# Determining whether a system is linear or not having the impulse response

I got many exercises where I have to determine the response of a system to an input signal

The only things i have are the impulse response and the input signal. I was thinking I could use the convolution to get the output, but to proceed that way I should know the system is linear and time invariant, none of which is specified

Is there a way to determine if a system is whether LTI or not by knowing the impulse response?

As an example I have this exercise right here

What is the response to the input signal $$x[n]=2^n u[n]$$

Knowing the impulse response is $$h[n] = u[n]$$

More than resolving this particular problem I'm concerned on how to proceed in general when I get those type of exercises

In general, the term "impulse response" doesn't mean much if the system is nonlinear, and a time-varying system will have an impulse response that depends on, well, time.

So if I tell you the impulse response is $$h[n] = u[n]$$, then that implies that the system is linear and time invariant. If I tell you that the impulse response is $$h(\tau, t) = \sin(\omega t) u(\tau)$$ then that implies that the system is linear and time varying.

If I tell you a system is nonlinear and then give you an impulse response then without defining what the term means in the context of what I'm saying I'm in error, because an impulse response implies the system obeys superposition, and that, in turn, implies linearity.

• I know I'm being pedantic, but I don't see why an impulse response implies superposition. A non-linear system has a perfectly well-defined impulse response, just like a linear system. I would put it like this instead: for the impulse response to be useful, the system must be linear; conversely, when someone gives you an impulse response $h(t)$ (or $h(\tau,t)$) it is assumed by convention that the system is LTI (or LTV).
– MBaz
Apr 7 at 15:47
• Agreed with @MBaz, I think these descriptions are more for the transfer function. Apr 7 at 16:11

A non-linear and/or time-varying system cannot be completely described by its response to an impulse.

(Linear time-varying systems can be described by a two-dimensional impulse response, but that's beyond the scope here.)

So, if you're given the impulse response of a system, you can be sure that the system is LTI, and, consequently, the output signal is given by the convolution of the input signal and the impulse response.