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Say the Fibonacci sequence is defined by:

$y(n) = y(n-1) + y(n-2)$

initial conditions: $y(0)=0, y(1)=1$

I incorporate those initial conditions as:

$y(n) = y(n-1) + y(n-2) + \delta(n-1)$

I compute the z-transform as:

$Y(z) = z^{-1}Y(z) + z^{-2}Y(z) + z^{-1}$

solve for $Y(z)$:

$Y(z) = \frac{z^{-1}}{1-z^{-1}-z^{-2}}$

Now I am having the hardest time taking the causal inverse z-transform. Any tips?

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Using partial fraction techniques and denoting the zeroes of $z^2-z-1$ by $\alpha_1$ and $\alpha_2$, we have

$$ Y(z) = \frac{z}{z^2-z-1} \implies \frac{Y(z)}{z} = \left(-\frac{1}{5}+\frac{2}{5}\alpha_1\right) \frac{1}{z-\alpha_1}+ \left(-\frac{1}{5}+\frac{2}{5}\alpha_2\right) \frac{1}{z-\alpha_2} $$

$$ \implies Y(z) = \beta_1 \frac{z}{z-\alpha_1}+\beta_2 \frac{z}{z-\alpha_2} $$

$$ \implies Y(z) = \beta_1\left(1-\frac{\alpha_1}{z}\right)^{-1}+\beta_2\left(1-\frac{\alpha_2}{z}\right)^{-1} $$

where $\beta_1 =\left(-\frac{1}{5}+\frac{2}{5}\alpha_1\right)$ and $\beta_2=\left(-\frac{1}{5}+\frac{2}{5}\alpha_2\right)$.

$$ \implies Y(z) = \beta_1 \sum_{n=0}^{\infty}\frac{\alpha_1^n}{z^n}+\beta_2 \sum_{n=0}^{\infty}\frac{\alpha_2^n}{z^n}$$

$$ \implies y(n)= \beta_1\alpha_1^n + \beta_2\alpha_2^n . $$

Note: You can use the inverse Z-transform formula which uses the contour integration

$$ y(n)=\frac{1}{2\pi i}\int_{C} z^{k-1}Y(z) dz .$$

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