Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we study a well-known and influential demosaicking algorithm based on alternating projections (AP), proposed by Gunturk, Altunbasak and Mersereau in 2002. Since its publication, the AP algorithm has been widely cited and compared against in a series of more recent papers in the demosaicking literature. Despite good performances, a limitation of the AP algorithm is its high computational complexity. We provide three main contributions in this paper. First, we present a rigorous analysis of the convergence property of the AP demosaicking algorithm, showing that it is a contraction mapping, with a unique fixed point. Second, we show that this fixed point is in fact the solution to a constrained quadratic minimization problem, thus establishing the optimality of the AP algorithm. Finally, using the tool of polyphase representation, we show how to obtain the results of the AP algorithm in a single step, implemented as linear filtering in the polyphase domain. Replacing the original iterative procedure by the proposed one-step solution leads to substantial computational savings, by about an order of magnitude in our experiments.