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From: glen herrmannsfeldt on 7 May 2010 21:03
Steve Pope <spope33(a)speedymail.org> wrote:
(snip)

> Yes, that is what you get. If you use four sample points, you get
> an interpolated point on the cubic polynomial that goes through
> them.

Is it similar to spline interpolation? The popular cubic splines
can also be done at higher orders, and that might be useful in
this case.

> A "Lagrangian interpolator" specifically is a calculation that
> uses a ratio of two polynomials, one of which is the derivative of
> the other, as in N/D in the above example. It's a way of computing
> a point on the target polynomial (e.g. on the cubic curve if you're
> interpolating from four points), without having to compute the
> coefficients of the polynomial.

> This shows up all over the place, such as in algebraic decoding
> when computing error values.

-- glen
From: Steve Pope on 7 May 2010 21:21
glen herrmannsfeldt <gah(a)ugcs.caltech.edu> wrote:

>Steve Pope <spope33(a)speedymail.org> wrote:

[Lagrangian interpolation]

>> Yes, that is what you get. If you use four sample points, you get
>> an interpolated point on the cubic polynomial that goes through
>> them.

>Is it similar to spline interpolation?

Spline interpolators are more complex and are designed to
avoid unaesthetic stuff like overshoots for some datasets.

They are similar in that both Lagrange and Spline interpolators
pass through the datapoints, as opposed to a least squares
curve which does not.


Steve
From: robert bristow-johnson on 7 May 2010 22:46
On May 7, 9:21 pm, spop...(a)speedymail.org (Steve Pope) wrote:
> glen herrmannsfeldt  <g...(a)ugcs.caltech.edu> wrote:
>
> >Steve Pope <spop...(a)speedymail.org> wrote:
>
> [Lagrangian interpolation]
>
> >> Yes, that is what you get.  If you use four sample points, you get
> >> an interpolated point on the cubic polynomial that goes through
> >> them.
> >Is it similar to spline interpolation?  
>
> Spline interpolators are more complex and are designed to
> avoid unaesthetic stuff like overshoots for some datasets.
>
> They are similar in that both Lagrange and Spline interpolators
> pass through the datapoints, as opposed to a least squares
> curve which does not.

but if the number of degrees of freedom and the number of data points
are the same (and they are both polynomials), don't they have to be
exactly the same?

i know about B-splines and Hermite polynomial interpolation, unlike
Lagrange, they might not go through all the points (maybe just the two
adjacent points to the left and right). but i have been trying to
understand what exactly they mean by a "cubic spline" if it isn't the
polynomial that hits 4 points, 2 on the left and 2 on the right.
maybe there is some edge conditions that are different.

r b-j
From: pnachtwey on 8 May 2010 01:30
On May 7, 7:46 pm, robert bristow-johnson <r...(a)audioimagination.com>
wrote:
> On May 7, 9:21 pm, spop...(a)speedymail.org (Steve Pope) wrote:
>
>
>
>
>
> > glen herrmannsfeldt  <g...(a)ugcs.caltech.edu> wrote:
>
> > >Steve Pope <spop...(a)speedymail.org> wrote:
>
> > [Lagrangian interpolation]
>
> > >> Yes, that is what you get.  If you use four sample points, you get
> > >> an interpolated point on the cubic polynomial that goes through
> > >> them.
> > >Is it similar to spline interpolation?  
>
> > Spline interpolators are more complex and are designed to
> > avoid unaesthetic stuff like overshoots for some datasets.
>
> > They are similar in that both Lagrange and Spline interpolators
> > pass through the datapoints, as opposed to a least squares
> > curve which does not.
>
> but if the number of degrees of freedom and the number of data points
> are the same (and they are both polynomials), don't they have to be
> exactly the same?
>
> i know about B-splines and Hermite polynomial interpolation, unlike
> Lagrange, they might not go through all the points (maybe just the two
> adjacent points to the left and right).  but i have been trying to
> understand what exactly they mean by a "cubic spline" if it isn't the
> polynomial that hits 4 points, 2 on the left and 2 on the right.
> maybe there is some edge conditions that are different.
>
> r b-j
Cubic splines are special cases in that the sum of second derivatives
squared is minimized for any function. The cubic spline is calculated
all at once whereas Lagrange interpolation can be calculated 4 points
at a time. The second derivative of Lagrange interpolation is not
guaranteed to be continuous whereas a cubic spline's second derivative
is continuous.

Peter Nachtwey
From: Rune Allnor on 8 May 2010 02:37
On 8 Mai, 00:46, "Alfred Bovin" <alf...(a)bovin.invalid> wrote:

> random offsets
> determined by unpredictable initial conditions. I can, however, get rather
> precise time stamps for every measurement.
>
> For example I can get readings
> from sensor1 at 0.0, 0.1, 0.2, 0.3, ... sec
> from sensor 2 at 0.0, 0.07, 0.17, 0.27, ... sec
> from sensor 3 at 0.08, 0.18, 0.28, 0.38, ... sec
>
> My idea is then to reconstruct the x(t)'s with standard windows sinc
> interpolation and then resample at 10 Hz at the same time points for all
> three signal. What will work, won't it?

You are addressing the wrong question. The problem here is not that
the three channels are sampled at different rates, but that they are
not synced up. Except for that first unpredictable delay, the rates
are 10 Hz all over. The time stamps don't matter, as there is no
telling how far down the instrumentation chain they were added
to the data points.

Instead of asking how to reseample three channels to the same rate,
you should ask how to sync up the three channels. There are at least
two ways of doing this:

1) Correlate the data from the various channels, and use relative
delays to match up shared features in the data
2) Investigate the measurement system to estimate system delays in
the
data flow.

Rune
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