# Determining if a 2D system is LSI

After a long time away from school, I'm a bit rusty and struggling with this question:

Determine if the following discrete system is LSI: $$y(m,n) = mn*x(m+n) + mn*x(m-n)$$

So here's what I've done so far (note that the '*' means regular multiplication, I just added for better readability IMO).

1) Test linearity:

Let $$r(m,n) = \alpha x_1(m,n) + \beta x_2(m,n)$$, so $$y(r(m,n)) = mn*(\alpha x_1(m+n) + \beta x_2(m+n)) + mn*(\alpha x_1(m-n) + \beta x_2(m-n))$$

Now, evaluate $$y(\alpha x_1(m,n)) + y(\beta x_2(m,n)) = mn* \alpha x_1(m+n) + mn* \alpha x_1(m-n) + mn* \beta x_2(m+n) + mn*\beta x_2(m-n)$$

If this is correct, then the system is linear.

2) Test for Shift-invariance: Let $$g(m,n) = y(m,n)$$, evaluate:

$$g(m-m',n-n') = (m-m')(n-n')x(m-m'+n-n') + (m-m')(n-n')x(m-m'-(n-n'))$$

Now evaluate:

$$y(m-m',n-n') = (m-m')(n-n')x(m-m'+n-n') + (m-m')(n-n')x(m-m'-n-n')$$

The only difference being the signal of the last $$n'$$, which makes it not Shift-Invariant.

It is my first time doing this process with a 2D system, so I'm really confused, especially with the system being on terms of $$x(m+n)$$ instead of $$x(m,n)$$

I'm following this tutorial

Thanks in advance for the help!