs? Or if a system is linear, then it is linear, no matter what input I apply to it? Is the linearity of the system independent of the input?

To me it seems like yes. It is independent. But I would want some proof, or some little logic that leads to this conclusion.

(and maybe besides the fact that a system is described by a differential equation relating the input and the output...so no matter what input you choose, the differential equation is the same)

We apply the unit step function to it. And we observe a response. And we conclude based on the response that the system is linear.

For example, we see that the response is s(n) = constant * u(n). So it is linear.

Based on this observation, can we conclude that the system is linear for all possible inputs? For example, if I apply the sine wave, would I obtain a response from which I can conclude too that the system is linear?

Put it in other words, are there systems that are linear for only particular inputs? Or if a system is linear, then it is linear, no matter what input I apply to it? Is the linearity of the system independent of the input?

To me it seems like yes. It is independent. But I would want some proof, or some little logic that leads to this conclusion.

(and maybe besides the fact that a system is described by a differential equation relating the input and the output...so no matter what input you choose, the differential equation is the same)

Last edited: