# Impulse response of recursive DT system

## Homework Statement

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$
Determine the impulse response $$h[n]$$ by calculating the zero-state response with $$x[n] = \delta [n]$$

## Homework Equations

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$

## The Attempt at a Solution

$$y[n] - 1.8\cos (\frac{\pi }{{16}})y[n - 1] + 0.81y[n - 2] = x[n] + 0.5x[n - 1]$$
$$y[n] = 1.8\cos (\frac{\pi }{{16}})y[n - 1] - 0.81y[n - 2] + x[n] + 0.5x[n - 1]$$
$$h[n] = 1.8\cos (\frac{\pi }{{16}})y[n - 1] - 0.81y[n - 2] + \delta [n] + 0.5\delta [n - 1]$$
The zero-state response means
$$\begin{array}{l} y[- 1] = 0 \\ y[- 2] = 0 \\ \end{array}$$
So my question is how can I write out the impulse response explicitly in one expression. I think I could calculate it by $$h[0],h[1],h[2].....$$ but I want to write it out as just one expression but the recursive terms$$y[n - 1]$$ and $$y[n - 2]$$ are kind of throwing me off.