More tables and math...
Let me go ahead and re-print the previous table of fourier transforms and add some more functions to it, that will simplify things.
remember, these functions are non-zero on the interval from [-a,a] and zero, elsewhere f(t) | F(w) 1 | 2/w*sin(aw) |t| | 2a/w*sin(aw) + 2/w^2*(cos(aw)-1) t^2 | 2a^2/w*sin(aw) + 4a/w^2*cos(aw) - 4/w^3*sin(aw) |t|^3 | 2a^3/w*sin(aw) + 6a^2/w^2*cos(aw) - 12a/w^3*sin(aw) - 12/w^4*(cos(aw)-1)
and a new set of functions that seem to be more useful, because terms cancel. I'll also include another column for the limit as w->0.
f(t) | lim(w->0) F(w) | F(w) |t| - a | -a^2 | 2/w^2*(cos(aw)-1) (|t| - a)^2 | 2/3*a^3 | 4a/w^2 - 4/w^3*sin(aw) (|t| - a)^3 | -1/2*a^4 | -6a^2/w^2 - 12/w^4*(cos(aw)-1) (|t| - a)^4 | 2/5*a^5 | 8a^3/w^2 - 48a/w^4 + 48/w^5*sin(aw) (|t| - a)^5 | -1/3*a^6 | -10a^4/w^2 + 120a^2/w^4 + 240/w^6*(cos(aw)-1) (|t| - a)^6 | 2/7*a^7 | 12a^5/w^2 - 240a^3/w^4 + 1440a/w^6 - 1440/w^7*sin(aw) (|t| - a)^7 | -1/4*a^8 | -14a^6/w^2 + 420a^4/w^4 - 5040a^2/w^6 - 10080/w^8*(cos(aw)-1)
There is a general form for these functions, but I'm struggling to put it in a good way. I will show the last 4 functions F(w) to show the pattern. (if anyone wants to continue this work to bigger and bigger polynomials, I hope this spares some headaches)
(|t| - a)^4 | 4*2*a^3/w^2 - 4!/(4-3)!*2*a/w^4 + 4!*2/w^5*sin(aw) (|t| - a)^5 | -5*2*a^4/w^2 + 5!/(5-3)!*2*a^2/w^4 + 5!*2/w^6*(cos(aw)-1) (|t| - a)^6 | 6*2*a^5/w^2 - 6!/(6-3)!*2*a^3/w^4 + 6!/(6-5)!*2*a/w^6 - 6!*2/w^7*sin(aw) (|t| - a)^7 | -7*2*a^6/w^2 + 7!/(7-3)!*2*a^4/w^4 - 7!/(7-5)!*2*a^2/w^6 - 7!*2/w^8*(cos(aw)-1)
okay, to business. We want to construct a 6-point polynomial, whose spectrum falls off at a rate of 1/w^5. I haven't exactly worked out the rationale, but this math is tedious and slow and haven't gotten much of a solid pattern established.
We will construct a fully constrained polynomial, using the functions (|t|-a)^4 and (|t|-a)^5 using a=1, a=2, and a=3.
