We consider the problem of minimizing the disturbance power at the output of the matched filter in a single antenna cognitive radar set-up. The aforementioned disturbance power can be shown to be an expectation of the slow-time ambiguity function (STAF) of the transmitted waveform over range-Doppler bins of interest. The design problem is known to yield a nonconvex quartic function of the transmit radar waveform. This STAF shaping problem becomes even more challenging in the presence of practical constraints on the transmit waveform such as the constant modulus constraint (CMC). Most existing approaches address the aforementioned challenges by suitably modifying or relaxing the design cost function and/or the CMC. In a departure from such methods, we develop a solution that involves direct optimization over the nonconvex complex circle manifold, i.e., the CMC set. We derive a new update strategy [quartic-gradient-descent (QGD)] that computes an exact gradient of the quartic cost and invokes principles of optimization over manifolds toward an iterative procedure with guarantees of monotonic cost function decrease and convergence. Experimentally, QGD can outperform state-of-the-art approaches for shaping the ambiguity function under the CMC while being computationally less expensive.

中文翻译：

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四次梯度下降用于可追踪的雷达慢时间模糊函数 (STAF) 整形

我们考虑在单天线认知雷达设置中最小化匹配滤波器输出端的干扰功率的问题。上述干扰功率可以显示为对感兴趣的距离多普勒区间上传输波形的慢时模糊度函数 (STAF) 的期望。已知设计问题会产生发射雷达波形的非凸四次函数。在存在对发射波形的实际约束（例如恒模约束 (CMC)）的情况下，STAF 整形问题变得更具挑战性。大多数现有方法通过适当修改或放宽设计成本函数和/或 CMC 来解决上述挑战。与此类方法不同，我们开发了一种解决方案，该解决方案涉及对非凸复圆流形的直接优化，即。例如，CMC 集。我们推导出一种新的更新策略 [quartic-gradient-descent (QGD)]，它计算四次成本的精确梯度，并调用流形上的优化原则，以保证单调成本函数减少和收敛的迭代过程。在实验上，QGD 可以胜过在 CMC 下塑造模糊函数的最先进方法，同时计算成本更低。