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.

中文翻译：

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乘法处理器中的交叉项衰减

乘法处理器结合欠采样子阵列的波束成形输出来估计空间功率谱。虽然乘法处理器需要更少的传感器来实现与传统线性处理器相同的分辨率，但它也会产生交叉项，从而降低频谱估计值。交叉项是当通过一个子阵列的栅瓣的信号与通过另一个子阵列的不同信号相互作用时形成的假峰。当信号不相关时，快照平均会减少交叉项。这封信量化了有效交叉项缓解所需的快照数量，说明了使用子阵列锥度来控制旁瓣电平。为了便于分析，这封信推导出了独立复数高斯随机变量乘积之和的概率密度函数和累积分布函数的闭式表达式。分析表明，交叉项以每十年快照平均 5 dB 的速率衰减。虽然子阵列锥形减少了旁瓣泄漏，但它也增加了交叉项缓解所需的快照数量。