To try to keep the terms clear, b4 is the coefficient of (|t| - 1)^4 on the interval [-1,1] b5 is the coefficient of (|t| - 1)^5 on the interval [-1,1] c4 is the coefficient of (|t| - 2)^4 on the interval [-2,2] c5 is the coefficient of (|t| - 2)^5 on the interval [-2,2] d4 is the coefficient of (|t| - 3)^4 on the interval [-3,3] d5 is the coefficient of (|t| - 3)^5 on the interval [-3,3]
We need to set the following constraints:
- 1/w^2 terms cancel
8*b4 - 10*b5 + 64*c4 - 160*c5 + 216*d4 - 810*d5 = 0
- 1/w^4 terms cancel
-48*b4 + 120*b5 - 96*c4 + 480*c5 -144*d4 + 1080*d5 = 0
- f(0)=1
b4 - b5 + 16*c4 - 32*c5 + 81*d4 - 243*d5 = 1
- f(1)=0
c4 - c5 + 16*d4 - 32*d5 = 0
- f(2)=0
d4 - d5 = 0
- lim(w->0, F(w)) = 1
2/5*b4 - 1/3*b5 + 64/5*c4 - 64/3*c5 + 486/5*d4 - 729/3*d5 = 1
Solve by linear algebra, b4= -125/64 b5= -67/64 c4= 29/32 c5= 5/32 d4= 3/64 d5= 3/64
f(t)= (|t| < 1) * [ -67/64*(|t| - 1)^5 - 125/64*(|t| - 1)^4 ] + (|t| < 2) * [ 5/32*(|t| - 2)^5 + 29/32*(|t| - 2)^4 ] + (|t| < 3) * [ 3/64*(|t| - 3)^5 + 3/64*(|t| - 3)^4 ]
F(w) = 1/w^5*( -375/4*sin(w) + 87/2*sin(2w) + 9/4*sin(3w)) + 1/w^6*(405/2 - 1005/4*cos(w) + 75/2*cos(2w) + 45/4*cos(3w))
okay, so now, we've set the spectrum and impulse response 1st, and we need to work backwards to find the polynomial interpolation. We need to re-write everything in terms of x on [0,1]
from the left: ----------------substitute t = -2 -x (coefficient of g[-2])
3/64*(| -2 - x| - 3)^5 + 3/64*(| -2 -x| - 3)^4) =3/64*(x-1)^5 + 3/64*(x-1)^4
----------------substitute t = -1 -x (coefficient of g[-1])
3/64*(| -1 - x| - 3)^5 + 3/64*(| -1 -x| - 3)^4) + 5/32*(| -1 -x| - 2)^5 + 29/32*(|-1 - x| - 2)^4
=3/64*(x-2)^5 + 3/64*(x-2)^4 + 5/32*(x-1)^5 + 29/32*(x-1)^4
----------------substitute t = -x (coefficient of g[0])
3/64*(| -x| - 3)^5 + 3/64*(| -x| - 3)^4) + 5/32*(| -x| - 2)^5 + 29/32*(| - x| - 2)^4 + 3/64*(| -x| - 3)^5 + 3/64*(| -x| - 3)^4 - 67/64*(| -x| - 1)^5 - 125/64*(| -x| - 1)^4
=3/64*(x-3)^5 + 3/64*(x-3)^4 + 5/32*(x-2)^5 + 29/32*(x-2)^4 - 67/64*(x-1)^5 - 125/64*(x-1)^4
---------------substitute t = 1 -x (coefficient of g[1])
3/64*(|1 -x| - 3)^5 + 3/64*(|1 -x| - 3)^4) + 5/32*(|1 -x| - 2)^5 + 29/32*(|1 - x| - 2)^4 + 3/64*(|1 -x| - 3)^5 + 3/64*(|1 -x| - 3)^4 - 67/64*(|1 -x| - 1)^5 - 125/64*(|1 -x| - 1)^4 =3/64*(-x-2)^5 + 3/64*(-x-2)^4 + 5/32*(-x-1)^5 + 29/32*(-x-1)^4 - 67/64*(-x)^5 - 125/64*(-x)^4
= -3/64*(x+2)^5 + 3/64*(x+2)^4 - 5/32*(x+1)^5 + 29/32*(x+1)^4 + 67/64*x^5 - 125/64*x^4
---------------substitute t = 2 -x (coefficient of g[2])
3/64*(| 2 - x| - 3)^5 + 3/64*(| 2 -x| - 3)^4) + 5/32*(| 2 -x| - 2)^5 + 29/32*(| 2 - x| - 2)^4 =3/64*(-x-1)^5 + 3/64*(-1x-1)^4 + 5/32*(-x)^5 + 29/32*(-x)^4
= -3/64*(x+1)^5 + 3/64*(x+1)^4 - 5/32*x^5 + 29/32*x^4
--------------substitute t = 3 -x (coefficient of g[3])
3/64*(| 3 - x| - 3)^5 + 3/64*(| 3 -x| - 3)^4) =3/64*(-x)^5 + 3/64*(-x)^4
= -3/64*x^5 + 3/64*x^4
I'm not sure yet how to condense it into code. I'll come back to it again on the weekend.
Chuck