Cross Term Decay in Multiplicative Processors

Multiplicative processors combine the beamformed outputs of undersampled subarrays to estimate the spatial power spectrum. While the multiplicative processor requires fewer sensors to achieve the same resolution as a conventional linear processor, it also produces cross terms that can degrade the spectral estimate. Cross terms are false peaks formed when a signal passing through the grating lobe of one subarray interacts with a different signal passing through the other subarray. Snapshot averaging reduces cross terms when the signals are uncorrelated. This letter quantifies the number of snapshots required for effective cross term mitigation, accounting for the use of subarray tapers to control sidelobe levels. To facilitate the analysis, the letter derives closed form expressions for the probability density function and cumulative distribution function of the sum of products of independent complex Gaussian random variables. The analysis demonstrates that cross terms decay at a rate of 5 dB per decade of snapshots averaged. While subarray tapering reduces sidelobe leakage, it also increases the number of snapshots required for cross term mitigation.