  Unitary matrix-valued functions of frequency are matrix all-pass systems,
since they preserve the norm of the input vector signals. Typically, such
systems are represented and analyzed using their unitary-matrix valued
frequency domain characteristics, although obtaining rational realizations for
matrix all-pass systems enables compact representations and efficient
implementations. However, an approach to obtain matrix all-pass filters that
satisfy phase constraints at certain frequencies was hitherto unknown. In this
paper, we present an interpolation strategy to obtain a rational matrix-valued
transfer function from frequency domain constraints for discrete-time matrix
all-pass systems. Using an extension of the Subspace Nevanlinna Pick
Interpolation Problem (SNIP), we design a construction for discrete-time matrix
all-pass systems that satisfy the desired phase characteristics. An innovation
that enables this is the extension of the SNIP to the boundary case to obtain
efficient time-domain implementations of matrix all-pass filters as matrix
linear constant coefficient difference equations, facilitated by a rational
(realizable) matrix transfer function. We also show that the derivative of
matrix phase constraints, related to the group delay at the interpolating
points, can be optimized to control the all-pass transfer matrices at the
unspecified frequencies. Simulations show that the proposed technique for
unitary matrix filter design performs as well as traditional DFT based
interpolation approaches, including Geodesic interpolation and the popular
Givens rotation based matrix parameterization.
