Algebraic Derivation

By inspection:

$$F_o(z) = z^{-N} \left\{ F_i(z) + z^{-2M}\left[F_i(z) + z^{-N} H_l(z)F_o(z)\right]\right\}$$

$$\begin{eqnarray*} \Rightarrow\quad H(z) \mathrel{\stackrel{\Delta}{=}}\frac{F_o(... ...}}\\ &=& \left(1+z^{-2M}\right)\frac{z^{-N}}{1-z^{-(2M+2N)}}\\ \end{eqnarray*}$$