Charles Henry czhenry at gmail.com
Wed Jul 2 01:39:33 CEST 2008

```On Sat, Jun 28, 2008 at 6:43 AM, cyrille henry
<cyrille.henry at la-kitchen.fr> wrote:

> ok, i'll try that.
> but i don't think adjusting the 2nd derivative is the best thing to do.
> for me, having a 6 point interpolation would be more important.

I put together a 6-point interpolation formula and analyzed it.  For
this I used a 5th degree polynomial, and 6 constraints:

(I want to change up the notation a bit, and not use the letters a, b,
c, etc... when switching to 6-point.  Y[-2],Y[-1],Y[0], Y[1], Y[2],
Y[3] are the points from the table. a5 is the coefficient of x^5, a4
is the coeff. of x^4, ... a0 is a constant term.  f(x) is the
interpolation polynomial.)

f(0)=Y[0]
f(1)=Y[1]
f'(0)= 1/12*Y[-2] - 2/3*Y[-1] + 2/3*Y[1] - 1/12*Y[2]
f'(1)= 1/12*Y[-1] - 2/3*Y[0]  + 2/3*Y[2] - 1/12*Y[3]
f''(0)= -1/12*Y[-2] + 4/3*Y[-1] - 5/2*Y[0] + 4/3*Y[1] - 1/12*Y[2]
f''(1)= -1/12*Y[-1] + 4/3*Y[0]  - 5/2*Y[1] + 4/3*Y[2] - 1/12*Y[3]

This uses improved approximations for the derivative.  One advantage
of going to 6-point interpolation is to get better numerical
derivatives.  These approximations of the 1st and 2nd derivatives are
accurate up to a higher frequency than before.  We can also continue
to increase the number of points arbitrarily, without necessarily
having to increase the degree of the polynomial.  The degree of the
polynomial is only determined by the number of constraints, not the
number of points.

The coefficients used in this scheme are

a0= Y[0]
a1= 1/12*Y[-2] - 2/3*Y[-1] + 2/3*Y[1] - 1/12*Y[2]
a2= -1/24*Y[-2] + 2/3*Y[-1] - 5/4*Y[0] + 2/3*Y[1] - 1/24*Y[2]
a3= -3/8*Y[-2] + 13/8*Y[-1] - 35/12*Y[0] + 11/4*Y[1] - 11/8*Y[2] + 7/24*Y[3]
a4= 13/24*Y[-2] - 8/3*Y[-1] + 21/4*Y[0] - 31/6*Y[1] + 61/24*Y[2] - 1/2*Y[3]
a5= -5/24*Y[-2] + 25/24*y[-1] - 25/12*Y[0] + 25/12*Y[1] - 25/24*Y[2] + 5/24*Y[3]

After that, I continued with the impulse response calculations and
spectral response calculations, which are a bit disappointing.  I'll
spare you the equations (for now) and post the graphs.  The new traces
for the 6-point interpolator are shown in green.  It's a little bit
hard to see, but the things to look for are the rate at which the
graph falls off and the locations of the peaks.  The 6-point function
has a flatter spectrum, which comes up closer to the Nyquist
frequency, and falls off faster.  These are the key characteristics of
the spectrum we want.  The green trace falls off according to 1/w^4,

You can see the impulse response in the first graph along with the
spectrum.  The log vs. dB scale is used same as before, and secondly,
I've posted a linear graph, so you can see the difference between
functions near the Nyquist frequency (x=pi).

It gives me some ideas for another 6-point scheme, more like
tabread4c~, which will fall off at a rate of 1/w^5 and have more
notches in the frequency response.  I'll work on it a bit, and see how
it goes.

Chuck
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