Proper rational function Taking a0 = 1, we have
This is called as proper rational function if aN ≠ 0 and M < N. This amounts to saying that the number of finite zeros is less than the number of finite poles. This condition is related to partial fraction expansion and has nothing to do with causality.
Example Give pole-zero plot for H(z) = 
Solution The denominator has roots or poles at
There is a zero at z = 0. Further, as the denominator degree is greater than numerator degree by 1 it is clear that H(∞) = 0, such that there is an additional zero at z = ∞.
In MATLAB the transfer function can be specified as a ratio of polynomials in z-1
Numerator coefficients, {bi, i = 0 to M} and denominator coefficients {ai, i = 0 to N} are specified as the 2 vectors b = [0, 1] and a = [1, -1, -1].
%Pole-zero plot
b = [0, 1]; a = [1, -1, -1]; zplane (b, a)
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