Impulse response of recursive DT system

  • Thread starter dionysian
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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.
 

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