Analysis of 2-D state-space periodically shift-variant discrete systems

Two-dimensional (2-D) periodically shift-variant (PSV) digital filters are considered. These filters have applications in processing video signals with cyclostationary noise, scrambling digital images, and 2-D multirate signal processing. The filters are formulated in the form of a Givone-Roesser (GR) state-space model with periodic coefficients. This PSV model is then presented in block form as a shift-invariant system that also has the same GR state-space form. This block form has reduced computations and ease of analysis. An algorithm is developed that transforms any given 2-D PSV GR system to its equivalent shift-invariant model. Invertibility of this model is an important consideration, especially in image scrambling applications. A condition is established for the invertibility of the shift-invariant model of the 2-D PSV system. Also, the inverse system can be easily computed from the original. The established results are illustrated with an example.