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Unitary matrix-valued functions of frequency are matrix all-pass systems,since they preserve the norm of the input vector signals. Typically, suchsystems are represented and analyzed using their unitary-matrix valuedfrequency domain characteristics, although obtaining rational realizations formatrix all-pass systems enables compact representations and efficientimplementations. However, an approach to obtain matrix all-pass filters thatsatisfy phase constraints at certain frequencies was hitherto unknown. In thispaper, we present an interpolation strategy to obtain a rational matrix-valuedtransfer function from frequency domain constraints for discrete-time matrixall-pass systems. Using an extension of the Subspace Nevanlinna PickInterpolation Problem (SNIP), we design a construction for discrete-time matrixall-pass systems that satisfy the desired phase characteristics. An innovationthat enables this is the extension of the SNIP to the boundary case to obtainefficient time-domain implementations of matrix all-pass filters as matrixlinear constant coefficient difference equations, facilitated by a rational(realizable) matrix transfer function. We also show that the derivative ofmatrix phase constraints, related to the group delay at the interpolatingpoints, can be optimized to control the all-pass transfer matrices at theunspecified frequencies. Simulations show that the proposed technique forunitary matrix filter design performs as well as traditional DFT basedinterpolation approaches, including Geodesic interpolation and the popularGivens rotation based matrix parameterization.

Design of Discrete-time Matrix All-Pass Filters Using Subspace Nevanlinna Pick Interpolation