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

On the Computation of Complex-valued Gradients with Application to Statistically Optimum Beamforming