login
Coefficients of "optimum L" polynomials L_n(ω^2) ordered by increasing powers.
0

%I #9 Feb 13 2023 09:00:05

%S 0,0,1,0,0,1,0,1,-3,3,0,0,3,-8,6,0,1,-8,28,-40,20,0,0,6,-40,105,-120,

%T 50,0,1,-15,105,-355,615,-525,175,0,0,10,-120,615,-1624,2310,-1680,

%U 490,0,1,-24,276,-1624,5376,-10416,11704,-7056,1764,0,0,15,-280

%N Coefficients of "optimum L" polynomials L_n(ω^2) ordered by increasing powers.

%C Used in the generation of "optimum L" (or Legendre-Papoulis) filters.

%D A. Papoulis, ”On Monotonic Response Filters,” Proc. IRE, 47, No. 2, Feb. 1959, 332-333 (correspondence section)

%H C. Bond, <a href="http://www.crbond.com/papers/optf2.pdf">Optimum “L” Filters: Polynomials, Poles and Circuit Elements</a>, 2004.

%H C. Bond, <a href="http://www.crbond.com/papers/lopt.pdf">Notes on “L” (Optimal) Filters</a>, 2011.

%H A. Papoulis, <a href="https://doi.org/10.1109/JRPROC.1958.286876">Optimum Filters with Monotonic Response</a>, Proc. IRE, 46, No. 3, March 1958, pp. 606-609.

%e Triangle begins:

%e 0;

%e 0, 1;

%e 0, 0, 1;

%e 0, 1, -3, 3;

%e 0, 0, 3, -8, 6;

%e 0, 1, -8, 28, -40, 20;

%e 0, 0, 6, -40, 105, -120, 50;

%e ...

%e So:

%e L_4(ω^2) = 0 + 0ω^2 + 3ω^4 - 8ω^6 + 6ω^8

%e L_5(ω^2) = 0 + 1ω^2 - 8ω^4 + 28ω^6 - 40ω^8 + 20ω^10

%Y Derived from A100258 and A060818.

%K sign,tabl

%O 0,9

%A _Jonathan Bright_, Jul 17 2014