| In a number of electrical engineering problems, so-called "crossing points" -- the instants at which two continuous-time signals cross each other -- are of interest. Often, particularly in applications using a Digital Signal Processor (DSP), only periodic samples along with a partial statistical characterization of the signals are available. In this situation, we are faced with the following problem: Given limited information about these signals, how can we efficiently and accurately estimate their crossing points?;We consider estimating the crossing points of a known function and a Gaussian random process, given uniformly-spaced, noisy samples of the random process for which the second-order statistics are assumed to be known. We derive the Maximum A-Posteriori (MAP) estimator, along with a Minimum Mean-Squared Error (MMSE) estimator which we show to be a computationally efficient approximation to the MAP estimator.;We also derive the Cramer-Rao bound (CRB) on estimator variance for the problem, which allows practical estimators to be evaluated against a best-case performance limit. We investigate several comparison estimators chosen from the literature. The structure of the MMSE estimator and comparison estimators is shown to be very similar, making the difference in computational expense between each technique largely dependent on the cost of evaluating various (generally non-linear) functions.;Simulations for both Pulse-Width and Click Modulation scenarios show the MMSE estimator performs very near to the Cramer-Rao bound and outperforms the alternative estimators selected from the literature.;For example, an audio amplifier typically receives its input from a digital source decoded into regular samples (e.g. from MP3, DVD, or CD audio), or obtained from a continuous-time signal using an analog-to-digital converter (ADC). In a switching amplifier based on Pulse-Width Modulation (PWM) or Click Modulation (CM), a signal derived from the sampled audio is compared against a deterministic reference waveform; the crossing points of these signals control a switching power stage. Crossing-point estimates must be accurate in order to preserve audio quality. They must also be simple to calculate, in order to minimize processing requirements and delays. |
