Re: Re: 2nd derivarive of an accumulating signal

Martin Blume!
"Heck" schrieb
snip

This is called numerical differentiation.

Books can give you formulas, mine [*] gives:

f''(x) = (f(x+h)-2f(x)+f(x-h))/h^2
(with three points f(x-h), f(x), f(x+h))

or

f''(x) = (-f(x+2h)+16f(x+h)-30f(x)+16f(x-h)-f(x-2h))/h^2
with 4 points.

Beware that measured data tend to fluctuate, differentiation
is like a highpass filter that accentuates these fluctuations.
You might want to low-pass filter your data first. OTOH this
might give you wrong results if you expect your acceleration
in the high frequency range.

[*] H.Stöcker, Taschenbuch mathematischer Formeln und moderner
Verfahren, Verlag Harri Deutsch


[...] and I have to have
this characteristic nailed down before I received 705 of them.

I mistyped. I meant to say 70%.


[...]
Err, this characteristic is likely to change at every sample,
you know that? And as they tell you in banking, "past performance
is no guarantee for future performance" :-)
You want to have a "single value that respresents" the 2nd
derivative?

Can someone suggest how I might achieve this? What mathematical
relationship can I exploit? What algorithm might I use? Perhaps
if I
were to iteratively fit a curve to the data. The curve won't be
pathological at all, but I can't know where or how it will
accelerate
or taper off.

If you can wait for 705 sample (and a wee bit of compuatation time)
and you know that you can fit your data to a known mathematical
formula (albeit with unknown parameters) you can define this formula
and let gnuplot fit the data and compute the parameters, e.g.
f(x)=a*log(d*x+b)+c
fit f(x) "exp.dat" via a,b,c,d
Beware that this:
- might take a long time
- the fit might be bad
But from then on you can analytically compute the 2nd derivative at
any point.


I suspect this is a common problem in many applications. Just
point
me into the right area. Thanks.


In the hope that the above helps ...
Martin

Thanks very much to you and to the others who answered. Numerical
differentiation is the term I need, I think. The formulae you cited
seem to fit the bill. They look like the simple definition of
derivatives, the difference between a measured point and an
infinitesimally larger or smaller neighbor.

Thank you in particular for the warnings about the potential
volatility and inconsistency of the samples and, consequently, the
need and the danger, the advantage versus the liability, of filtering
the samples.
.

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