Determining if a system is Linear

Question

Having $h[n]=u_0[n]+0.8u_0[n-1]+1.6u_0[n-2]$ and $x[n]=(u_1[n]-u_1[n-3])$

The goal is to determine if the system $y[n]=h[n]*x[n]$ is linear.

I know that I would have to test it for Homogenity and Additivity in order to determine if the system is linear or not, but the actual proccess testing it is getting me confused.

My professor did it the following way:

$$y_1[n]=h[n]*x_1[n]$$

$$y_2[n]=h[n]*x_2[n]$$

$$h[n]*(\alpha x_1[n]+\beta x_2[n])$$

$$y[n]=h[n]*\alpha x_1[n]+h[n]*\beta x_2[n]$$

Wasn't this insufficient to prove linearity? Wouldn't we have to test it with the actual values of $x[n]$ and $h[n]$?

Answer 1

To show that a system

$$y[n] = T\{x[n]\} $$

is linear, you have to show that it's additive and homogeneous which is codified in

$$T\{ a ~x_1[n] + b ~ x_2[n] \} = a ~T\{x_1[n]\} + b ~T\{x_2[n] \} $$ for all complex $a,b$ and all $x_1,x_2$.

The property set above is satisfied for a system given by : $$y[n] = T\{x[n]\} = \sum_{k=-\infty}^{\infty} h[k] x[n-k] = h[n] \star x[n] $$ where $\star$ is the convolution operator and $h[n]$ is some sequence to be defined (incidentally it will be the impulse response of the given LTI system).

Of course you didn't have to prove it, as it's already known that if the input-output relation of a system is given by convolution sum; i.e. $y[n] = h[n] \star x[n]$, then it's by definition an LTI (linear and time-invariant) system. Indeed input output relation of any LTI system is given by the convolution sum...