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H-infinity optimal causal IIR inverse filter for the cubic spline

H-infinity optimal causal IIR inverse filter for the cubic spline

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This computes the H-infinity optimal causal filter (indirect B-spline filter) for the cubic spline.

psicb(d);
function [psi gopt] = psicb(d);

% Computes the H-infinity optimal causal filter (indirect B-spline filter)
% for the cubic spline.
% 
% [INPUT]
% d: delay
%
% [OUTPUT]
% psi: the optimal filter psi(z) in state space representation
% gopt: optimal value

% Sampling period
T = 1;

%alpha,beta
alpha=-2+sqrt(3);
beta=-2-sqrt(3);

%phi
phi=tf(conv([1,-alpha],[1,-beta]),[6,0,0],T);

%E
E=tf(1/beta^d,1,T);

%psi
z=tf('z',T);
psi0=1/phi*((1/z)^d-E);
G=-6*z^2/beta^d/z^d/(z-alpha);
H=tf(0,1,T);
for k=0:d-1
    H=H+beta^(d-1-k)*z^k;
end
psi=G*H;

%gopt
gopt = abs(beta)^(-d);

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