We illustrate some recent results on exact solutions to discrete-time l1-norm minimization problems with convolution constraints. A fixed-point property for this class of problems is introduced. The convolution constraints can be interpreted as a dynamic system with initial conditions. We show by construction that optimal solutions with a rational Z-transform exist for any initial conditions satisfying the fixed-point property. Some fixed-point initial conditions satisfy a further stability property. If there exists a stable fixed point, then for any initial condition in some neighbourhood of the fixed point an optimal solution can be constructed having a rational Z-transform.