A velocity is estimated as the slope of a straight line fitted by least squares to…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A velocity is estimated as the slope of a straight line fitted by least squares to

m

readings spaced

\tau

apart and centred on the time of interest. Each reading has independent noise of standard deviation

\sigma

, and the underlying curve has a constant third derivative

f'''

. The slope has a noise term

\sigma\big/\bigl[\tau\sqrt{m(m^2-1)/12}\bigr]

and a truncation term

(3m^2-7)\,\tau^2|f'''|/120

, which add in quadrature in the mean-square error. Let

W=m\tau

be the window width. ((a)) For

m\gg1

write both terms in terms of

W

\sigma

\tau

and

|f'''|

, keeping only the leading power of

W

in each. ((b)) Show that the window width that minimises the rms error satisfies

W^7=14400\,\sigma^2\tau/f'''^2

. ((c)) Take

\sigma=1\,\mathrm{mm}

and

|f'''|=1\,\mathrm{m\,s^{-3}}

. Find the best

W

and the rms error of the velocity at

100\,\mathrm{Hz}

and at

1\,\mathrm{kHz}

, and say how the best error scales with the sampling interval

\tau

. Compare with the

6.0\,\mathrm{mm\,s^{-1}}

of the best single difference.

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