[Introduction]  [Signal arithmetic]  [Signals and noise]   [Smoothing]   [Differentiation]  [Peak Sharpening]  [Harmonic analysis]   [Fourier convolution]  [Fourier deconvolution]  [Fourier filter]  [Wavelets]   [Peak area measurement]  [Linear Least Squares]  [Multicomponent Spectroscopy]  [Iterative Curve Fitting]  [Hyperlinear quantitative absorption spectrophotometry] [Appendix and Case Studies]  [Peak Finding and Measurement]  [iPeak]   [iSignal]  [Peak Fitters]   [iFilter]  [iPower]  [List of downloadable software]  [Interactive tools]

### Appendix E: Ensemble averaging patterns in a continuous signal.

Ensemble averaging is a powerful method of reducing the effect of random noise in experimental signals, when it can be applied. The idea is that the signal is repeated, preferably a large number of times, and all the repeats are averaged. The signal builds up, and the noise gradually averages towards zero, as the number of repeats increases.

An important requirement is that the repeats be aligned or synchronized, so that in the absence of random noise, the repeated signals would line up exactly. There are two ways of managing this:

(a) the signal repeats are triggered by some external event and the data acquisition can use that trigger to synchronize the signals, or
(b) the signal itself has some feature that can be used to detect each repeat, whenever it occurs.

The first method (a) has the advantage that the signal-to-noise (S/N) ratio can be arbitrarily low and the average signal will still gradually emerge from the noise if the number of repeats is large enough. However, not every experiment has a reliable external trigger.

The second method (b) can be used to average repeated patterns in one continuous signal without an external trigger that corresponds to each repeat, but the signal must then contain some feature (for example, a peak) with a signal-to-noise ratio large enough to detect reliably in each repeat. This method can be used even when the signal patters occur at random intervals, when the timing of the repetitions is not of interest. The interactive peak detector iPeak 6 has a built-in ensemble averaging function (Shift-E) can compute the average of all the repeating waveforms. It works by detecting a single peak in each repeat in order to synchronize the repeats.

The Matlab script iPeakEnsembleAverageDemo.m (on http://tinyurl.com/cey8rwh) demonstrates this idea, with a signal that contains a repeated underlying pattern of two overlapping Gaussian peaks, 12 points apart, with a 2:1 height ratio, both of width 12. These patterns occur a random intervals, and the noise level is about 10% of the average peak height. Using iPeak (above left), you adjust the peak detection controls to detect only one peak in each repeat pattern, zoom in to isolate any one of those repeat patterns, and press Shift-E. In this case there are about 60 repeats, so the expected signal-to-noise (S/N) ratio improvement is sqrt(60) = 7.7. You can save the averaged pattern (above right) into the Matlab workspace as "EA" by typing

then curve-fit this averaged pattern to a 2-Gaussian model using the peakfit.m function (figure on the right):

>> peakfit([1:length(EA);EA],40,60,2,1,0,10)

Position     Height     Width    Area

32.54       13.255     12.003   169.36

44.72       6.7916     12.677    91.648

You will see a big improvement in the accuracy of the peak separation, height ratio and width, compared to fitting a single pattern in the original x,y signal:

>> peakfit([x;y],16352,60,2,1,0,10)

This page is part of "A Pragmatic Introduction to Signal Processing", created and maintained by Prof. Tom O'Haver , Department of Chemistry and Biochemistry, The University of Maryland at College Park. Comments, suggestions and questions should be directed to Prof. O'Haver at toh@umd.edu. Updated July, 2022.