[PD] better tabread4~
cyrille henry
cyrille.henry at la-kitchen.fr
Tue Jul 8 18:35:51 CEST 2008
hello Chuck,
i tested this. (and commited)
i think tabread6c~ is a bit better than tabread4c~. but differences are more smaller
thx
Cyrille
Charles Henry a écrit :
> 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,
> compared to 1/w^3 for tabread4c~ and 1/w^2 for tabread4~
>
> 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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