[PD] better tabread4~

[cyrille henry]

ok, cool

now, it would also be nice to have a good band limited table reader...

cyrille


Charles Henry a écrit :
> 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
> <cyrille.henry at la-kitchen.fr> 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
>>> <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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