Do Causal Discrete-time systems have proper transfer functions?
Note that a stable and causal continuous-time transfer function does not need to be strictly proper but only proper, i.e. the degree of the numerator does not exceed the degree of the denominator, but numerator and denominator degree can be equal. E.g.

$$H(s)=\frac{as^2+bs+c}{s^2+ds+e}$$

can represent a causal and stable system, as long as its poles are in the left half of the complex $s$-plane.

For discrete-time systems the same is true. A transfer function for which the degree of the numerator is greater than the degree of the denominator has at least one pole at infinity. Since causal and stable discrete-time systems must have all their poles inside the unit circle, such a system can't be causal and stable. However, as is the case with continuous-time systems, equal degrees of numerator and denominator are possible.

Assume a causal and stable discrete-time system is described by the following linear difference equation with constant coefficients:

$$y[n]+a_1y[n-1]+\ldots+a_Ny[n-N]=b_0x[n]+b_1x[n-1]+\ldots+b_Mx[n-M]\tag{1}$$

The corresponding transfer function is

$$H(z)=\frac{\sum_{m=0}^{M}b_mz^{-m}}{1+\sum_{n=1}^Na_nz^{-n}}\tag{2}$$

Note that there are of course no restrictions on the choices of $M$ and $N$. Nevertheless, the transfer function $H(z)$ is always proper. Assuming $M&gt;N$ gives

$$H(z)=\frac{b_0z^M+b_1z^{M-1}+\ldots+b_M}{z^M+a_1z^{M-1}+\ldots+a_Nz^{M-N}}\tag{3}$$

Obviously, the transfer function given by (3) is proper. $M&gt;N$ just means that the last few coefficients of the denominator polynomial are zero. A very similar argument holds for $M&lt;N$.