  This report describes the computation of gradients by algorithmic
differentiation for statistically optimum beamforming operations. Especially
the derivation of complex-valued functions is a key component of this approach.
Therefore the real-valued algorithmic differentiation is extended via the
complex-valued chain rule. In addition to the basic mathematic operations the
derivative of the eigenvalue problem with complex-valued eigenvectors is one of
the key results of this report. The potential of this approach is shown with
experimental results on the CHiME-3 challenge database. There, the beamforming
task is used as a front-end for an ASR system. With the developed derivatives a
joint optimization of a speech enhancement and speech recognition system w.r.t.
the recognition optimization criterion is possible.
