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{{Unreferenced|date=December 2009}}
In [[control theory]], a '''proper transfer function''' is a [[transfer function]] in which the [[degree (mathematics)|degree]] of the numerator does not exceed the degree of the denominator.
A '''strictly proper''' [[transfer function]] is a transfer function where the [[Degree (mathematics)|degree]] of the numerator is [[less than]] the degree of the denominator.
==Example==
The following transfer function
:<math> \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}}</math>
because▼
is '''proper''', because
:<math> \deg(\textbf{N}(s)) = 4 \leq \deg(\textbf{D}(s)) = 4 </math>.
The following transfer function however, is '''not proper'''▼
but is '''not strictly proper''', because
:<math> \deg(\textbf{N}(s)) = 4 \nless \deg(\textbf{D}(s)) = 4 </math>.
The following transfer function is '''not proper''' (or strictly proper)
:<math> \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}}</math>
because
:<math> \deg(\textbf{N}(s)) = 4 \nleq \deg(\textbf{D}(s)) = 3 </math>.
:<math> \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}}</math>
▲because
:<math> \deg(\textbf{N}(s)) = 3 < \deg(\textbf{D}(s)) = 4 </math>.
==Implications==
A proper transfer function will never grow unbounded as the frequency approaches infinity
:<math> |\textbf{G}(\pm j\infty)| < \infty </math>
A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes):
:<math> \textbf{G}(\pm j\infty) = 0 </math>
Also, the integral of the real part of a strictly proper transfer function is zero.
{{DEFAULTSORT:Proper Transfer Function}}
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