Randomized Signal Processing with Continuous Frames

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.