  In many multi-microphone algorithms, an estimate of the relative transfer
functions (RTFs) of the desired speaker is required. Recently, a
computationally efficient RTF vector estimation method was proposed for
acoustic sensor networks, assuming that the spatial coherence (SC) of the noise
component between a local microphone array and multiple external microphones is
low. Aiming at optimizing the output signal-to-noise ratio (SNR), this method
linearly combines multiple RTF vector estimates, where the complex-valued
weights are computed using a generalized eigenvalue decomposition (GEVD). In
this paper, we perform a theoretical bias analysis for the SC-based RTF vector
estimation method with multiple external microphones. Assuming a certain model
for the noise field, we derive an analytical expression for the weights,
showing that the optimal model-based weights are real-valued and only depend on
the input SNR in the external microphones. Simulations with real-world
recordings show a good accordance of the GEVD-based and the model-based
weights. Nevertheless, the results also indicate that in practice, estimation
errors occur which the model-based weights cannot account for.
