- #1

- 53

- 1

## Homework Statement

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

Determine the impulse response [tex]h[n][/tex] by calculating the zero-state response with [tex]x[n] = \delta [n][/tex]

## Homework Equations

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

## The Attempt at a Solution

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

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

[tex]h[n] = 1.8\cos (\frac{\pi }{{16}})y[n - 1] - 0.81y[n - 2] + \delta [n] + 0.5\delta [n - 1][/tex]

The zero-state response means

[tex]\begin{array}{l}

y[- 1] = 0 \\

y[- 2] = 0 \\

\end{array}[/tex]

So my question is how can I write out the impulse response explicitly in one expression. I think I could calculate it by [tex]h[0],h[1],h[2].....[/tex] but I want to write it out as just one expression but the recursive terms[tex]y[n - 1][/tex] and [tex]y[n - 2][/tex] are kind of throwing me off.