  Binaural reproduction is rapidly becoming a topic of great interest in the
research community, especially with the surge of new and popular devices, such
as virtual reality headsets, smart glasses, and head-tracked headphones. In
order to immerse the listener in a virtual or remote environment with such
devices, it is essential to generate realistic and accurate binaural signals.
This is challenging, especially since the microphone arrays mounted on these
devices are typically composed of an arbitrarily-arranged small number of
microphones, which impedes the use of standard audio formats like Ambisonics,
and provides limited spatial resolution. The binaural signal matching (BSM)
method was developed recently to overcome these challenges. While it produced
binaural signals with low error using relatively simple arrays, its performance
degraded significantly when head rotation was introduced. This paper aims to
develop the BSM method further and overcome its limitations. For this purpose,
the method is first analyzed in detail, and a design framework that guarantees
accurate binaural reproduction for relatively complex acoustic environments is
presented. Next, it is shown that the BSM accuracy may significantly degrade at
high frequencies, and thus, a perceptually motivated extension to the method is
proposed, based on a magnitude least-squares (MagLS) formulation. These
insights and developments are then analyzed with the help of an extensive
simulation study of a simple six-microphone semi-circular array. It is further
shown that the BSM-MagLS method can be very useful in compensating for head
rotations with this array. Finally, a listening experiment is conducted with a
four-microphone array on a pair of glasses in a reverberant speech environment
and including head rotations, where it is shown that BSM-MagLS can indeed
produce binaural signals with a high perceived quality.
