Example
The following transfer function: : \textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}} is proper, because : \deg(\textbf{N}(s)) = 4 \leq \deg(\textbf{D}(s)) = 4 . is biproper, because : \deg(\textbf{N}(s)) = 4 = \deg(\textbf{D}(s)) = 4 . but is not strictly proper, because : \deg(\textbf{N}(s)) = 4 \nless \deg(\textbf{D}(s)) = 4 . The following transfer function is not proper (or strictly proper) : \textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{s^{4} + n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}} because : \deg(\textbf{N}(s)) = 4 \nleq \deg(\textbf{D}(s)) = 3 . A not proper transfer function can be made proper, even strictly proper, by using the method of long division. The following transfer function is strictly proper : \textbf{G}(s) = \frac{\textbf{N}(s)}{\textbf{D}(s)} = \frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}} because : \deg(\textbf{N}(s)) = 3 . ==Implications==
Implications
A proper transfer function will never grow unbounded as the frequency approaches infinity: : |\textbf{G}(\pm j\infty)| A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes): : \textbf{G}(\pm j\infty) = 0 Also, the integral of the real part of a strictly proper transfer function is zero. ==References==