A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the form
$g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in time-frequency analysis
and digital signal processing. A principal problem is to find a dual system
$\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each function $f$ can be written as $f = \sum
< f, \gamma_{m,n} > g_{m,n}$. The mathematical theory usually addresses this problem
in infinite dimensions (typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have
to operate with a finite-dimensional model. Exploiting the link between the frame operator
and Laurent operators with matrix-valued symbol, we apply the finite section method to show
that the dual functions obtained by solving a finite-dimensional problem converge to the
dual functions of the original infinite-dimensional problem in $l_2(Z)$. For compactly
supported $g_{m,n}$ (FIR filter banks) we prove an exponential rate of convergence and
derive explicit expressions for the involved constants. Further we investigate under which
conditions one can replace the discrete model of the finite section method by the periodic
discrete model, which is used in many numerical procedures. Again we provide explicit
estimates for the speed of convergence. Some remarks on tight frames complete the paper.