How to calculate general formula for this recurrence?

Question

The recurrences are $$F_n = a F_{n - 1} + b G_{n - 2}$$ $$G_n = cG_{n - 1} + d F_{n - 2}$$ $$H_n = e F_{n} + f G_n$$ where $a, b, c, d, e, f$ are constants.

How do I calculate $H_n$ in terms of only constants?

Answer 1

the z-transform of a sequence is defined like this: $$ \mathcal Z\{ x[n]\}=X(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n} $$ then using this property of Z-transform you get(you can see all information needed for deep introduction to z-transform in https://en.wikipedia.org/wiki/Z-transform ) : $$\mathcal Z\{ x[n]\}=X(z) \Rightarrow \mathcal Z\{ x[n-k]\}=z^{-k}X(z)$$ then your system of equations change's into this: $$F_n = a F_{n - 1} + b G_{n - 2}\Rightarrow F(z)(1-az^{-1})=G(z)(bz^{-2})$$ $$G_n = cG_{n - 1} + d F_{n - 2}\Rightarrow G(z)(1-cz^{-1})=F(z)(dz^{-2})$$ $$H_n = e F_{n} + f G_n\Rightarrow H(z)=eF(z)+fG(z)$$