Hey, Cyrille, I kind of thought so... we are quickly running into the law of diminishing returns. I was up late, last night, working on the analysis some more. I think I can have another 6-point version with better characteristics tonight.
Chuck On Tue, Jul 8, 2008 at 11:35 AM, cyrille henry <[EMAIL PROTECTED]> wrote: > 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 >> <[EMAIL PROTECTED]> 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 >> >> >> ------------------------------------------------------------------------ >> >> >> ------------------------------------------------------------------------ >> >> >> ------------------------------------------------------------------------ >> >> _______________________________________________ >> [email protected] mailing list >> UNSUBSCRIBE and account-management -> >> http://lists.puredata.info/listinfo/pd-list > _______________________________________________ [email protected] mailing list UNSUBSCRIBE and account-management -> http://lists.puredata.info/listinfo/pd-list
