# Show that the system is linear?

1. Feb 11, 2008

1. The problem statement, all variables and given/known data
Is the following system linear:
$$y(n) = x(n) + 0.8y(n-1)$$

3. The attempt at a solution
I'm really stuck on this one.

My idea was to assume that the system is casual, find the impulse response and then use the fact that: $$y(n)=x(n)*h(n)$$. However, if I assume that the output is the convolution of the impulse response and the input, I have ALREADY ASSUMED (and not proven) that the system is linear.

So I am lost.

I can't wrap my brain around the fact that the output depends on the input and the previous output.

Another idea I had is that if I show that I input a sinusoid and I get a scaled and phase shifted sinusoid, then the system is LTI.

Any help?

2. Feb 12, 2008

### EnumaElish

What if, assuming y(0) = x(0), you wrote:

y(n) = x(n) + 0.8y(n-1) = x(n) + 0.8x(n-1)... + (0.8)^n x(0),

would that help?