Counter-example to show that this system is non-linear?

I have a discrete-time system y[n] = x[n]*x[n-1]

I need to show that this system is non-linear by using a counter example rather than by disproving with algebra and the properties of additivity and homogeneity.

I think that the system is homogeneous although not additive, but I can't think of an example to show this. Would someone help me out please?

• Hint: try with constant signals. – MBaz Sep 30 '15 at 21:46
• It's not even homogeneous. Really, nearly any two signals will work as a counter example. There's nothing complicated about constructing a counter example. Just take something and plug it in. – Jazzmaniac Sep 30 '15 at 22:03
• Hmm when I tried to do the algebra to see if it was homogeneous I got a*x[n] -> a*y[n], I guess I screwed up somewhere. I'm really just not familiar with how this works. Say I take a unit impulse and put it in, then I get a signal of 0 out. Then I shift the unit impulse to the right and put it in and still get a signal of 0 out. I wish I had an example to get me started. – Austin Sep 30 '15 at 23:27
• You are testing shift invariance with your example, not linearity. – Peter K. Sep 30 '15 at 23:43

OK. Let's try homogeneity again: $$y[n] = x[n]\cdot x[n-1]$$ for input $x[n]$.
For input $a\cdot x[n]$ we get $$y'[n] = a\cdot x[n] \cdot a x[n-1] = a^2 \cdot x[n]\cdot x[n-1] \not = ay[n]$$ so homogeneity doesn't apply.
Let's try: $$x[n] = 2$$ so that $$y[n] = 4$$ Then let's try twice that: $$x'[n] = 2\cdot x[n] = 4$$ So that $$y'[n] = 16 \not= 2\cdot y[n]$$