Discrete time system that is homogenous but not additive

[Talltunetalk]

I have just started working with discrete time signals, more specifically various system properties. I am wondering if their is a discrete time system that is homogeneous but not additive? This is basically testing the linearity of a signal with the additive and homogeneous criteria.

 

[jbunniii]

What does homogeneous mean in this context? Simple examples of non-additive systems would be $x \mapsto x^2$, $x \mapsto \sin(x)$ and so forth.

 

[Talltunetalk]

T{x₁[n] +x₂[n]} = T{x₁[n]} + T{x₂[n]}

It is using linear to algebra to see if the system is linear. It has to be both have the additivity and homogeneous/scaling property

 

[jbunniii]

What if you define something like
$$T(x[n]) = \begin{cases}x[n] & \text{ if }x[0] \neq x[1] \\
0 & \text{ if }x[0] = x[1]\end{cases}$$
In other words, $T$ maps the sequence $\{x[n]\}$ to itself if $x[0] \neq x[1]$, and it maps $\{x[n]\}$ to the zero sequence if $x[0] = x[1]$.

Verification that $T$ is homogeneous is straightforward.

Case 1: $c = 0$. Then clearly $cx[0] = cx[1]$, so $T(cx[n]) = 0 = cT(x[n])$.

Case 2: $c \neq 0$ and $x[0] \neq x[1]$. Then $cx[0] \neq cx[1]$, so $T(cx[n]) = cx[n] = cT(x[n])$.

Case 3: $c \neq 0$ and $x[0] = x[1]$. Then $cx[0] = cx[1]$, so $T(cx[n]) = 0 = cT(x[n])$.

On the other hand, $T$ is not additive. For example, let
$$x_1[n] = \begin{cases}1 & \text{ if }n = 0 \\
0 & \text{ if }n \neq 0\end{cases}$$
$$x_2[n] = \begin{cases}1 & \text{ if }n = 1 \\
0 & \text{ if }n \neq 1\end{cases}$$
Then $T$ maps $\{x_1[n]\}$ to itself, and $\{x_2[n]\}$ to itself, but $T$ maps $\{x_1[n] + x_2[n]\}$ to the zero sequence.