Consider a ay filter. We can define six transfer functions: H(z), H(2), H3(2), H4(2), H5(2), and H6(2) (predicted position, predicted velocity, predicted acceleration, smoothed position, smoothed velocity, and smoothed acceleration). Each transfer function has the form

The denominator remains the same for all six transfer functions. Compute all the relevant coefficients for each transfer function.

H(z) -1 a3+ a2z+az -1 -2 -2 -3 1+b+bz + boz