For a System Transfer funcition What is the condition that the degree of denominator and numerator will always be same?

Question

If the impulse response begins at n=0,then is it true that the degree of denominator and numerator will always be same? How can we justify this condition?

Answer 1

Let $h[n]$ be an impulse response with $h[n]=0, \ \forall n<0$, such that $H(z)=\mathcal{Z}\{h[n]\}$ also exists.

The transfer function is $$H(z)=h[0]+h[1]z^{-1}+h[2]z^{-2}+\cdots$$ Combining the above terms, the numerator and denomerator have equal degrees. For example when $h[n]$ has $N+1$ nonzero terms: $$H(z)=\frac{h[0]z^{N}+h[1]z^{N-1}+h[2]z^{N-2}+\cdots +h[N]}{z^N}$$