Linearity and shift-invariance of 2-D system on lattice

Question

I know how to check a 1-D system for these conditions, but am confused about translating this to a 2-D system over a lattice. The system $H$ is define as:

$$w[\mathbf x] = H\left\{u[\mathbf x]\right\} = \sum_{\mathbf y \in \mathbf Q} \varphi[\mathbf y] u[\mathbf x - \mathbf y],\quad\text{with}\quad\begin{cases} &\mathbf x \in \Lambda \subset R^2,\\ &\mathbf Q = \left\{\mathbf q \in \Lambda : \lVert\mathbf q\rVert \le \rho\right\},\\ &\quad \rho \in R, \ 0 < \rho < \infty \end{cases}$$

I have mainly done problems with very simple 1-D systems. I know that the summation part represents convolution and that the check for linearity is:

$$H\left\{af_{1}[\mathbf x] + bf_{2}[\mathbf x]\right\} = aH\left\{f_{1}[\mathbf x]\right\} + bH\left\{f_{2}[\mathbf x]\right\}$$

and the check for shift invariance is:

$$w[\mathbf x - \mathbf y] = H\left\{f[\mathbf x-\mathbf y]\right\}$$

However I can't figure out how to actually do these checks on this particular system.

Answer 1

Well, actually you only need to insert the system $H$ into the definition of linearity and check, if it holds:

$$\begin{align}H\{af_1[x]+bf_2[x]\}&=\sum_y\phi[y](af_1[x-y]+bf_2[x-y])\\ &= a\sum_y\phi[y]f_1[x-y]+b\sum_y\phi[y]f_2[x-y]\\ &=aH\{f_1[x]\}+bH\{f_2[x]\}\end{align}$$

So, the linearity holds. There is nothing special about the fact that the system is 2-dimensional or similar. You just follow the rules for the calculation, and you end up with the statement of linearity.

A similar thing can be carried out for the shift-invariance.