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
