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Suposse I have to solve this system:

$$ \left( \begin{array}{cccc} r_g(0) & r_g(1) & \cdots & r_g(N-1) \\ r_g(1) & r_g(0) & \cdots & r_g(N-2) \\ \vdots & \vdots & \ddots & \vdots \\ r_g(N-1) & r_g(N-2) & \cdots & r_g(0) \end{array} \right) \left( \begin{array}{c} h(0) \\ h(1)\\ \vdots \\ h(N-1) \end{array} \right) = \left( \begin{array}{c} g(0) \\ 0\\ \vdots \\ 0 \end{array} \right) $$

Where the unkown variables are $h(0), h(1), \cdots, h(N-1)$. Moreover, $g(n)$ is:

$$g(n) = \alpha^{n+1}u(n) - \alpha^{n-1}u(n-1) \quad \text{with } \left \lbrace \begin{array}{c} |\alpha| < 1 \\ n = 0,1,\cdots,\infty \end{array} \right.$$

$u(n)$ is a function that gives $1$ for $n \geq 0 $ and $0$ for $n < 0$. On the other hand, the definition of $r_g(n)$ is:

$$r_g(k) = \sum_{n=0}^{\infty}g(n)g(n-k)$$

If I didn't make any mistake, this results:

$$r_g(k) = \frac{\alpha^k (\alpha^2+1)^2}{1-\alpha^2}$$

Thus, the system I have to solve is:

$$ \frac{(\alpha^2+1)^2}{1-\alpha^2} \left( \begin{array}{cccc} 1 & \alpha & \cdots & \alpha^{N-1} \\ \alpha & 1 & \cdots & \alpha^{N-2} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha^{N-1} & \alpha^{N-2} & \cdots & 1 \end{array} \right) \left( \begin{array}{c} h(0) \\ h(1)\\ \vdots \\ h(N-1) \end{array} \right) = \left( \begin{array}{c} \alpha \\ 0\\ \vdots \\ 0 \end{array} \right) $$

The problem is that I need an analytical expression for $h(n)$ but is a $NxN$ matrix where the coefficient $N$ is unknown.

I hope someone can help me. Thank you for your answers.

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