All-Pass Filters for Mirroring Pairs of Complex-Conjugated Roots of Rational Matrix Functions
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In this note, we construct real-valued all-pass filters for mirroring pairs of complex-conjugated determinantal roots of a matrix polynomial. This problem appears, e.g., when proving the spectral factorization theorem, or more recently in the literature on possibly non-invertible or possibly non-causal vector autoregressive moving average (VARMA) models. In general, it is not obvious whether the all-pass filter (and as a consequence the all-pass transformed matrix polynomial with real-valued coefficients) which mirrors complex-conjugated roots at the unit circle is real-valued. Naive constructions result in complex-valued all-pass filters which implies that the real-valued parameter space (usually relevant for estimation) is left.
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