This paper focuses on signal processing tasks in which the signal is transformed from the signal space to a higher dimensional coefficient space (also called phase space) using a continuous frame, processed in the coefficient space, and synthesized to an output signal. We show how to approximate such methods, termed phase space signal processing methods, using a Monte Carlo method. As opposed to standard discretizations of continuous frames, based on sampling discrete frames from the continuous system, the proposed Monte Carlo method is directly a quadrature approximation of the continuous frame. We show that the Monte Carlo method allows working with highly redundant continuous frames, since the number of samples required for a certain accuracy is proportional to the dimension of the signal space, and not to the dimension of the phase space. Moreover, even though the continuous frame is highly redundant, the Monte Carlo samples are spread uniformly, and hence represent the coefficient space more faithfully than standard frame discretizations.
翻译:本文侧重于信号处理任务,其中将信号从信号空间转换为使用连续框架、在系数空间中处理和合成成输出信号的更高维系数空间(也称为相位空间)的信号处理任务。我们展示了如何使用蒙特卡洛方法来近似此类方法,称为相位空间信号处理方法。与基于连续系统离散框架取样的连续框架标准离散相比,拟议的蒙特卡洛方法直接是连续框架的二次近距离。我们显示蒙特卡洛方法允许使用高度冗余的连续框架,因为某种精确度所需的样本数量与信号空间的尺寸成正比,而不是与相位空间的尺寸成比例。此外,尽管连续框架非常多余,但蒙特卡洛样本分布一致,因此代表的系数空间比标准框架离散更加忠实。