Transfer Function

The Transfer Function $H(z)$ of an LTI system is the Z Transform of the Impulse Response

$$H(z)=\mathcal{Z}\{h[n]\}=\frac{Y(z)}{X(z)}=\frac{B(z^{-1})}{A(z^{-1})}$$

where $B$ and $A$ are polynomials with coefficients determined by the system’s DT LTI System. It completely characterizes the system’s input-output relationship in the z-domain. The poles of $H(z)$ determine the system’s fundamental modes.

Intuition

For a system $Y=HX$, where $H$ represents the operator the describes the entire discrete system between $X$ and $Y$, you can also express said relationship as $H=\frac{Y}{X}$. For example, for an Accumulator, you have $(1-\mathcal{R})Y=X$, so $H=\frac{1}{1-\mathcal{R}}$, and the equation in terms of $H$ is $H=\frac{1}{1-\mathcal{R}}=\frac{Y}{X}$. We care about the Transfer Function because it describes the entire system; how the system transfers the input signal to the output signal. The impulse response does too, but it and the transfer function are isomorphic via the Z Transform, so this naturally follows.

$$\begin{array}{rl}h[n]& :\mathbb{Z}\to \mathbb{C}\\ H(z)& :\mathbb{C}\to \mathbb{C}\end{array}$$

• When you use the impulse response function, you use Discrete Convolution, whereas with the transfer function, you use multiplication.
• The order of the polynomial of the transfer function tells you the order of the system
• If the polynomial is in the numerator, you have a Feed-Forward system. If the polynomial is in the denominator, then you have a Feedback system.
  • Sometimes you can compute the Operator Reciprocal of a denominator polynomial as equivalent to a numerator polynomial infinite series, meaning that some feedback systems are equivalent to infinitely large feedforward systems. This is reminiscent of Loop Unrolling / Tail Call Optimization on a Recursive function.