Conditional averaging of overlapping pulses

Abstract

Conditional averaging is a signal processing method used to study turbulent fluctuations in a variety of fields. The method, in its simplest form, works by finding peaks in a signal that fulfill a certain size threshold. Equally sized excerpts of the signal around every peak are then cut out and averaged. This yields the average shape of the events that fulfill the condition. Based on the peak finding within the method one also obtains the amplitudes and waiting times between the conditional events. the aim of this thesis is to test if these statistics can be used to estimate underlying properties of the signals we are looking at. We use signals created by superposing pulses with the same shape at different times and with different amplitudes decided by input probability distributions, and compare the inputs used to make the signals with the outputs of conditional averaging. By changing the input distributions, we can alter the degree of pulse overlap within the signals, and thus see for which degrees of pulse overlap conditional averaging successfully reproduces the underlying statistics. We will also investigate the methods robustness in the face of noise, studying how well different conditions recover the underlying pulses, while also attempting to establish if additional conditions aimed at reducing noise effects are necessary. Our results conclude that conditional averaging generally works well when predicting the shape of the underlying pulses, even in the face of pulse overlap. Noise severely affects the pulse shape estimates without any additional noise-reducing conditions. The noise reducing condition that we investigate here is one where we enforce a minimum distance between peaks. This additional condition results in greatly mitigating the effects of noise., leaving us to conclude that this it should always be used in addition to the size threshold condition to achieve the most robust results. The amplitude distribution estimates work well if there are not too many overlapping structures. However, the estimate break down even in the case of a moderate degree of pulse overlap, defined roughly as when there is on average one pulse arrival per pulse duration time in the signal. The estimates resemble the tail of the signals probability distribution largely independent of input amplitudes. Comparing the estimates from different input waiting time distributions, we conclude that the method fails to predict the underlying distribution, even in the case of little pulse overlap. We loosely define this as one pulse arrival per ten pulse durations on average within the signal. When pulses overlap more often, the method predicts exponentially distributed waiting times independent of input distribution. This leads to the main conclusion of the thesis; that the use of conditional averaging should in general be limited to estimating the average shape of underlying events. Finally, the use of conditional averaging in previous works is also discussed. We find that the conclusions based on the average pulse shape estimate are valid, as authors use the method in regimes of pulse overlap where this estimate is still accurate. Contrary to this, conclusions made from amplitude and waiting time distributions are often made at degrees of pulse overlap where we have demonstrated the method to give erroneous results. This might lead authors to make conclusions based on incorrect information.