Transfer Function $H(s)$

A transfer function is the Laplace transform of the Impulse Response of a LTI with a single input and single output when you set the initial conditions to zero.

Informally

The transfer function of a system is the ratio of the Laplace transform of its output and its input, assuming zero initial conditions.

Transfer functions allow us to connect several systems in series by performing convolution through simple multiplication.

We often write the transfer function as $H(s)=\frac{1}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$

Abstract

Transfer functions are a Laplace domain representation of the system.

Definition 1: Zero-State and Zero-Input Responses and the Transfer function

Given a linear DE with constant coefficients of the form $\frac{d^n}{d{t}^n}y+a_{n-1}\frac{d^{n-1}}{d^{n-1}}y+\cdots +a_{0}y=f(t)$ along with the needed initial conditions $y(0),y^{\prime }(0),\dots ,y^{(n-1)}(0)$, the Laplace transform of the equation will be of the form $\mathbf{Y}(\mathbf{s})=\frac{\mathbf{F}(\mathbf{s})}{{\mathbf{s}}^{\mathbf{n}}+{\mathbf{a}}_{\mathbf{n}-1}{\mathbf{s}}^{\mathbf{n}-1}+\cdots +{\mathbf{a}}_0}+\frac{\mathbf{g}({\mathbf{a}}_0,...,{\mathbf{a}}_{\mathbf{n}-1},\mathbf{y}(0),...,\mathbf{y}(\mathbf{n}-1)(0),\mathbf{s})}{{\mathbf{s}}^{\mathbf{n}}+{\mathbf{a}}_{\mathbf{n}-1}{\mathbf{s}}^{\mathbf{n}-1}+\cdots +{\mathbf{a}}_0}$ where $g$ is some function of the ICs and coefficients.

• The zero-input response is $\frac{g(a_0,...,a_{n-1},y(0),...,y(n-1)(0),s)}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$ . This is the response of the system the DE models to the initial conditions (i.e. when the forcing term is 0).
• The zero-state response is $\frac{F(s)}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$ . This is the response of the system the DE models to the forcing term (i.e. when the ICs are all $0$).
• The transfer function is $\frac{1}{s^n+a_{n-1}{s}^{n-1}+\cdots +a_0}$. This function determines the zero-input and the effect of the forcing term.

Theorem

The zero-state response of a linear DE is the convolution of the input (i.e. forcing term) with the system’s Impulse Response.

Transfer functions are super convenient because they enable us to compose multiple LTI Systems together!

Related

• LTI System
• Standard Second Order System