In the laboratory, scientists collect noisy data.  Often, the clean signal or function is unknown as well as the noise variance.  When studying the dynamics of a mechanical system, physicists measure displacements and velocities.  The displacement data consists of discrete points, sampled at equal intervals of time.  The velocities are then approximated using finite differences of the displacement data.  Unfortunately, this process introduces much noise, so much, in fact, that the velocity estimates are of little practical value.

Given a data set of the displacement {xn}, where n is an integer, of a system sampled at time intervals of length Dt, wavelet analysis can be used to find a smooth function that approximates the displacement data.

In examining the properties of dry friction, a block-spring system was constructed.  The displacement x(t) of the block and the force f(t) on the block due to friction were recorded every 0.005 seconds as the block slid back and forth on the surface.  Researchers investigated whether experimental laboratory data would match a modified model of friction shown in Figure 1 (left).  The velocity of the system was approximated by a common difference method.  These velocities were used to produce a v-f plot shown in Figure 1 (right).

Researchers then attempted to approximate the velocity function using digital filtering, and finally wavelets.  In the third approach, N4(t) cubic spline interpolation was used to estimate the displacement and force functions.  The velocity function was calculated as the first time derivative of the displacement function.  Results of all three methods are shown in Figure 2 [7].

Figure 2:   Estimates for the velocity of the block,
obtained from the use of three different methods [7].
 
 

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