# 4.6: Transfer Function - General Definition

- Page ID
- 7647

For any LTI, single-input-single-output (SISO) physical system, we denote the input as \(u(t)\) and the output as \(x(t)\). For an \(n\)^{th} order system, in general, the input and output are related by an ODE of the form

\[a_{1} \frac{d^{n} x}{d t^{n}}+a_{2} \frac{d^{n-1} x}{d t^{n-1}}+\ldots+a_{n+1} x=b_{1} \frac{d^{m} u}{d t^{m}}+b_{2} \frac{d^{m-1} u}{d t^{m-1}}+\ldots+b_{m+1} u\label{eqn:4.21}\]

\(a_{1}, \ldots, a_{n+1}\) and \(b_{1}, \ldots, b_{m+1}\) are constants (with the numbering system keyed to MATLAB notation), and \(m\) \(\leq\) \(n\). Also, we assume that the system is *stable*, which is defined more precisely in Section 4.7. Taking the Laplace transform of the ODE, with all ICs equal to zero, gives

\[\left(a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}\right) L[x(t)]_{I C_s=0}=\left(b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}\right) L[u(t)]\label{eqn:4.22}\]

Then, from Equation \(\ref{eqn:4.22}\), the system *transfer function*, defined to be the ratio of the output transform to the input transform, with zero ICs, is the ratio of two polynomials,

\[T F(s) \equiv \frac{L[x(t)]_{I C_s=0}}{L[u(t)]}=\frac{b_{1} s^{m}+b_{2} s^{m-1}+\ldots+b_{m+1}}{a_{1} s^{n}+a_{2} s^{n-1}+\ldots+a_{n+1}}\label{eqn:4.23}\]

It is appropriate to state here (without proof) that the transfer function of any **physically realizable** system has \(m\) \(\leq\) \(n\), *i.e.*, **the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial**. The condition \(m\) \(\leq\) \(n\) makes the transfer function **causal**, which means that the current (in time) output of the system is dependent only upon past and present (**not** future) values of the input. In general, the future values of an input cannot be predicted, so it is logical that a real system and its transfer function must be causal. See Bélanger, 1995, page 440.

Note also from Equation \(\ref{eqn:4.23}\) that, if given \(TF(s)\) and input \(u(t)\), we can express the transform of the output with zero initial conditions as

\[\left.L[x(t)]\right|_{I C s=0}=T F(s) \times L[u(t)]\label{eqn:4.24}\]