Phase Characteristics for the Stability of 2-D Quarter-Plane Recursive Digital All-Pass Filters

In this brief, the monotone phase-response property of a 2-D causal digital all-pass filter (DAF) with real coefficients or complex coefficients in the quarter plane (QP) support region is presented. Regarding the circumstance of real coefficients, we also prove that the previously proposed stability criterion on the viewpoint of unwrapped phase by Pei and Shyu is necessary and sufficient for 2-D separable DAFs but is only sufficient for QP DAFs. The resultant property possesses the advantage of increasing the freedom of phase design over the previously proposed one. A remarkable application of the presented property is choosing an appropriate specification for the desired phase response of a 2-D DAF design. A design example is given to show this application, and the stability is also inspected by the 2-D spectral factorization algorithm and the root maps for confirmation.