Cross-spectral matrix diagonal reconstruction

In some cases, measured cross-spectral matrices (CSM's) from a microphone array will be contaminated by severe incoherent noise signals in the individual channels. A typical example is flow noise when measuring in a wind tunnel. Assuming stationary signals and performing long-time averaging, the contamination will be concentrated on the diagonal of the CSM. When the CSM is used for traditional frequency-domain beamforming, the use of the diagonal can be avoided - a technique referred to as Diagonal Removal (DR). DR is effective at suppressing the effects of the contamination, but it also has some side effects, such as underestimation of source levels. With other beamforming algorithms, such as Functional Beamforming, and in connection with acoustic holography, however, the diagonal of the CSM is needed. The present paper describes a method for removal of incoherent noise contamination from the CSM diagonal. The method formulates the problem as a Semidefinite Program, which is a convex optimization problem that can be solved very efficiently and with guaranteed convergence properties. A numerical study focuses on the question, whether the Semidefinite Program formulation will provide in all cases the desired output, i.e. the diagonal with precisely the contamination removed. Results from measurements are also presented.