How to exploit periodicity to reduce noise of a signal?

100 periods have been collected from a 3 dimensional periodic signal. The wavelength slightly varies. The noise of the wavelength follows Gaussian distribution with zero mean. A good estimate of the wavelength is known, that is not an issue here. The noise of the amplitude may not be Gaussian and may be contaminated with outliers.

How can I compute a single period that approximates 'best' all of the collected 100 periods?

Time-series, ARMA, ARIMA, Kalman Filter, autoregression and autocorrelation seem to be keywords here.

UPDATE 1: I have no idea how time-series models work. Are they prepared for varying wavelengths开发者_如何学Go? Can they handle non-smooth true signals? If a time-series model is fitted, can I compute a 'best estimate' for a single period? How?

UPDATE 2: A related question is this. Speed is not an issue in my case. Processing is done off-line, after all periods have been collected.

Origin of the problem: I am measuring acceleration during human steps at 200 Hz. After that I am trying to double integrate the data to get the vertical displacement of the center of gravity. Of course the noise introduces a HUGE error when you integrate twice. I would like to exploit periodicity to reduce this noise. Here is a crude graph of the actual data (y: acceleration in g, x: time in second) of 6 steps corresponding to 3 periods (1 left and 1 right step is a period):

My interest is now purely theoretical, as http://jap.physiology.org/content/39/1/174.abstract gives a pretty good recipe what to do.

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We have used wavelets for noise suppression with similar signal measured from cows during walking. I'm don't think the noise is so much of a problem here and the biggest peaks represent actual changes in the acceleration during walking.

I suppose that the angle of the leg and thus accelerometer changes during your experiment and you need to account for that in order to calculate the distance i.e you need to know what is the orientation of the accelerometer in each time step. See e.g this technical note for one to account for angle.

If you need get accurate measures of the position the best solution would be to get an accelerometer with a magnetometer, which also measures orientation. Something like this should work: http://www.sparkfun.com/products/10321.

EDIT: I have looked into this a bit more in the last few days because a similar project is in my to do list as well... We have not used gyros in the past, but we are doing so in the next project.

The inaccuracy in the positioning doesn't come from the white noise, but from the inaccuracy and drift of the gyro. And the error then accumulates very quickly due to the double integration. Intersense has a product called Navshoe, that addresses this problem by zeroing the error after each step (see this paper). And this is a good introduction to inertial navigation.

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Periodic signal without noise has the following property:

f(a) = f(a+k), where k is the wavelength.

Next bit of information that is needed is that your signal is composed of separate samples. Every bit of information you've collected are based on samples, which are values of f() function. From 100 samples, you can get the mean value:

1/n * sum(s_i), where i is in range [0..n-1] and n = 100.

This needs to be done for every dimension of your data. If you use 3d data, it will be applied 3 times. Result would be (x,y,z) points. You can find value of s_i from the periodic signal equation simply by doing

s_i(a).x = f(a+k*i).x
s_i(a).y = f(a+k*i).y
s_i(a).z = f(a+k*i).z

If the wavelength is not accurate, this will give you additional source of error or you'll need to adjust it to match the real wavelength of each period. Since

k*i = k+k+...+k

if the wavelength varies, you'll need to use k_1+k_2+k_3+...+k_i instead of k*i. Unfortunately with errors in wavelength, there will be big problems keeping this k_1..k_i chain in sync with the actual data. You'd actually need to know how to regognize the starting position of each period from your actual data. Possibly need to mark them by hand.

Now, all the mean values you calculated would be functions like this:

m(a) :: R->(x,y,z)

Now this is a curve in 3d space. More complex error models will be left as an excersize for the reader.

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If you have a copy of Curve Fitting Toolbox, localized regression might be a good choice.

1. Curve Fitting Toolbox supports both lowess and loess localized regression models for curve and curve fitting.

2. There is an option for robust localized regression

The following blog post shows how to use cross validation to estimate an optimzal spaning parameter for a localized regression model, as well as techniques to estimate confidence intervals using a bootstrap.

http://blogs.mathworks.com/loren/2011/01/13/data-driven-fitting/